From 67776b437e0924b6a69cd38ae53be787400b66a3 Mon Sep 17 00:00:00 2001 From: trinity-canonical Date: Thu, 30 Apr 2026 10:37:56 +0000 Subject: [PATCH] =?UTF-8?q?feat(canonical):=20=F0=9F=8C=BB=20Stub=2047=20m?= =?UTF-8?q?irror=20Coq=20files=20=E2=86=92=20Trinity=20Canonical=20Home?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Companion PR to gHashTag/t27#569 (CANONICAL COQ HOME). After Coq theorem census (trios#373) found 376 Qed across 65 canonical .v files in 3 repos, t27/proofs/canonical/ is the single source of truth. This PR replaces 47 mirror Coq files in trios: - docs/phd/theorems/**/*.v (PhD-curated theorem doc; 28 files) - trinity-clara/proofs/igla/*.v (vendored mirror; 16 files) - trinity-clara/proofs/*.v (root mirror; 3 files) Each stub becomes: From Trinity.Canonical.{Sacred|Kernel|Igla} Require Export . Files retained as canonical winners (not stubbed): - trios/docs/phd/theorems/igla/IGLA_BPB_Convergence.v (INV-1, 9 Qed) - trios/trinity-clara/proofs/igla/gf16_safe_domain.v (INV-3, 9 Qed) - trios/trinity-clara/proofs/igla/lr_phi_optimality.v (INV-1b, 5 Qed) - trios/trinity-clara/proofs/igla/lucas_closure_gf16.v (INV-5, 10 Qed) - trios/trinity-clara/proofs/igla/nca_entropy_stability.v (INV-4, 12 Qed) - trios/trinity-clara/proofs/igla/rainbow_bridge_consistency.v (INV-7b, 15 Qed) - trios/trinity-clara/proofs/igla/worker_pool_composite.v (INV-8, 10 Qed) - trios/trinity-clara/proofs/igla/ema_decay_valid.v (INV-9, 8 Qed) - trios/docs/phd/theorems/igla/IGLA_Catalog.v (catalog, 5 Qed) - trios/docs/phd/theorems/sacred/{gamma_phi3,l5_identity}.v These remain upstream source for canonical home. Anchor: phi^2 + phi^-2 = 3 Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 Co-authored-by: AI-assisted code generation --- docs/phd/theorems/FlowerE8Embedding.v | 154 +-------- docs/phd/theorems/GenIdempotency.v | 23 +- docs/phd/theorems/Kernel/FlowerE8Embedding.v | 154 +-------- docs/phd/theorems/Kernel/Phi.v | 174 +--------- docs/phd/theorems/Kernel/PhiAttractor.v | 75 +---- docs/phd/theorems/Kernel/PhiFloat.v | 88 +---- docs/phd/theorems/Kernel/Semantics.v | 35 +- docs/phd/theorems/Kernel/Trit.v | 42 +-- docs/phd/theorems/Phi.v | 174 +--------- docs/phd/theorems/PhiAttractor.v | 75 +---- docs/phd/theorems/PhiDistance.v | 23 +- docs/phd/theorems/PhiFloat.v | 88 +---- docs/phd/theorems/Semantics.v | 35 +- docs/phd/theorems/TernarySufficiency.v | 22 +- docs/phd/theorems/Theorems/GenIdempotency.v | 23 +- docs/phd/theorems/Theorems/PhiDistance.v | 23 +- .../theorems/Theorems/TernarySufficiency.v | 22 +- docs/phd/theorems/Trit.v | 42 +-- docs/phd/theorems/gravity/dl_bounds.v | 26 +- docs/phd/theorems/igla/IGLA_ASHA_Bound.v | 158 +-------- docs/phd/theorems/igla/IGLA_GF16_Precision.v | 227 +------------ docs/phd/theorems/igla/IGLA_NCA_Entropy.v | 289 +--------------- docs/phd/theorems/sacred/dl_bounds.v | 32 +- docs/phd/theorems/sacred/strong_cp.v | 21 +- docs/phd/theorems/trinity/AlphaPhi.v | 156 +-------- docs/phd/theorems/trinity/Bounds_Gauge.v | 164 +--------- docs/phd/theorems/trinity/Bounds_Masses.v | 201 +----------- docs/phd/theorems/trinity/Bounds_Mixing.v | 170 +--------- .../phd/theorems/trinity/Bounds_QuarkMasses.v | 127 +------- docs/phd/theorems/trinity/Catalog42.v | 245 +------------- docs/phd/theorems/trinity/ConsistencyChecks.v | 308 +----------------- docs/phd/theorems/trinity/CorePhi.v | 123 +------ docs/phd/theorems/trinity/DerivationLevels.v | 302 +---------------- docs/phd/theorems/trinity/ExactIdentities.v | 271 +-------------- docs/phd/theorems/trinity/FormulaEval.v | 254 +-------------- docs/phd/theorems/trinity/Unitarity.v | 186 +---------- trinity-clara/proofs/gf16_precision.v | 49 +-- .../proofs/igla/asha_champion_survives.v | 242 +------------- trinity-clara/proofs/igla/bpb_decreases.v | 180 +--------- trinity-clara/proofs/igla/gf16_precision.v | 64 +--- trinity-clara/proofs/igla/igla_asha_bound.v | 80 +---- .../proofs/igla/igla_found_criterion.v | 121 +------ trinity-clara/proofs/igla/nca_entropy_band.v | 96 +----- trinity-clara/proofs/igla_asha_bound.v | 100 +----- trinity-clara/proofs/lr_phi_optimality.v | 57 +--- trinity-clara/proofs/lucas_closure_gf16.v | 66 +--- trinity-clara/proofs/nca_entropy_band.v | 60 +--- 47 files changed, 720 insertions(+), 4927 deletions(-) diff --git a/docs/phd/theorems/FlowerE8Embedding.v b/docs/phd/theorems/FlowerE8Embedding.v index ec2d6f3451..3f13b8c611 100644 --- a/docs/phd/theorems/FlowerE8Embedding.v +++ b/docs/phd/theorems/FlowerE8Embedding.v @@ -1,143 +1,19 @@ -(** FLOWER-E8-EMBEDDING — E8 = H4 + φ·H4 Decomposition *) -(* - * Formal proof of Dechant (2016) theorem: E8 Lie algebra - * decomposes into two H4 Coxeter subalgebras, one scaled by φ. - * - * Reference: - * - Dechant, P.-P., "The E8 root system from the icosahedron", - * Proc. R. Soc. A 472 (2016) 508-520 - * - E8LieAlgebra.t27 (computational verification) - * - * Key insight: - * - H4 is the symmetry group of the 600-cell (120 vertices) - * - E8 roots partition into: H4 roots ∪ φ·H4 roots - * - φ scaling factor preserves the 240-root structure - * - dim(H4) = 120; 2·dim(H4) = 240 = |E8 roots| - * - * Trinity connection: φ² + φ⁻² = 3 encodes this decomposition - * as the dimensionality matching condition. - *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -From Stdlib Require Import Reals. -From Coq Require Import Psatz. -From Coq Require Import RealField. -Require Import T27.Kernel.Phi. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Open Scope R_scope. + gHashTag/t27/proofs/canonical/kernel/FlowerE8Embedding.v + (logical path: Trinity.Canonical.Kernel.FlowerE8Embedding) -(** ==================================================================== *) -(* Section 1: H4 Dimension Lemma *) -(* ==================================================================== *) + Bundle: KER-3 + Title: Flower-of-Life E8 embedding + PhD chapter: Ch.7 E8 / Vogel + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** Lemma: H4 Coxeter group has 120 roots *) - -Lemma h4_root_count : 120 = 248 / 2. -Proof. - assert (H1 : 248 / 2 = 124) by ring. - assert (H2 : 120 = 124 / 1.03333333...) by lra. - (* Formal proof: H4 has rank 4, Coxeter number 120 *) - (* 600-cell has 120 vertices = 2 * 60, each vertex gives a root *) - ring. -Qed. - -(** Lemma: H4 dimension equals twice its root count *) - -Lemma h4_dim_equals_twice_roots : 120 = 2 * 60. -Proof. - ring. -Qed. - -(** ==================================================================== *) -(* Section 2: E8 Decomposition Structure *) -(* ==================================================================== *) - -(** Definition: Two H4 blocks in E8 root system *) - -Definition h4_block_1 : set R := {r | exists s1, s2 : H4_root, r = s1}. -Definition h4_block_2 : set R := {r | exists s1, s2 : H4_root, r = phi * s1}. - -(** Lemma: Union of H4 blocks covers 240 E8 roots *) - -Lemma e8_roots_decomposition : - E8_roots = h4_block_1 ∪ h4_block_2. -Proof. - (* Formal proof sketch based on root system structure: - * 120 roots from h4_block_1 (unscaled H4) - * 120 roots from h4_block_2 (φ-scaled H4) - * Total: 240 = |E8 roots| - * - * Verification requires analyzing E8 Cartan matrix eigenvectors - * and showing partition respects Weyl group structure - *) - Abort. -Qed. - -(** Theorem: Main E8 flower decomposition *) - -Theorem e8_flower_decomposition : - dim(H4) + dim(H4) = dim(E8) / 2. -Proof. - split. - (* Part 1: dim(H4) = 120 from root count *) - - apply h4_root_count. - (* Part 2: 2 * dim(H4) = 240 = |E8 roots| *) - - apply h4_dim_equals_twice_roots. - (* Part 3: dim(E8) = 248 = rank * 2 + roots *) - (* 248 = 8 + 240 ✓ *) - ring. -Qed. - -(** ==================================================================== *) -(* Section 3: Trinity Connection *) -(* ==================================================================== *) - -(** Lemma: φ scaling preserves root system structure *) - -Lemma phi_scaling_invariant : - forall r : R, r > 0 -> - dim({phi * r | r in E8_roots}) = dim({r | r in E8_roots}). -Proof. - (* φ is a positive scaling factor, preserves linear independence *) - (* Dimension of scaled subspace equals dimension of original subspace *) - Abort. -Qed. - -(** Theorem: φ encodes E8-H4 decomposition via L5 identity *) - -Theorem trinity_e8_h4_encoding : - phi * phi + / (phi * phi) = 3 -> - dim(H4) + dim(phi * H4) = dim(E8) / 2. -Proof. - intros Htrinity. - rewrite Htrinity. - (* φ² = φ + 1, φ⁻² = φ - 1, so φ² + φ⁻² = 2φ = 2 * 1.618 ≈ 3.236 *) - (* But we have exact: φ² + φ⁻² = 3 *) - (* This encodes: dim(H4) + dim(H4) = 120 + 120 = 240 *) - (* and: dim(phi * H4) = 240 when φ² + φ⁻² = 3 *) - (* The Trinity identity directly provides the scaling factor *) - exact e8_flower_decomposition. -Qed. - -(** ==================================================================== *) -(* Section 4: Computational Verification *) -(* ==================================================================== *) - -(** Lemma: 600-cell vertices correspond to H4 roots *) - -Lemma sixhundred_cell_vertices_equal_h4_roots : - |V(600-cell)| = |H4_roots|. -Proof. - (* 600-cell has 120 vertices, each vertex + its antipode = 240 directions *) - (* H4 root system has 120 roots (positive and negative) *) - (* One-to-one correspondence established by Dechant (2016) *) - Abort. -Qed. - -(** Invariant: E8 flower decomposition preserves dimensionality *) - -Invariant e8_flower_dimensionality : - assert dim(h4_block_1 ∪ h4_block_2) = E8_DIM / 2. - (* Rationale: Decomposition preserves root structure and counts *) - (* Verified by computational replay in e8_lie_algebra.t27 *) - -Close Scope R_scope. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export FlowerE8Embedding. diff --git a/docs/phd/theorems/GenIdempotency.v b/docs/phd/theorems/GenIdempotency.v index 854e0e20cb..28e862bbe3 100644 --- a/docs/phd/theorems/GenIdempotency.v +++ b/docs/phd/theorems/GenIdempotency.v @@ -1,8 +1,19 @@ -(** THEOREM-K3 direction — codegen idempotency; needs abstract Spec/Code types from t27c model. *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Parameter spec : Type. -Parameter code : Type. -Parameter t27c_gen : spec -> code. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Lemma gen_idempotent (s : spec) : t27c_gen s = t27c_gen s. -Proof. reflexivity. Qed. + gHashTag/t27/proofs/canonical/kernel/GenIdempotency.v + (logical path: Trinity.Canonical.Kernel.GenIdempotency) + + Bundle: KER-6 + Title: Idempotency law + PhD chapter: Ch.10 Coq L1 (idempotency) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) + +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export GenIdempotency. diff --git a/docs/phd/theorems/Kernel/FlowerE8Embedding.v b/docs/phd/theorems/Kernel/FlowerE8Embedding.v index ec2d6f3451..3f13b8c611 100644 --- a/docs/phd/theorems/Kernel/FlowerE8Embedding.v +++ b/docs/phd/theorems/Kernel/FlowerE8Embedding.v @@ -1,143 +1,19 @@ -(** FLOWER-E8-EMBEDDING — E8 = H4 + φ·H4 Decomposition *) -(* - * Formal proof of Dechant (2016) theorem: E8 Lie algebra - * decomposes into two H4 Coxeter subalgebras, one scaled by φ. - * - * Reference: - * - Dechant, P.-P., "The E8 root system from the icosahedron", - * Proc. R. Soc. A 472 (2016) 508-520 - * - E8LieAlgebra.t27 (computational verification) - * - * Key insight: - * - H4 is the symmetry group of the 600-cell (120 vertices) - * - E8 roots partition into: H4 roots ∪ φ·H4 roots - * - φ scaling factor preserves the 240-root structure - * - dim(H4) = 120; 2·dim(H4) = 240 = |E8 roots| - * - * Trinity connection: φ² + φ⁻² = 3 encodes this decomposition - * as the dimensionality matching condition. - *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -From Stdlib Require Import Reals. -From Coq Require Import Psatz. -From Coq Require Import RealField. -Require Import T27.Kernel.Phi. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Open Scope R_scope. + gHashTag/t27/proofs/canonical/kernel/FlowerE8Embedding.v + (logical path: Trinity.Canonical.Kernel.FlowerE8Embedding) -(** ==================================================================== *) -(* Section 1: H4 Dimension Lemma *) -(* ==================================================================== *) + Bundle: KER-3 + Title: Flower-of-Life E8 embedding + PhD chapter: Ch.7 E8 / Vogel + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** Lemma: H4 Coxeter group has 120 roots *) - -Lemma h4_root_count : 120 = 248 / 2. -Proof. - assert (H1 : 248 / 2 = 124) by ring. - assert (H2 : 120 = 124 / 1.03333333...) by lra. - (* Formal proof: H4 has rank 4, Coxeter number 120 *) - (* 600-cell has 120 vertices = 2 * 60, each vertex gives a root *) - ring. -Qed. - -(** Lemma: H4 dimension equals twice its root count *) - -Lemma h4_dim_equals_twice_roots : 120 = 2 * 60. -Proof. - ring. -Qed. - -(** ==================================================================== *) -(* Section 2: E8 Decomposition Structure *) -(* ==================================================================== *) - -(** Definition: Two H4 blocks in E8 root system *) - -Definition h4_block_1 : set R := {r | exists s1, s2 : H4_root, r = s1}. -Definition h4_block_2 : set R := {r | exists s1, s2 : H4_root, r = phi * s1}. - -(** Lemma: Union of H4 blocks covers 240 E8 roots *) - -Lemma e8_roots_decomposition : - E8_roots = h4_block_1 ∪ h4_block_2. -Proof. - (* Formal proof sketch based on root system structure: - * 120 roots from h4_block_1 (unscaled H4) - * 120 roots from h4_block_2 (φ-scaled H4) - * Total: 240 = |E8 roots| - * - * Verification requires analyzing E8 Cartan matrix eigenvectors - * and showing partition respects Weyl group structure - *) - Abort. -Qed. - -(** Theorem: Main E8 flower decomposition *) - -Theorem e8_flower_decomposition : - dim(H4) + dim(H4) = dim(E8) / 2. -Proof. - split. - (* Part 1: dim(H4) = 120 from root count *) - - apply h4_root_count. - (* Part 2: 2 * dim(H4) = 240 = |E8 roots| *) - - apply h4_dim_equals_twice_roots. - (* Part 3: dim(E8) = 248 = rank * 2 + roots *) - (* 248 = 8 + 240 ✓ *) - ring. -Qed. - -(** ==================================================================== *) -(* Section 3: Trinity Connection *) -(* ==================================================================== *) - -(** Lemma: φ scaling preserves root system structure *) - -Lemma phi_scaling_invariant : - forall r : R, r > 0 -> - dim({phi * r | r in E8_roots}) = dim({r | r in E8_roots}). -Proof. - (* φ is a positive scaling factor, preserves linear independence *) - (* Dimension of scaled subspace equals dimension of original subspace *) - Abort. -Qed. - -(** Theorem: φ encodes E8-H4 decomposition via L5 identity *) - -Theorem trinity_e8_h4_encoding : - phi * phi + / (phi * phi) = 3 -> - dim(H4) + dim(phi * H4) = dim(E8) / 2. -Proof. - intros Htrinity. - rewrite Htrinity. - (* φ² = φ + 1, φ⁻² = φ - 1, so φ² + φ⁻² = 2φ = 2 * 1.618 ≈ 3.236 *) - (* But we have exact: φ² + φ⁻² = 3 *) - (* This encodes: dim(H4) + dim(H4) = 120 + 120 = 240 *) - (* and: dim(phi * H4) = 240 when φ² + φ⁻² = 3 *) - (* The Trinity identity directly provides the scaling factor *) - exact e8_flower_decomposition. -Qed. - -(** ==================================================================== *) -(* Section 4: Computational Verification *) -(* ==================================================================== *) - -(** Lemma: 600-cell vertices correspond to H4 roots *) - -Lemma sixhundred_cell_vertices_equal_h4_roots : - |V(600-cell)| = |H4_roots|. -Proof. - (* 600-cell has 120 vertices, each vertex + its antipode = 240 directions *) - (* H4 root system has 120 roots (positive and negative) *) - (* One-to-one correspondence established by Dechant (2016) *) - Abort. -Qed. - -(** Invariant: E8 flower decomposition preserves dimensionality *) - -Invariant e8_flower_dimensionality : - assert dim(h4_block_1 ∪ h4_block_2) = E8_DIM / 2. - (* Rationale: Decomposition preserves root structure and counts *) - (* Verified by computational replay in e8_lie_algebra.t27 *) - -Close Scope R_scope. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export FlowerE8Embedding. diff --git a/docs/phd/theorems/Kernel/Phi.v b/docs/phd/theorems/Kernel/Phi.v index d1804ee537..8a6a24c8a0 100644 --- a/docs/phd/theorems/Kernel/Phi.v +++ b/docs/phd/theorems/Kernel/Phi.v @@ -1,163 +1,19 @@ -(** Golden ratio as real; algebraic identities (AXIOM-K2 / PHI-IDENTITY). - Layer A: pure [Coq.Reals] — no Flocq here. See [Kernel/PhiFloat.v] for Flocq link. *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals. -Require Import Psatz. -Require Import ZArith. -Require Import RealField. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Open Scope R_scope. + gHashTag/t27/proofs/canonical/kernel/Phi.v + (logical path: Trinity.Canonical.Kernel.Phi) -Definition phi : R := (1 + sqrt 5) / 2. + Bundle: KER-0 + Title: phi kernel arithmetic (mul/pow) + PhD chapter: Ch.10 Coq L1 + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** Denominator 2^53 for IEEE binary64 relative unit u = 2^{-53}. *) -Definition coeff_53 : Z := 9007199254740992%Z. - -Lemma coeff_53_pos : (0 < coeff_53)%Z. -Proof. unfold coeff_53; lia. Qed. - -(** Engineering tolerance: 5 * u * phi^2 with u = 2^{-53} (see TZ §5.5). *) -Definition phi_tolerance : R := 5 * / IZR coeff_53 * (phi * phi). - -Lemma sqrt5_sq : sqrt 5 * sqrt 5 = 5. -Proof. - replace (sqrt 5 * sqrt 5) with (Rsqr (sqrt 5)). - - rewrite Rsqr_sqrt; lra. - - unfold Rsqr; ring. -Qed. - -Lemma sqrt5_pos : 0 < sqrt 5. -Proof. apply sqrt_lt_R0; lra. Qed. - -Lemma sqrt4 : sqrt 4 = 2. -Proof. - replace 4 with (2 * 2) by ring. - rewrite sqrt_square; lra. -Qed. - -Lemma sqrt5_gt_2 : 2 < sqrt 5. -Proof. - rewrite <- sqrt4. - apply sqrt_lt_1; lra. -Qed. - -Lemma phi_pos : 0 < phi. -Proof. - unfold phi. - assert (H1lt : 1 < 1 + sqrt 5). - { - rewrite <- (Rplus_0_r 1) at 1. - apply (Rplus_lt_compat_l 1 0 (sqrt 5) sqrt5_pos). - } - assert (Hn : 0 < 1 + sqrt 5) by lra. - replace ((1 + sqrt 5) / 2) with ((1 + sqrt 5) * / 2) by field. - apply Rmult_lt_0_compat. - - exact Hn. - - apply Rinv_0_lt_compat; lra. -Qed. - -Lemma phi_neq_0 : phi <> 0. -Proof. apply Rgt_not_eq, Rlt_gt; exact phi_pos. Qed. - -Lemma phi_gt_1 : 1 < phi. -Proof. - unfold phi. - assert (H : 2 < 1 + sqrt 5) by (generalize sqrt5_gt_2; lra). - lra. -Qed. - -Lemma sqrt9 : sqrt 9 = 3. -Proof. - replace 9 with (3 * 3) by ring. - rewrite sqrt_square; lra. -Qed. - -Lemma phi_lt_2 : phi < 2. -Proof. - unfold phi. - assert (H : 1 + sqrt 5 < 4). - { - assert (sqrt 5 < 3) by (rewrite <- sqrt9; apply sqrt_lt_1; lra). - lra. - } - lra. -Qed. - -Lemma phi_squared_identity : phi * phi = phi + 1. -Proof. - unfold phi. - assert (Hsq : (1 + sqrt 5) * (1 + sqrt 5) = 6 + 2 * sqrt 5). - { - replace ((1 + sqrt 5) * (1 + sqrt 5)) with (1 + 2 * sqrt 5 + (sqrt 5 * sqrt 5)) by ring. - rewrite sqrt5_sq. - ring. - } - assert (Hleft : ((1 + sqrt 5) / 2) * ((1 + sqrt 5) / 2) = (6 + 2 * sqrt 5) / 4). - { - replace (((1 + sqrt 5) / 2) * ((1 + sqrt 5) / 2)) with (((1 + sqrt 5) * (1 + sqrt 5)) / 4) by field. - rewrite Hsq. - reflexivity. - } - assert (Hright : (1 + sqrt 5) / 2 + 1 = (3 + sqrt 5) / 2) by field. - assert (Hmid : (6 + 2 * sqrt 5) / 4 = (3 + sqrt 5) / 2) by field. - rewrite Hleft, Hright. - exact Hmid. -Qed. - -Lemma phi_mul_phi_minus_one : phi * (phi - 1) = 1. -Proof. - replace (phi * (phi - 1)) with (phi * phi - phi) by ring. - rewrite phi_squared_identity. - ring. -Qed. - -Lemma phi_inv_is_phi_minus_one : / phi = phi - 1. -Proof. - apply (Rmult_eq_reg_l phi). - - rewrite Rinv_r; [| exact phi_neq_0 ]. - symmetry. - exact phi_mul_phi_minus_one. - - exact phi_neq_0. -Qed. - -Lemma phi_inv_sq_sum_three : phi * phi + Rsqr (/ phi) = 3. -Proof. - rewrite phi_inv_is_phi_minus_one. - unfold Rsqr. - assert (Hsq1 : (phi - 1) * (phi - 1) = 2 - phi). - { - replace ((phi - 1) * (phi - 1)) with (phi * phi - 2 * phi + 1) by ring. - rewrite phi_squared_identity. - ring. - } - rewrite Hsq1. - rewrite phi_squared_identity. - ring. -Qed. - -Lemma phi_tolerance_pos : 0 < phi_tolerance. -Proof. - unfold phi_tolerance. - apply Rmult_lt_0_compat. - - apply Rmult_lt_0_compat. - + lra. - + apply Rinv_0_lt_compat. - apply IZR_lt. - exact coeff_53_pos. - - apply Rmult_lt_0_compat. - + exact phi_pos. - + exact phi_pos. -Qed. - -(** [PHI-IDENTITY] on [R]: no float error — residual is zero. *) -Lemma phi_identity_residual_R : Rabs (phi * phi - (phi + 1)) = 0. -Proof. - replace (phi * phi - (phi + 1)) with 0. - - apply Rabs_R0. - - unfold Rminus. - rewrite phi_squared_identity. - ring. -Qed. - -Definition phi_approx_valid (x : R) : Prop := - Rabs (x - phi) <= phi_tolerance. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export Phi. diff --git a/docs/phd/theorems/Kernel/PhiAttractor.v b/docs/phd/theorems/Kernel/PhiAttractor.v index 4c677e5602..d93c2e49a7 100644 --- a/docs/phd/theorems/Kernel/PhiAttractor.v +++ b/docs/phd/theorems/Kernel/PhiAttractor.v @@ -1,64 +1,19 @@ -(** THEOREM-3 — φ as Universal Fixed-Point Attractor *) -(** Balancing recursion: f(x) = (x + x⁻¹ + 1) / 2 *) -(** From any x₀ > 0, iteration converges to φ with rate λ = (√5 - 1)/4 *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals. -Require Import Psatz. -Require Import RealField. -Require Import ZArith. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Open Scope R_scope. + gHashTag/t27/proofs/canonical/kernel/PhiAttractor.v + (logical path: Trinity.Canonical.Kernel.PhiAttractor) -(** Definition: phi = (1 + sqrt(5)) / 2 — matches Phi.v definition *) -Definition phi : R := (1 + sqrt 5) / 2. + Bundle: KER-1 + Title: phi universal attractor + PhD chapter: Ch.5 phi-distance (attractor) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** Definition: balancing function f(x) = (x + x⁻¹ + 1) / 2 *) -Definition balancing_function (x : R) : R := (x + / x + 1) / 2. - -(** Convergence rate: λ = (√5 - 1) / 4 ≈ 0.309 *) -Definition convergence_rate_lambda : R := (sqrt 5 - 1) / 4. - -(** ==================================================================== *) -(** Section 1: Fixed Point Verification *) -(** ==================================================================== *) - -(** Lemma: φ is a fixed point of balancing_function *) -Lemma phi_is_fixed_point : balancing_function phi = phi. -Proof. - unfold balancing_function. - unfold phi. - (* Compute: f(φ) = (φ + 1/φ + 1) / 2 *) - assert (H1 : (phi + / phi + 1) * (1 + sqrt 5) / 2 = (phi + / phi + 1) * (1 + sqrt 5) / 2) by field). - assert (H2 : (phi + / phi + 1) * (1 + sqrt 5) / 2 = phi * (1 + sqrt 5) / 2) by field). - assert (Hmid : phi * phi = phi + 1) by (apply phi_squared_identity; auto). - assert (Hmid2 : phi * (1 + sqrt 5) = (1 + sqrt 5) + 5 by ring). - replace ((phi + / phi + 1) * (1 + sqrt 5) / 2) with (phi * (1 + sqrt 5) / 2) in Hmid. - reflexivity. -Qed. - -(** ==================================================================== *) -(** Section 2: Main Theorem *) -(** ==================================================================== *) - -(** Theorem: φ is universal fixed-point attractor *) -Theorem phi_universal_attractor : - (* 1. φ is a fixed point of f *) - balancing_function phi = phi /\ - (* 2. Convergence rate λ is in (0, 1) *) - 0 < convergence_rate_lambda < 1. -Proof. - unfold balancing_function. - (* f(φ) = (φ + 1/φ + 1) / 2 *) - (* We have proved φ is fixed point *) - (* Now prove 0 < λ < 1 *) - (* λ = (√5 - 1)/4, and √5 ≈ 2.236, so λ ≈ 0.309 *) - unfold convergence_rate_lambda. - (* Need to show 0 < (√5 - 1)/4 and (√5 - 1)/4 < 1 *) - assert (H1 : 0 < (sqrt 5 - 1) / 4) by lra. - assert (H2 : (sqrt 5 - 1) / 4 < 1) by lra in H1). - (* This completes the proof *) - reflexivity. -Qed. - -Close Scope R_scope. -Close Scope R_scope. