feat(spectral): add TCS Model Theorem for spectral bounds

CHANGELOG.md

@@ -5,6 +5,70 @@ All notable changes to GIFT Core will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
+## [3.3.12] - 2026-01-26
+
+### Summary
+
+**TCS Spectral Bounds Model Theorem!** New `GIFT.Spectral.NeckGeometry` and `GIFT.Spectral.TCSBounds` modules formalize the spectral bounds for Twisted Connected Sum manifolds with cylindrical neck.
+
+### Added
+
+- **Spectral/NeckGeometry.lean** - TCS manifold structure and hypotheses:
+  - `TCSManifold`: K = M₁ ∪_N M₂ with cylindrical neck N ≅ Y × [0, L]
+  - `BoundedNeckVolume` (H2): Vol(N) ∈ [v₀, v₁]
+  - `BlockCheegerBound` (H4): h(Mᵢ \ N) ≥ h₀
+  - `BalancedBlocks` (H5): Vol(Mᵢ) ∈ [1/4, 3/4]
+  - `ProductNeckMetric` (H3), `NeckMinimality` (H6): Geometric axioms
+  - `TCSHypotheses`: Complete hypothesis bundle
+  - `L₀`: Threshold neck length 2v₀/h₀ with `L₀_pos` proven
+  - `typical_parameters`: Algebraic verification via native_decide
+
+- **Spectral/TCSBounds.lean** - Model Theorem for spectral bounds:
+  - `c₁`, `c₂_robust`, `c₂_symmetric`: Bound constants
+  - `spectral_upper_bound`: λ₁ ≤ 16v₁/((1-v₁)L²) via Rayleigh quotient
+  - `spectral_lower_bound`: λ₁ ≥ v₀²/L² via Cheeger inequality
+  - `tcs_spectral_bounds`: Main theorem combining both bounds
+  - `spectral_gap_scales_as_inverse_L_squared`: λ₁ = Θ(1/L²)
+  - `tcs_bounds_certificate`: Complete algebraic verification
+
+- **Spectral.lean** - Updated re-exports for new modules
+
+### Fixed
+
+- **Lean toolchain compatibility**: Updated to v4.27.0 (stable)
+- **Dependency pinning**: mathlib v4.27.0, doc-gen4 v4.27.0, checkdecls master
+- **Type annotations**: Explicit ℚ annotations for native_decide in conjunctions
+
+### Mathematical Significance
+
+The Model Theorem establishes:
+```
+For TCS manifold K with neck length L > L₀ satisfying (H1)-(H6):
+    v₀²/L² ≤ λ₁(K) ≤ 16v₁/((1-v₁)L²)
+```
+
+**Key insight**: The spectral gap λ₁ scales as 1/L² for long-necked TCS manifolds.
+
+For K7 with L² ~ H*:
+- λ₁ ~ 1/H* = 1/99
+- Combined with dim(G₂) = 14 gives λ₁ = 14/99 (universal spectral law)
+
+### Module Structure
+
+| Module | Content | Status |
+|--------|---------|--------|
+| `NeckGeometry.lean` | TCS structure, hypotheses H1-H6 | Structures + axioms |
+| `TCSBounds.lean` | Model Theorem, bound constants | Axioms + proven algebraic |
+
+### Relation Count
+
+| Module | New Relations |
+|--------|---------------|
+| NeckGeometry | 4 |
+| TCSBounds | 7 |
+| **Total v3.3.12** | **11** |
+| **Cumulative** | **~276** |
+
 ## [3.3.11] - 2026-01-24
 
 ### Summary

CLAUDE.md

@@ -322,6 +322,9 @@ python -c "from gift_core import *; print(GAMMA_GIFT)"
 | `exp (y * log x)` vs `exp (log x * y)` | `rpow_def_of_pos` gives `exp(log x * y)` | Match multiplication order (see §34) |
 | `norm_num` proves bound is false | Arithmetic error (e.g. 5.33 > 5.329692) | Double-check calculations before coding (see §35) |
 | `nlinarith` fails on products | Can't handle `a < b → a*c < b*c` | Use `mul_lt_mul_of_pos_right` explicitly (see §36) |
+| `native_decide` evaluates False on ℚ | Type annotation only on first conjunct | Annotate each conjunct: `(16 : ℚ) * ...` (see §47) |
+| `revision not found` for checkdecls | Wrong branch name | Use `rev = "master"` not `"main"` (see §48) |
+| `no such file` for Mathlib imports | Mathlib version mismatch | Pin mathlib to match lean-toolchain (see §48) |
 
 ### 6. Namespace Conflicts in Lean 4
 
@@ -1604,9 +1607,73 @@ This is a docstring.
 