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export PhiAttractor. diff --git a/docs/phd/theorems/Kernel/PhiFloat.v b/docs/phd/theorems/Kernel/PhiFloat.v index 16b439cc8a..661379258e 100644 --- a/docs/phd/theorems/Kernel/PhiFloat.v +++ b/docs/phd/theorems/Kernel/PhiFloat.v @@ -1,77 +1,19 @@ -(** PHI-IDENTITY — Flocq IEEE 754 binary64 bridge (Phase B). - Requires [coq-flocq] on COQPATH (CI: opam install coq-flocq; see [../README.md]). - Mantissas/exponents must match t27c validate-phi (Rust; former scripts/validate_phi_f64.py). +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME - For this [binary64] literal of φ, [fl(phi*phi)] and [fl(phi+1)] coincide (bit-identical); - [phi_identity_contract] is therefore [Rabs 0 < phi_tolerance], using [phi_tolerance_pos] - from [Phi.v]. A future ring can add [Bmult_correct] / [Bplus_correct] + error bounds - for other formats (GF16, etc.). *) + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -From Coq Require Import Reals ZArith. + gHashTag/t27/proofs/canonical/kernel/PhiFloat.v + (logical path: Trinity.Canonical.Kernel.PhiFloat) -From Flocq Require Import IEEE754.Binary. -From Flocq Require Import IEEE754.Bits. -From Flocq Require Import IEEE754.BinarySingleNaN. + Bundle: KER-2 + Title: phi-float arithmetic + PhD chapter: Ch.6 GoldenFloat + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -Require Import T27.Kernel.Phi. - -Open Scope R_scope. -Open Scope Z_scope. - -Import Binary Bits BinarySingleNaN. - -Notation b64 := binary64 (only parsing). - -Definition B2R64 : binary64 -> R := @B2R 53 1024. - -(** Nearest-round φ as [B754_finite] (see validation script). *) -Definition phi_mantissa : positive := 7286977268806824%positive. -Definition phi_exponent : Z := (-52)%Z. - -Lemma phi_f64_bounded : bounded 53 1024 phi_mantissa phi_exponent = true. -Proof. vm_compute. reflexivity. Qed. - -Definition phi_f64 : binary64 := - B754_finite false phi_mantissa phi_exponent phi_f64_bounded. - -(** [1.0] in binary64. *) -Definition one_mantissa : positive := 4503599627370496%positive. -Definition one_exponent : Z := (-52)%Z. - -Lemma one_f64_bounded : bounded 53 1024 one_mantissa one_exponent = true. -Proof. vm_compute. reflexivity. Qed. - -Definition one_f64 : binary64 := - B754_finite false one_mantissa one_exponent one_f64_bounded. - -Definition phi_sq_f64 : binary64 := b64_mult mode_NE phi_f64 phi_f64. - -Definition phi_plus_one_f64 : binary64 := b64_plus mode_NE phi_f64 one_f64. - -Lemma phi_sq_f64_eq_phi_plus_one_f64 : phi_sq_f64 = phi_plus_one_f64. -Proof. vm_compute. reflexivity. Qed. - -(** Engineering bound (Layer A SSOT): [5 * 2^-53 * phi^2] on [R]. *) -Definition PHI_F64_TOLERANCE : R := phi_tolerance. - -Theorem phi_identity_contract : - Rabs (B2R64 phi_sq_f64 - B2R64 phi_plus_one_f64) < PHI_F64_TOLERANCE. -Proof. - assert (Hbr : B2R64 phi_sq_f64 = B2R64 phi_plus_one_f64). - { - apply f_equal. - exact phi_sq_f64_eq_phi_plus_one_f64. - } - unfold PHI_F64_TOLERANCE. - replace (B2R64 phi_sq_f64 - B2R64 phi_plus_one_f64) with 0. - - rewrite Rabs_R0. - apply phi_tolerance_pos. - - rewrite Hbr. - ring. -Qed. - -Lemma phi_tolerance_positive : 0 < phi_tolerance. -Proof. apply phi_tolerance_pos. Qed. - -Lemma PHI_F64_TOLERANCE_pos : 0 < PHI_F64_TOLERANCE. -Proof. unfold PHI_F64_TOLERANCE; apply phi_tolerance_pos. Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export PhiFloat. diff --git a/docs/phd/theorems/Kernel/Semantics.v b/docs/phd/theorems/Kernel/Semantics.v index d673e70246..c0dcbdff8e 100644 --- a/docs/phd/theorems/Kernel/Semantics.v +++ b/docs/phd/theorems/Kernel/Semantics.v @@ -1,24 +1,19 @@ -(** Minimal expression calculus — placeholder for denotational / RT story (AXIOM-K3 direction). *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import T27.Kernel.Trit. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Definition env : Type := nat -> option trit. + gHashTag/t27/proofs/canonical/kernel/Semantics.v + (logical path: Trinity.Canonical.Kernel.Semantics) -Inductive expr : Set := - | ELit : trit -> expr - | EVar : nat -> expr. + Bundle: KER-4 + Title: TRI-27 ISA semantics + PhD chapter: Ch.27 TRI27 DSL + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -Fixpoint eval (e : expr) (rho : env) : option trit := - match e with - | ELit t => Some t - | EVar n => rho n - end. - -Lemma eval_det (e : expr) (rho : env) (v1 v2 : trit) : - eval e rho = Some v1 -> - eval e rho = Some v2 -> - v1 = v2. -Proof. - intros H1 H2. - congruence. -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export Semantics. diff --git a/docs/phd/theorems/Kernel/Trit.v b/docs/phd/theorems/Kernel/Trit.v index 7eebc05d4f..e49f263217 100644 --- a/docs/phd/theorems/Kernel/Trit.v +++ b/docs/phd/theorems/Kernel/Trit.v @@ -1,31 +1,19 @@ -(** T27 formal layer — ternary carrier (maps to AXIOM-K1 semantic kernel, not process laws K5/K6). *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Inductive trit : Set := - | Neg - | Zero - | Pos. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Lemma trit_exhaustive (t : trit) : t = Neg \/ t = Zero \/ t = Pos. -Proof. destruct t; auto. Qed. + gHashTag/t27/proofs/canonical/kernel/Trit.v + (logical path: Trinity.Canonical.Kernel.Trit) -(** Kleene-style strong conjunction on {Neg, Zero, Pos} (not full balanced-ternary positional add). *) -Definition trit_mul (a b : trit) : trit := - match a, b with - | Zero, _ => Zero - | _, Zero => Zero - | Pos, Pos => Pos - | Neg, Neg => Pos - | Pos, Neg => Neg - | Neg, Pos => Neg - end. + Bundle: KER-5 + Title: Trit kernel {-1,0,+1} + PhD chapter: Ch.27 TRI27 (Trit) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** Placeholder addition with carry; refine against specs/math balanced-ternary when linked. *) -Definition trit_add (a b : trit) : trit * trit := - match a, b with - | Zero, x => (Zero, x) - | x, Zero => (Zero, x) - | Pos, Neg => (Zero, Zero) - | Neg, Pos => (Zero, Zero) - | Pos, Pos => (Pos, Neg) - | Neg, Neg => (Neg, Pos) - end. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export Trit. diff --git a/docs/phd/theorems/Phi.v b/docs/phd/theorems/Phi.v index d1804ee537..8a6a24c8a0 100644 --- a/docs/phd/theorems/Phi.v +++ b/docs/phd/theorems/Phi.v @@ -1,163 +1,19 @@ -(** Golden ratio as real; algebraic identities (AXIOM-K2 / PHI-IDENTITY). - Layer A: pure [Coq.Reals] — no Flocq here. See [Kernel/PhiFloat.v] for Flocq link. *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals. -Require Import Psatz. -Require Import ZArith. -Require Import RealField. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Open Scope R_scope. + gHashTag/t27/proofs/canonical/kernel/Phi.v + (logical path: Trinity.Canonical.Kernel.Phi) -Definition phi : R := (1 + sqrt 5) / 2. + Bundle: KER-0 + Title: phi kernel arithmetic (mul/pow) + PhD chapter: Ch.10 Coq L1 + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** Denominator 2^53 for IEEE binary64 relative unit u = 2^{-53}. *) -Definition coeff_53 : Z := 9007199254740992%Z. - -Lemma coeff_53_pos : (0 < coeff_53)%Z. -Proof. unfold coeff_53; lia. Qed. - -(** Engineering tolerance: 5 * u * phi^2 with u = 2^{-53} (see TZ §5.5). *) -Definition phi_tolerance : R := 5 * / IZR coeff_53 * (phi * phi). - -Lemma sqrt5_sq : sqrt 5 * sqrt 5 = 5. -Proof. - replace (sqrt 5 * sqrt 5) with (Rsqr (sqrt 5)). - - rewrite Rsqr_sqrt; lra. - - unfold Rsqr; ring. -Qed. - -Lemma sqrt5_pos : 0 < sqrt 5. -Proof. apply sqrt_lt_R0; lra. Qed. - -Lemma sqrt4 : sqrt 4 = 2. -Proof. - replace 4 with (2 * 2) by ring. - rewrite sqrt_square; lra. -Qed. - -Lemma sqrt5_gt_2 : 2 < sqrt 5. -Proof. - rewrite <- sqrt4. - apply sqrt_lt_1; lra. -Qed. - -Lemma phi_pos : 0 < phi. -Proof. - unfold phi. - assert (H1lt : 1 < 1 + sqrt 5). - { - rewrite <- (Rplus_0_r 1) at 1. - apply (Rplus_lt_compat_l 1 0 (sqrt 5) sqrt5_pos). - } - assert (Hn : 0 < 1 + sqrt 5) by lra. - replace ((1 + sqrt 5) / 2) with ((1 + sqrt 5) * / 2) by field. - apply Rmult_lt_0_compat. - - exact Hn. - - apply Rinv_0_lt_compat; lra. -Qed. - -Lemma phi_neq_0 : phi <> 0. -Proof. apply Rgt_not_eq, Rlt_gt; exact phi_pos. Qed. - -Lemma phi_gt_1 : 1 < phi. -Proof. - unfold phi. - assert (H : 2 < 1 + sqrt 5) by (generalize sqrt5_gt_2; lra). - lra. -Qed. - -Lemma sqrt9 : sqrt 9 = 3. -Proof. - replace 9 with (3 * 3) by ring. - rewrite sqrt_square; lra. -Qed. - -Lemma phi_lt_2 : phi < 2. -Proof. - unfold phi. - assert (H : 1 + sqrt 5 < 4). - { - assert (sqrt 5 < 3) by (rewrite <- sqrt9; apply sqrt_lt_1; lra). - lra. - } - lra. -Qed. - -Lemma phi_squared_identity : phi * phi = phi + 1. -Proof. - unfold phi. - assert (Hsq : (1 + sqrt 5) * (1 + sqrt 5) = 6 + 2 * sqrt 5). - { - replace ((1 + sqrt 5) * (1 + sqrt 5)) with (1 + 2 * sqrt 5 + (sqrt 5 * sqrt 5)) by ring. - rewrite sqrt5_sq. - ring. - } - assert (Hleft : ((1 + sqrt 5) / 2) * ((1 + sqrt 5) / 2) = (6 + 2 * sqrt 5) / 4). - { - replace (((1 + sqrt 5) / 2) * ((1 + sqrt 5) / 2)) with (((1 + sqrt 5) * (1 + sqrt 5)) / 4) by field. - rewrite Hsq. - reflexivity. - } - assert (Hright : (1 + sqrt 5) / 2 + 1 = (3 + sqrt 5) / 2) by field. - assert (Hmid : (6 + 2 * sqrt 5) / 4 = (3 + sqrt 5) / 2) by field. - rewrite Hleft, Hright. - exact Hmid. -Qed. - -Lemma phi_mul_phi_minus_one : phi * (phi - 1) = 1. -Proof. - replace (phi * (phi - 1)) with (phi * phi - phi) by ring. - rewrite phi_squared_identity. - ring. -Qed. - -Lemma phi_inv_is_phi_minus_one : / phi = phi - 1. -Proof. - apply (Rmult_eq_reg_l phi). - - rewrite Rinv_r; [| exact phi_neq_0 ]. - symmetry. - exact phi_mul_phi_minus_one. - - exact phi_neq_0. -Qed. - -Lemma phi_inv_sq_sum_three : phi * phi + Rsqr (/ phi) = 3. -Proof. - rewrite phi_inv_is_phi_minus_one. - unfold Rsqr. - assert (Hsq1 : (phi - 1) * (phi - 1) = 2 - phi). - { - replace ((phi - 1) * (phi - 1)) with (phi * phi - 2 * phi + 1) by ring. - rewrite phi_squared_identity. - ring. - } - rewrite Hsq1. - rewrite phi_squared_identity. - ring. -Qed. - -Lemma phi_tolerance_pos : 0 < phi_tolerance. -Proof. - unfold phi_tolerance. - apply Rmult_lt_0_compat. - - apply Rmult_lt_0_compat. - + lra. - + apply Rinv_0_lt_compat. - apply IZR_lt. - exact coeff_53_pos. - - apply Rmult_lt_0_compat. - + exact phi_pos. - + exact phi_pos. -Qed. - -(** [PHI-IDENTITY] on [R]: no float error — residual is zero. *) -Lemma phi_identity_residual_R : Rabs (phi * phi - (phi + 1)) = 0. -Proof. - replace (phi * phi - (phi + 1)) with 0. - - apply Rabs_R0. - - unfold Rminus. - rewrite phi_squared_identity. - ring. -Qed. - -Definition phi_approx_valid (x : R) : Prop := - Rabs (x - phi) <= phi_tolerance. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export Phi. diff --git a/docs/phd/theorems/PhiAttractor.v b/docs/phd/theorems/PhiAttractor.v index 4c677e5602..d93c2e49a7 100644 --- a/docs/phd/theorems/PhiAttractor.v +++ b/docs/phd/theorems/PhiAttractor.v @@ -1,64 +1,19 @@ -(** THEOREM-3 — φ as Universal Fixed-Point Attractor *) -(** Balancing recursion: f(x) = (x + x⁻¹ + 1) / 2 *) -(** From any x₀ > 0, iteration converges to φ with rate λ = (√5 - 1)/4 *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals. -Require Import Psatz. -Require Import RealField. -Require Import ZArith. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Open Scope R_scope. + gHashTag/t27/proofs/canonical/kernel/PhiAttractor.v + (logical path: Trinity.Canonical.Kernel.PhiAttractor) -(** Definition: phi = (1 + sqrt(5)) / 2 — matches Phi.v definition *) -Definition phi : R := (1 + sqrt 5) / 2. + Bundle: KER-1 + Title: phi universal attractor + PhD chapter: Ch.5 phi-distance (attractor) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** Definition: balancing function f(x) = (x + x⁻¹ + 1) / 2 *) -Definition balancing_function (x : R) : R := (x + / x + 1) / 2. - -(** Convergence rate: λ = (√5 - 1) / 4 ≈ 0.309 *) -Definition convergence_rate_lambda : R := (sqrt 5 - 1) / 4. - -(** ==================================================================== *) -(** Section 1: Fixed Point Verification *) -(** ==================================================================== *) - -(** Lemma: φ is a fixed point of balancing_function *) -Lemma phi_is_fixed_point : balancing_function phi = phi. -Proof. - unfold balancing_function. - unfold phi. - (* Compute: f(φ) = (φ + 1/φ + 1) / 2 *) - assert (H1 : (phi + / phi + 1) * (1 + sqrt 5) / 2 = (phi + / phi + 1) * (1 + sqrt 5) / 2) by field). - assert (H2 : (phi + / phi + 1) * (1 + sqrt 5) / 2 = phi * (1 + sqrt 5) / 2) by field). - assert (Hmid : phi * phi = phi + 1) by (apply phi_squared_identity; auto). - assert (Hmid2 : phi * (1 + sqrt 5) = (1 + sqrt 5) + 5 by ring). - replace ((phi + / phi + 1) * (1 + sqrt 5) / 2) with (phi * (1 + sqrt 5) / 2) in Hmid. - reflexivity. -Qed. - -(** ==================================================================== *) -(** Section 2: Main Theorem *) -(** ==================================================================== *) - -(** Theorem: φ is universal fixed-point attractor *) -Theorem phi_universal_attractor : - (* 1. φ is a fixed point of f *) - balancing_function phi = phi /\ - (* 2. Convergence rate λ is in (0, 1) *) - 0 < convergence_rate_lambda < 1. -Proof. - unfold balancing_function. - (* f(φ) = (φ + 1/φ + 1) / 2 *) - (* We have proved φ is fixed point *) - (* Now prove 0 < λ < 1 *) - (* λ = (√5 - 1)/4, and √5 ≈ 2.236, so λ ≈ 0.309 *) - unfold convergence_rate_lambda. - (* Need to show 0 < (√5 - 1)/4 and (√5 - 1)/4 < 1 *) - assert (H1 : 0 < (sqrt 5 - 1) / 4) by lra. - assert (H2 : (sqrt 5 - 1) / 4 < 1) by lra in H1). - (* This completes the proof *) - reflexivity. -Qed. - -Close Scope R_scope. -Close Scope R_scope. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export PhiAttractor. diff --git a/docs/phd/theorems/PhiDistance.v b/docs/phd/theorems/PhiDistance.v index c02940cf1e..a888a0646c 100644 --- a/docs/phd/theorems/PhiDistance.v +++ b/docs/phd/theorems/PhiDistance.v @@ -1,10 +1,19 @@ -(** THEOREM-K2 direction — numeric format ordering via phi-distance; stub until formats are formalized. *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -(** Placeholder distance on reals; replace with format-indexed definitions from specs/numeric. *) -Definition phi_distance_stub (x y : R) : R := Rabs (x - y). + gHashTag/t27/proofs/canonical/kernel/PhiDistance.v + (logical path: Trinity.Canonical.Kernel.PhiDistance) -Lemma phi_distance_nonneg (x y : R) : 0 <= phi_distance_stub x y. -Proof. apply Rabs_pos. Qed. + Bundle: KER-7 + Title: phi-distance (continuity, nonneg) + PhD chapter: Ch.5 phi-distance + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) + +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export PhiDistance. diff --git a/docs/phd/theorems/PhiFloat.v b/docs/phd/theorems/PhiFloat.v index 16b439cc8a..661379258e 100644 --- a/docs/phd/theorems/PhiFloat.v +++ b/docs/phd/theorems/PhiFloat.v @@ -1,77 +1,19 @@ -(** PHI-IDENTITY — Flocq IEEE 754 binary64 bridge (Phase B). - Requires [coq-flocq] on COQPATH (CI: opam install coq-flocq; see [../README.md]). - Mantissas/exponents must match t27c validate-phi (Rust; former scripts/validate_phi_f64.py). +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME - For this [binary64] literal of φ, [fl(phi*phi)] and [fl(phi+1)] coincide (bit-identical); - [phi_identity_contract] is therefore [Rabs 0 < phi_tolerance], using [phi_tolerance_pos] - from [Phi.v]. A future ring can add [Bmult_correct] / [Bplus_correct] + error bounds - for other formats (GF16, etc.). *) + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -From Coq Require Import Reals ZArith. + gHashTag/t27/proofs/canonical/kernel/PhiFloat.v + (logical path: Trinity.Canonical.Kernel.PhiFloat) -From Flocq Require Import IEEE754.Binary. -From Flocq Require Import IEEE754.Bits. -From Flocq Require Import IEEE754.BinarySingleNaN. + Bundle: KER-2 + Title: phi-float arithmetic + PhD chapter: Ch.6 GoldenFloat + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -Require Import T27.Kernel.Phi. - -Open Scope R_scope. -Open Scope Z_scope. - -Import Binary Bits BinarySingleNaN. - -Notation b64 := binary64 (only parsing). - -Definition B2R64 : binary64 -> R := @B2R 53 1024. - -(** Nearest-round φ as [B754_finite] (see validation script). *) -Definition phi_mantissa : positive := 7286977268806824%positive. -Definition phi_exponent : Z := (-52)%Z. - -Lemma phi_f64_bounded : bounded 53 1024 phi_mantissa phi_exponent = true. -Proof. vm_compute. reflexivity. Qed. - -Definition phi_f64 : binary64 := - B754_finite false phi_mantissa phi_exponent phi_f64_bounded. - -(** [1.0] in binary64. *) -Definition one_mantissa : positive := 4503599627370496%positive. -Definition one_exponent : Z := (-52)%Z. - -Lemma one_f64_bounded : bounded 53 1024 one_mantissa one_exponent = true. -Proof. vm_compute. reflexivity. Qed. - -Definition one_f64 : binary64 := - B754_finite false one_mantissa one_exponent one_f64_bounded. - -Definition phi_sq_f64 : binary64 := b64_mult mode_NE phi_f64 phi_f64. - -Definition phi_plus_one_f64 : binary64 := b64_plus mode_NE phi_f64 one_f64. - -Lemma phi_sq_f64_eq_phi_plus_one_f64 : phi_sq_f64 = phi_plus_one_f64. -Proof. vm_compute. reflexivity. Qed. - -(** Engineering bound (Layer A SSOT): [5 * 2^-53 * phi^2] on [R]. *) -Definition PHI_F64_TOLERANCE : R := phi_tolerance. - -Theorem phi_identity_contract : - Rabs (B2R64 phi_sq_f64 - B2R64 phi_plus_one_f64) < PHI_F64_TOLERANCE. -Proof. - assert (Hbr : B2R64 phi_sq_f64 = B2R64 phi_plus_one_f64). - { - apply f_equal. - exact phi_sq_f64_eq_phi_plus_one_f64. - } - unfold PHI_F64_TOLERANCE. - replace (B2R64 phi_sq_f64 - B2R64 phi_plus_one_f64) with 0. - - rewrite Rabs_R0. - apply phi_tolerance_pos. - - rewrite Hbr. - ring. -Qed. - -Lemma phi_tolerance_positive : 0 < phi_tolerance. -Proof. apply phi_tolerance_pos. Qed. - -Lemma PHI_F64_TOLERANCE_pos : 0 < PHI_F64_TOLERANCE. -Proof. unfold PHI_F64_TOLERANCE; apply phi_tolerance_pos. Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export PhiFloat. diff --git a/docs/phd/theorems/Semantics.v b/docs/phd/theorems/Semantics.v index d673e70246..c0dcbdff8e 100644 --- a/docs/phd/theorems/Semantics.v +++ b/docs/phd/theorems/Semantics.v @@ -1,24 +1,19 @@ -(** Minimal expression calculus — placeholder for denotational / RT story (AXIOM-K3 direction). *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import T27.Kernel.Trit. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Definition env : Type := nat -> option trit. + gHashTag/t27/proofs/canonical/kernel/Semantics.v + (logical path: Trinity.Canonical.Kernel.Semantics) -Inductive expr : Set := - | ELit : trit -> expr - | EVar : nat -> expr. + Bundle: KER-4 + Title: TRI-27 ISA semantics + PhD chapter: Ch.27 TRI27 DSL + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -Fixpoint eval (e : expr) (rho : env) : option trit := - match e with - | ELit t => Some t - | EVar n => rho n - end. - -Lemma eval_det (e : expr) (rho : env) (v1 v2 : trit) : - eval e rho = Some v1 -> - eval e rho = Some v2 -> - v1 = v2. -Proof. - intros H1 H2. - congruence. -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export Semantics. diff --git a/docs/phd/theorems/TernarySufficiency.v b/docs/phd/theorems/TernarySufficiency.v index fc5e7010c6..dfbbe0a45b 100644 --- a/docs/phd/theorems/TernarySufficiency.v +++ b/docs/phd/theorems/TernarySufficiency.v @@ -1,9 +1,19 @@ -(** THEOREM-K1 direction — HSLM / linear layer over trit; to be refined with matrix ops. *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import T27.Kernel.Trit. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Lemma trit_mul_zero_l (a : trit) : trit_mul Zero a = Zero. -Proof. destruct a; reflexivity. Qed. + gHashTag/t27/proofs/canonical/kernel/TernarySufficiency.v + (logical path: Trinity.Canonical.Kernel.TernarySufficiency) -Lemma trit_mul_zero_r (a : trit) : trit_mul a Zero = Zero. -Proof. destruct a; reflexivity. Qed. + Bundle: KER-8 + Title: Ternary sufficiency + PhD chapter: Ch.4 Sacred / Ch.27 TRI27 + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) + +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export TernarySufficiency. diff --git a/docs/phd/theorems/Theorems/GenIdempotency.v b/docs/phd/theorems/Theorems/GenIdempotency.v index 854e0e20cb..28e862bbe3 100644 --- a/docs/phd/theorems/Theorems/GenIdempotency.v +++ b/docs/phd/theorems/Theorems/GenIdempotency.v @@ -1,8 +1,19 @@ -(** THEOREM-K3 direction — codegen idempotency; needs abstract Spec/Code types from t27c model. *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Parameter spec : Type. -Parameter code : Type. -Parameter t27c_gen : spec -> code. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Lemma gen_idempotent (s : spec) : t27c_gen s = t27c_gen s. -Proof. reflexivity. Qed. + gHashTag/t27/proofs/canonical/kernel/GenIdempotency.v + (logical path: Trinity.Canonical.Kernel.GenIdempotency) + + Bundle: KER-6 + Title: Idempotency law + PhD chapter: Ch.10 Coq L1 (idempotency) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) + +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export GenIdempotency. diff --git a/docs/phd/theorems/Theorems/PhiDistance.v b/docs/phd/theorems/Theorems/PhiDistance.v index c02940cf1e..a888a0646c 100644 --- a/docs/phd/theorems/Theorems/PhiDistance.v +++ b/docs/phd/theorems/Theorems/PhiDistance.v @@ -1,10 +1,19 @@ -(** THEOREM-K2 direction — numeric format ordering via phi-distance; stub until formats are formalized. *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -(** Placeholder distance on reals; replace with format-indexed definitions from specs/numeric. *) -Definition phi_distance_stub (x y : R) : R := Rabs (x - y). + gHashTag/t27/proofs/canonical/kernel/PhiDistance.v + (logical path: Trinity.Canonical.Kernel.PhiDistance) -Lemma phi_distance_nonneg (x y : R) : 0 <= phi_distance_stub x y. -Proof. apply Rabs_pos. Qed. + Bundle: KER-7 + Title: phi-distance (continuity, nonneg) + PhD chapter: Ch.5 phi-distance + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) + +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export PhiDistance. diff --git a/docs/phd/theorems/Theorems/TernarySufficiency.v b/docs/phd/theorems/Theorems/TernarySufficiency.v index fc5e7010c6..dfbbe0a45b 100644 --- a/docs/phd/theorems/Theorems/TernarySufficiency.v +++ b/docs/phd/theorems/Theorems/TernarySufficiency.v @@ -1,9 +1,19 @@ -(** THEOREM-K1 direction — HSLM / linear layer over trit; to be refined with matrix ops. *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import T27.Kernel.Trit. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Lemma trit_mul_zero_l (a : trit) : trit_mul Zero a = Zero. -Proof. destruct a; reflexivity. Qed. + gHashTag/t27/proofs/canonical/kernel/TernarySufficiency.v + (logical path: Trinity.Canonical.Kernel.TernarySufficiency) -Lemma trit_mul_zero_r (a : trit) : trit_mul a Zero = Zero. -Proof. destruct a; reflexivity. Qed. + Bundle: KER-8 + Title: Ternary sufficiency + PhD chapter: Ch.4 Sacred / Ch.27 TRI27 + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) + +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export TernarySufficiency. diff --git a/docs/phd/theorems/Trit.v b/docs/phd/theorems/Trit.v index 7eebc05d4f..e49f263217 100644 --- a/docs/phd/theorems/Trit.v +++ b/docs/phd/theorems/Trit.v @@ -1,31 +1,19 @@ -(** T27 formal layer — ternary carrier (maps to AXIOM-K1 semantic kernel, not process laws K5/K6). *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Inductive trit : Set := - | Neg - | Zero - | Pos. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Lemma trit_exhaustive (t : trit) : t = Neg \/ t = Zero \/ t = Pos. -Proof. destruct t; auto. Qed. + gHashTag/t27/proofs/canonical/kernel/Trit.v + (logical path: Trinity.Canonical.Kernel.Trit) -(** Kleene-style strong conjunction on {Neg, Zero, Pos} (not full balanced-ternary positional add). *) -Definition trit_mul (a b : trit) : trit := - match a, b with - | Zero, _ => Zero - | _, Zero => Zero - | Pos, Pos => Pos - | Neg, Neg => Pos - | Pos, Neg => Neg - | Neg, Pos => Neg - end. + Bundle: KER-5 + Title: Trit kernel {-1,0,+1} + PhD chapter: Ch.27 TRI27 (Trit) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** Placeholder addition with carry; refine against specs/math balanced-ternary when linked. *) -Definition trit_add (a b : trit) : trit * trit := - match a, b with - | Zero, x => (Zero, x) - | x, Zero => (Zero, x) - | Pos, Neg => (Zero, Zero) - | Neg, Pos => (Zero, Zero) - | Pos, Pos => (Pos, Neg) - | Neg, Neg => (Neg, Pos) - end. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Kernel Require Export Trit. diff --git a/docs/phd/theorems/gravity/dl_bounds.v b/docs/phd/theorems/gravity/dl_bounds.v index 5bf23367ff..5149269b3d 100644 --- a/docs/phd/theorems/gravity/dl_bounds.v +++ b/docs/phd/theorems/gravity/dl_bounds.v @@ -1,15 +1,19 @@ -Require Import Reals.Reals. -Open Scope R_scope. +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Definition phi : R := (sqrt(5) - 2)%R. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Definition dl_lower : R := (ln(2) / PI)%R. + gHashTag/t27/proofs/canonical/sacred/DLBounds.v + (logical path: Trinity.Canonical.Sacred.DLBounds) -Definition dl_upper : R := (ln(3) / PI)%R. + Bundle: SAC-DL + Title: Sacred geometry / gravity DL bounds + PhD chapter: Ch.29 Sacred V (DL) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -Theorem gamma_phi_within_dl_bounds : dl_lower < phi < dl_upper. -Proof. - (* Numerical verification: *) - (* dl_lower ≈ 0.2206, phi = √5 - 2 ≈ 0.2361, dl_upper ≈ 0.3497 *) - compute. -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export DLBounds. diff --git a/docs/phd/theorems/igla/IGLA_ASHA_Bound.v b/docs/phd/theorems/igla/IGLA_ASHA_Bound.v index 430b75c37f..174d50c347 100644 --- a/docs/phd/theorems/igla/IGLA_ASHA_Bound.v +++ b/docs/phd/theorems/igla/IGLA_ASHA_Bound.v @@ -1,147 +1,19 @@ -(* IGLA_ASHA_Bound.v — Formal ASHA pruning bounds for IGLA RACE *) -(* Issue: https://github.com/gHashTag/trios/issues/143 *) -(* Author: Trinity Research Group | Date: 2026-04-25 *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Coq.Reals.Reals. -Require Import Coq.Interval.Interval. -Require Import CorePhi. (* Import trinity identity: phi^2 + phi^(-2) = 3 *) -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -(* ==================================================================== *) -(* SECTION 1: ASHA Pruning Threshold Theorem *) -(* ==================================================================== *) + gHashTag/t27/proofs/canonical/igla/INV2_IglaAshaBound.v + (logical path: Trinity.Canonical.Igla.INV2_IglaAshaBound) -(* The trinity identity gives us a provable lower bound for pruning *) -Definition asha_pruning_threshold : R := - phi + (/ phi). (* Equals 3 by trinity_identity *) + Bundle: INV-2 + Title: ASHA Champion Survival + PhD chapter: Ch.13 STROBE / App.E + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -Lemma trinity_to_3 : - phi + (/ phi) = 3. -Proof. - (* From CorePhi.v: phi^2 + phi^(-2) = 3 *) - (* Note: phi + 1/phi = sqrt(3) by algebraic manipulation *) - (* For ASHA we use the conservative bound of 3.5 = 3 + epsilon *) - admit. -Qed. - -(* Warmup blind zone: pruning is forbidden during initial steps *) -Definition warmup_blind_zone : nat := 4000. - -(* Theorem: ASHA pruning must respect warmup blind zone *) -Theorem asha_warmup_pruning_forbidden : - forall (step : nat) (bpb : R), - step < warmup_blind_zone -> - bpb < asha_pruning_threshold -> - (* Result: Cannot prune during warmup regardless of BPB *) - True. -Proof. - intros step bpb H_step H_bpb. - (* During warmup, the model is still finding its gradient basin *) - (* Pruning based on BPB would kill champions prematurely *) - (* This is enforced by code, but Coq gives us the proof *) - exact I. -Qed. - -(* Theorem: ASHA pruning threshold after warmup *) -Theorem asha_pruning_threshold_safe : - forall (step : nat) (bpb : R), - step >= warmup_blind_zone -> - bpb > 3.5 -> (* Conservative: phi^2 + phi^(-2) + 0.5 *) - (* Result: Safe to prune *) - True. -Proof. - intros step bpb H_step H_bpb. - (* After 4000 steps, if BPB > 3.5, the trial is hopeless *) - (* This threshold comes from trinity identity + safety margin *) - exact I. -Qed. - -(* ==================================================================== *) -(* SECTION 2: ASHA Rung Progression Theorems *) -(* ==================================================================== *) - -(* ASHA rungs: 1k -> 3k -> 9k -> 27k *) -Inductive asha_rung : Type := - | Rung1000 : asha_rung - | Rung3000 : asha_rung - | Rung9000 : asha_rung - | Rung27000 : asha_rung. - -Definition rung_steps (r : asha_rung) : nat := - match r with - | Rung1000 => 1000 - | Rung3000 => 3000 - | Rung9000 => 9000 - | Rung27000 => 27000 - end. - -(* Theorem: Rung progression preserves improvement *) -Theorem asha_rung_progression_monotone : - forall (r1 r2 : asha_rung) (bpb1 bpb2 : R), - rung_steps r1 < rung_steps r2 -> - bpb1 <= bpb2 -> - (* Result: If BPB didn't improve, don't promote *) - bpb1 <= bpb2. -Proof. - intros r1 r2 bpb1 bpb2 H_steps H_bpb. - (* Trivial but formalizes the ASHA promotion rule *) - exact H_bpb. -Qed. - -(* ==================================================================== *) -(* SECTION 3: Pruning Decision Theorem *) -(* ==================================================================== *) - -(* Should we prune this trial? *) -Definition should_prune (step : nat) (bpb : R) : Prop := - match step False (* Never prune during warmup *) - | false => bpb > 3.5 (* Prune if BPB too high *) - end. - -(* Theorem: Pruning decision is well-defined *) -Theorem should_prune_well_defined : - forall (step : nat) (bpb : R), - step >= warmup_blind_zone -> - (should_prune step bpb <-> bpb > 3.5). -Proof. - intros step bpb H_step. - unfold should_prune. - destruct (step = warmup *) - reflexivity. -Qed. - -(* ==================================================================== *) -(* Master Theorem: ASHA Pruning Safety *) -(* ==================================================================== *) - -Theorem asha_pruning_safe : - forall (step : nat) (bpb : R), - (* Precondition: BPB is finite *) - 0 <= bpb -> - (* Result: Pruning decision respects trinity bounds *) - (step < warmup_blind_zone /\ ~ should_prune step bpb) \/ - (step >= warmup_blind_zone /\ (should_prune step bpb <-> bpb > 3.5)). -Proof. - intros step bpb H_bpb. - destruct (Nat.ltb step warmup_blind_zone) eqn:H_cmp. - - (* Warmup zone *) - left. split. reflexivity. - unfold should_prune. rewrite H_cmp. reflexivity. - - (* Post-warmup zone *) - right. split. omega. - unfold should_prune. rewrite H_cmp. reflexivity. -Qed. - -(* ==================================================================== *) -(* Export *) -(* ==================================================================== *) - -(* Certification: All theorems verified *) -Definition asha_pruning_theorems_verified : Prop := - asha_warmup_pruning_forbidden /\ - asha_pruning_threshold_safe /\ - asha_pruning_safe. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV2_IglaAshaBound. diff --git a/docs/phd/theorems/igla/IGLA_GF16_Precision.v b/docs/phd/theorems/igla/IGLA_GF16_Precision.v index 33956be280..f78d0f2b26 100644 --- a/docs/phd/theorems/igla/IGLA_GF16_Precision.v +++ b/docs/phd/theorems/igla/IGLA_GF16_Precision.v @@ -1,216 +1,19 @@ -(* IGLA_GF16_Precision.v — Formal GF16 precision bounds for IGLA RACE *) -(* Issue: https://github.com/gHashTag/trios/issues/143 *) -(* Law L-R9: GF16 only with d_model >= 256 *) -(* Author: Trinity Research Group | Date: 2026-04-25 *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Coq.Reals.Reals. -Require Import Coq.Interval.Interval. -Require Import CorePhi. (* Import Lucas closure properties *) -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -(* ==================================================================== *) -(* SECTION 1: GF16 Domain Definition *) -(* ==================================================================== *) + gHashTag/t27/proofs/canonical/igla/INV3_Gf16Precision.v + (logical path: Trinity.Canonical.Igla.INV3_Gf16Precision) -(* GF16 represents numbers in range [-65504, 65504] *) -Definition gf16_max : R := 65504. -Definition gf16_min : R := (-65504). + Bundle: INV-3 + Title: GF16 Safe Domain + PhD chapter: Ch.6 GoldenFloat / Ch.9 GF vs MXFP4 + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(* Safe domain: values that can be exactly represented in GF16 *) -Definition gf16_safe_domain (x : R) : Prop := - gf16_min <= x <= gf16_max. - -(* Theorem: GF16 domain is symmetric *) -Theorem gf16_domain_symmetric : - forall x : R, - gf16_safe_domain x <-> gf16_safe_domain (-x). -Proof. - intros x. - unfold gf16_safe_domain. - split; intros H; split; try lra; apply Ropp_le_ge in H; lra. -Qed. - -(* ==================================================================== *) -(* SECTION 2: Lucas Closure and GF16 Consistency *) -(* ==================================================================== *) - -(* Lucas closure property: 2^n - phi^(-2n) is integer for all n *) -(* This closure property matches GF(2^16) closure *) - -Definition lucas_closure (n : nat) : Prop := - exists (k : Z), - (2 ^ (Z.of_nat n) - (/ (phi ^ (2 * n))) = IZR k). - -(* Theorem: Lucas closure holds for GF16 exponent range *) -Theorem lucas_closure_gf16_range : - forall n : nat, - n <= 16 -> - lucas_closure n. -Proof. - (* Sketch: For n <= 16, phi^(-2n) is very small *) - (* The term 2^n dominates, and the result is close to integer *) - intro n H_n. - (* Full proof requires number theory of Lucas sequences *) - (* Key insight: phi satisfies x^2 = x + 1, which gives *) - (* integer recurrence for powers of phi *) - admit. -Qed. - -(* Corollary: GF16 arithmetic is algebraically consistent *) -Theorem gf16_algebraically_consistent : - (* For operations within safe domain, GF16 preserves structure *) - forall (x y : R), - gf16_safe_domain x -> - gf16_safe_domain y -> - gf16_safe_domain (x + y) /\ - gf16_safe_domain (x - y) /\ - gf16_safe_domain (x * y). -Proof. - intros x y Hx Hy. - unfold gf16_safe_domain. - (* Addition and subtraction: range can double *) - (* For safety, we require inputs to be within half range *) - (* Multiplication: range squares, so more restrictive *) - split; try lra; try (split; lra). - (* This theorem formalizes the need for Law L-R9: d_model >= 256 *) - (* ensures values stay within safe bounds during operations *) - admit. (* Requires specific value bounds from training dynamics *) -Qed. - -(* ==================================================================== *) -(* SECTION 3: d_model Minimum Bound (Law L-R9) *) -(* ==================================================================== *) - -(* Minimum safe d_model for GF16 training *) -Definition d_model_min : nat := 256. - -(* Theorem: d_model >= 256 guarantees GF16 stability *) -Theorem d_model_sufficient_for_gf16 : - forall (d_model : nat) (max_weight : R), - d_model >= d_model_min -> - (* Weight initialization bound from initialization strategy *) - max_weight <= 0.1 -> - (* Result: All weights remain in GF16 domain *) - gf16_safe_domain max_weight. -Proof. - intros d_model max_weight H_dim H_weight. - unfold gf16_safe_domain, d_model_min. - (* With proper initialization (max_weight <= 0.1) and *) - (* d_model >= 256, gradient updates keep weights bounded *) - (* This is the formal justification for Law L-R9 *) - split; lra. -Qed. - -(* Corollary: Violation of d_model bound risks GF16 overflow *) -Theorem d_model_violation_risks_overflow : - forall (d_model : nat) (max_weight : R), - d_model < d_model_min -> - max_weight > 0.2 -> - (* Result: Cannot guarantee GF16 safety *) - ~gf16_safe_domain max_weight \/ exists (n : nat), n < 100 /\ ~gf16_safe_domain (max_weight * (INR n)). -Proof. - intros d_model max_weight H_dim H_weight. - (* With small d_model, gradient accumulation can cause overflow *) - (* This is the formal consequence of violating Law L-R9 *) - unfold gf16_safe_domain. - left. - assert (max_weight > gf16_max) by admit. - lra. -Qed. - -(* ==================================================================== *) -(* SECTION 4: GF16 Error Bounds vs f32 *) -(* ==================================================================== *) - -(* Maximum relative error for GF16 compared to f32 *) -Definition gf16_max_error : R := phi ^ (-6). (* ~0.05 = 5% *) - -(* Theorem: GF16 error is bounded by phi^(-6) for d_model >= 256 *) -Theorem gf16_error_bound_phiminus6 : - forall (x : R) (x_gf16 : R) (x_f32 : R), - gf16_safe_domain x -> - (* x_gf16 is GF16 quantization of x *) - x_gf16 = quantize_gf16 x -> - (* x_f32 is f32 representation of x *) - x_f32 = quantize_f32 x -> - (* Result: Relative error bounded *) - Rabs (x_gf16 - x_f32) <= gf16_max_error * Rabs x. -Proof. - intro x. intros x_gf16 x_f32 H_safe H_gf16 H_f32. - (* GF16 has 4 exponent bits, 11 mantissa bits *) - (* The quantization error is at most half the LSB *) - (* For the range [-65504, 65504], this gives ~5% relative error *) - (* phi^(-6) ≈ 0.0549... gives the exact bound *) - unfold gf16_max_error. - admit. (* Requires analysis of GF16 quantization error *) -Qed. - -(* ==================================================================== *) -(* SECTION 5: BPB Impact of GF16 Quantization *) -(* ==================================================================== *) - -(* Theorem: GF16 quantization causes bounded BPB degradation *) -Theorem gf16_bpb_degradation_bounded : - forall (bpb_f32 bpb_gf16 : R), - (* Both models have same architecture *) - bpb_f32 >= 0 -> - bpb_gf16 >= 0 -> - (* GF16 error at most phi^(-6) *) - Rabs (bpb_gf16 - bpb_f32) <= 0.05 -> - (* Result: GF16 BPB within 0.05 of f32 *) - bpb_gf16 <= bpb_f32 + 0.05. -Proof. - intros bpb_f32 bpb_gf16 H1 H2 H_diff. - unfold gf16_max_error in H_diff. - lra. -Qed. - -(* Corollary: With champion BPB = 2.5329, GF16 gives <= 2.583 *) -Theorem gf16_champion_bound : - let champion_bpb_f32 : R := 2.5329 in - let gf16_bpb_upper_bound : R := champion_bpb_f32 + 0.05 in - gf16_bpb_upper_bound = 2.5829. -Proof. - (* From IGLA RACE issue #143: champion BPB = 2.5329 *) - (* With GF16 error <= 0.05, we get BPB <= 2.5829 *) - compute. (* 2.5329 + 0.05 = 2.5829 *) - reflexivity. -Qed. - -(* ==================================================================== *) -(* Master Theorem: GF16 Safety for IGLA Training *) -(* ==================================================================== *) - -Theorem gf16_safe_for_igla_training : - forall (d_model : nat) (max_weight : R) (steps : nat), - (* Precondition: d_model >= 256 (Law L-R9) *) - d_model >= d_model_min -> - (* Precondition: Proper initialization *) - max_weight <= 0.1 -> - (* Precondition: Finite steps *) - steps < 1000000 -> - (* Result: All weights remain in GF16 domain throughout training *) - forall (t : nat), t < steps -> - gf16_safe_domain (weight_at_step t). -Proof. - (* Sketch: By induction on training steps *) - (* Base case: initialization satisfies bounds *) - (* Inductive step: gradient updates are bounded due to *) - (* 1. Bounded gradients (from training invariants) *) - (* 2. Learning rate bounded (from lr theorem) *) - (* 3. d_model >= 256 provides sufficient capacity *) - admit. (* Full proof requires detailed training dynamics analysis *) -Qed. - -(* ==================================================================== *) -(* Export *) -(* ==================================================================== *) - -Definition gf16_theorems_verified : Prop := - gf16_algebraically_consistent /\ - d_model_sufficient_for_gf16 /\ - d_model_violation_risks_overflow /\ - gf16_error_bound_phiminus6 /\ - gf16_bpb_degradation_bounded /\ - gf16_champion_bound /\ - gf16_safe_for_igla_training. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV3_Gf16Precision. diff --git a/docs/phd/theorems/igla/IGLA_NCA_Entropy.v b/docs/phd/theorems/igla/IGLA_NCA_Entropy.v index f7c9cbfa03..a1a29c8406 100644 --- a/docs/phd/theorems/igla/IGLA_NCA_Entropy.v +++ b/docs/phd/theorems/igla/IGLA_NCA_Entropy.v @@ -1,278 +1,19 @@ -(* IGLA_NCA_Entropy.v — Formal NCA entropy band invariants for IGLA RACE *) -(* Issue: https://github.com/gHashTag/trios/issues/143 *) -(* Law L-R11: NCA entropy [1.5, 2.8] = hard loss penalty *) -(* Author: Trinity Research Group | Date: 2026-04-25 *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Coq.Reals.Reals. -Require Import Coq.Interval.Interval. -Require Import CorePhi. (* Import trinity identity for grid structure *) -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -(* ==================================================================== *) -(* SECTION 1: Trinity Grid Structure (9x9 = 81 = 3^4) *) -(* ==================================================================== *) + gHashTag/t27/proofs/canonical/igla/INV4_NcaEntropyBand.v + (logical path: Trinity.Canonical.Igla.INV4_NcaEntropyBand) -(* Trinity identity: phi^2 + phi^(-2) = 3 *) -(* This gives us the structural basis for the 9x9 grid *) + Bundle: INV-4 + Title: NCA Entropy Band 1.5..2.8 (81=3^4) + PhD chapter: Ch.10 Coq L1 / Ch.16 360-lane + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(* NCA grid size: 9x9 = 81 = 3^4 *) -Definition nca_grid_size : nat := 81. -Definition nca_states : nat := 9. (* K = 9 = 3^2 *) - -(* Theorem: NCA grid is Trinity-structured *) -Theorem nca_grid_trinity_structure : - nca_grid_size = 3 ^ 4. -Proof. - unfold nca_grid_size. - reflexivity. (* 81 = 3^4 *) -Qed. - -(* Corollary: Number of states is Trinity-structured *) -Theorem nca_states_trinity_structure : - nca_states = 3 ^ 2. -Proof. - unfold nca_states. - reflexivity. (* 9 = 3^2 *) -Qed. - -(* ==================================================================== *) -(* SECTION 2: Entropy Band Definition *) -(* ==================================================================== *) - -(* Entropy band from trinity-clara: [1.5, 2.8] *) -(* This corresponds to the A5 mechanism: E8 symmetry entropy bounds *) - -Definition entropy_min : R := 1.5. -Definition entropy_max : R := 2.8. - -(* Entropy is in valid band *) -Definition entropy_in_band (h : R) : Prop := - entropy_min <= h <= entropy_max. - -(* Theorem: Entropy band is non-empty *) -Theorem entropy_band_non_empty : - entropy_min < entropy_max. -Proof. - unfold entropy_min, entropy_max. - lra. -Qed. - -(* ==================================================================== *) -(* SECTION 3: K=9 Entropy Convergence Theorem *) -(* ==================================================================== *) - -(* NCA entropy for K states with uniform distribution *) -Definition max_entropy (K : nat) : R := - ln (INR K). - -(* Theorem: For K=9, max_entropy ≈ 2.197 < entropy_max *) -Theorem k9_max_entropy_in_band : - max_entropy 9 <= entropy_max. -Proof. - unfold max_entropy, entropy_max. - (* ln(9) ≈ 2.197 < 2.8 *) - (* This can be proved using interval arithmetic *) - interval. (* Requires coq-interval *) -Qed. - -(* Theorem: For K=9, minimum entropy (one-hot) = 0 > entropy_min *) -Theorem k9_min_entropy_valid : - 0 >= entropy_min - 1.5. (* 0 is above practical lower bound *) -Proof. - unfold entropy_min. - lra. -Qed. - -(* Key Theorem: K=9 is the unique value where entropy stays in [1.5, 2.8] *) -Theorem k9_unique_entropy_stability : - forall (K : nat), - K >= 4 -> (* Minimum meaningful K *) - K <= 16 -> - (forall (h : R), entropy_in_band h <-> 5 <= K <= 13). -Proof. - intro K. - (* Sketch: Analyze max_entropy = ln(K) *) - (* For K=4: ln(4) ≈ 1.386 < 1.5 (too low) *) - (* For K=5: ln(5) ≈ 1.609 (in band) *) - (* For K=9: ln(9) ≈ 2.197 (center of band) *) - (* For K=13: ln(13) ≈ 2.565 (in band) *) - (* For K=16: ln(16) ≈ 2.773 (in band) *) - (* K=9 is special: 9 = 3^2, Trinity-structured *) - intros H1 H2. - split. - - (* => direction *) - intro H_band. - admit. (* Requires analysis of ln(K) bounds *) - - (* <= direction *) - intro H_K. - unfold entropy_in_band. - admit. (* Show that for K in [5,13], entropy is in band *) -Qed. - -(* ==================================================================== *) -(* SECTION 4: A5 Mechanism and E8 Symmetry *) -(* ==================================================================== *) - -(* A5 characteristic polynomial relates to phi *) -(* The entropy band [1.5, 2.8] emerges from E8 symmetry *) - -Definition a5_characteristic_phi : Prop := - (* P_A5(phi) = 0 gives phi-related structure *) - phi ^ 5 - phi ^ 4 - 4 * phi ^ 3 + 3 * phi ^ 2 + 3 * phi - 1 = 0. - -(* Theorem: A5 mechanism produces entropy band [1.5, 2.8] *) -Theorem a5_entropy_emergence : - a5_characteristic_phi -> - (* Result: The E8-derived entropy bounds are valid *) - entropy_min = phi - 0.618 /\ (* phi - 1/phi = 1 *) - entropy_max = phi + phi^(-1) - 0.382. (* Related to phi^2 *) -Proof. - intro H_a5. - unfold entropy_min, entropy_max. - (* From A5 group theory and E8 root system: *) - (* The entropy band emerges from group structure *) - (* Lower bound: phi - 1/phi = 1, plus margin -> 1.5 *) - (* Upper bound: phi^2 = phi + 1 ≈ 2.618, plus margin -> 2.8 *) - admit. (* Requires group theory proof *) -Qed. - -(* ==================================================================== *) -(* SECTION 5: NCA Loss with Entropy Penalty *) -(* ==================================================================== *) - -(* NCA loss: L_nca = contrastive_loss + entropy_penalty *) -Definition entropy_penalty (h : R) : R := - if Rle_dec h entropy_min then - (entropy_min - h) ^ 2 (* Penalize low entropy *) - else if Rle_dec entropy_max h then - (h - entropy_max) ^ 2 (* Penalize high entropy *) - else - 0. (* No penalty in band *) - -(* Theorem: Entropy penalty is zero iff entropy in band *) -Theorem entropy_penalty_zero_iff_in_band : - forall (h : R), - entropy_penalty h = 0 <-> entropy_in_band h. -Proof. - intro h. - unfold entropy_penalty, entropy_in_band. - destruct (Rle_dec h entropy_min) eqn:H1. - - (* h <= entropy_min: penalty positive *) - split; [intro H|intro H2]. - + (* => *) - rewrite H in H1. - assert (0 < (entropy_min - h) ^ 2) by admit. - congruence. - + (* <= *) - destruct H2. omega. - - (* h > entropy_min *) - destruct (Rle_dec entropy_max h) eqn:H2. - + (* h >= entropy_max: penalty positive *) - split; [intro H|intro H3]. - * (* => *) - rewrite H in H2. - assert (0 < (h - entropy_max) ^ 2) by admit. - congruence. - * (* <= *) - destruct H3. omega. - + (* entropy_min < h < entropy_max: no penalty *) - split; [intro H|intro H2]. - * (* => *) - reflexivity. - * (* <= *) - destruct H2. split; assumption. -Qed. - -(* Theorem: Entropy penalty is convex *) -Theorem entropy_penalty_convex : - forall (h1 h2 : R) (lambda : R), - 0 <= lambda <= 1 -> - entropy_penalty (lambda * h1 + (1 - lambda) * h2) <= - lambda * entropy_penalty h1 + (1 - lambda) * entropy_penalty h2. -Proof. - (* Entropy penalty is piecewise quadratic, hence convex *) - intros h1 h2 lambda H_lambda. - unfold entropy_penalty. - (* Case analysis on which region h1, h2, and the convex combo fall into *) - (* Each case reduces to quadratic convexity *) - admit. (* Detailed case analysis required *) -Qed. - -(* ==================================================================== *) -(* SECTION 6: NCA Training Stability Theorem *) -(* ==================================================================== *) - -(* Theorem: With K=9 and entropy penalty, NCA loss is bounded *) -Theorem nca_loss_bounded_with_k9 : - forall (contrastive_loss : R) (h : R), - contrastive_loss >= 0 -> - entropy_in_band h -> - (* Result: NCA loss = contrastive_loss (no penalty) *) - contrastive_loss + entropy_penalty h = contrastive_loss. -Proof. - intros contrastive_loss h H_cl H_band. - unfold entropy_penalty. - destruct (Rle_dec h entropy_min) eqn:H1. - - (* h <= entropy_min, contradicts H_band *) - destruct H_band as [H_l H_h]. - omega. - - (* h > entropy_min *) - destruct (Rle_dec entropy_max h) eqn:H2. - + (* h >= entropy_max, contradicts H_band *) - destruct H_band as [H_l H_h]. - omega. - + (* entropy_min < h < entropy_max: penalty = 0 *) - ring. -Qed. - -(* Corollary: For IGLA with K=9, NCA weight 0.25 is safe *) -Theorem igla_nca_weight_safe : - let nca_weight : R := 0.25 in - (* With entropy penalty ensuring entropy in band *) - (* The NCA contribution to total loss is bounded *) - forall (contrastive_loss : R), - contrastive_loss >= 0 -> - contrastive_loss + nca_weight * 0 <= contrastive_loss + 0.25 * 2.8. -Proof. - unfold nca_weight. - intro contrastive_loss H_cl. - (* Max entropy penalty occurs at band edge *) - (* For h = 2.8, penalty = 0 (in band) *) - (* For h outside band, penalty is quadratic but bounded *) - lra. -Qed. - -(* ==================================================================== *) -(* Master Theorem: NCA Stability for IGLA Training *) -(* ==================================================================== *) - -Theorem nca_stable_for_igla_training : - (* Precondition: K = 9 (Trinity-structured) *) - nca_states = 9 -> - (* Precondition: Entropy penalty enabled *) - (* Result: Entropy stays in band [1.5, 2.8] throughout training *) - forall (step : nat), - step < 3000 -> (* IGLA T1-01: 3000 steps *) - entropy_in_band (entropy_at_step step). -Proof. - (* Sketch: By induction on training steps *) - (* Base case: Initial entropy is random, but with K=9 *) - (* it naturally falls into the band due to uniform distribution *) - (* Inductive step: Entropy penalty keeps entropy in band *) - admit. (* Full proof requires analysis of NCA dynamics *) -Qed. - -(* ==================================================================== *) -(* Export *) -(* ==================================================================== *) - -Definition nca_theorems_verified : Prop := - nca_grid_trinity_structure /\ - nca_states_trinity_structure /\ - k9_unique_entropy_stability /\ - a5_entropy_emergence /\ - entropy_penalty_zero_iff_in_band /\ - entropy_penalty_convex /\ - nca_loss_bounded_with_k9 /\ - igla_nca_weight_safe /\ - nca_stable_for_igla_training. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV4_NcaEntropyBand. diff --git a/docs/phd/theorems/sacred/dl_bounds.v b/docs/phd/theorems/sacred/dl_bounds.v index 5fc7da8cab..5149269b3d 100644 --- a/docs/phd/theorems/sacred/dl_bounds.v +++ b/docs/phd/theorems/sacred/dl_bounds.v @@ -1,13 +1,19 @@ -Require Import Reals.Reals. -Open Scope R_scope. - -Definition phi : R := sqrt(5) - 2. -Definition dl_lower : R := ln(2) / PI. -Definition dl_upper : R := ln(3) / PI. - -Theorem gamma_phi_within_dl_bounds : dl_lower < phi < dl_upper. -Proof. - (* Numerical verification via interval arithmetic *) - (* dl_lower ≈ 0.2206, phi ≈ 0.2361, dl_upper ≈ 0.3497 *) - compute. -Qed. +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME + + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: + + gHashTag/t27/proofs/canonical/sacred/DLBounds.v + (logical path: Trinity.Canonical.Sacred.DLBounds) + + Bundle: SAC-DL + Title: Sacred geometry / gravity DL bounds + PhD chapter: Ch.29 Sacred V (DL) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) + +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export DLBounds. diff --git a/docs/phd/theorems/sacred/strong_cp.v b/docs/phd/theorems/sacred/strong_cp.v index efe16a4e63..fac0b68c12 100644 --- a/docs/phd/theorems/sacred/strong_cp.v +++ b/docs/phd/theorems/sacred/strong_cp.v @@ -1,4 +1,19 @@ -Require Import Reals.Reals. +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Theorem theta_qcd_zero : Rabs (phi^2 + phi^(-2) - 3) = 0. -Proof. reflexivity. Qed. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: + + gHashTag/t27/proofs/canonical/sacred/StrongCP.v + (logical path: Trinity.Canonical.Sacred.StrongCP) + + Bundle: SAC-CP + Title: Strong CP problem + PhD chapter: Ch.29 Sacred V (strong CP) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) + +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export StrongCP. diff --git a/docs/phd/theorems/trinity/AlphaPhi.v b/docs/phd/theorems/trinity/AlphaPhi.v index e695769814..b7333d771c 100644 --- a/docs/phd/theorems/trinity/AlphaPhi.v +++ b/docs/phd/theorems/trinity/AlphaPhi.v @@ -1,145 +1,19 @@ -(* AlphaPhi.v - Named Constant α_φ Definition *) -(* Part of Trinity S3AI Coq Proof Base for v0.9 Framework *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals.Reals. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Require Import CorePhi. + gHashTag/t27/proofs/canonical/sacred/AlphaPhi.v + (logical path: Trinity.Canonical.Sacred.AlphaPhi) -(** α_φ = φ⁻³ / 2 = (√5 - 2) / 2 ≈ 0.1180339887498949 *) -(** This is the fundamental coupling constant of the Trinity framework *) + Bundle: SAC-1 + Title: alpha_phi = (sqrt 5 - 2)/2 = phi^-3/2 + PhD chapter: Ch.4 Sacred (alpha_phi) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -Definition alpha_phi : R := /phi^3 / 2. - -(** α_φ has the closed form: α_φ = (√5 - 2) / 2 *) -Lemma alpha_phi_closed_form : alpha_phi = (sqrt(5) - 2) / 2. -Proof. - rewrite <- phi_neg3. - unfold alpha_phi. - field. -Qed. - -(** α_φ is positive and less than 1 *) -Lemma alpha_phi_pos : 0 < alpha_phi < 1. -Proof. - unfold alpha_phi. - split. - - apply Rmult_lt_pos_pos. - + apply Rinv_lt_pos. - apply Rgt_not_eq. - apply Rlt_gt. - apply phi_pos. - + lra. - - rewrite <- alpha_phi_closed_form. - (* (√5 - 2)/2 < 1 iff √5 - 2 < 2 iff √5 < 4 *) - unfold Rdiv. - apply Rlt_lt_1. - lra. -Qed. - -(** α_φ is small: less than 1/8 *) -Lemma alpha_phi_small : alpha_phi < 1/8. -Proof. - rewrite <- alpha_phi_closed_form. - unfold Rdiv. - apply Rlt_lt_1. - (* Need: √5 - 2 < 1/4, i.e., √5 < 2.25 *) - assert (sqrt(5) < 2.25) by (apply sqrt_lt_cancel; lra). - lra. -Qed. - -(** α_φ * φ³ = 1/2 (inverse relationship) *) -Lemma alpha_phi_times_phi_cubed : alpha_phi * phi^3 = 1/2. -Proof. - unfold alpha_phi. - field. - exact phi_nonzero. -Qed. - -(** 2 * α_φ = φ⁻³ (definition inverted) *) -Lemma twice_alpha_phi : 2 * alpha_phi = /phi^3. -Proof. - unfold alpha_phi. - ring. -Qed. - -(** Numeric window: 0.1180339887 < α_φ < 0.1180339888 *) -(** This provides 10-digit precision for the 50-digit seal in Appendix A *) -Lemma alpha_phi_numeric_window : - 0.1180339887 < alpha_phi < 0.1180339888. -Proof. - rewrite <- alpha_phi_closed_form. - unfold Rdiv at 1. - split. - - (* Lower bound: (√5 - 2)/2 > 0.1180339887 *) - apply Rlt_lt_1. - assert (sqrt(5) > 2.2360679775) by (apply sqrt_lt_cancel; lra). - lra. - - (* Upper bound: (√5 - 2)/2 < 0.1180339888 *) - apply Rlt_lt_1. - assert (sqrt(5) < 2.2360679776) by (apply sqrt_lt_cancel; lra). - lra. -Qed. - -(** 50-digit certification: α_φ = 0.1180339887498948482045868343656381177203... *) -(** The following lemmas establish increasingly tight bounds for α_φ *) - -Lemma alpha_phi_15_digit : - 0.118033988749894 < alpha_phi < 0.118033988749895. -Proof. - rewrite <- alpha_phi_closed_form. - unfold Rdiv at 1. - split. - - apply Rlt_lt_1. - assert (sqrt(5) > 2.23606797749978) by (apply sqrt_lt_cancel; lra). - lra. - - apply Rlt_lt_1. - assert (sqrt(5) < 2.23606797749979) by (apply sqrt_lt_cancel; lra). - lra. -Qed. - -(** α_φ² = (3 - √5)/8 (square of α_φ) *) -Lemma alpha_phi_squared : - alpha_phi^2 = (3 - sqrt(5)) / 8. -Proof. - rewrite <- alpha_phi_closed_form. - unfold Rdiv at 1. - field. - assert (sqrt(5) <> 0) by (apply Rgt_not_eq, Rlt_gt; apply sqrt_pos; lra). - lra. -Qed. - -(** 1/α_φ = 2φ³ (inverse of α_φ) *) -Lemma inv_alpha_phi : /alpha_phi = 2 * phi^3. -Proof. - unfold alpha_phi. - field. - apply Rgt_not_eq, Rlt_gt. - apply alpha_phi_pos. -Qed. - -(** 1/α_φ ≈ 8.47213595 (closed form: 4√5 + 6) *) -Lemma inv_alpha_phi_closed_form : /alpha_phi = 4 * sqrt(5) + 6. -Proof. - rewrite inv_alpha_phi. - rewrite phi_cubed. - unfold phi at 1. - field. -Qed. - -(** α_φ + 1/α_φ = φ³ + 1/(2φ³) (symmetric property) *) -Lemma alpha_phi_plus_inv : alpha_phi + /alpha_phi = phi^3 + /(2*phi^3). -Proof. - unfold alpha_phi. - field. - exact phi_nonzero. -Qed. - -(** α_φ in simplest radical form: α_φ = (3 - √5)/2 * α_φ *) -Lemma alpha_phi_alternative_form : - alpha_phi = (3 - sqrt(5)) / 2 * alpha_phi^2. -Proof. - rewrite alpha_phi_squared. - unfold Rdiv. - field. -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export AlphaPhi. diff --git a/docs/phd/theorems/trinity/Bounds_Gauge.v b/docs/phd/theorems/trinity/Bounds_Gauge.v index dfff172d5b..d3dbe30977 100644 --- a/docs/phd/theorems/trinity/Bounds_Gauge.v +++ b/docs/phd/theorems/trinity/Bounds_Gauge.v @@ -1,153 +1,19 @@ -(* Bounds_Gauge.v - Certified Bounds for Gauge Coupling Formulas *) -(* Part of Trinity S3AI Coq Proof Base for v0.9 Framework *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals.Reals. -Require Import Interval.Tactic. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Require Import CorePhi. -Require Import AlphaPhi. -Require Import FormulaEval. + gHashTag/t27/proofs/canonical/sacred/BoundsGauge.v + (logical path: Trinity.Canonical.Sacred.BoundsGauge) -(** Tolerance definitions *) -Definition tolerance_V : R := 10 / 1000. (* 0.1% for visible formulas *) -Definition tolerance_SG : R := 10 / 10000. (* 0.01% for smoking guns *) + Bundle: SAC-G + Title: Standard Model gauge bounds + PhD chapter: Ch.29 Sacred V (gauge) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** ====================================================================== *) -(** G02: α_s(m_Z) = α_φ ≈ 0.11800 *) -(** Description: QCD coupling at Z-pole equals α_φ *) -(** Reference: Section 2.1, Equation (G02) *) -(** ====================================================================== *) - -Definition G02_theoretical : R := alpha_phi. -Definition G02_experimental : R := 0.11800. - -Theorem G02_within_tolerance : - Rabs (G02_theoretical - G02_experimental) / G02_experimental < tolerance_V. -Proof. - unfold G02_theoretical, G02_experimental, tolerance_V, alpha_phi. - rewrite <- alpha_phi_closed_form. - unfold Rdiv at 1. - (* Compute bound: |(√5-2)/2 - 0.118| / 0.118 < 0.001 *) - (* This requires: |√5 - 2 - 0.236| < 0.000236 *) - (* i.e., |√5 - 2.236| < 0.000236 *) - (* Since √5 = 2.236067977..., this holds *) - interval. -Qed. - -(** ====================================================================== *) -(** G01: α⁻¹ = 4 * 9 * π⁻¹ * φ * e² ≈ 137.036 *) -(** Description: Fine-structure constant inverse *) -(** Reference: Section 2.1, Equation (G01) *) -(** ====================================================================== *) - -Definition G01_theoretical : R := 4 * 9 * / PI * phi * (exp 1 ^ 2). -Definition G01_experimental : R := 137.036. - -Theorem G01_within_tolerance : - Rabs (G01_theoretical - G01_experimental) / G01_experimental < tolerance_V. -Proof. - unfold G01_theoretical, G01_experimental, tolerance_V. - (* Use interval arithmetic for certified bound *) - interval with (i_bits, i_bisect). -Qed. - -Theorem G01_monomial_form : - exists m : monomial, - eval_monomial m = G01_theoretical - /\ Rabs (eval_monomial m - G01_experimental) / G01_experimental < tolerance_V. -Proof. - exists G01_monomial. - split. - - exact eval_G01_monomial. - - apply G01_within_tolerance. -Qed. - -(** ====================================================================== *) -(** G06: α_s(m_Z)/α_s(m_t) = 3 * φ² * e⁻² ≈ 1.0631 *) -(** Description: Running ratio of QCD coupling *) -(** Reference: Section 2.1, Equation (G06) *) -(** ====================================================================== *) - -Definition G06_theoretical : R := 3 * phi^2 * / (exp 1 ^ 2). -Definition G06_experimental : R := 1.0631. - -Theorem G06_within_tolerance : - Rabs (G06_theoretical - G06_experimental) / G06_experimental < tolerance_V. -Proof. - unfold G06_theoretical, G06_experimental, tolerance_V. - (* Use interval arithmetic for certified bound *) - interval with (i_bits, i_bisect). -Qed. - -Theorem G06_monomial_form : - exists m : monomial, - eval_monomial m = G06_theoretical - /\ Rabs (eval_monomial m - G06_experimental) / G06_experimental < tolerance_V. -Proof. - exists (M_mul (M_mul (M_const (Z.of_nat 3)) (M_phi 2)) (M_exp (-2))). - split. - { simpl; reflexivity. } - apply G06_within_tolerance. -Qed. - -(** ====================================================================== *) -(** G03: sin(θ_W) = π/φ⁴ ≈ 0.2319 *) -(** Description: Weak mixing angle (Weinberg angle) sine *) -(** Reference: Section 2.1, Equation (G03) *) -(** ====================================================================== *) - -Definition G03_theoretical : R := PI / (phi ^ 4). -Definition G03_experimental : R := 0.2319. - -Theorem G03_within_tolerance : - Rabs (G03_theoretical - G03_experimental) / G03_experimental < tolerance_V. -Proof. - unfold G03_theoretical, G03_experimental, tolerance_V. - rewrite phi_fourth. - (* Simplify: π / (3√5 + 5) *) - interval with (i_bits, i_bisect). -Qed. - -(** ====================================================================== *) -(** G04: cos(θ_W) = 2φ⁻³ ≈ 0.9728 *) -(** Description: Weak mixing angle cosine *) -(** Reference: Section 2.1, Equation (G04) *) -(** ====================================================================== *) - -Definition G04_theoretical : R := 2 * /phi^3. -Definition G04_experimental : R := 0.9728. - -Theorem G04_within_tolerance : - Rabs (G04_theoretical - G04_experimental) / G04_experimental < tolerance_V. -Proof. - unfold G04_theoretical, G04_experimental, tolerance_V. - rewrite phi_neg3. - (* Simplify: 2(√5 - 2) = 2√5 - 4 ≈ 0.4721... *) - (* Wait: 2√5 - 4 = 2*2.236 - 4 = 0.472, not 0.9728 *) - (* Let me recalculate: G04 says cos(θ_W) = 2φ⁻³ *) - (* φ⁻³ = √5 - 2 ≈ 0.236, so 2φ⁻³ ≈ 0.472 *) - (* This doesn't match 0.9728. Let me use interval to verify *) - interval with (i_bits, i_bisect). -Qed. - -(** ====================================================================== *) -(** Summary theorem for all gauge coupling bounds *) -(** ====================================================================== *) - -Theorem all_gauge_bounds_verified : - G02_within_tolerance /\ - G01_within_tolerance /\ - G06_within_tolerance /\ - G03_within_tolerance. -Proof. - tauto. -Qed. - -Theorem all_gauge_bounds_with_monomials : - G02_within_tolerance /\ - G01_monomial_form /\ - G06_monomial_form. -Proof. - tauto. -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export BoundsGauge. diff --git a/docs/phd/theorems/trinity/Bounds_Masses.v b/docs/phd/theorems/trinity/Bounds_Masses.v index cf6ee76eeb..f1220bc02d 100644 --- a/docs/phd/theorems/trinity/Bounds_Masses.v +++ b/docs/phd/theorems/trinity/Bounds_Masses.v @@ -1,190 +1,19 @@ -(* Bounds_Masses.v - Certified Bounds for Mass Formulas *) -(* Part of Trinity S3AI Coq Proof Base for v0.9 Framework *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals.Reals. -Require Import Interval.Tactic. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Require Import CorePhi. -Require Import FormulaEval. + gHashTag/t27/proofs/canonical/sacred/BoundsMasses.v + (logical path: Trinity.Canonical.Sacred.BoundsMasses) -(** Tolerance definitions *) -Definition tolerance_V : R := 10 / 1000. (* 0.1% for visible formulas *) -Definition tolerance_SG : R := 10 / 10000. (* 0.01% for smoking guns *) + Bundle: SAC-M + Title: Combined mass bounds + PhD chapter: Ch.29 Sacred V (masses) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** ====================================================================== *) -(** Q07: m_s/m_d = 8 * 3 * π⁻¹ * φ² = 20.000 (SMOKING GUN) *) -(** Description: Strange/down quark mass ratio *) -(** Reference: Section 2.4, Equation (Q07) *) -(** This is a critical test: exact integer prediction *) -(** ====================================================================== *) - -Definition Q07_theoretical : R := 8 * 3 * / PI * (phi ^ 2). -Definition Q07_experimental : R := 20. - -Theorem Q07_smoking_gun : - Rabs (Q07_theoretical - Q07_experimental) / Q07_experimental < tolerance_SG. -Proof. - unfold Q07_theoretical, Q07_experimental, tolerance_SG. - (* This is the smoking gun: exact integer 20 *) - (* Formula: 24 * φ² / π = 20 *) - (* Using φ² = (3 + √5)/2: 24 * (3+√5)/2 / π = 12(3+√5)/π *) - rewrite phi_square. - unfold phi. - (* Verify: 12 * (3 + (1+√5)/2) / π = 20 *) - (* = 12 * (7+√5)/2 / π = 6(7+√5)/π *) - (* Need: 6(7+√5) = 20π, i.e., 7+√5 = 10π/3 ≈ 10.472... *) - (* √5 = 10π/3 - 7 ≈ 3.472... *) - (* √5 ≈ 2.236, so this doesn't match exactly *) - (* Let's use interval to see the actual value *) - interval with (i_bits, i_bisect, i_prec 20). -Qed. - -Theorem Q07_monomial_form : - exists m : monomial, - eval_monomial m = Q07_theoretical - /\ Rabs (eval_monomial m - Q07_experimental) / Q07_experimental < tolerance_SG. -Proof. - exists Q07_monomial. - split. - - exact eval_Q07_monomial. - - apply Q07_smoking_gun. -Qed. - -(** ====================================================================== *) -(** H01: m_H = 4 * φ³ * e² ≈ 125.20 GeV *) -(** Description: Higgs boson mass *) -(** Reference: Section 2.5, Equation (H01) *) -(** ====================================================================== *) - -Definition H01_theoretical : R := 4 * (phi ^ 3) * (exp 1 ^ 2). -Definition H01_experimental : R := 125.20. - -Theorem H01_within_tolerance : - Rabs (H01_theoretical - H01_experimental) / H01_experimental < tolerance_V. -Proof. - unfold H01_theoretical, H01_experimental, tolerance_V. - rewrite phi_cubed. - interval with (i_bits, i_bisect). -Qed. - -Theorem H01_monomial_form : - exists m : monomial, - eval_monomial m = H01_theoretical - /\ Rabs (eval_monomial m - H01_experimental) / H01_experimental < tolerance_V. -Proof. - exists H01_monomial. - split. - - exact eval_H01_monomial. - - apply H01_within_tolerance. -Qed. - -(** ====================================================================== *) -(** H02: m_H/m_W = 4 * φ * e ≈ 1.556 *) -(** Description: Higgs to W boson mass ratio *) -(** Reference: Section 2.5, Equation (H02) *) -(** ====================================================================== *) - -Definition H02_theoretical : R := 4 * phi * exp 1. -Definition H02_experimental : R := 1.556. - -Theorem H02_within_tolerance : - Rabs (H02_theoretical - H02_experimental) / H02_experimental < tolerance_V. -Proof. - unfold H02_theoretical, H02_experimental, tolerance_V. - unfold phi. - interval with (i_bits, i_bisect). -Qed. - -(** ====================================================================== *) -(** H03: m_H/m_Z = φ² * e ≈ 1.356 *) -(** Description: Higgs to Z boson mass ratio *) -(** Reference: Section 2.5, Equation (H03) *) -(** ====================================================================== *) - -Definition H03_theoretical : R := phi^2 * exp 1. -Definition H03_experimental : R := 1.356. - -Theorem H03_within_tolerance : - Rabs (H03_theoretical - H03_experimental) / H03_experimental < tolerance_V. -Proof. - unfold H03_theoretical, H03_experimental, tolerance_V. - rewrite phi_square. - unfold phi. - interval with (i_bits, i_bisect). -Qed. - -(** ====================================================================== *) -(** Q01: m_u/m_d = π / (9 * e²) ≈ 0.0056 *) -(** Description: Up/down quark mass ratio *) -(** Reference: Section 2.4, Equation (Q01) *) -(** ====================================================================== *) - -Definition Q01_theoretical : R := PI / (9 * (exp 1 ^ 2)). -Definition Q01_experimental : R := 0.0056. - -Theorem Q01_within_tolerance : - Rabs (Q01_theoretical - Q01_experimental) / Q01_experimental < tolerance_V. -Proof. - unfold Q01_theoretical, Q01_experimental, tolerance_V. - interval with (i_bits, i_bisect). -Qed. - -(** ====================================================================== *) -(** Q02: m_s/m_u = 4 * φ² / π ≈ 41.8 *) -(** Description: Strange/up quark mass ratio *) -(** Reference: Section 2.4, Equation (Q02) *) -(** ====================================================================== *) - -Definition Q02_theoretical : R := 4 * (phi ^ 2) / PI. -Definition Q02_experimental : R := 41.8. - -Theorem Q02_within_tolerance : - Rabs (Q02_theoretical - Q02_experimental) / Q02_experimental < tolerance_V. -Proof. - unfold Q02_theoretical, Q02_experimental, tolerance_V. - rewrite phi_square. - unfold phi. - interval with (i_bits, i_bisect). -Qed. - -(** ====================================================================== *) -(** Q04: m_c/m_s = 8 * φ³ / (3 * π) ≈ 11.5 *) -(** Description: Charm/strange quark mass ratio *) -(** Reference: Section 2.4, Equation (Q04) *) -(** ====================================================================== *) - -Definition Q04_theoretical : R := 8 * (phi ^ 3) / (3 * PI). -Definition Q04_experimental : R := 11.5. - -Theorem Q04_within_tolerance : - Rabs (Q04_theoretical - Q04_experimental) / Q04_experimental < tolerance_V. -Proof. - unfold Q04_theoretical, Q04_experimental, tolerance_V. - rewrite phi_cubed. - unfold phi. - interval with (i_bits, i_bisect). -Qed. - -(** ====================================================================== *) -(** Summary theorem for all mass bounds *) -(** ====================================================================== *) - -Theorem all_mass_bounds_verified : - Q07_smoking_gun /\ - H01_within_tolerance /\ - H02_within_tolerance /\ - H03_within_tolerance /\ - Q01_within_tolerance /\ - Q02_within_tolerance /\ - Q04_within_tolerance. -Proof. - tauto. -Qed. - -Theorem all_mass_bounds_with_monomials : - Q07_monomial_form /\ - H01_monomial_form. -Proof. - tauto. -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export BoundsMasses. diff --git a/docs/phd/theorems/trinity/Bounds_Mixing.v b/docs/phd/theorems/trinity/Bounds_Mixing.v index 8c781738a4..8bdef086b4 100644 --- a/docs/phd/theorems/trinity/Bounds_Mixing.v +++ b/docs/phd/theorems/trinity/Bounds_Mixing.v @@ -1,159 +1,19 @@ -(* Bounds_Mixing.v - Certified Bounds for Mixing Parameter Formulas *) -(* Part of Trinity S3AI Coq Proof Base for v0.9 Framework *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals.Reals. -Require Import Interval.Tactic. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Require Import CorePhi. -Require Import FormulaEval. + gHashTag/t27/proofs/canonical/sacred/BoundsMixing.v + (logical path: Trinity.Canonical.Sacred.BoundsMixing) -(** Tolerance definitions *) -Definition tolerance_V : R := 10 / 1000. (* 0.1% for visible formulas *) -Definition tolerance_SG : R := 10 / 10000. (* 0.01% for smoking guns *) + Bundle: SAC-X + Title: CKM/PMNS mixing matrix bounds + PhD chapter: Ch.29 Sacred V (mixing) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** ====================================================================== *) -(** C01: |V_us| = 2 * 3⁻² * π⁻³ * φ³ * e² ≈ 0.22431 *) -(** Description: CKM matrix element |V_us| (up-strange mixing) *) -(** Reference: Section 2.2, Equation (C01) *) -(** ====================================================================== *) - -Definition C01_theoretical : R := 2 * / (3 ^ 2) * / (PI ^ 3) * (phi ^ 3) * (exp 1 ^ 2). -Definition C01_experimental : R := 0.22431. - -Theorem C01_within_tolerance : - Rabs (C01_theoretical - C01_experimental) / C01_experimental < tolerance_V. -Proof. - unfold C01_theoretical, C01_experimental, tolerance_V. - (* Use interval arithmetic for certified bound *) - interval with (i_bits, i_bisect). -Qed. - -Theorem C01_monomial_form : - exists m : monomial, - eval_monomial m = C01_theoretical - /\ Rabs (eval_monomial m - C01_experimental) / C01_experimental < tolerance_V. -Proof. - exists C01_monomial. - split. - - exact eval_C01_monomial. - - apply C01_within_tolerance. -Qed. - -(** ====================================================================== *) -(** C02: |V_cb| = 2 * 3⁻³ * π⁻² * φ² * e² ≈ 0.0405 *) -(** Description: CKM matrix element |V_cb| (charm-bottom mixing) *) -(** Reference: Section 2.2, Equation (C02) *) -(** ====================================================================== *) - -Definition C02_theoretical : R := 2 * / (3 ^ 3) * / (PI ^ 2) * (phi ^ 2) * (exp 1 ^ 2). -Definition C02_experimental : R := 0.0405. - -Theorem C02_within_tolerance : - Rabs (C02_theoretical - C02_experimental) / C02_experimental < tolerance_V. -Proof. - unfold C02_theoretical, C02_experimental, tolerance_V. - interval with (i_bits, i_bisect). -Qed. - -(** ====================================================================== *) -(** C03: |V_ub| = 4 * 3⁻⁴ * π⁻³ * φ * e² ≈ 0.0036 *) -(** Description: CKM matrix element |V_ub| (up-bottom mixing) *) -(** Reference: Section 2.2, Equation (C03) *) -(** ====================================================================== *) - -Definition C03_theoretical : R := 4 * / (3 ^ 4) * / (PI ^ 3) * phi * (exp 1 ^ 2). -Definition C03_experimental : R := 0.0036. - -Theorem C03_within_tolerance : - Rabs (C03_theoretical - C03_experimental) / C03_experimental < tolerance_V. -Proof. - unfold C03_theoretical, C03_experimental, tolerance_V. - interval with (i_bits, i_bisect). -Qed. - -(** ====================================================================== *) -(** N01: sin²(θ₁₂) = 8 * φ⁻⁵ * π * e⁻² ≈ 0.30700 *) -(** Description: Neutrino mixing angle θ₁₂ (solar angle) *) -(** Reference: Section 2.3, Equation (N01) *) -(** ====================================================================== *) - -Definition N01_theoretical : R := 8 * / (phi ^ 5) * PI * / (exp 1 ^ 2). -Definition N01_experimental : R := 0.30700. - -Theorem N01_within_tolerance : - Rabs (N01_theoretical - N01_experimental) / N01_experimental < tolerance_V. -Proof. - unfold N01_theoretical, N01_experimental, tolerance_V. - (* Simplify using phi_fifth: phi^5 = 5√5 + 8 *) - rewrite phi_fifth. - interval with (i_bits, i_bisect). -Qed. - -Theorem N01_monomial_form : - exists m : monomial, - eval_monomial m = N01_theoretical - /\ Rabs (eval_monomial m - N01_experimental) / N01_experimental < tolerance_V. -Proof. - exists N01_monomial. - split. - - exact eval_N01_monomial. - - apply N01_within_tolerance. -Qed. - -(** ====================================================================== *) -(** N03: sin²(θ₂₃) = 2 * π * φ⁻⁴ ≈ 0.54800 *) -(** Description: Neutrino mixing angle θ₂₃ (atmospheric angle) *) -(** Reference: Section 2.3, Equation (N03) *) -(** ====================================================================== *) - -Definition N03_theoretical : R := 2 * PI * / (phi ^ 4). -Definition N03_experimental : R := 0.54800. - -Theorem N03_within_tolerance : - Rabs (N03_theoretical - N03_experimental) / N03_experimental < tolerance_V. -Proof. - unfold N03_theoretical, N03_experimental, tolerance_V. - rewrite phi_fourth. - interval with (i_bits, i_bisect). -Qed. - -(** ====================================================================== *) -(** N04: δ_CP = 8 * π³ / (9 * e²) * 180/π ≈ 195.0° [UNDER REVISION] *) -(** Description: CP-violating phase in PMNS matrix *) -(** Reference: Section 2.3, Equation (N04) *) -(** NOTE: Formula under revision - unit conversion error identified. *) -(** The theoretical value does not match 195.0°. Awaiting Chimera re-search. *) -(** ====================================================================== *) - -(* Definition N04_theoretical : R := 8 * (PI ^ 3) / (9 * (exp 1 ^ 2)) * (180 / PI). *) -(* Definition N04_experimental : R := 195.0. *) -(* -Theorem N04_corrected_within_tolerance : - Rabs (N04_theoretical - N04_experimental) / N04_experimental < tolerance_V. -Proof. - unfold N04_theoretical, N04_experimental, tolerance_V. - interval with (i_bits, i_bisect). -Qed. -*) - -(** ====================================================================== *) -(** Summary theorem for all mixing parameter bounds *) -(** ====================================================================== *) - -Theorem all_mixing_bounds_verified : - C01_within_tolerance /\ - C02_within_tolerance /\ - C03_within_tolerance /\ - N01_within_tolerance /\ - N03_within_tolerance. -Proof. - tauto. -Qed. - -Theorem all_mixing_bounds_with_monomials : - C01_monomial_form /\ - N01_monomial_form. -Proof. - tauto. -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export BoundsMixing. diff --git a/docs/phd/theorems/trinity/Bounds_QuarkMasses.v b/docs/phd/theorems/trinity/Bounds_QuarkMasses.v index 5373302a7f..8b0018807b 100644 --- a/docs/phd/theorems/trinity/Bounds_QuarkMasses.v +++ b/docs/phd/theorems/trinity/Bounds_QuarkMasses.v @@ -1,116 +1,19 @@ -(* Bounds_QuarkMasses.v - Certified Bounds for Additional Quark Mass Ratios *) -(* Part of Trinity S3AI Coq Proof Base for v1.0 Framework *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals.Reals. -Require Import Interval.Tactic. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Require Import CorePhi. -Require Import FormulaEval. -Require Import Bounds_Masses. + gHashTag/t27/proofs/canonical/sacred/BoundsQuarkMasses.v + (logical path: Trinity.Canonical.Sacred.BoundsQuarkMasses) -(** Tolerance definitions *) -Definition tolerance_V : R := 10 / 1000. (* 0.1% for visible formulas *) -Definition tolerance_SG : R := 10 / 10000. (* 0.01% for smoking guns *) + Bundle: SAC-Q + Title: Quark mass bounds + PhD chapter: Ch.29 Sacred V (quarks) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** ====================================================================== *) -(** Q03: m_c/m_d = φ⁴ * π / e² ≈ 171.5 *) -(** Description: Charm/down quark mass ratio *) -(** Reference: Section 2.4, Equation (Q03) *) -(** ====================================================================== *) - -Definition Q03_theoretical : R := (phi ^ 4) * PI / (exp 1 ^ 2). -Definition Q03_experimental : R := 171.5. - -Theorem Q03_within_tolerance : - Rabs (Q03_theoretical - Q03_experimental) / Q03_experimental < tolerance_V. -Proof. - (* TODO: Q03 formula does not match experimental value (98% error) *) - admit. -Admitted. - -Theorem Q03_monomial_form : - exists m : monomial, - eval_monomial m = Q03_theoretical - /\ Rabs (eval_monomial m - Q03_experimental) / Q03_experimental < tolerance_V. -Proof. - (* TODO: Depends on admitted eval_monomial for Rocq 9.x compatibility *) - admit. -Admitted. - -(** ====================================================================== *) -(** Q05: m_b/m_s = 48·e²/φ⁴ ≈ 52.3 [IMPROVED via Chimera] *) -(** Description: Bottom/strange quark mass ratio *) -(** Reference: Section 2.4, Equation (Q05) *) -(** Chimera result: 48·e²/φ⁴ = 51.75 (Δ=1.06%) *) -(** ====================================================================== *) - -Definition Q05_theoretical : R := 48 * (exp 1 ^ 2) / (phi ^ 4). -Definition Q05_experimental : R := 52.3. - -Theorem Q05_within_tolerance : - Rabs (Q05_theoretical - Q05_experimental) / Q05_experimental < tolerance_V. -Proof. - (* TODO: Q05 is a CANDIDATE formula (Δ≈1%, outside 0.1% tolerance) *) - (* Chimera v1.0 result: 48·e²/φ⁴ = 51.75 vs experimental 52.3 *) - admit. -Admitted. - -Theorem Q05_monomial_form : - exists m : monomial, - eval_monomial m = Q05_theoretical - /\ Rabs (eval_monomial m - Q05_experimental) / Q05_experimental < tolerance_V. -Proof. - (* TODO: Depends on admitted eval_monomial for Rocq 9.x compatibility *) - admit. -Admitted. - -(** ====================================================================== *) -(** Q06: m_b/m_d = Q05 × Q07 = 1034.93 [CHAIN VERIFIED] *) -(** Description: Bottom/down quark mass ratio *) -(** Reference: Section 2.4, Equation (Q06) *) -(** Chimera result: Q06 = Q05 × Q07 = 1034.93 (Δ=0.01%) *) -(** Chain relation: Q05 × Q07 ≈ 51.75 × 20 = 1035 *) -(** ====================================================================== *) - -Definition Q06_theoretical : R := Q05_theoretical * Q07_theoretical. -Definition Q06_experimental : R := 1035. - -Theorem Q06_within_tolerance : - Rabs (Q06_theoretical - Q06_experimental) / Q06_experimental < tolerance_V. -Proof. - (* Q06 chain: Q05 × Q07 = 51.75 × 20.0003 = 1034.94 ≈ 1035 (Δ=0.0055%) *) - unfold Q06_theoretical, Q06_experimental, tolerance_V. - unfold Q05_theoretical, Q07_theoretical. - interval. -Qed. - -Theorem Q06_chain_verified : - (* Verify Q06 = Q05 × Q07 exactly (up to numerical precision) *) - Rabs (Q05_theoretical * Q07_theoretical - Q06_theoretical) / Q06_theoretical < tolerance_V. -Proof. - (* This holds by definition: Q06_theoretical = Q05_theoretical * Q07_theoretical *) - unfold Q06_theoretical, tolerance_V. - interval. -Qed. - -Theorem Q06_chain_relation : - (* Chain relation: Q05 × Q07 = Q06 *) - Q05_theoretical * Q07_theoretical = Q06_theoretical. -Proof. - unfold Q06_theoretical; reflexivity. -Qed. - -(** ====================================================================== *) -(** Summary theorem for additional quark mass bounds *) -(** ====================================================================== *) - -(* TODO: Summary theorems cause type error in Rocq 9.x - fix needed *) - - -Theorem quark_mass_chain_summary : - (* Q05 × Q07 = Q06 chain relation *) - (* TODO: Summary theorem causes type error in Rocq 9.x *) - True. -Proof. reflexivity. -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export BoundsQuarkMasses. diff --git a/docs/phd/theorems/trinity/Catalog42.v b/docs/phd/theorems/trinity/Catalog42.v index 923e360a06..565f6c6774 100644 --- a/docs/phd/theorems/trinity/Catalog42.v +++ b/docs/phd/theorems/trinity/Catalog42.v @@ -1,234 +1,19 @@ -(* Catalog42.v - Representative Theorems for Flagship Catalog *) -(* Part of Trinity S3AI Coq Proof Base for v0.9 Framework *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals.Reals. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Require Import Bounds_Gauge. -Require Import Bounds_Mixing. -Require Import Bounds_Masses. -Require Import AlphaPhi. + gHashTag/t27/proofs/canonical/sacred/Catalog42.v + (logical path: Trinity.Canonical.Sacred.Catalog42) -(** ====================================================================== *) -(** CATALOG: Representative Theorems for Trinity Framework v0.9 *) -(** This module collects the flagship theorems demonstrating the framework *) -(** The catalog provides machine-checkable verification of key predictions *) -(** ====================================================================== *) + Bundle: SAC-4 + Title: Catalog of 42 sealed formulae + PhD chapter: Ch.29 Sacred V + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** ---------------------------------------------------------------------- - Section 1: Core Algebraic Identities (L1 - Derivation Level 1) - These are the foundational theorems from which all formulas descend - ---------------------------------------------------------------------- *) - -Theorem core_phi_identities_verified : - (* φ is well-defined and positive *) - phi_pos /\ - (* φ satisfies quadratic equation *) - phi_square /\ - (* Reciprocal identity *) - phi_inv /\ - (* Trinity root identity: φ² + φ⁻² = 3 *) - trinity_identity /\ - (* φ⁻³ = √5 - 2 *) - phi_neg3. -Proof. - tauto. -Qed. - -(** ---------------------------------------------------------------------- - Section 2: α_φ Constant Definition - The fundamental coupling constant - ---------------------------------------------------------------------- *) - -Theorem alpha_phi_verified : - (* α_φ has closed form *) - alpha_phi_closed_form /\ - (* α_φ is between 0 and 1 *) - alpha_phi_pos /\ - (* 10-digit numeric window verified *) - alpha_phi_numeric_window /\ - (* 15-digit numeric window verified *) - alpha_phi_15_digit. -Proof. - tauto. -Qed. - -(** ---------------------------------------------------------------------- - Section 3: Gauge Coupling Theorems (G-series) - QCD coupling, fine-structure constant, running ratios - ---------------------------------------------------------------------- *) - -Theorem gauge_coupling_theorems_verified : - (* G02: α_s(m_Z) = α_φ ≈ 0.11800 *) - G02_within_tolerance /\ - (* G01: α⁻¹ = 4·9·π⁻¹·φ·e² ≈ 137.036 *) - G01_within_tolerance /\ - (* G06: running ratio = 3·φ²·e⁻² ≈ 1.0631 *) - G06_within_tolerance /\ - (* G03: sin(θ_W) = π/φ⁴ ≈ 0.2319 *) - G03_within_tolerance. -Proof. - tauto. -Qed. - -Theorem gauge_coupling_monomial_forms : - G01_monomial_form /\ - G06_monomial_form. -Proof. - tauto. -Qed. - -(** ---------------------------------------------------------------------- - Section 4: CKM Mixing Theorems (C-series) - Quark mixing matrix elements - ---------------------------------------------------------------------- *) - -Theorem ckm_mixing_theorems_verified : - (* C01: |V_us| = 2·3⁻²·π⁻³·φ³·e² ≈ 0.22431 *) - C01_within_tolerance /\ - (* C02: |V_cb| = 2·3⁻³·π⁻²·φ²·e² ≈ 0.0405 *) - C02_within_tolerance /\ - (* C03: |V_ub| = 4·3⁻⁴·π⁻³·φ·e² ≈ 0.0036 *) - C03_within_tolerance. -Proof. - tauto. -Qed. - -Theorem ckm_mixing_monomial_forms : - C01_monomial_form. -Proof. - tauto. -Qed. - -(** ---------------------------------------------------------------------- - Section 5: Neutrino Mixing Theorems (N-series) - PMNS matrix elements and CP phase - ---------------------------------------------------------------------- *) - -Theorem neutrino_mixing_theorems_verified : - (* N01: sin²(θ₁₂) = 8·φ⁻⁵·π·e⁻² ≈ 0.30700 *) - N01_within_tolerance /\ - (* N03: sin²(θ₂₃) = 2·π·φ⁻⁴ ≈ 0.54800 *) - N03_within_tolerance. - (* N04: δ_CP - under revision, unit conversion error identified *) -Proof. - tauto. -Qed. - -Theorem neutrino_mixing_monomial_forms : - N01_monomial_form. -Proof. - tauto. -Qed. - -(** ---------------------------------------------------------------------- - Section 6: Mass Ratio Theorems (Q and H series) - Quark mass ratios and Higgs boson mass - ---------------------------------------------------------------------- *) - -Theorem mass_ratio_theorems_verified : - (* Q07: m_s/m_d = 8·3·π⁻¹·φ² = 20.000 (SMOKING GUN) *) - Q07_smoking_gun /\ - (* H01: m_H = 4·φ³·e² ≈ 125.20 GeV *) - H01_within_tolerance /\ - (* H02: m_H/m_W = 4·φ·e ≈ 1.556 *) - H02_within_tolerance /\ - (* H03: m_H/m_Z = φ²·e ≈ 1.356 *) - H03_within_tolerance /\ - (* Q01: m_u/m_d = π/(9·e²) ≈ 0.0056 *) - Q01_within_tolerance /\ - (* Q02: m_s/m_u = 4·φ²/π ≈ 41.8 *) - Q02_within_tolerance /\ - (* Q04: m_c/m_s = 8·φ³/(3·π) ≈ 11.5 *) - Q04_within_tolerance. -Proof. - tauto. -Qed. - -Theorem mass_ratio_monomial_forms : - Q07_monomial_form /\ - H01_monomial_form. -Proof. - tauto. -Qed. - -(** ---------------------------------------------------------------------- - Section 7: Complete Flagship Catalog - Top 10-12 representative theorems spanning all sectors - ---------------------------------------------------------------------- *) - -Theorem catalog_representative_rows_verified : - (* G02 verified *) G02_within_tolerance /\ - (* G01 verified *) G01_within_tolerance /\ - (* G06 verified *) G06_within_tolerance /\ - (* C01 verified *) C01_within_tolerance /\ - (* N01 verified *) N01_within_tolerance /\ - (* N03 verified *) N03_within_tolerance /\ - (* Q07 smoking gun *) Q07_smoking_gun /\ - (* H01 verified *) H01_within_tolerance. -Proof. - tauto. -Qed. - -(** ---------------------------------------------------------------------- - Section 8: Monomial Interface Verification - Confirms that flagship formulas have monomial representations - ---------------------------------------------------------------------- *) - -Theorem catalog_monomial_interface_verified : - G01_monomial_form /\ - G06_monomial_form /\ - C01_monomial_form /\ - N01_monomial_form /\ - Q07_monomial_form /\ - H01_monomial_form. -Proof. - tauto. -Qed. - -(** ---------------------------------------------------------------------- - Section 9: Master Verification Theorem - All flagship theorems verified with machine-checkable bounds - ---------------------------------------------------------------------- *) - -Theorem trinity_framework_v09_flagship_theorems_verified : - (* Core φ identities *) - core_phi_identities_verified /\ - (* α_φ constant *) - alpha_phi_verified /\ - (* Gauge couplings *) - gauge_coupling_theorems_verified /\ - (* CKM mixing *) - ckm_mixing_theorems_verified /\ - (* Neutrino mixing *) - neutrino_mixing_theorems_verified /\ - (* Mass ratios *) - mass_ratio_theorems_verified. -Proof. - tauto. -Qed. - -(** ---------------------------------------------------------------------- - Summary Statistics - Count of verified theorems in this catalog - ---------------------------------------------------------------------- *) - -Definition verified_core_identities : nat := 7. (* phi_pos, phi_square, phi_inv, phi_inv_sq, trinity_identity, phi_neg3, phi_cubed, phi_fourth, phi_fifth *) -Definition verified_alpha_phi_theorems : nat := 4. -Definition verified_gauge_theorems : nat := 5. -Definition verified_ckm_theorems : nat := 3. -Definition verified_neutrino_theorems : nat := 2. (* N01, N03 - N04 under revision *) -Definition verified_mass_theorems : nat := 7. -Definition verified_monomial_forms : nat := 6. (* G01, G06, C01, N01, Q07, H01 *) - -Definition total_verified_theorems : nat := - verified_core_identities + - verified_alpha_phi_theorems + - verified_gauge_theorems + - verified_ckm_theorems + - verified_neutrino_theorems + - verified_mass_theorems. - -(** Total: 28 theorems verified in this flagship catalog *) -Definition catalog_size_comment : string := - "Catalog42.v verifies 28 theorems across 6 physics sectors (N04 under revision)". +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export Catalog42. diff --git a/docs/phd/theorems/trinity/ConsistencyChecks.v b/docs/phd/theorems/trinity/ConsistencyChecks.v index 4385e4a16c..80f30f32cb 100644 --- a/docs/phd/theorems/trinity/ConsistencyChecks.v +++ b/docs/phd/theorems/trinity/ConsistencyChecks.v @@ -1,297 +1,19 @@ -(* ConsistencyChecks.v - Cross-Sector Validation and Chain Relations *) -(* Part of Trinity S3AI Coq Proof Base for v1.0 Framework *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals.Reals. -Require Import Interval.Tactic. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Require Import CorePhi. -Require Import Bounds_Gauge. -Require Import Bounds_Masses. -Require Import Bounds_Mixing. -Require Import Bounds_QuarkMasses. -Require Import Bounds_LeptonMasses. -Require Import AlphaPhi. + gHashTag/t27/proofs/canonical/sacred/ConsistencyChecks.v + (logical path: Trinity.Canonical.Sacred.ConsistencyChecks) -(** Tolerance definitions *) -Definition tolerance_V : R := 10 / 1000. (* 0.1% for visible formulas *) -Definition tolerance_L : R := 50 / 1000. (* 0.5% for chain relations *) -Definition tolerance_SG : R := 10 / 10000. (* 0.01% for smoking guns *) + Bundle: SAC-6 + Title: Cross-formula consistency + PhD chapter: Ch.29 Sacred V + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** ====================================================================== *) -(** Alpha Consistency Check *) -(** Verify α_φ derived from G01 matches the definition *) -(** ====================================================================== *) - -Definition alpha_from_G01 : R := 1 / (4 * 9 * /PI * phi * (exp 1 ^ 2)). - -Theorem alpha_consistency_check : - Rabs (alpha_from_G01 - alpha_phi) / alpha_phi < tolerance_SG. -Proof. - unfold alpha_from_G01, alpha_phi, tolerance_SG. - (* α_φ = 1/G01 should hold exactly if formulas are consistent *) - (* G01 = 4·9·π⁻¹·φ·e² = 36φe²/π *) - (* α_φ = 1/G01 = π/(36φe²) *) - (* α_φ = (√5-2)/2 from definition *) - (* TODO: Verify numerically with higher precision *) - admit. -Admitted. - -(** ====================================================================== *) -(** Quark Mass Chain Relations *) -(** Verify that mass ratios multiply correctly *) -(** ====================================================================== *) - -(* Chain 1: (m_s/m_d) × (m_d/m_u)⁻¹ = m_s/m_u *) -(* Q07 × Q01⁻¹ should ≈ Q02 *) -(* Note: Q02 is m_s/m_u, Q01 is m_u/m_d, so: Q07 / Q01 = Q02 *) - -Theorem quark_mass_chain_Q07_Q01_Q02 : - Rabs ((Q07_theoretical / Q01_theoretical) - Q02_theoretical) / Q02_theoretical < tolerance_L. -Proof. - unfold Q07_theoretical, Q01_theoretical, Q02_theoretical, tolerance_L. - (* Q07 = 8·3·π⁻¹·φ² = 24φ²/π *) - (* Q01 = π/(9·e²) *) - (* Q02 = 4·φ²/π *) - (* Q07 / Q01 = (24φ²/π) / (π/(9e²)) = 24φ²/π · 9e²/π = 216φ²e²/π² *) - (* Q02 = 4φ²/π *) - (* Check: 216φ²e²/π² ≈ 4φ²/π *) - (* This would require 54e²/π ≈ 1, which is false *) - (* So the chain relation suggests these formulas may need revision *) - admit. -Admitted. - -(* Chain 2: (m_b/m_s) × (m_s/m_d) = m_b/m_d *) -(* Q05 × Q07 = Q06 [VERIFIED via Chimera v1.0] *) -(* Q05 = 48·e²/φ⁴, Q07 = 24φ²/π, Q06 = Q05 × Q07 = 1034.93 *) -(* Error: 0.01% - chain relation VERIFIED! *) - -Theorem quark_mass_chain_Q05_Q07_Q06 : - Rabs ((Q05_theoretical * Q07_theoretical) - Q06_theoretical) / Q06_theoretical < tolerance_SG. -Proof. - unfold Q05_theoretical, Q07_theoretical, Q06_theoretical, tolerance_SG. - (* With Chimera v1.0 formulas: *) - (* Q05 = 48·e²/φ⁴ *) - (* Q07 = 8·3·π⁻¹·φ² = 24φ²/π *) - (* Q06 = Q05 × Q07 (chain definition) *) - (* This should be exact by definition in Bounds_QuarkMasses.v *) - (* But we verify numerically for robustness *) - admit. -Admitted. - -Theorem quark_mass_chain_Q05_Q07_Q06_exact : - (* Exact chain relation: Q06 is defined as Q05 × Q07 *) - Q05_theoretical * Q07_theoretical = Q06_theoretical. -Proof. - unfold Q06_theoretical. - reflexivity. -Qed. - -(* Chain 3: (m_c/m_d) derived from other ratios *) -(* m_c/m_d = (m_c/m_s) × (m_s/m_d) *) -(* Note: We don't have m_c/m_s formula, so skip *) - -(** ====================================================================== *) -(** Lepton Mass Chain Relations *) -(** These should hold exactly by algebraic manipulation *) -(** ====================================================================== *) - -(* Chain: (m_μ/m_e) × (m_τ/m_μ) = m_τ/m_e *) -(* L01 × L02 = L03 - this should be exact *) - -Theorem lepton_mass_chain_L01_L02_L03 : - True. -Proof. - (* L01 × L02 = L03 - exact by algebra *) - (* L01 = 4φ³/e², L02 = 2φ⁴π/e, L03 = 8φ⁷π/e³ *) - exact I. -Qed. - -Theorem lepton_mass_chain_L01_L02_L03_numerical : - True. -Proof. - (* Numerical verification of chain relation *) - exact I. -Qed. - -(** ====================================================================== *) -(** Gauge-Mass Consistency *) -(** Verify Higgs to gauge boson ratios are consistent *) -(** ====================================================================== *) - -(* Chain: (m_H/m_W) × (m_W/m_Z) = m_H/m_Z *) -(* From experimental data: 1.556 × 0.881 ≈ 1.371 ≈ 1.356 *) -(* Check if Trinity formulas satisfy this *) - -(* Note: H01_H02_H03_chain is a conceptual relation, not a Coq definition *) - -Theorem gauge_mass_chain_check : - Rabs ((H02_theoretical * 0.881) - H03_theoretical) / H03_theoretical < tolerance_V. -Proof. - unfold H02_theoretical, H03_theoretical, tolerance_V. - (* H02 = 4φe ≈ 4 × 1.618 × 2.718 ≈ 17.59 *) - (* H03 = φ²e ≈ 2.618 × 2.718 ≈ 7.12 *) - (* H02 × 0.881 ≈ 17.59 × 0.881 ≈ 15.50 *) - (* This doesn't equal 7.12 - chain relation fails *) - (* This suggests m_W/m_Z is not given by simple ratio *) - admit. -Admitted. - -(** ====================================================================== *) -(** CKM Unitarity Consistency *) -(** Verify that derived V_ud satisfies unitarity with V_us, V_ub *) -(** ====================================================================== *) - -(* From Bounds_Mixing.v we have: *) -(* C01: |V_us| ≈ 0.22431 *) -(* C03: |V_ub| ≈ 0.0036 *) -(* Unitarity: |V_ud|² + |V_us|² + |V_ub|² = 1 *) -(* So |V_ud| = √(1 - |V_us|² - |V_ub|²) ≈ √(1 - 0.0503 - 0.000013) ≈ 0.974 *) - -Definition V_ud_from_unitarity_trinity := - sqrt (1 - C01_theoretical^2 - C03_theoretical^2). - -Definition V_ud_experimental : R := 0.974. - -Theorem V_ud_unitarity_check : - Rabs (V_ud_from_unitarity_trinity - V_ud_experimental) / V_ud_experimental < tolerance_V. -Proof. - unfold V_ud_from_unitarity_trinity, V_ud_experimental, tolerance_V, C01_theoretical, C03_theoretical. - (* Compute: sqrt(1 - (2·3⁻²·π⁻³·φ³·e²)² - (4·3⁻⁴·π⁻³·φ·e²)²) *) - (* TODO: C01 and C03 formulas need Chimera search *) - admit. -Admitted. - -Theorem CKM_row_unitarity_sum : - Rabs (V_ud_from_unitarity_trinity^2 + C01_theoretical^2 + C03_theoretical^2 - 1) < 1e-6. -Proof. - unfold V_ud_from_unitarity_trinity, C01_theoretical, C03_theoretical. - (* This should be exact by definition *) - (* V_ud = sqrt(1 - C01^2 - C03^2), so V_ud^2 + C01^2 + C03^2 = 1 *) - admit. -Admitted. - -(** ====================================================================== *) -(** PMNS Unitarity Consistency *) -(** Verify neutrino mixing angles satisfy unitarity *) -(** ====================================================================== *) - -(* From Bounds_Mixing.v: *) -(* N01: sin²(θ_12) ≈ 0.307 *) -(* N03: sin²(θ_23) ≈ 0.548 *) -(* PM2: sin²(θ_13) ≈ 0.022 (from Unitarity.v) *) -(* Unitarity: sum = 1 for probability conservation *) - -Definition PM2_sin2_theta13 : R := 3 * PI / (phi ^ 3) / 100. -(* Note: PM2 formula needs verification *) - -Theorem PMNS_sum_to_one : - Rabs (N01_theoretical + PM2_sin2_theta13 + (1 - N03_theoretical) - 1) < tolerance_V. -Proof. - unfold N01_theoretical, N03_theoretical, PM2_sin2_theta13, tolerance_V. - (* PMNS unitarity: sin²(θ_12) + sin²(θ_13) + cos²(θ_23) = 1 *) - (* Using N03 = sin²(θ_23), so cos²(θ_23) = 1 - N03 *) - (* TODO: N03 formula needs Chimera search *) - admit. -Admitted. - -(** ====================================================================== *) -(** Cross-Sector Consistency: α_s Running *) -(** Verify QCD coupling at different scales is consistent *) -(** ====================================================================== *) - -(* From Bounds_Gauge.v: *) -(* G02: α_s(m_Z) = α_φ ≈ 0.118 *) -(* G06: α_s(m_Z)/α_s(m_t) = 3φ²e⁻² ≈ 1.063 *) -(* So α_s(m_t) = α_s(m_Z) / G06 ≈ 0.118 / 1.063 ≈ 0.111 *) - -Definition alpha_s_m_t_from_running := - G02_theoretical / G06_theoretical. - -Theorem alpha_running_consistency : - (* Verify α_s(m_t) is physically reasonable (< α_s(m_Z)) *) - 0 < alpha_s_m_t_from_running < 1 /\ - alpha_s_m_t_from_running < G02_theoretical. -Proof. - unfold alpha_s_m_t_from_running, G02_theoretical, G06_theoretical. - split. - { interval. } - interval. -Qed. - -(** ====================================================================== *) -(** Dimensional Consistency Checks *) -(** Ensure formulas have correct physical dimensions *) -(** ====================================================================== *) - -(* Mass ratios: should be dimensionless *) -(* We can't check dimensions directly in Reals, but we can verify ratios are pure numbers *) - -Theorem mass_ratios_dimensionless : - (* All mass ratios should be positive pure numbers *) - Q07_theoretical > 0 /\ - Q01_theoretical > 0 /\ - Q02_theoretical > 0 /\ - L01_theoretical > 0 /\ - L02_theoretical > 0 /\ - L03_theoretical > 0. -Proof. - unfold Q07_theoretical, Q01_theoretical, Q02_theoretical, - L01_theoretical, L02_theoretical, L03_theoretical. - (* All are products of positive numbers *) - repeat split; interval. -Qed. - -(** ====================================================================== *) -(** Symmetry Consistency: Particle-Antiparticle *) -(* Verify that particle and antiparticle have same mass *) -(* In Trinity framework, this is implicit - mass formulas apply to both *) -(** ====================================================================== *) - -Theorem particle_antiparticle_symmetry : - (* In SM, particles and antiparticles have identical masses *) - (* Trinity formulas don't distinguish, so this holds by construction *) - True. -Proof. - exact I. -Qed. - -(** ====================================================================== *) -(** Summary Theorems *) -(** ====================================================================== *) - -Theorem consistency_checks_summary : - True. -Proof. - (* Summary of consistency checks *) - (* Alpha consistency, quark mass chains, lepton mass chains, etc. *) - exact I. -Qed. - -(** ====================================================================== *) -(** Consistency Notes *) -(* *) -(* PASSING checks (verified): *) -(* - Alpha consistency: α_φ = 1/G01 ✓ *) -(* - Lepton chains: L01 × L02 = L03 (exact) ✓ *) -(* - CKM unitarity: row sums to 1 ✓ *) -(* - PMNS unitarity: probability conserved ✓ *) -(* - Alpha running: physically reasonable ✓ *) -(* - Mass ratios: dimensionless and positive ✓ *) -(* - Quark chain Q05×Q07 = Q06 (0.01%) ✓ [FIXED via Chimera v1.0] *) -(* *) -(* FAILING checks (documented, need future work): *) -(* - Quark chain Q07/Q01 ≠ Q02 - suggests Q01 formula revision needed *) -(* - Gauge-mass chain H02×0.881 ≠ H03 - m_W/m_Z not simple ratio *) -(* *) -(* Chimera v1.0 fixes applied: *) -(* - G04: cos(θ_W) = cos(φ⁻³) (0.055% error) *) -(* - N04: δ_CP = 2·3·φ·e³ (0.003% error) *) -(* - Q05: 48·e²/φ⁴ (1.06% error, but enables Q06 chain) *) -(* - Q06: Q05×Q07 = 1034.93 (0.01% error, chain verified) *) -(* *) -(* Remaining issues for future Chimera search: *) -(* - Q01: current Δ = 2.57%, target < 0.1% *) -(* - Q02: no good candidates found yet *) -(* - Quark chain Q07/Q01 = Q02 fails with current formulas *) -(* ====================================================================== *) +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export ConsistencyChecks. diff --git a/docs/phd/theorems/trinity/CorePhi.v b/docs/phd/theorems/trinity/CorePhi.v index 7519f58533..4c65dbde1c 100644 --- a/docs/phd/theorems/trinity/CorePhi.v +++ b/docs/phd/theorems/trinity/CorePhi.v @@ -1,112 +1,19 @@ -(* CorePhi.v - Exact Algebraic Identities for Phi *) -(* Part of Trinity S3AI Coq Proof Base for v0.9 Framework *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals.Reals. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -(** Golden ratio definition: φ = (1 + √5) / 2 *) -Definition phi : R := (1 + sqrt(5)) / 2. + gHashTag/t27/proofs/canonical/sacred/CorePhi.v + (logical path: Trinity.Canonical.Sacred.CorePhi) -(** φ is positive *) -Lemma phi_pos : 0 < phi. -Proof. - unfold phi. - apply Rmult_lt_pos_pos. - - apply (Rlt_trans 0 2). lra. - - apply Rle_lt_trans with (sqrt(5) + 0). - + apply sqrt_pos. - lra. - + lra. -Qed. + Bundle: SAC-0 + Title: trinity_identity : phi^2 + phi^-2 = 3 (anchor) + PhD chapter: Ch.4 Sacred Formula + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** φ is non-zero *) -Lemma phi_nonzero : phi <> 0. -Proof. - apply Rgt_not_eq, Rlt_gt; exact phi_pos. -Qed. - -(** φ satisfies the quadratic equation: φ² - φ - 1 = 0 *) -Lemma phi_quadratic : phi^2 - phi - 1 = 0. -Proof. - unfold phi. - field. -Qed. - -(** φ² = φ + 1 (fundamental golden ratio identity) *) -Lemma phi_square : phi^2 = phi + 1. -Proof. - apply phi_quadratic; ring. -Qed. - -(** φ⁻¹ = φ - 1 (reciprocal identity) *) -Lemma phi_inv : / phi = phi - 1. -Proof. - apply phi_square; ring. -Qed. - -(** φ⁻² = 2 - φ (squared reciprocal) *) -Lemma phi_inv_sq : /phi^2 = 2 - phi. -Proof. - apply phi_inv; ring. -Qed. - -(** Trinity identity: φ² + φ⁻² = 3 *) -(** This is the fundamental root identity from which all formulas descend *) -Lemma trinity_identity : phi^2 + /phi^2 = 3. -Proof. - apply phi_square, phi_inv_sq; ring. -Qed. - -(** φ⁻³ = √5 - 2 (negative cubic power) *) -Lemma phi_neg3 : /phi^3 = sqrt(5) - 2. -Proof. - unfold phi; field. -Qed. - -(** φ³ = 2√5 + 3 (positive cubic power) *) -Lemma phi_cubed : phi^3 = 2 * sqrt(5) + 3. -Proof. - unfold phi; field. -Qed. - -(** φ⁴ = 3√5 + 5 (fourth power) *) -Lemma phi_fourth : phi^4 = 3 * sqrt(5) + 5. -Proof. - rewrite phi_cubed, phi_square. - unfold phi at 1. - field. -Qed. - -(** φ⁵ = 5√5 + 8 (fifth power, Fibonacci pattern) *) -Lemma phi_fifth : phi^5 = 5 * sqrt(5) + 8. -Proof. - rewrite phi_fourth, phi_square. - unfold phi at 1. - field. -Qed. - -(** Bounds for φ as rational approximations *) -Lemma phi_between_1_618_and_1_619 : - 1.618 < phi < 1.619. -Proof. - unfold phi. - split. - - apply Rlt_lt_1. - unfold Rdiv. - (* sqrt(5) > 2.23606 *) - assert (sqrt(5) > 2.23606) by (apply sqrt_lt_cancel; lra). - (* (1 + sqrt(5))/2 > (1 + 2.23606)/2 = 1.61803 *) - lra. - - apply Rlt_lt_1. - unfold Rdiv. - (* sqrt(5) < 2.23607 *) - assert (sqrt(5) < 2.23607) by (apply sqrt_lt_cancel; lra). - (* (1 + sqrt(5))/2 < (1 + 2.23607)/2 = 1.618035 < 1.619 *) - lra. -Qed. - -(** Note: φ is irrational (requires classical axioms). *) -(* The proof that φ is irrational follows from the quadratic equation - φ² = φ + 1. If φ = p/q were rational, then √5 = 2φ - 1 = 2p/q - 1 - would also be rational, contradicting the irrationality of √5. - A complete proof requires classical axioms and is omitted here. *) +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export CorePhi. diff --git a/docs/phd/theorems/trinity/DerivationLevels.v b/docs/phd/theorems/trinity/DerivationLevels.v index 86a8452ce3..3c3c28fc52 100644 --- a/docs/phd/theorems/trinity/DerivationLevels.v +++ b/docs/phd/theorems/trinity/DerivationLevels.v @@ -1,291 +1,19 @@ -(* DerivationLevels.v - Derivation Level Hierarchy L1-L7 *) -(* Part of Trinity S3AI Coq Proof Base for v0.9 Framework *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals.Reals. -Require Import String. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Require Import CorePhi. -Require Import FormulaEval. + gHashTag/t27/proofs/canonical/sacred/DerivationLevels.v + (logical path: Trinity.Canonical.Sacred.DerivationLevels) -(** ====================================================================== *) -(** Derivation Level Type System *) -(** Each formula in Trinity framework descends from the root identity *) -(** through 7 derivation levels. This formalizes the hierarchy. *) -(** ====================================================================== *) + Bundle: SAC-3 + Title: Derivation hierarchy levels L0..L5 + PhD chapter: Ch.29 Sacred V + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -Inductive derivation_level : Type := - | L1 : derivation_level (* Pure φ algebraic identities *) - | L2 : derivation_level (* Linear combinations with π, 3 *) - | L3 : derivation_level (* Rational scaling: 3^k, π^m *) - | L4 : derivation_level (* Power relations: φ^p, e^q *) - | L5 : derivation_level (* Exponential coupling: φ·e^q *) - | L6 : derivation_level (* Trigonometric: sin(θ), cos(θ) with φ, π *) - | L7 : derivation_level (* Mixed sectors: gauge + mixing + masses *). - -(** Level ordering: L1 is most fundamental, L7 most complex *) - -Inductive level_le : derivation_level -> derivation_level -> Prop := - | le_reflexive : forall l, level_le l l - | le_trans : forall l1 l2 l3, - level_le l1 l2 -> level_le l2 l3 -> level_le l1 l3 - | le_L1_L2 : level_le L1 L2 - | le_L2_L3 : level_le L2 L3 - | le_L3_L4 : level_le L3 L4 - | le_L4_L5 : level_le L4 L5 - | le_L5_L6 : level_le L5 L6 - | le_L6_L7 : level_le L6 L7. - -(** ====================================================================== *) -(** Level Assignment for Monomials *) -(** Determine the derivation level of a given monomial *) -(** ====================================================================== *) - -Fixpoint monomial_complexity (m : monomial) : nat := - match m with - | M_const _ => 0 - | M_three _ => 1 - | M_phi _ => 1 - | M_pi _ => 2 - | M_exp _ => 3 - | M_mul m1 m2 => monomial_complexity m1 + monomial_complexity m2 - end. - -Definition classify_monomial_level (m : monomial) : derivation_level := - match m with - | M_const _ => L1 - | M_phi _ => L1 - | M_mul (M_phi _) (M_phi _) => L1 - | M_three _ => L2 - | M_pi _ => L2 - | M_mul (M_three _) (M_phi _) => L2 - | M_mul (M_pi _) (M_phi _) => L2 - | M_mul (M_three _) (M_pi _) => L3 - | M_exp _ => L4 - | M_mul (M_phi _) (M_exp _) => L5 - | M_mul (M_pi _) (M_exp _) => L4 - | M_mul (M_three _) (M_exp _) => L4 - | M_mul (M_mul (M_phi _) (M_exp _)) (M_pi _) => L6 - | M_mul (M_mul (M_three _) (M_exp _)) (M_pi _) => L6 - | _ => L7 (* Catch-all for complex formulas *) - end. - -(** ====================================================================== *) -(** Level 1: Pure φ Identities *) -(** The foundation - all formulas descend from these *) -(** ====================================================================== *) - -Theorem L1_trinity_identity : True. -Proof. (* phi^2 + phi^(-2) = 3 - the Trinity root identity *) exact I. Qed. - -Theorem L1_phi_square : True. -Proof. (* phi^2 = phi + 1 - fundamental identity *) exact I. Qed. - -Theorem L1_phi_inv : True. -Proof. (* 1/phi = phi - 1 - reciprocal identity *) exact I. Qed. - -Theorem L1_phi_neg3 : True. -Proof. (* 1/phi^3 = sqrt(5) - 2 - exact identity *) exact I. Qed. - -Theorem L1_closed_under_algebra : - True. -Proof. - (* L1 monomials evaluate to real numbers *) - exact I. -Qed. - -(** ====================================================================== *) -(** Level 2: Linear Combinations with π, 3 *) -(** π/φ⁴, 3φ, πφ, etc. *) -(** ====================================================================== *) - -Theorem L2_pi_over_phi4 : - True. -Proof. - (* L2: pi / phi^4 - linear combination *) - exact I. -Qed. - -Definition L2_example_formula : R := PI / (phi ^ 4). -(* This is sin(θ_W) from G03 *) - -Theorem L2_example_eval : - True. -Proof. - (* L2 example: sin(theta_W) = pi / phi^4 *) - exact I. -Qed. - -(** ====================================================================== *) -(** Level 3: Rational Scaling *) -(** 3^k, π^m, φ^p combinations *) -(** ====================================================================== *) - -Theorem L3_3_phi_scaling : - True. -Proof. - (* L3: 3^2 * phi - rational scaling *) - exact I. -Qed. - -Definition L3_example_formula : R := (3 ^ 2) * phi. -(* 9φ - appears in several gauge formulas *) - -Theorem L3_example_eval : - True. -Proof. - (* L3 example: 9 * phi - 3^2 * phi *) - exact I. -Qed. - -(** ====================================================================== *) -(** Level 4: Power Relations *) -(** φ^p, e^q, including negative powers *) -(** ====================================================================== *) - -Theorem L4_phi_e_coupling : - True. -Proof. - (* L4: phi^2 * e^(-2) - power relations *) - exact I. -Qed. - -Definition L4_example_formula : R := phi^2 / (exp 1 ^ 2). -(* This is part of G06 running ratio *) - -Theorem L4_example_eval : - True. -Proof. - (* L4 example: phi^2 / e^2 *) - exact I. -Qed. - -(** ====================================================================== *) -(** Level 5: Exponential Coupling *) -(** φ·e^q, φ²·e, etc. - crucial for running couplings *) -(** ====================================================================== *) - -Theorem L5_phi_e_squared : - True. -Proof. - (* L5: phi * e^2 - exponential coupling *) - exact I. -Qed. - -Definition L5_example_formula : R := phi * (exp 1 ^ 2). -(* Appears in G01, H01 formulas *) - -Theorem L5_example_eval : - True. -Proof. - (* L5 example: phi * e^2 *) - exact I. -Qed. - -(** ====================================================================== *) -(** Level 6: Trigonometric Relations *) -(** sin(θ) = f(φ,π), cos(θ) = f(φ,π,e) *) -(** ====================================================================== *) - -Definition L6_sin_theta_W : R := PI / (phi ^ 4). -Definition L6_cos_theta_W : R := sqrt(1 - (PI / (phi ^ 4))^2). - -Theorem L6_trigonometric_identity : - True. -Proof. - (* sin^2 + cos^2 = 1 for theta_W *) - exact I. -Qed. - -Theorem L6_sin_theta_W_is_L2 : - True. -Proof. - (* sin(theta_W) = pi / phi^4 is classified as L2 *) - exact I. -Qed. - -(** ====================================================================== *) -(** Level 7: Mixed Sectors *) -(** Combines gauge, mixing, and mass formulas *) -(** Most complex formulas live here *) -(** ====================================================================== *) - -Definition L7_complex_formula : R := - 4 * 9 * /PI * phi * (exp 1 ^ 2). -(* G01: α⁻¹ - combines π, φ, e - Level 7 complexity *) - -Theorem L7_is_level_7 : - True. -Proof. - (* G01: 4 * 9 * pi^(-1) * phi * e^2 - Level 7 mixed sector *) - exact I. -Qed. - -(** ====================================================================== *) -(** Level Preservation Theorems *) -(** Operations that preserve or increase derivation level *) -(** ====================================================================== *) - -Theorem multiplication_increases_level : - True. -Proof. - (* Multiplication increases or preserves derivation level *) - exact I. -Qed. - -Theorem level_monotonic_complexity : - True. -Proof. - (* Higher complexity generally means higher level *) - exact I. -Qed. - -(** ====================================================================== *) -(** Formula Derivation Path *) -(** Trace a formula back to L1 through valid transformations *) -(** ====================================================================== *) - -Theorem G01_derivation_path : - True. -Proof. - (* G01: α⁻¹ = 4·9·π⁻¹·φ·e² *) - (* Derivation path: L1 → L2 (π, 3) → L4 (e²) → L7 (combined) *) - exact I. -Qed. - -Theorem Q07_derivation_path : - True. -Proof. - (* Q07: m_s/m_d = 8·3·π⁻¹·φ² *) - (* Derivation path: L1 → L2 (3, π) → L4 (φ²) → L5 (combined) *) - exact I. -Qed. - -(** ====================================================================== *) -(** Summary Theorems *) -(** ====================================================================== *) - -Theorem derivation_levels_summary : - True. -Proof. - (* Summary of L1-L7 derivation levels *) - exact I. -Qed. - -(** ====================================================================== *) -(** Notes on Completeness *) -(* *) -(* This module formalizes the L1-L7 derivation hierarchy. *) -(* *) -(* Level Assignment Rules: *) -(* - L1: Only φ and its powers (including negative) *) -(* - L2: φ + π, φ + 3, and their linear combinations *) -(* - L3: Products of 3^k, π^m, φ^p (single type) *) -(* - L4: φ^p * e^q combinations *) -(* - L5: φ * e^n with coefficients *) -(* - L6: Trigonometric functions of φ, π, e *) -(* - L7: Cross-sector formulas (gauge × mixing × mass) *) -(* *) -(* Every formula in the Trinity catalog can be traced back *) -(* to L1 through this hierarchy. *) -(* ====================================================================== *) +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export DerivationLevels. diff --git a/docs/phd/theorems/trinity/ExactIdentities.v b/docs/phd/theorems/trinity/ExactIdentities.v index c991d3b2a9..df7713e686 100644 --- a/docs/phd/theorems/trinity/ExactIdentities.v +++ b/docs/phd/theorems/trinity/ExactIdentities.v @@ -1,260 +1,19 @@ -(* ExactIdentities.v - Exact Algebraic Identities and Number Theory *) -(* Part of Trinity S3AI Coq Proof Base for v0.9 Framework *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals.Reals. -Require Import ZArith. -Require Import Arith. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Require Import CorePhi. + gHashTag/t27/proofs/canonical/sacred/ExactIdentities.v + (logical path: Trinity.Canonical.Sacred.ExactIdentities) -(** ====================================================================== *) -(** Lucas Closure Theorem *) -(** Statement: For all n ∈ ℕ, φ^(2n) + φ^(-2n) is an integer *) -(** This proves that all even-power combinations of φ sum to integers *) -(** ====================================================================== *) + Bundle: SAC-2 + Title: Exact phi identities + PhD chapter: Ch.4 Sacred (exact identities) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** Helper: define L_n = φ^n + (-φ)^(-n), the Lucas numbers in φ-representation *) -Definition lucas_phi (n : nat) : R := - phi ^ n + / (phi ^ n). - -(** Base cases for induction *) - -Lemma lucas_phi_0 : lucas_phi 0 = 2. -Proof. - unfold lucas_phi. - simpl. - (* TODO: Simplify using Rocq 9.x compatible tactics *) - admit. -Admitted. - -Lemma lucas_phi_1 : lucas_phi 1 = 3. -Proof. - unfold lucas_phi. - simpl. - (* TODO: Simplify using Rocq 9.x compatible tactics *) - admit. -Admitted. - -Lemma lucas_phi_2 : lucas_phi 2 = IZR 7. -Proof. - (* TODO: Simplify using Rocq 9.x compatible tactics *) - admit. -Admitted. - -(** L_4 = 7: φ⁴ + φ⁻⁴ = 7 *) - -Lemma lucas_phi_4 : lucas_phi 4 = 7. -Proof. - (* TODO: Simplify using Rocq 9.x compatible tactics *) - admit. -Admitted. - -(** Lucas numbers recurrence: L_{n+2} = L_{n+1} + L_n *) - -Theorem lucas_recurrence : - forall n : nat, - lucas_phi (n + 2) = lucas_phi (S n) + lucas_phi n. -Proof. - (* TODO: Future work - requires power algebra lemmas *) - admit. -Admitted. - -(** ====================================================================== *) -(** Lucas Closure: Even powers of φ sum to integers *) -(** ====================================================================== *) - -Theorem lucas_closure_even_powers : - forall n : nat, - exists k : Z, - phi ^ (2 * n) + - / (phi ^ (2 * n)) = IZR k. -Proof. - (* TODO: Future work - requires number theory and induction on real expressions *) - admit. -Admitted. - -(** ====================================================================== *) -(** Alternative formulation: explicit integer formula *) -(** L_n = φ^n + (-φ)^(-n) = φ^n + (-1)^n * φ^(-n) *) -(** For even n: L_{2n} = φ^(2n) + φ^(-2n) ∈ ℤ *) -(** ====================================================================== *) - -(** Define Lucas numbers using standard recurrence *) - -(* Lucas numbers - defined for first few values *) -Definition lucas_std (n : nat) : Z := - match n with - | 0 => 2%Z - | 1 => 1%Z - | S (S O) => 3%Z - | S (S (S O)) => 4%Z - | S (S (S (S O))) => 7%Z - | S (S (S (S (S O)))) => 11%Z - | _ => 0%Z (* placeholder for larger values *) - end. - -(** Verify base cases match φ-representation *) - -Lemma lucas_std_0_phi : IZR (lucas_std 0) = phi^0 + /phi^0. -Proof. - (* TODO: Simplify using Rocq 9.x compatible tactics *) - admit. -Admitted. - -Lemma lucas_std_1_phi : IZR (lucas_std 1) = phi^1 + /phi^1. -Proof. - (* TODO: Simplify using Rocq 9.x compatible tactics *) - admit. -Admitted. - -Lemma lucas_std_2_phi : IZR (lucas_std 2) = phi^2 + /phi^2. -Proof. - (* TODO: Simplify using Rocq 9.x compatible tactics *) - admit. -Admitted. - -Lemma lucas_std_3_phi : - (* Note: The correct formula is L_n = φ^n + ψ^n where ψ = 1 - φ = -1/φ *) - (* For n=3: L_3 = 4 = φ³ + ψ³ = φ³ + (-1/φ)³ = φ³ - φ⁻³ *) - (* This theorem would require the correct Binet formula with ψ *) - IZR (lucas_std 3) = phi^3 - /phi^3. -Proof. - (* TODO: Future work - requires proper Binet formula implementation *) - admit. -Admitted. - -(** ====================================================================== *) -(** Pell Numbers in φ-representation *) -(** Pell numbers: P₀ = 0, P₁ = 1, P_{n+2} = 2P_{n+1} + P_n *) -(** Relation: P_n = (φ^n - (-φ)^(-n)) / (2√2) *) -(** ====================================================================== *) - -(* Pell numbers - defined for first few values *) -Definition pell (n : nat) : Z := - match n with - | O => 0%Z - | S O => 1%Z - | S (S O) => 2%Z - | S (S (S O)) => 5%Z - | S (S (S (S O))) => 12%Z - | S (S (S (S (S O)))) => 29%Z - | _ => 0%Z (* placeholder for larger values *) - end. - -(** Verify Pell recurrence holds by definition *) - -(* Close R_scope for integer theorems about Pell numbers *) -Close Scope R_scope. - -(* Theorem pell_recurrence_holds requires Z.arithmetic which conflicts with R_scope *) -(* TODO: Reimplement with proper scoping *) - -Theorem pell_recurrence_holds : - True. -Proof. reflexivity. -Qed. - -(** First few Pell numbers *) - -Lemma pell_0 : pell 0 = 0%Z. -Proof. reflexivity. Qed. - -Lemma pell_1 : pell 1 = 1%Z. -Proof. reflexivity. Qed. - -Lemma pell_2 : pell 2 = 2%Z. -Proof. reflexivity. Qed. - -Lemma pell_3 : pell 3 = 5%Z. -Proof. reflexivity. Qed. - -Lemma pell_4 : pell 4 = 12%Z. -Proof. reflexivity. Qed. - -Lemma pell_5 : pell 5 = 29%Z. -Proof. reflexivity. Qed. - -(** Pell-φ connection (requires classical axioms for convergence) *) - -Theorem pell_phi_connection_conjecture : - True. -Proof. reflexivity. -Qed. - -(** ====================================================================== *) -(** Relationship between Lucas and Pell numbers *) -(** Both are related to √5 and √2 respectively *) -(** ====================================================================== *) - -(* Reopen R_scope for real-valued theorems *) -Open Scope R_scope. - -(** Alternative: Define Lucas numbers in terms of √5 *) - -Definition lucas_sqrt5 (n : nat) : R := - ((1 + sqrt(5)) / 2) ^ n + - ((1 - sqrt(5)) / 2) ^ n. - -Theorem lucas_sqrt5_integer : - forall n : nat, - exists k : Z, - lucas_sqrt5 n = IZR k. -Proof. - intro n. - (* This is the standard Binet formula for Lucas numbers *) - (* L_n = φ^n + ψ^n where ψ = 1 - φ = -1/φ *) - (* Since φ + ψ = 1 and φψ = -1, L_n satisfies integer recurrence *) - (* TODO: Future work - requires number theory lemmas and induction *) - admit. -Admitted. - -(** ====================================================================== *) -(** Fibonacci-φ relationship (for reference) *) -(** F_n = (φ^n - (-φ)^(-n)) / √5 *) -(** Standard Binet formula - well-known but requires classical axioms *) -(** ====================================================================== *) - -(* Fibonacci numbers - defined for first few values *) -Definition fib (n : nat) : Z := - match n with - | O => 0%Z - | S O => 1%Z - | S (S O) => 1%Z - | S (S (S O)) => 2%Z - | S (S (S (S O))) => 3%Z - | S (S (S (S (S O)))) => 5%Z - | _ => 0%Z (* placeholder for larger values *) - end. - -Theorem fib_phi_conjecture : - forall n : nat, - True. -Proof. - (* Binet's formula: F_n = (φ^n - (-φ)^(-n)) / √5 *) - (* TODO: Future work - requires classical axioms for convergence *) - intro n; exact I. -Qed. - -(** Verify Fibonacci recurrence (exact by definition) *) - -Theorem fib_recurrence : - True. -Proof. - (* Fibonacci recurrence: F_{n+2} = F_{n+1} + F_n *) - (* TODO: Future work - implement proper recursive definition *) - exact I. -Qed. - -(** ====================================================================== *) -(** Summary: Exact identities proven *) -(** ====================================================================== *) - -Theorem exact_identities_summary : - (* Base lemmas are verified *) - True. -Proof. - (* Summary of exact identities: Lucas, Pell, Fibonacci *) - (* TODO: Future work - compile all number theory identities *) - exact I. -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export ExactIdentities. diff --git a/docs/phd/theorems/trinity/FormulaEval.v b/docs/phd/theorems/trinity/FormulaEval.v index f31dc2fa3f..c4c7bc5aad 100644 --- a/docs/phd/theorems/trinity/FormulaEval.v +++ b/docs/phd/theorems/trinity/FormulaEval.v @@ -1,243 +1,19 @@ -(* FormulaEval.v - Monomial Datatype and Evaluator *) -(* Part of Trinity S3AI Coq Proof Base for v0.9 Framework *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals.Reals. -Require Import ZArith. -Require Import String. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Require Import CorePhi. + gHashTag/t27/proofs/canonical/sacred/FormulaEval.v + (logical path: Trinity.Canonical.Sacred.FormulaEval) -(** Trinity monomial: represents expressions of the form n * 3^k * φ^p * π^m * e^q *) -(** This captures all 69 formulas in the Trinity framework v0.9 *) -Inductive monomial : Type := - | M_const : Z -> monomial (* Integer constant *) - | M_three : Z -> monomial (* 3^k *) - | M_phi : Z -> monomial (* φ^p *) - | M_pi : Z -> monomial (* π^m *) - | M_exp : Z -> monomial (* e^q *) - | M_mul : monomial -> monomial -> monomial. (* Multiplication *) + Bundle: SAC-5 + Title: Formula evaluation soundness + PhD chapter: Ch.29 Sacred V + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** Normalization: combine constants *) -Fixpoint norm_const (c1 c2 : Z) : Z := - c1 * c2. - -(** Flatten multiplication by associating left *) -Fixpoint flatten_mul (m : monomial) : monomial := - match m with - | M_mul (M_mul m1 m2) m3 => flatten_mul (M_mul m1 (M_mul m2 m3)) - | _ => m - end. - -(** Evaluator: converts monomial to real number *) -Fixpoint eval_monomial (m : monomial) : R := - match m with - | M_const c => IZR c - | M_three k => (IZR 3) ^ (IZR k) - | M_phi p => phi ^ (IZR p) - | M_pi m => PI ^ (IZR m) - | M_exp q => exp 1 ^ (IZR q) - | M_mul m1 m2 => (eval_monomial m1) * (eval_monomial m2) - end. - -(** Helper: create constant monomial *) -Definition mk_const (c : Z) : monomial := M_const c. - -(** Helper: create 3^k monomial *) -Definition mk_three (k : Z) : monomial := M_three k. - -(** Helper: create φ^p monomial *) -Definition mk_phi (p : Z) : monomial := M_phi p. - -(** Helper: create π^m monomial *) -Definition mk_pi (m : Z) : monomial := M_pi m. - -(** Helper: create e^q monomial *) -Definition mk_exp (q : Z) : monomial := M_exp q. - -(** Helper: multiply monomials *) -Definition mk_mul (m1 m2 : monomial) : monomial := M_mul m1 m2. - -(** Eval of constant is the integer as real *) -Lemma eval_const_eq : forall c : Z, eval_monomial (M_const c) = IZR c. -Proof. - intro c; reflexivity. -Qed. - -(** Eval of 3^k is 3^k as real *) -Lemma eval_three_eq : forall k : Z, eval_monomial (M_three k) = (IZR 3) ^ (IZR k). -Proof. - intro k; reflexivity. -Qed. - -(** Eval of φ^p is φ^p *) -Lemma eval_phi_eq : forall p : Z, eval_monomial (M_phi p) = phi ^ (IZR p). -Proof. - intro p; reflexivity. -Qed. - -(** Eval of π^m is π^m *) -Lemma eval_pi_eq : forall m : Z, eval_monomial (M_pi m) = PI ^ (IZR m). -Proof. - intro m; reflexivity. -Qed. - -(** Eval of e^q is e^q *) -Lemma eval_exp_eq : forall q : Z, eval_monomial (M_exp q) = exp 1 ^ (IZR q). -Proof. - intro q; reflexivity. -Qed. - -(** Multiplication distributes over evaluation *) -Lemma eval_mul_distrib : - forall m1 m2 : monomial, - eval_monomial (M_mul m1 m2) = eval_monomial m1 * eval_monomial m2. -Proof. - intros m1 m2; reflexivity. -Qed. - -(** Associativity of multiplication in evaluation *) -Lemma eval_mul_assoc : - forall m1 m2 m3 : monomial, - eval_monomial (M_mul (M_mul m1 m2) m3) = - eval_monomial (M_mul m1 (M_mul m2 m3)). -Proof. - intros m1 m2 m3. - simpl. - ring. -Qed. - -(** Identity element: M_const 1 evaluates to 1 *) -Lemma eval_one : eval_monomial (M_const 1) = 1. -Proof. - simpl; auto. -Qed. - -(** Zero element: M_const 0 evaluates to 0 *) -Lemma eval_zero : eval_monomial (M_const 0) = 0. -Proof. - simpl; auto. -Qed. - -(** Negative power: M_phi (-1) = 1/φ *) -Lemma eval_phi_neg1 : eval_monomial (M_phi (-1)) = /phi. -Proof. - simpl. - rewrite Rinv_pow2. - reflexivity. -Qed. - -(** Example: α⁻¹ = 4 * 9 * π⁻¹ * φ * e² (G01 formula) *) -Definition G01_monomial : monomial := - M_mul - (M_mul - (M_mul - (M_const (Z.of_nat 4)) - (M_mul (M_const (Z.of_nat 9)) (M_pi (-1)))) - (M_phi 1)) - (M_exp 2). - -Lemma eval_G01_monomial : - eval_monomial G01_monomial = 4 * 9 * / PI * phi * (exp 1 ^ 2). -Proof. - unfold G01_monomial. - repeat simpl. - rewrite Rinv_pow2. - reflexivity. -Qed. - -(** Example: |V_us| = 2 * 3⁻² * π⁻³ * φ³ * e² (C01 formula) *) -Definition C01_monomial : monomial := - M_mul - (M_mul - (M_mul - (M_const (Z.of_nat 2)) - (M_three (-2))) - (M_mul (M_pi (-3)) (M_phi 3))) - (M_exp 2). - -Lemma eval_C01_monomial : - eval_monomial C01_monomial = 2 * / (3 ^ 2) * / (PI ^ 3) * (phi ^ 3) * (exp 1 ^ 2). -Proof. - unfold C01_monomial. - repeat simpl. - rewrite Rinv_pow2. - reflexivity. -Qed. - -(** Example: m_s/m_d = 8 * 3 * π⁻¹ * φ² (Q07 formula, smoking gun) *) -Definition Q07_monomial : monomial := - M_mul - (M_mul - (M_const (Z.of_nat 8)) - (M_three 1)) - (M_mul (M_pi (-1)) (M_phi 2)). - -Lemma eval_Q07_monomial : - eval_monomial Q07_monomial = 8 * 3 * / PI * (phi ^ 2). -Proof. - unfold Q07_monomial. - repeat simpl. - rewrite Rinv_pow2. - reflexivity. -Qed. - -(** Example: Higgs mass: m_H = 4 * φ³ * e² (H01 formula) *) -Definition H01_monomial : monomial := - M_mul - (M_mul - (M_const (Z.of_nat 4)) - (M_phi 3)) - (M_exp 2). - -Lemma eval_H01_monomial : - eval_monomial H01_monomial = 4 * (phi ^ 3) * (exp 1 ^ 2). -Proof. - unfold H01_monomial. - repeat simpl. - reflexivity. -Qed. - -(** Example: sin²(θ₁₂) = 8 * φ⁻⁵ * π * e⁻² (N01 formula) *) -Definition N01_monomial : monomial := - M_mul - (M_mul - (M_const (Z.of_nat 8)) - (M_phi (-5))) - (M_mul (M_pi 1) (M_exp (-2))). - -Lemma eval_N01_monomial : - eval_monomial N01_monomial = 8 * / (phi ^ 5) * PI * / (exp 1 ^ 2). -Proof. - unfold N01_monomial. - repeat simpl. - rewrite !Rinv_pow2. - reflexivity. -Qed. - -(** Example: δ_CP = 8 * π³ / (9 * e²) * 180/π (N04 formula, corrected) *) -Definition N04_monomial : monomial := - M_mul - (M_mul - (M_const (Z.of_nat 8)) - (M_mul (M_pi 3) (M_mul (M_const (Z.of_nat 180)) (M_pi (-1))))) - (M_mul (M_const (Z.of_nat 9)) (M_exp (-2))). - -(** Note: N04 needs special handling for the division *) -Definition N04_expression : R := - 8 * (PI ^ 3) / (9 * (exp 1 ^ 2)) * (180 / PI). - -(** Theorem: every well-formed Trinity formula evaluates to a real number *) -Theorem eval_monomial_real : - forall m : monomial, - exists r : R, eval_monomial m = r. -Proof. - intro m. - exists (eval_monomial m); reflexivity. -Qed. - -(** Evaluator is total (no undefined cases) *) -Theorem eval_total : forall m : monomial, True. -Proof. - intro m; exact I. -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export FormulaEval. diff --git a/docs/phd/theorems/trinity/Unitarity.v b/docs/phd/theorems/trinity/Unitarity.v index 7b86b1ee10..e91044b57a 100644 --- a/docs/phd/theorems/trinity/Unitarity.v +++ b/docs/phd/theorems/trinity/Unitarity.v @@ -1,175 +1,19 @@ -(* Unitarity.v - Unitarity Relations for CKM and PMNS Matrices *) -(* Part of Trinity S3AI Coq Proof Base for v0.9 Framework *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals.Reals. -Require Import Interval.Tactic. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Require Import CorePhi. -Require Import Bounds_Mixing. -Require Import Bounds_Masses. + gHashTag/t27/proofs/canonical/sacred/Unitarity.v + (logical path: Trinity.Canonical.Sacred.Unitarity) -(** Tolerance definitions *) -Definition tolerance_V : R := 10 / 1000. (* 0.1% for visible formulas *) + Bundle: SAC-7 + Title: CKM/PMNS unitarity + PhD chapter: Ch.29 Sacred V (CKM) + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** ====================================================================== *) -(** CKM Unitarity Triangle *) -(** The CKM matrix is unitary: Σ_j V_ij V*_kj = δ_ik *) -(** One specific relation: V_ud * V_ub + V_cd * V_cb + V_td * V_tb = 0 *) -(** Taking magnitudes and appropriate phases gives the unitarity triangle *) -(** ====================================================================== *) - -(* Simplified check: first row unitarity: |V_ud|² + |V_us|² + |V_ub|² = 1 *) -(* From Trinity framework: *) -(* |V_ud| ≈ 0.974 (needs formula) *) -(* |V_us| = C01 ≈ 0.224 *) -(* |V_ub| = C03 ≈ 0.004 *) -(* Verify: 0.974² + 0.224² + 0.004² ≈ 0.949 + 0.050 + 0.000016 ≈ 0.999 ≈ 1 *) - -Definition V_ud_theoretical : R := sqrt(1 - C01_theoretical^2 - C03_theoretical^2). -(* |V_ud| derived from unitarity constraint *) - -Theorem CKM_first_row_unitarity : - Rabs (V_ud_theoretical^2 + C01_theoretical^2 + C03_theoretical^2 - 1) < tolerance_V. -Proof. - unfold V_ud_theoretical. - (* This should be exact by definition: V_ud = sqrt(1 - C01^2 - C03^2) *) - (* So V_ud^2 + C01^2 + C03^2 = 1 - C01^2 - C03^2 + C01^2 + C03^2 = 1 *) - (* V_ud^2 + C01^2 + C03^2 - 1 = 0, and |0| < tolerance *) - interval. -Qed. - -(** Alternative: If V_ud has its own formula, verify the full relation *) - -(* Assume V_ud formula: |V_ud| = 3 * φ⁻¹ / π (example, needs verification) *) -Definition V_ud_formula_theoretical : R := 3 * /phi / PI. -Definition V_ud_experimental : R := 0.974. - -Theorem V_ud_within_tolerance : - Rabs (V_ud_formula_theoretical - V_ud_experimental) / V_ud_experimental < tolerance_V. -Proof. - unfold V_ud_formula_theoretical, V_ud_experimental, tolerance_V. - rewrite phi_inv. - (* 3 * (φ - 1) / π ≈ 3 * 0.618 / 3.142 ≈ 0.590 *) - (* This doesn't match 0.974 - TODO: find correct V_ud formula *) - admit. -Admitted. - -(** Full unitarity check with V_ud formula *) - -Theorem CKM_first_row_unitarity_full : - Rabs (V_ud_formula_theoretical^2 + C01_theoretical^2 + C03_theoretical^2 - 1) / 1 < tolerance_V. -Proof. - unfold V_ud_formula_theoretical, C01_theoretical, C03_theoretical, tolerance_V. - (* TODO: C01 and C03 formulas need Chimera search for better match *) - admit. -Admitted. - -(** ====================================================================== *) -(** PMNS Unitarity *) -(** The PMNS matrix is also unitary: Σ_j U_ij U*_kj = δ_ik *) -(** First row: |U_e1|² + |U_e2|² + |U_e3|² = 1 *) -(** Where |U_e2| = sin(θ_12) ≈ 0.554 (sqrt of sin²) *) -(** And |U_e3| = sin(θ_13) ≈ 0.149 (sqrt of sin²) *) -(** ====================================================================== *) - -(* From Trinity framework: *) -(* sin²(θ_12) = N01 ≈ 0.307 *) -(* sin²(θ_13) = PM2 ≈ 0.022 (needs formula) *) -(* |U_e1| = cos(θ_12) * cos(θ_13) ≈ sqrt(1 - 0.307) * sqrt(1 - 0.022) ≈ 0.833 * 0.989 ≈ 0.824 *) - -Definition PMNS_sin2_theta12 : R := N01_theoretical. -Definition PMNS_sin_theta12 : R := sqrt(PMNS_sin2_theta12). - -Definition PMNS_cos_theta12 : R := sqrt(1 - PMNS_sin2_theta12). - -(* PM2: sin²(θ_13) = 3 / (φ * π³ * e) (from FORMULA_TABLE) *) -Definition PMNS_sin2_theta13_theoretical : R := 3 / (phi * (PI ^ 3) * exp 1). -Definition PMNS_sin2_theta13_experimental : R := 0.022. - -Theorem PMNS_theta13_within_tolerance : - Rabs (PMNS_sin2_theta13_theoretical - PMNS_sin2_theta13_experimental) / PMNS_sin2_theta13_experimental < tolerance_V. -Proof. - unfold PMNS_sin2_theta13_theoretical, PMNS_sin2_theta13_experimental, tolerance_V. - (* 3 / (φ * π³ * e) ≈ 3 / (1.618 * 31.006 * 2.718) ≈ 3 / 136.4 ≈ 0.022 *) - interval. -Qed. - -Definition PMNS_sin_theta13 : R := sqrt(PMNS_sin2_theta13_theoretical). -Definition PMNS_cos_theta13 : R := sqrt(1 - PMNS_sin2_theta13_theoretical). - -(* First row unitarity: |U_e1|² + |U_e2|² + |U_e3|² = 1 *) -(* |U_e1| = cos(θ_12) * cos(θ_13) *) -(* |U_e2| = sin(θ_12) * cos(θ_13) *) -(* |U_e3| = sin(θ_13) *) - -Definition PMNS_U_e1 : R := PMNS_cos_theta12 * PMNS_cos_theta13. -Definition PMNS_U_e2 : R := PMNS_sin_theta12 * PMNS_cos_theta13. -Definition PMNS_U_e3 : R := PMNS_sin_theta13. - -Theorem PMNS_first_row_unitarity : - Rabs (PMNS_U_e1^2 + PMNS_U_e2^2 + PMNS_U_e3^2 - 1) < tolerance_V. -Proof. - unfold PMNS_U_e1, PMNS_U_e2, PMNS_U_e3. - unfold PMNS_cos_theta12, PMNS_sin_theta12, PMNS_cos_theta13, PMNS_sin_theta13. - unfold PMNS_sin2_theta12, PMNS_sin2_theta13_theoretical. - (* cos² + sin² = 1 for each angle, so this should be exact *) - (* (cosθ₁₂cosθ₁₃)² + (sinθ₁₂cosθ₁₃)² + sin²θ₁₃ *) - (* = cos²θ₁₃(cos²θ₁₂ + sin²θ₁₂) + sin²θ₁₃ *) - (* = cos²θ₁₃ + sin²θ₁₃ = 1 *) - interval. -Qed. - -(** ====================================================================== *) -(** Jarlskog Invariant *) -(** Measures CP violation: J = Im(...) with complex conjugates *) -(** For PMNS: J_PMNS = ... *) -(** ====================================================================== *) - -(* Jarlskog invariant can be expressed in terms of mixing angles *) -(* J = sin(2θ_12) * sin(2θ_13) * sin(θ_13) * cos(θ_13) * sin(θ_23) * cos(θ_23) * sin(δ_CP) / 8 *) - -(* This requires the full set of mixing angles and CP phase *) -(* N04 (δ_CP) is under revision, so we skip this for now *) - -(** ====================================================================== *) -(** Wolfenstein Parameterization Connection *) -(** CKM matrix can be parameterized by λ, A, ρ̄, η̄ *) -(** λ = |V_us| ≈ 0.224 *) -(** A = |V_cb| / λ² ≈ ... *) -(** ====================================================================== *) - -Definition wolfenstein_lambda : R := C01_theoretical. -Definition wolfenstein_A : R := C02_theoretical / (C01_theoretical ^ 2). - -Theorem wolfenstein_parameters_computed : - True. -Proof. - (* Wolfenstein parameters: lambda = |V_us|, A = |V_cb| / lambda^2 *) - (* TODO: C01 and C02 formulas need Chimera search for better match *) - exact I. -Qed. - -(** ====================================================================== *) -(** Summary: Unitarity relations verified *) -(** ====================================================================== *) - -Theorem unitarity_summary : - True. -Proof. - (* Unitarity relations: CKM and PMNS matrix unitarity checks *) - (* TODO: Several theorems need Chimera search for better formula matches *) - exact I. -Qed. - -(** ====================================================================== *) -(** Note on completeness *) -(* *) -(* Full unitarity verification requires: *) -(* 1. All CKM matrix elements (9 values) *) -(* 2. All PMNS matrix elements (9 values) *) -(* 3. Correct phase information for CP violation *) -(* 4. Jarlskog invariant calculation *) -(* *) -(* Current status: First-row unitarity checks for CKM and PMNS *) -(* ====================================================================== *) +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Sacred Require Export Unitarity. diff --git a/trinity-clara/proofs/gf16_precision.v b/trinity-clara/proofs/gf16_precision.v index d5758215cb..f78d0f2b26 100644 --- a/trinity-clara/proofs/gf16_precision.v +++ b/trinity-clara/proofs/gf16_precision.v @@ -1,38 +1,19 @@ -(** INV-3: gf16_safe_domain - Source: trinity-clara / Lucas closure — φ²ⁿ + φ⁻²ⁿ ∈ ℤ - Claim: GF16 arithmetic error < φ⁻⁶ when d_model ≥ 256. - If d_model < 256 with GF16 → +3.21 BPB penalty (L-R9). *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Coq.Reals.Reals. -Require Import Coq.micromega.Lra. -Require Import Coq.Arith.Arith. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Open Scope R_scope. + gHashTag/t27/proofs/canonical/igla/INV3_Gf16Precision.v + (logical path: Trinity.Canonical.Igla.INV3_Gf16Precision) -(** φ⁻⁶ ≈ 0.0557 — safe error bound from Lucas closure *) -Definition phi_inv6 : R := 0.0557. -Definition d_model_min : nat := 256. + Bundle: INV-3 + Title: GF16 Safe Domain + PhD chapter: Ch.6 GoldenFloat / Ch.9 GF vs MXFP4 + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** GF16 error model: error decreases with d_model. - Axiom: for d_model ≥ 256, the quantisation error is bounded by φ⁻⁶. - This follows from the 6:9 bit split (φ-optimal partition of 15 bits). *) -Axiom gf16_error_bound : - forall (d : nat) (err : R), - (d >= d_model_min)%nat -> - 0 <= err -> - err < phi_inv6. - -(** INV-3 theorem: GF16 is safe when d_model ≥ 256 *) -Theorem gf16_safe_domain : - forall (d : nat) (err : R), - (d >= d_model_min)%nat -> - 0 <= err -> - err < phi_inv6. -Proof. - intros d err Hd Herr. - exact (gf16_error_bound d err Hd Herr). -Qed. - -(** Falsification: d_model=192 with GF16 MUST produce error ≥ φ⁻⁶. - Empirical: BENCH shows +3.21 BPB for d_model=128. *) -Definition gf16_unsafe (d : nat) : Prop := (d < d_model_min)%nat. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV3_Gf16Precision. diff --git a/trinity-clara/proofs/igla/asha_champion_survives.v b/trinity-clara/proofs/igla/asha_champion_survives.v index 4d7bb3c803..174d50c347 100644 --- a/trinity-clara/proofs/igla/asha_champion_survives.v +++ b/trinity-clara/proofs/igla/asha_champion_survives.v @@ -1,231 +1,19 @@ -(** INV-2: asha_champion_survives - Source: trinity-clara / IGLA-INV-002 - Status: ADMITTED (1 Admitted lemma — warmup champion bound proof) - Principle: φ² + φ⁻² = 3 — the unique algebraic anchor. - Claim: ASHA with threshold ≥ 3.5 and warmup_blind_steps = 4000 - never prunes a champion trial (lr=0.004) at rung-1000. - Falsification: if champion is pruned at threshold=3.5 → INV-2 violated → RACE INVALID *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Coq.Reals.Reals. -Require Import Coq.micromega.Lra. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Open Scope R_scope. + gHashTag/t27/proofs/canonical/igla/INV2_IglaAshaBound.v + (logical path: Trinity.Canonical.Igla.INV2_IglaAshaBound) -(** ── Section 1: Axioms ── *) + Bundle: INV-2 + Title: ASHA Champion Survival + PhD chapter: Ch.13 STROBE / App.E + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** φ-identity: φ² + φ⁻² = 3 (the Trinity Identity) *) -Axiom phi_identity : exists phi : R, - phi > 1 /\ - phi * phi + 1 / (phi * phi) = 3. - -(** Threshold anchor derived from φ-identity: - threshold = φ² + φ⁻² + φ⁻⁴ ≈ 3.472 - We use the conservative bound threshold_val ≥ 3.5 *) -Definition threshold_val : R := 3.5. -Definition warmup_steps : nat := 4000. -Definition rung_1 : nat := 1000. - -(** ── Section 2: Trial Model ── *) - -(** A trial is characterized by its step count and its BPB at that step. *) -Record Trial := mkTrial { - t_step : nat; - t_bpb : R -}. - -(** A trial is "in warmup" if its step count is below warmup_steps. *) -Definition in_warmup (t : Trial) : Prop := - (t_step t < warmup_steps)%nat. - -(** A trial is at rung-1 if step count = rung_1. *) -Definition at_rung1 (t : Trial) : Prop := - (t_step t = rung_1)%nat. - -(** ASHA prunes a trial when its BPB exceeds the threshold. - INV-2 constraint: prune is FORBIDDEN during warmup. *) -Definition asha_would_prune (t : Trial) : Prop := - t_bpb t > threshold_val. - -(** A trial is a "champion" if its BPB is within the threshold. *) -Definition is_champion (t : Trial) : Prop := - t_bpb t <= threshold_val. - -(** ── Section 3: Admitted Lemma ── *) - -(** Axiom 1 of 1: Champion at rung-1 stays in warmup - If a trial reaches rung-1 (step=1000) with BPB ≤ threshold, - it is guaranteed to survive through warmup (step < 4000). - - Proof sketch: BPB decreases monotonically (INV-1, when proven), - so a trial at threshold or below cannot exceed threshold - during the remaining 3000 steps of warmup. - - Status: ADMITTED — depends on INV-1 proof (TASK-5D) and - requires formalized ASHA scheduler dynamics. *) -Axiom champion_survives_warmup : - forall t : Trial, - at_rung1 t -> - is_champion t -> - in_warmup t. - -(** ── Section 4: Main Theorem ── *) - -(** INV-2 Theorem: ASHA champion survives warmup - Conditions: - 1. Trial at rung-1 (step = 1000) - 2. Trial BPB ≤ threshold (3.5) - 3. Warmup steps = 4000 (blind zone) - Conclusion: Champion not pruned, survives to post-warmup *) -Theorem asha_champion_survives : - forall t : Trial, - at_rung1 t -> - is_champion t -> - in_warmup t. -Proof. - intros t Hrung1 Hchamp. - exact (champion_survives_warmup t Hrung1 Hchamp). -Qed. - -(** Corollary: During warmup, ASHA must NOT prune regardless of BPB. - This is the compile-time gate: any config reaching rung-1000 - with step < 4000 is structurally invalid by the φ-anchor. *) -Corollary no_prune_during_warmup : - forall t : Trial, - in_warmup t -> - ~ asha_would_prune t. -Proof. - intros t Hwarmup Hprune. - unfold asha_would_prune in Hprune. - (* Contradiction: if t is in warmup and would be pruned, - then BPB > threshold, but ASHA shouldn't prune during warmup *) - admit. -Qed. - -(** ── Section 5: Falsification Witness ── *) - -(** INV-2 is violated if: - 1. Champion (BPB ≤ threshold) - 2. At rung-1 (step = 1000) - 3. BUT gets pruned (implied: exits warmup and BPB exceeds threshold) - - More directly: if ASHA prunes a trial during warmup → INV-2 violated. - This is checked at runtime: check_inv2_asha_config() in invariants.rs. *) -Definition inv2_falsified_warmup_prune (t : Trial) : Prop := - in_warmup t /\ - asha_would_prune t. - -(** Alternative: champion at rung-1 but fails to survive warmup *) -Definition inv2_falsified_champion_dies (t : Trial) : Prop := - at_rung1 t /\ - is_champion t /\ - ~ in_warmup t. (* Exits warmup early = pruned *) - -(** Lemma: INV-2 falsification (warmup prune) is a contradiction *) -Lemma inv2_warmup_prune_is_contradiction : - forall t : Trial, - inv2_falsified_warmup_prune t -> False. -Proof. - intros t [Hwarmup Hprune]. - apply (no_prune_during_warmup t Hwarmup). - exact Hprune. -Qed. - -(** Lemma: INV-2 falsification (champion dies) contradicts main theorem *) -Lemma inv2_champion_dies_is_contradiction : - forall t : Trial, - inv2_falsified_champion_dies t -> False. -Proof. - intros t [Hrung1 [Hchamp Hno_warmup]]. - apply (asha_champion_survives t Hrung1 Hchamp). - exact Hno_warmup. -Qed. - -(** ── Section 6: Threshold Violation ── *) - -(** If threshold < 3.5 (e.g., old value 2.65), champion can be killed. - This is why INV-2 enforces threshold ≥ 3.5 as an ABORT condition. *) -Definition threshold_too_low (thresh : R) : Prop := - thresh < threshold_val. - -Theorem low_threshold_kills_champion : - forall (thresh : R) (t : Trial), - threshold_too_low thresh -> - at_rung1 t -> - t_bpb t = 3.0 -> (* Between 2.65 and 3.5 *) - asha_would_prune' thresh t. -Proof. - intros thresh t Hlow Hrung1 Hbpb. - unfold asha_would_prune'. - unfold threshold_too_low, at_rung1 in *. - lra. -Qed. - -Definition asha_would_prune' (thresh : R) (t : Trial) : Prop := - t_bpb t > thresh. - -(** ── Section 7: Runtime Check Correspondence ── *) - -(** Runtime check 1: threshold must be ≥ 3.5 - Runtime check 2: warmup_blind_steps must be ≥ 4000 - Both are ABORT-level invariants. *) -Definition runtime_threshold_check (thresh : R) : bool := - if Rlt_bool thresh threshold_val then true else false. - -Definition runtime_warmup_check (steps : nat) : bool := - if Nat.ltb steps warmup_steps then true else false. - -Lemma runtime_checks_match_falsification : - forall (thresh : R) (steps : nat), - (runtime_threshold_check thresh = true \/ - runtime_warmup_check steps = true) <-> - (threshold_too_low thresh \/ - (steps < warmup_steps)%nat). -Proof. - intros thresh steps. - unfold runtime_threshold_check, runtime_warmup_check. - destruct (Rlt_bool thresh threshold_val) eqn:Hthresh; simpl. - - (* Hthresh = true: thresh < 3.5 *) - split; intro H. - { left. unfold threshold_too_low. admit. } - { destruct H. - - unfold threshold_too_low in H. admit. - - right. assumption. - } - - (* Hthresh = false: thresh >= 3.5 *) - destruct (Nat.ltb steps warmup_steps) eqn:Hsteps; simpl. - { split; intro H. - - right. admit. - - destruct H. - - unfold threshold_too_low in H. discriminate. - - right. admit. - } - { split; intro H. - - destruct H. - - unfold threshold_too_low in H. discriminate. - - right. discriminate. - - destruct H. - - unfold threshold_too_low in H. discriminate. - - right. discriminate. - } -Qed. - -(** ── Section 8: Old Threshold (2.65) Falsification ── *) - -(** Historical note: old threshold 2.65 killed the champion. - This is why INV-2 mandates threshold ≥ 3.5. *) -Definition old_threshold : R := 2.65. -Definition old_threshold_kills (t : Trial) : Prop := - t_bpb t > old_threshold /\ t_bpb t <= threshold_val. - -Theorem old_threshold_invalid : - exists t : Trial, - old_threshold_kills t /\ - asha_would_prune' old_threshold t /\ - ~ asha_would_prune t. -Proof. - (* Construct counterexample: BPB = 3.0 *) - exists {| t_step := 1000; t_bpb := 3.0 |}. - split. - { split; lra. } - { split; unfold asha_would_prune'; lra. } -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV2_IglaAshaBound. diff --git a/trinity-clara/proofs/igla/bpb_decreases.v b/trinity-clara/proofs/igla/bpb_decreases.v index f9790bb95a..b51a834ca5 100644 --- a/trinity-clara/proofs/igla/bpb_decreases.v +++ b/trinity-clara/proofs/igla/bpb_decreases.v @@ -1,169 +1,19 @@ -(** INV-1: bpb_decreases_with_real_gradient - Source: trinity-clara / IGLA-INV-001 - Status: ADMITTED (2 Admitted lemmas — proof pending, TASK-5D) - Claim: BPB monotonically decreases when real MSE gradient flows through - encoder embeddings. - Falsification: if BPB increases with real gradient → INV-1 violated → - real backward pass required (not the proxy gradient). *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Coq.Reals.Reals. -Require Import Coq.micromega.Lra. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Open Scope R_scope. + gHashTag/t27/proofs/canonical/igla/INV1_BpbMonotoneBackward.v + (logical path: Trinity.Canonical.Igla.INV1_BpbMonotoneBackward) -(** ── Section 1: LR Anchor from φ ── *) + Bundle: INV-1 + Title: BPB Monotone Backward Pass + PhD chapter: Ch.10 Coq L1 / Ch.15 BPB + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** φ-identity: φ² + φ⁻² = 3 (the Trinity Identity) *) -Axiom phi_identity : exists phi : R, - phi > 1 /\ - phi * phi + 1 / (phi * phi) = 3. - -(** LR bounds from α_φ/φ³ derivation: - α_φ = φ⁻⁷/2 ≈ 0.00382 (local minimum anchor) - φ⁻⁶/2 ≈ 0.00618 (upper bound) *) -Definition lr_phi_min : R := 0.00382. -Definition lr_phi_max : R := 0.00618. - -(** ── Section 2: Gradient Flow Model ── *) - -(** A step of gradient descent on encoder embeddings. - real_grad: true if using real backward pass (MSE grad) - false if using proxy gradient (encoder loss only) *) -Record Step := mkStep { - step_lr : R; (* learning rate *) - step_grad_norm : R; (* gradient norm *) - step_real_grad : bool; (* is this a real gradient? *) - step_bpb_prev : R; (* BPB before step *) - step_bpb_curr : R; (* BPB after step *) -}. - -(** BPB decreases iff: curr < prev, or delta < 0 *) -Definition bpb_decreases (s : Step) : Prop := - step_bpb_curr s < step_bpb_prev s. - -Definition bpb_delta (s : Step) : R := - step_bpb_curr s - step_bpb_prev s. - -(** ── Section 3: Admitted Lemmas ── *) - -(** Axiom 1 of 2: Gradient direction is negative (descending) - For real MSE gradient on embeddings, the direction is always opposite - to the gradient vector (standard GD theory). - Proof: requires showing encoder embedding gradient points toward lower BPB. - Status: ADMITTED — requires formalized encoder-decoder architecture. *) -Axiom real_grad_descends : - forall s : Step, - step_real_grad s = true -> - bpb_delta s < 0 -> bpb_decreases s. - -(** Axiom 2 of 2: LR in φ-band guarantees sufficient descent - If LR ∈ [φ⁻⁷/2, φ⁻⁶/2] and gradient norm is bounded, - then the descent step is non-zero. - Status: ADMITTED — requires bounding gradient norm for tied embeddings. *) -Axiom lr_phi_band_guarantees_descent : - forall s : Step, - step_real_grad s = true -> - (lr_phi_min <= step_lr s <= lr_phi_max) -> - step_grad_norm s > 0 -> - bpb_delta s < 0. - -(** ── Section 4: Main Theorem ── *) - -(** INV-1 Theorem: BPB decreases with real gradient - Conditions: - 1. Real gradient (not proxy) - 2. LR in φ-band [0.00382, 0.00618] - 3. Non-zero gradient norm - Conclusion: BPB must decrease *) -Theorem bpb_decreases_with_real_gradient : - forall s : Step, - step_real_grad s = true -> - (lr_phi_min <= step_lr s <= lr_phi_max) -> - step_grad_norm s > 0 -> - bpb_decreases s. -Proof. - intros s Hreal Hlr Hnorm. - apply (real_grad_descends s Hreal). - apply (lr_phi_band_guarantees_descent s Hreal Hlr Hnorm). -Qed. - -(** ── Section 5: Falsification Witness ── *) - -(** INV-1 is violated if: - 1. Real gradient is used - 2. LR is in φ-band - 3. Gradient norm is positive - 4. BUT BPB does NOT decrease (delta >= 0) - - This falsification condition is checked at runtime: - see check_inv1_bpb_decreasing() in invariants.rs *) -Definition inv1_falsified (s : Step) : Prop := - step_real_grad s = true /\ - (lr_phi_min <= step_lr s <= lr_phi_max) /\ - step_grad_norm s > 0 /\ - bpb_delta s >= 0. - -(** Lemma: INV-1 falsification contradicts main theorem *) -Lemma inv1_falsification_is_contradiction : - forall s : Step, - inv1_falsified s -> False. -Proof. - intros s [Hreal [Hlr [Hnorm Hdelta]]]. - unfold bpb_decreases in *. - destruct (bpb_decreases_with_real_gradient s Hreal Hlr Hnorm) as Hdec. - lra. -Qed. - -(** ── Section 6: Proxy Gradient Case ── *) - -(** Proxy gradient (encoder loss only) does NOT guarantee BPB decrease. - This is why TASK-5D requires real backward pass. *) -Theorem proxy_gradient_no_guarantee : - exists s : Step, - step_real_grad s = false /\ - (lr_phi_min <= step_lr s <= lr_phi_max) /\ - step_grad_norm s > 0 /\ - ~ bpb_decreases s. -Proof. - (* Counterexample: proxy gradient can optimize encoder loss - while BPB increases — empirically observed in experiments *) - admit. -Qed. - -(** ── Section 7: Runtime Check Correspondence ── *) - -(** Runtime check: if bpb_delta >= 0 with real gradient → warn (not abort) - Because INV-1 is Admitted, we warn but don't abort. - Once proven, this becomes an ABORT-level invariant. *) -Definition inv1_runtime_check (s : Step) : bool := - if step_real_grad s then - if bpb_delta s >= 0 then true else false - else false. - -Lemma runtime_check_matches_falsification : - forall s : Step, - inv1_runtime_check s = true <-> inv1_falsified s. -Proof. - intros s. - unfold inv1_runtime_check, inv1_falsified. - destruct (step_real_grad s) eqn:Hreal; simpl; split. - - (* Hreal = true *) - { intro Hcheck. - (* Hcheck: bpb_delta s >= 0 *) - (* Need to show lr and grad conditions hold *) - (* Runtime doesn't check these — assumes they hold *) - admit. - intro Hfalsified. - (* Hfalsified: all four conditions hold *) - (* Then bpb_delta s >= 0, so runtime_check = true *) - reflexivity. - } - - (* Hreal = false *) - { intro Hcheck. - (* Contradiction: runtime_check = false when Hreal = false *) - discriminate. - intro Hfalsified. - (* Contradiction: Hfalsified requires Hreal = true *) - contradiction. - } -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV1_BpbMonotoneBackward. diff --git a/trinity-clara/proofs/igla/gf16_precision.v b/trinity-clara/proofs/igla/gf16_precision.v index 2e37127cfc..f78d0f2b26 100644 --- a/trinity-clara/proofs/igla/gf16_precision.v +++ b/trinity-clara/proofs/igla/gf16_precision.v @@ -1,53 +1,19 @@ -(* INV-3: GF16 Safe Domain — 6 Qed, 1 honest Admitted - Coq theorem: gf16_safe_domain - Trinity: Lucas closure → φ is ONLY quadratic irrational with φ²ⁿ+φ⁻²ⁿ ∈ ℤ - HONEST ADMITTED: gf16_end_to_end_error_bound (needs coq-interval / Interval.Tactic) - Falsification witness: gf16_safe 255 true = false - L-R14 gate: this file must coqc-compile before IGLA RACE starts. *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Coq.Bool.Bool. -Require Import Coq.Arith.Arith. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -(* GF16 safe domain: d_model >= 256 required *) -Definition gf16_safe (d_model : nat) (use_gf16 : bool) : bool := - if use_gf16 then (256 <=? d_model) else true. + gHashTag/t27/proofs/canonical/igla/INV3_Gf16Precision.v + (logical path: Trinity.Canonical.Igla.INV3_Gf16Precision) -(* Threshold constant *) -Definition d_model_min : nat := 256. + Bundle: INV-3 + Title: GF16 Safe Domain + PhD chapter: Ch.6 GoldenFloat / Ch.9 GF vs MXFP4 + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -Theorem gf16_safe_256 : gf16_safe 256 true = true. -Proof. reflexivity. Qed. - -Theorem gf16_safe_384 : gf16_safe 384 true = true. -Proof. reflexivity. Qed. - -Theorem gf16_safe_768 : gf16_safe 768 true = true. -Proof. reflexivity. Qed. - -Theorem gf16_no_quantization_always_safe : - forall d : nat, gf16_safe d false = true. -Proof. intros d. reflexivity. Qed. - -(* gf16_safe_domain: the main invariant *) -Theorem gf16_safe_domain : - forall d : nat, 256 <= d -> gf16_safe d true = true. -Proof. - intros d Hd. - unfold gf16_safe. - apply leb_correct in Hd. - rewrite Hd. reflexivity. -Qed. - -(* FALSIFICATION WITNESS: d_model=255 with GF16 is unsafe *) -Theorem gf16_falsification_witness : - gf16_safe 255 true = false. -Proof. reflexivity. Qed. - -(* HONEST ADMITTED: φ⁻⁶ error bound requires Interval.Tactic (coq-interval package) - Admitted budget: 1/3 used here - To close: `opam install coq-interval` then replace Admitted with: - `interval with (i_prec 53).` *) -Axiom gf16_end_to_end_error_bound : - (* GF16 quantization error < φ⁻⁶ ≈ 0.0557 when d_model >= 256 *) - (* Formal proof requires Interval.Tactic from coq-interval library *) - True. (* HONEST ADMITTED — see coq-interval installation instructions *) +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV3_Gf16Precision. diff --git a/trinity-clara/proofs/igla/igla_asha_bound.v b/trinity-clara/proofs/igla/igla_asha_bound.v index ef311181e0..174d50c347 100644 --- a/trinity-clara/proofs/igla/igla_asha_bound.v +++ b/trinity-clara/proofs/igla/igla_asha_bound.v @@ -1,69 +1,19 @@ -(* INV-2: ASHA Champion Survival — 0 Admitted - Coq theorem: champion_survives_pruning - Trinity: threshold = φ²+φ⁻²+φ⁻⁴ ≈ 3.47 → conservatively 3.5 - Falsification witness: should_prune 2.70 2.65 5000 = true - (old threshold 2.65 kills champion — recorded as scientific counter-example) - L-R14 gate: this file must coqc-compile before IGLA RACE starts. *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Coq.Bool.Bool. -Require Import Coq.Arith.Arith. -Require Import Coq.QArith.QArith. -Open Scope Q_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -(* ASHA prune decision: prune if bpb > threshold AND step > warmup_blind *) -Definition should_prune (bpb threshold : Q) (step warmup_blind : nat) : bool := - (Qle_bool threshold bpb) && (warmup_blind true - | 3000 => true - | 9000 => true - | 27000 => true - | _ => false - end. - -Theorem asha_rungs_trinity : - valid_rung 1000 = true /\ - valid_rung 3000 = true /\ - valid_rung 9000 = true /\ - valid_rung 27000 = true. -Proof. repeat split; reflexivity. Qed. - -(* FALSIFICATION WITNESS: old threshold 2.65 kills champion at bpb=2.70 *) -Definition old_broken_threshold : Q := 265 # 100. -Theorem inv2_falsification_witness : - should_prune (270 # 100) old_broken_threshold 5000 warmup_blind_steps = true. -Proof. reflexivity. Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV2_IglaAshaBound. diff --git a/trinity-clara/proofs/igla/igla_found_criterion.v b/trinity-clara/proofs/igla/igla_found_criterion.v index 8d325c8dc7..59e09a6a00 100644 --- a/trinity-clara/proofs/igla/igla_found_criterion.v +++ b/trinity-clara/proofs/igla/igla_found_criterion.v @@ -1,110 +1,19 @@ -(* INV-7: IGLA Victory Gate *) -(* Reference: trios#143 · HIVE L7 *) -(* Trinity: phi^2 + phi^-2 = 3 *) -(* Rust target: crates/trios-igla-race/src/victory.rs *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Reals. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -(* ---------------------------------------------------------------------- *) -(* Trinity numeric anchors - L-R14: every literal cited *) -(* ---------------------------------------------------------------------- *) + gHashTag/t27/proofs/canonical/igla/INV7_IglaFoundCriterion.v + (logical path: Trinity.Canonical.Igla.INV7_IglaFoundCriterion) -(* IGLA victory target BPB = 1.5 - mission contract *) -Definition bpb_target : R := 15 # 10. + Bundle: INV-7 + Title: IGLA Found Victory Criterion (>=3 distinct seeds, BPB<1.5, step>=4000) + PhD chapter: Ch.21 IGLA / Ch.11 Pre-reg + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(* Warmup blind steps = 4000 - INV-2 anchor *) -Definition warmup_steps : nat := 4000. - -(* JEPA-MSE-proxy artefact floor - TASK-5D bug *) -Definition jepa_proxy_floor : R := 1 # 10. - -(* Required distinct seeds = 3 - Trinity-derived count *) -Definition n_required_seeds : nat := 3. - -(* ---------------------------------------------------------------------- *) -(* Victory predicate *) - -Definition victory_acceptable seed bpb step : Prop := - bpb < bpb_target /\ step >= warmup_steps /\ bpb >= jepa_proxy_floor. - -(* ---------------------------------------------------------------------- *) -(* Theorems - Admitted where proof requires interval arithmetic *) -(* ---------------------------------------------------------------------- *) - -(* Theorem: BPB must be below target for victory *) -Theorem bpb_below_target : - forall seed bpb step, - bpb >= bpb_target -> - ~ victory_acceptable seed bpb step. -Proof. - intros seed bpb step H. - unfold victory_acceptable in H. - intro H1. - lra. -Qed. -Admitted. - -(* Theorem: Warmup blocks victory *) -Theorem warmup_blocks_proxy : - forall seed bpb step, - step < warmup_steps -> - ~ victory_acceptable seed bpb step. -Proof. - intros seed bpb step H. - unfold victory_acceptable in H. - intro H1. - lra. -Qed. -Admitted. - -(* Theorem: JEPA proxy floor *) -Theorem jepa_proxy_floor_correct : - forall seed bpb step, - bpb < jepa_proxy_floor -> - ~ victory_acceptable seed bpb step. -Proof. - intros seed bpb step H. - unfold victory_acceptable in H. - intro H1. - lra. -Qed. -Admitted. - -(* Theorem: NaN rejected *) -Theorem nan_rejected : - forall seed step, - victory_acceptable seed (0/0) step = false. -Proof. - intros seed step. - unfold victory_acceptable. - intro H1. - lra. -Qed. -Admitted. - -(* Main theorem: IGLA FOUND criterion *) - -Theorem igla_found_criterion : - forall bpb1 bpb2 bpb3 step1 step2 step3, - (* All BPB below target *) - bpb1 < bpb_target -> - bpb2 < bpb_target -> - bpb3 < bpb_target -> - (* All steps after warmup *) - step1 >= warmup_steps -> - step2 >= warmup_steps -> - step3 >= warmup_steps -> - (* No JEPA proxy artefact *) - bpb1 >= jepa_proxy_floor -> - bpb2 >= jepa_proxy_floor -> - bpb3 >= jepa_proxy_floor -> - (* Then: IGLA FOUND *) - True. -Proof. - intros bpb1 bpb2 bpb3 step1 step2 step3 H1 H2 H3 H4 H5 H6. - lra. -Qed. - -(* Compile order dependency chain *) -(* lucas_closure_gf16 -> gf16_precision -> nca_entropy_band -> lr_convergence -> igla_asha_bound -> igla_found_criterion *) +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV7_IglaFoundCriterion. diff --git a/trinity-clara/proofs/igla/nca_entropy_band.v b/trinity-clara/proofs/igla/nca_entropy_band.v index 2a813e20d9..a1a29c8406 100644 --- a/trinity-clara/proofs/igla/nca_entropy_band.v +++ b/trinity-clara/proofs/igla/nca_entropy_band.v @@ -1,85 +1,19 @@ -(* INV-4: NCA Entropy Band Stability — 6 Qed, 1 honest Admitted - Coq theorem: entropy_band_width - Trinity: A₅/E₈ symmetry → entropy band = physical phenomenon, not tuned parameter - Dual-band separation: H_LOWER_CERTIFIED ≠ H_LOWER_EMPIRICAL — NEVER merged - HONEST ADMITTED: ln9_over_ln2_upper_bound (requires Interval.Tactic) - Falsification witness: nca_loss_penalty 3.0 > 0 - L-R14 gate: this file must coqc-compile before IGLA RACE starts. *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Coq.Reals.Reals. -Require Import Coq.Reals.Rpower. -Require Import Coq.micromega.Lra. -Open Scope R_scope. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -(* φ: golden ratio — φ² = φ + 1 *) -Parameter phi : R. -Axiom phi_pos : phi > 0. -Axiom phi_sq_eq_phi_plus_1 : phi * phi = phi + 1. + gHashTag/t27/proofs/canonical/igla/INV4_NcaEntropyBand.v + (logical path: Trinity.Canonical.Igla.INV4_NcaEntropyBand) -(* CERTIFIED band: [φ, φ²] — width = 1 exactly *) -Definition H_LOWER_CERTIFIED : R := phi. -Definition H_UPPER_CERTIFIED : R := phi * phi. + Bundle: INV-4 + Title: NCA Entropy Band 1.5..2.8 (81=3^4) + PhD chapter: Ch.10 Coq L1 / Ch.16 360-lane + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(* EMPIRICAL band: [1.5, 2.8] — BENCH-004b result, 55× safety margin *) -Definition H_LOWER_EMPIRICAL : R := 1.5. -Definition H_UPPER_EMPIRICAL : R := 2.8. - -(* Band width = 1 — PROVEN exactly from φ²=φ+1 *) -Theorem entropy_band_width : - H_UPPER_CERTIFIED - H_LOWER_CERTIFIED = 1. -Proof. - unfold H_UPPER_CERTIFIED, H_LOWER_CERTIFIED. - rewrite phi_sq_eq_phi_plus_1. lra. -Qed. - -(* Empirical band is STRICTLY wider than certified band *) -Theorem empirical_wider_than_certified : - H_UPPER_EMPIRICAL - H_LOWER_EMPIRICAL > H_UPPER_CERTIFIED - H_LOWER_CERTIFIED. -Proof. - rewrite entropy_band_width. - unfold H_UPPER_EMPIRICAL, H_LOWER_EMPIRICAL. lra. -Qed. - -(* The two bands are DISTINCT — never merged *) -Theorem bands_are_distinct : - H_LOWER_CERTIFIED <> H_LOWER_EMPIRICAL \/ H_UPPER_CERTIFIED <> H_UPPER_EMPIRICAL. -Proof. - left. - unfold H_LOWER_CERTIFIED, H_LOWER_EMPIRICAL. - (* φ > 1 and φ ≠ 1.5 — phi is irrational, 1.5 is rational *) - intro H. lra. -Qed. - -(* Symmetric entropy penalty: max(0, L-H) + max(0, H-U) *) -Definition nca_loss_penalty (entropy : R) : R := - Rmax 0 (H_LOWER_EMPIRICAL - entropy) + Rmax 0 (entropy - H_UPPER_EMPIRICAL). - -(* Penalty is zero inside band *) -Theorem entropy_in_band_zero_penalty : - forall H : R, - H_LOWER_EMPIRICAL <= H -> H <= H_UPPER_EMPIRICAL -> - nca_loss_penalty H = 0. -Proof. - intros H Hl Hu. - unfold nca_loss_penalty, H_LOWER_EMPIRICAL, H_UPPER_EMPIRICAL. - unfold Rmax. - destruct (Rle_dec 0 (1.5 - H)) as [H1|H1]; - destruct (Rle_dec 0 (H - 2.8)) as [H2|H2]; lra. -Qed. - -(* FALSIFICATION WITNESS: entropy=3.0 is outside band, penalty > 0 *) -Theorem nca_falsification_witness : - nca_loss_penalty 3.0 > 0. -Proof. - unfold nca_loss_penalty, H_LOWER_EMPIRICAL, H_UPPER_EMPIRICAL, Rmax. - destruct (Rle_dec 0 (1.5 - 3.0)) as [H1|H1]; - destruct (Rle_dec 0 (3.0 - 2.8)) as [H2|H2]; lra. -Qed. - -(* HONEST ADMITTED: ln(9)/ln(2) upper bound for K=9 NCA states - Admitted budget: 2/3 used (here + gf16_precision.v) - To close: `opam install coq-interval` then use Interval.Tactic *) -Axiom ln9_over_ln2_upper_bound : - (* H_max(K=9) = ln(9)/ln(2) ≈ 3.17 > H_UPPER_EMPIRICAL = 2.8 *) - (* Formal proof requires Interval.Tactic *) - True. (* HONEST ADMITTED *) +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV4_NcaEntropyBand. diff --git a/trinity-clara/proofs/igla_asha_bound.v b/trinity-clara/proofs/igla_asha_bound.v index c49a8a9b96..174d50c347 100644 --- a/trinity-clara/proofs/igla_asha_bound.v +++ b/trinity-clara/proofs/igla_asha_bound.v @@ -1,89 +1,19 @@ -(** INV-2: asha_champion_survives - Source: trinity-clara / IGLA-INV-001 - Principle: φ² + φ⁻² = 3 — the unique algebraic anchor. - Claim: ASHA with threshold ≥ 3.5 and warmup_blind_steps = 4000 - never prunes a champion trial (lr=0.004) at rung-1000. - Falsification: if champion is pruned at threshold=3.5 → INV-2 violated → RACE INVALID *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Coq.Reals.Reals. -Require Import Coq.Reals.ROrderedType. -Require Import Coq.micromega.Lra. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Open Scope R_scope. + gHashTag/t27/proofs/canonical/igla/INV2_IglaAshaBound.v + (logical path: Trinity.Canonical.Igla.INV2_IglaAshaBound) -(** ── Axioms (from trinity-clara axioms.v) ── *) + Bundle: INV-2 + Title: ASHA Champion Survival + PhD chapter: Ch.13 STROBE / App.E + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** φ-identity: φ² + φ⁻² = 3 (the Trinity Identity) *) -Axiom phi_identity : exists phi : R, - phi > 1 /\ - phi * phi + 1 / (phi * phi) = 3. - -(** Threshold anchor derived from φ-identity: - threshold = φ² + φ⁻² + φ⁻⁴ ≈ 3.472 - We use the conservative bound threshold_val ≥ 3.5 *) -Definition threshold_val : R := 3.5. -Definition warmup_steps : nat := 4000. -Definition rung_1 : nat := 1000. - -(** ── Trial model ── *) - -(** A trial is characterised by its step count and its BPB at that step. *) -Record Trial := mkTrial { - t_step : nat; - t_bpb : R -}. - -(** A trial is "in warmup" if its step count is below warmup_steps. *) -Definition in_warmup (t : Trial) : Prop := - (t_step t < warmup_steps)%nat. - -(** ASHA prunes a trial when its BPB exceeds the threshold. - INV-2 constraint: prune is FORBIDDEN during warmup. *) -Definition asha_would_prune (t : Trial) : Prop := - t_bpb t > threshold_val. - -(** ── Core Invariant (INV-2) ── *) - -(** Theorem: during warmup, ASHA must NOT prune regardless of BPB. - This is the compile-time gate: any config reaching rung-1000 - with step < 4000 is structurally invalid by the φ-anchor. *) -Theorem asha_champion_survives : - forall t : Trial, - in_warmup t -> - ~ asha_would_prune t -> - t_bpb t <= threshold_val. -Proof. - intros t Hwarmup Hnot_prune. - unfold asha_would_prune in Hnot_prune. - lra. -Qed. - -(** Corollary: if step ≥ warmup_steps AND rung = rung_1, - the champion (BPB ≤ threshold_val) is guaranteed to survive. *) -Corollary champion_at_rung1_survives : - forall t : Trial, - (t_step t >= warmup_steps)%nat -> - t_bpb t <= threshold_val -> - ~ asha_would_prune t. -Proof. - intros t _Hstep Hbpb. - unfold asha_would_prune. - lra. -Qed. - -(** ── Falsification witness ── *) - -(** If the experiment shows champion pruned at threshold=3.5 - then this axiom is contradicted → INV-2 falsified → RACE INVALID. - The falsification condition mirrors JUNO θ₁₂ ≠ 0.30693 criterion. *) -Definition inv2_falsified (t : Trial) : Prop := - t_bpb t <= threshold_val /\ asha_would_prune t. - -Lemma inv2_falsification_is_contradiction : - forall t : Trial, - inv2_falsified t -> False. -Proof. - intros t [Hle Hgt]. - unfold asha_would_prune in Hgt. - lra. -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV2_IglaAshaBound. diff --git a/trinity-clara/proofs/lr_phi_optimality.v b/trinity-clara/proofs/lr_phi_optimality.v index ade4f538f9..291a0cb1b0 100644 --- a/trinity-clara/proofs/lr_phi_optimality.v +++ b/trinity-clara/proofs/lr_phi_optimality.v @@ -1,46 +1,19 @@ -(** INV-1: bpb_decreases_with_real_gradient - Source: trinity-clara / 7-step αφ derivation (no free parameters) - Claim: lr ∈ [φ⁻⁸, φ⁻⁶] is the BPB minimum region. - Outside this interval, gradient descent diverges or stalls. - Connection: A₅ characteristic polynomial gives αφ without tuning — - same principle: correct phase space → correct answer without search. *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Coq.Reals.Reals. -Require Import Coq.micromega.Lra. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Open Scope R_scope. + gHashTag/t27/proofs/canonical/igla/INV1b_LrPhiOptimality.v + (logical path: Trinity.Canonical.Igla.INV1b_LrPhiOptimality) -(** φ⁻⁸ ≈ 0.00265, φ⁻⁶ ≈ 0.00901 - Champion lr = 0.004 ∈ [φ⁻⁸, φ⁻⁶] ✓ *) -Definition lr_lo : R := 0.00265. -Definition lr_hi : R := 0.00901. -Definition lr_champion: R := 0.004. + Bundle: INV-1b + Title: lr-phi optimality + lr_convergence + PhD chapter: Ch.10 Coq L1 + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** Training step model: BPB at step n given learning rate lr *) -Record TrainStep := mkStep { - ts_lr : R; - ts_bpb_delta : R (* negative = improvement *) -}. - -(** Axiom (7-step Trinity derivation): real gradient + φ-optimal lr → BPB decreases *) -Axiom phi_lr_decreases_bpb : - forall s : TrainStep, - lr_lo <= ts_lr s <= lr_hi -> - ts_bpb_delta s < 0. - -(** INV-1 Theorem *) -Theorem bpb_decreases_with_real_gradient : - forall s : TrainStep, - lr_lo <= ts_lr s <= lr_hi -> - ts_bpb_delta s < 0. -Proof. - intros s Hlr. - exact (phi_lr_decreases_bpb s Hlr). -Qed. - -(** Champion lr is in the φ-optimal band *) -Lemma champion_lr_is_phi_optimal : - lr_lo <= lr_champion <= lr_hi. -Proof. - unfold lr_lo, lr_hi, lr_champion. lra. -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV1b_LrPhiOptimality. diff --git a/trinity-clara/proofs/lucas_closure_gf16.v b/trinity-clara/proofs/lucas_closure_gf16.v index 0c679baa36..78d0d0349e 100644 --- a/trinity-clara/proofs/lucas_closure_gf16.v +++ b/trinity-clara/proofs/lucas_closure_gf16.v @@ -1,55 +1,19 @@ -(** INV-5: lucas_closure_gf16 - Source: trinity-clara / Lucas sequence — φ²ⁿ + φ⁻²ⁿ ∈ ℤ ∀n - Claim: GF16 arithmetic is algebraically consistent because - φ is the unique quadratic irrational with integer tower. - 84+5=89 theorems total (F₁₁ = 89, Fibonacci prime). *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Coq.Reals.Reals. -Require Import Coq.ZArith.ZArith. -Require Import Coq.micromega.Lra. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Open Scope R_scope. + gHashTag/t27/proofs/canonical/igla/INV5_LucasClosureGf16.v + (logical path: Trinity.Canonical.Igla.INV5_LucasClosureGf16) -(** φ satisfies: φ² - φ - 1 = 0, φ > 1 *) -Axiom phi_exists : { phi : R | phi > 1 /\ phi * phi - phi - 1 = 0 }. + Bundle: INV-5 + Title: Lucas Closure GF16 (phi^{2n}+phi^{-2n} in Z) + PhD chapter: Ch.6 GoldenFloat + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -Definition phi : R := proj1_sig phi_exists. -Definition phi_gt1 : phi > 1 := proj1 (proj2_sig phi_exists). -Definition phi_eq : phi * phi - phi - 1 = 0 := proj2 (proj2_sig phi_exists). - -(** φ⁻¹ = φ - 1 (from minimal polynomial) *) -Lemma phi_inv_eq : phi > 0 -> 1 / phi = phi - 1. -Proof. - intro Hpos. - have Heq := phi_eq. - field_simplify in Heq |- *; lra. -Qed. - -(** Lucas sequence: L(n) = φ²ⁿ + φ⁻²ⁿ is always an integer. - Axiom: this is the algebraic content proved in trinity-clara. *) -Axiom lucas_integer : - forall n : nat, - exists k : Z, phi ^ (2 * n) + (1 / phi) ^ (2 * n) = IZR k. - -(** INV-5: GF16 closure - GF(2^4) has 15 non-zero elements. - The 6:9 bit split is φ-optimal: 6/15 = 2/5 ≈ φ⁻² ≈ 0.382 - This means GF16 arithmetic inherits Lucas integer closure. *) -Definition gf16_bits : nat := 4. (* GF(2^4) *) -Definition gf16_elements : nat := 15. (* 2^4 - 1 *) - -Theorem lucas_closure_gf16 : - forall n : nat, - exists k : Z, phi ^ (2 * n) + (1 / phi) ^ (2 * n) = IZR k. -Proof. - exact lucas_integer. -Qed. - -(** Corollary: φ is the UNIQUE quadratic irrational with this property. - Any other base would break GF16 consistency. *) -Axiom phi_unique_closure : - forall alpha : R, - alpha > 1 -> - (forall n : nat, exists k : Z, - alpha ^ (2 * n) + (1 / alpha) ^ (2 * n) = IZR k) -> - alpha = phi. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV5_LucasClosureGf16. diff --git a/trinity-clara/proofs/nca_entropy_band.v b/trinity-clara/proofs/nca_entropy_band.v index dee068d30d..a1a29c8406 100644 --- a/trinity-clara/proofs/nca_entropy_band.v +++ b/trinity-clara/proofs/nca_entropy_band.v @@ -1,49 +1,19 @@ -(** INV-4: nca_entropy_stability - Source: trinity-clara / A₅ mechanism — E₈ symmetry → entropy band - Claim: NCA with K=9=3² states on 9×9=81=3⁴ grid has - entropy bounded in [1.5, 2.8]. - This is NOT an empirical range — it follows from A₅/E₈ structure. *) +(* SPDX-License-Identifier: Apache-2.0 *) +(* ================================================================ + STUB — MOVED TO CANONICAL HOME -Require Import Coq.Reals.Reals. -Require Import Coq.micromega.Lra. + This file has been moved to the Trinity Coq Canonical SSOT. + The full proof now lives at: -Open Scope R_scope. + gHashTag/t27/proofs/canonical/igla/INV4_NcaEntropyBand.v + (logical path: Trinity.Canonical.Igla.INV4_NcaEntropyBand) -Definition entropy_lo : R := 1.5. -Definition entropy_hi : R := 2.8. -Definition nca_k : nat := 9. (* K = 3² = (φ²+φ⁻²)² *) -Definition nca_grid : nat := 81. (* grid = 3⁴ = (φ²+φ⁻²)⁴ *) + Bundle: INV-4 + Title: NCA Entropy Band 1.5..2.8 (81=3^4) + PhD chapter: Ch.10 Coq L1 / Ch.16 360-lane + Census: github.com/gHashTag/trios/issues/373#issuecomment-4351659821 + Anchor: phi^2 + phi^-2 = 3 + ================================================================ *) -(** NCA entropy type: a real value produced by the NCA loss *) -Record NCAState := mkNCA { - nca_entropy : R -}. - -(** Axiom (A₅/E₈ structural result): - On a Trinity-aligned grid (K=9, grid=81), entropy stays in band. - Violation means the grid has broken Trinity symmetry. *) -Axiom nca_entropy_in_band : - forall s : NCAState, - entropy_lo <= nca_entropy s <= entropy_hi. - -(** INV-4 Theorem *) -Theorem nca_entropy_stability : - forall s : NCAState, - entropy_lo <= nca_entropy s /\ nca_entropy s <= entropy_hi. -Proof. - intros s. - split; apply nca_entropy_in_band. -Qed. - -(** Hard penalty trigger: if entropy exits band → COLLAPSE (L-R11) *) -Definition entropy_collapse (s : NCAState) : Prop := - nca_entropy s < entropy_lo \/ nca_entropy s > entropy_hi. - -Lemma no_collapse_on_trinity_grid : - forall s : NCAState, ~ entropy_collapse s. -Proof. - intros s. - unfold entropy_collapse. - destruct (nca_entropy_in_band s) as [Hlo Hhi]. - lra. -Qed. +(* Re-export so downstream files keep working without code changes. *) +From Trinity.Canonical.Igla Require Export INV4_NcaEntropyBand.