 **Rule**: Always place `import` statements at the very top of the file, before any `/-! ... -/` docstrings.
 
+### 47. Type Annotations in Conjunctions for native_decide
+
+**Problem**: In a conjunction `A ∧ B ∧ C`, type annotations only apply to their immediate expression, not to other conjuncts.
+
+```lean
+-- BAD - native_decide evaluates as False (integer division!)
+theorem foo :
+    ((1 : ℚ) / 2) ^ 2 = 1 / 4 ∧      -- ✓ This is ℚ
+    16 * (1 / 2) / (1 - 1 / 2) = 16 ∧ -- ✗ This defaults to ℕ!
+    2 * (1 / 2) / 1 = 1 := by         -- ✗ Also ℕ!
+  native_decide  -- FAILS: 16 * 0 / (1 - 0) ≠ 16
+
+-- GOOD - annotate each conjunct's first number
+theorem foo :
+    ((1 : ℚ) / 2) ^ 2 = 1 / 4 ∧
+    (16 : ℚ) * (1 / 2) / (1 - 1 / 2) = 16 ∧
+    (2 : ℚ) * (1 / 2) / 1 = 1 := by
+  native_decide  -- WORKS!
+```
+
+**Rule**: For each conjunct involving division or fractions, annotate the first literal with `(n : ℚ)`.
+
+### 48. Lean Toolchain and Dependency Version Compatibility
+
+**Problem**: Mathlib file structure changes between versions. Imports like `Mathlib.Data.Nat.Basic` may not exist in older Mathlib.
+
+**Key versions (as of v3.3.12):**
+
+| Component | Version | Notes |
+|-----------|---------|-------|
+| `lean-toolchain` | `leanprover/lean4:v4.27.0` | Stable release |
+| `mathlib` | `v4.27.0` | Must match toolchain |
+| `doc-gen4` | `v4.27.0` | Version-tagged |
+| `checkdecls` | `master` | Uses `master`, NOT `main` |
+
+**Common issues:**
+
+1. **Mathlib paths don't exist**: You're using a Mathlib version where files were reorganized
+   - Solution: Update lean-toolchain AND mathlib to matching versions
+
+2. **checkdecls revision not found**: The `lean4.X.Y` tag doesn't exist
+   - Solution: Use `rev = "master"` (checkdecls uses master branch, not version tags for newer Lean)
+
+3. **Batteries compilation errors with rc versions**: Release candidates may have bugs
+   - Solution: Use stable versions (e.g., v4.27.0 not v4.28.0-rc1)
+
+**lakefile.toml pattern:**
+```toml
+[[require]]
+name = "mathlib"
+git  = "https://github.com/leanprover-community/mathlib4.git"
+rev  = "v4.27.0"  # Match lean-toolchain
+
+[[require]]
+name = "checkdecls"
+git  = "https://github.com/PatrickMassot/checkdecls.git"
+rev  = "master"   # NOT "main"!
+
+[[require]]
+name = "«doc-gen4»"
+git  = "https://github.com/leanprover/doc-gen4"
+rev  = "v4.27.0"  # Match lean-toolchain
+```
+
 ---
 
-### Axiom Status (v3.3.11)
+### Axiom Status (v3.3.12)
 
 **Numerical Bounds - COMPLETE! (0 remaining):**
 - ✓ All Taylor series bounds proven
@@ -1615,6 +1682,13 @@ This is a docstring.
 - `CompactManifold`, `MassGap`, `spectral_theorem_discrete`
 - `universal_spectral_law`, `CheegerConstant`, `cheeger_inequality`
 
+**TCS Spectral Bounds (NEW in v3.3.12):**
+- `ProductNeckMetric` - Product metric g|_N = dt² + g_Y
+- `NeckMinimality` - Area(Γ) ≥ Area(Y) for separating hypersurfaces
+- `spectral_upper_bound` - Rayleigh quotient bound λ₁ ≤ c₂/L²
+- `spectral_lower_bound` - Cheeger-based bound λ₁ ≥ c₁/L²
+- `neck_dominates` - For L > L₀, neck controls Cheeger constant
+
 **Zeta Correspondences - New axioms:**
 - `gamma : ℕ+ → ℝ` - Riemann zeta zeros (empirical)
 - `gamma_positive`, `gamma_increasing` - Basic properties
@@ -1626,4 +1700,4 @@ This is a docstring.
 
 ---
 
-*Last updated: 2026-01-24 - V3.3.11: Monster Dimension via Coxeter Numbers*
+*Last updated: 2026-01-26 - V3.3.12: TCS Spectral Bounds Model Theorem*

Lean/GIFT/Spectral.lean

@@ -11,7 +11,7 @@ This module formalizes the spectral gap result:
 
 The key insight: the mass gap is determined by TOPOLOGY, not dynamics.
 
-## Contents (v3.3.10)
+## Contents (v3.3.12)
 
 ### Spectral Theory Foundation
 - `SpectralTheory`: Laplacian, spectral theorem, mass gap definition
@@ -23,6 +23,10 @@ The key insight: the mass gap is determined by TOPOLOGY, not dynamics.
 - `UniversalLaw`: λ₁ × H* = dim(G₂), the KEY theorem
 - `MassGapRatio`: The 14/99 theorem (algebraic)
 
+### TCS Spectral Bounds (NEW in v3.3.12)
+- `NeckGeometry`: TCS manifold structure and hypotheses (H1)-(H6)
+- `TCSBounds`: Model Theorem - λ₁ ~ 1/L² for TCS manifolds
+
 ### Applications
 - `CheegerInequality`: Cheeger-Buser bounds
 - `YangMills`: Connection to Clay Millennium Prize
@@ -32,9 +36,10 @@ The key insight: the mass gap is determined by TOPOLOGY, not dynamics.
 - Joyce, D.D. (2000). Compact Manifolds with Special Holonomy
 - Cheeger, J. (1970). A lower bound for the smallest eigenvalue of the Laplacian
 - Jaffe, A. & Witten, E. (2000). Yang-Mills Existence and Mass Gap
-- GIFT Framework v3.3.10: Yang-Mills spectral gap from K7 topology
+- Kovalev, A. (2003). Twisted connected sums and special Riemannian holonomy
+- GIFT Framework v3.3.12: TCS spectral bounds
 
-Version: 2.0.0
+Version: 2.1.0
 -/
 
 -- Spectral theory foundations
@@ -47,6 +52,10 @@ import GIFT.Spectral.G2Manifold
 import GIFT.Spectral.UniversalLaw
 import GIFT.Spectral.MassGapRatio
 
+-- TCS Spectral Bounds (NEW)
+import GIFT.Spectral.NeckGeometry
+import GIFT.Spectral.TCSBounds
+
 -- Applications
 import GIFT.Spectral.CheegerInequality
 import GIFT.Spectral.YangMills
@@ -119,6 +128,42 @@ export MassGapRatio (
   mass_gap_ratio_certified
 )
 
+-- ============================================================================
+-- RE-EXPORTS: NECK GEOMETRY (TCS)
+-- ============================================================================
+
+export NeckGeometry (
+  TCSManifold
+  BoundedNeckVolume
+  ProductNeckMetric
+  BlockCheegerBound
+  BalancedBlocks
+  NeckMinimality
+  TCSHypotheses
+  L₀
+  L₀_pos
+  typical_parameters
+  neck_geometry_certificate
+)
+
+-- ============================================================================
+-- RE-EXPORTS: TCS BOUNDS
+-- ============================================================================
+
+export TCSBounds (
+  c₁
+  c₁_pos
+  c₂_robust
+  c₂_robust_pos
+  c₂_symmetric
+  spectral_upper_bound
+  spectral_lower_bound
+  tcs_spectral_bounds
+  spectral_gap_scales_as_inverse_L_squared
+  typical_tcs_bounds_algebraic
+  tcs_bounds_certificate
+)
+
 -- ============================================================================
 -- RE-EXPORTS: CHEEGER INEQUALITY
 -- ============================================================================
@@ -168,6 +213,8 @@ Spectral/
 ├── G2Manifold.lean         # G₂ holonomy, K7
 ├── UniversalLaw.lean       # λ₁ × H* = 14
 ├── MassGapRatio.lean       # 14/99 algebraic
+├── NeckGeometry.lean       # TCS structure, hypotheses (H1)-(H6)
+├── TCSBounds.lean          # Model Theorem: λ₁ ~ 1/L²
 ├── CheegerInequality.lean  # Cheeger-Buser bounds
 └── YangMills.lean          # Clay Prize connection
 ```
@@ -180,6 +227,10 @@ Spectral/
 | `K7_spectral_law` | λ₁ × 99 = 14 | Heat kernel + trace formula |
 | `K7_cheeger_constant` | h(K7) = 14/99 | Isoperimetric analysis |
 | `GIFT_mass_gap_relation` | Δ = λ₁ × Λ_QCD | M-theory compactification |
+| `ProductNeckMetric` | Product metric on TCS neck | Differential geometry |
+| `NeckMinimality` | Isoperimetric bound on neck | Coarea formula |
+| `spectral_upper_bound` | Rayleigh quotient bound | L² space formalization |
+| `neck_dominates` | Neck controls Cheeger | Cut classification |
 -/
 
 end GIFT.Spectral