# CFP-MCFP Lean 4 Theorem Inventory

**Source File:** `lean4/CFP_MCFP_Complete_Canonical.lean`  
**Version:** 6.0.0 | **Date:** 2026-05-05  
**Status:** 0 sorry | 54 axioms | 209+ theorems

---

## Core CFP Framework (Part I-III)

| Theorem | Line | Statement |
|---------|------|-----------|
| `flat_is_admissible` | 73 | `Regime.flat.isAdmissible = true` |
| `balance_zero_admissible` | 95 | `b.isZero = true → b.isAdmissible` |
| `cfp_flat_torsionfree_admissible` | 122 | `K=0 ∧ T=0 ∧ Ω=0 → Admissible` |
| `complement_involution` | 153 | `C.isInvolution = true` |

---

## Number Theory (Part IV)

| Theorem | Line | Statement |
|---------|------|-----------|
| `rh_spectral_implies_zeros` | 200 | `spectralCondition ∧ areEquivalent → zeroOnCriticalLine` |
| `lambda_zero_from_cfp` | 232 | `Λ.derivedFromCFP → Λ.value = 0` |

---

## κ-Spectrum Category (Part VIII)

| Theorem | Line | Statement |
|---------|------|-----------|
| `kappa_cat_left_id` | 486 | Left identity for κ-morphisms |
| `kappa_cat_right_id` | 491 | Right identity for κ-morphisms |
| `kappa_cat_assoc` | 496 | Associativity of composition |
| `langlands_preserves_kappa` | 510 | Langlands correspondence preserves κ-spectrum |

---

## Formalization Tasks (Part IX)

| Theorem | Line | Statement |
|---------|------|-----------|
| `fibonacci_minimal_obstruction` | 552 | Φ_F is minimal-obstruction morphism (FORM-001) |
| `fibonacci_preserves_balance` | 558 | Φ_F preserves balance: Ω = 0 |
| `scale_invariant_admissibility` | 575 | Admissibility conserved within regime (FORM-002) |
| `ramanujan_emergence` | 599 | ℛₛ emerges from CFP-MCFP under scale lifting (FORM-003) |

---

## Physics Critical (Part X)

| Theorem | Line | Statement |
|---------|------|-----------|
| `renorm_bec_connection` | 641 | Renormalizability connects to BEC via CFP |
| `prime_uv_complete_nonrenorm` | 653 | Prime system is UV complete but non-renormalizable |
| `pilot_bec_prime_unification` | 672 | Pilot waves, BEC, and primes unify via CFP |
| `lifting_not_extension` | 691 | Theory of Lifting ≠ Theory of Everything |
| `tol_explains_emergence` | 696 | ToL explains emergence without closure |

---

## k-Color Darboux (Part XI-XIII)

| Theorem | Line | Statement |
|---------|------|-----------|
| `prime_indexed_hierarchy` | 752 | Prime-indexed hierarchy: p < q |
| `gauge_dim_3` | 784 | `gaugeDimension 3 = 3` |
| `gauge_dim_5` | 785 | `gaugeDimension 5 = 15` |
| `gauge_dim_7` | 786 | `gaugeDimension 7 = 35` |
| `embedding_chain_exists` | 804 | 𝒟₂ ↪ 𝒟₃ ↪ 𝒟₅ ↪ 𝒟₇ exists |
| `prime_gauge_conjecture_dim` | 827 | SU(3)+SU(2)+U(1) = 12 dimensions |
| `prime_gauge_hierarchy` | 833 | 2→U(1), 3→SU(2), 5→SU(4) |

---

## L-Function Computations (Part XIV)

| Theorem | Line | Statement |
|---------|------|-----------|
| `zeta_constant_kappa` | 889 | `∀ k, zetaKappaSpectrum.kappa k = 1` |
| `dirichlet_kappa_is_chi` | 890 | Dirichlet κ equals character |
| `modular_kappa_is_coefficient` | 892 | Modular κ equals Fourier coefficient |

---

## Transcendental κ-Spectrum (Part XV)

| Theorem | Line | Statement |
|---------|------|-----------|
| `exp_unbounded_kappa` | 943 | Exponential has unbounded κ |
| `log_bounded_kappa` | 944 | Logarithm has bounded κ |
| `bessel_bounded_kappa` | 945 | Bessel functions have bounded κ |
| `exp_is_transcendental` | 949 | exp(x) is transcendental |
| `gamma_is_transcendental` | 951 | Γ(s) is transcendental |

---

## Ribbon Surfaces (Part XVI)

| Theorem | Line | Statement |
|---------|------|-----------|
| `self_intersection_creates_regime` | 983 | Self-intersection creates regime transition |
| `ribbon_boundary_is_3manifold` | 986 | Ribbon boundary is S¹ × S² |

---

## Regime Transition (Part XVII)

| Theorem | Line | Statement |
|---------|------|-----------|
| `transition_preserves_total_kappa` | 1023 | Transition preserves total κ |
| `critical_at_zero_kappa` | 1026 | Critical point at κ = 0 |
| `hysteresis_is_symmetric` | 1032 | Hysteresis loop is symmetric |

---

## Universal Darboux Limit (Part XVIII)

| Theorem | Line | Statement |
|---------|------|-----------|
| `prime_tower_increasing` | 1056 | Prime tower: p_n < p_{n+1} |
| `universal_contains_all_finite` | 1083 | Universal algebra contains all finite |
| `colimit_exists` | 1089 | Colimit exists for consistent tower |

---

## Spectral Gap (Part XIX)

| Theorem | Line | Statement |
|---------|------|-----------|
| `gapped_implies_stable` | 1120 | Spectral gap implies stability |
| `mass_gap_from_kappa` | 1130 | Mass gap from κ-structure |

---

## Dark Matter (Part XX)

| Theorem | Line | Statement |
|---------|------|-----------|
| `dark_matter_balance` | 1162 | `hiddenKappa + visibleKappa = 0` |
| `cfp_explains_rotation_curves` | 1173 | CFP explains galaxy rotation curves |

---

## Arithmetic Operators (Part XXXI)

| Theorem | Line | Statement |
|---------|------|-----------|
| `complement_not_involution` | 1523 | R(R(f)) ≠ f in general |
| `complement_torsion_not_negative` | 1530 | R(T) ≠ -T (physical requirement) |
| `asymptotic_freedom_basic` | 1558 | Coupling decreases with k |

---

## Prime vs Composite Separation (Part XXXII)

| Theorem | Line | Statement |
|---------|------|-----------|
| `dminus_delta1` | 1599 | D⁻ on delta1 structure |
| `torsion_prime_restricted` | 1604 | Prime torsion has restricted support |
| `prime_composite_separation` | 1611 | **MAIN THEOREM**: Primes vs composites separate |
| `observable_prime` | 1626 | O_p ≤ N² for primes |
| `observable_composite` | 1636 | O_k ≥ 0 for composites |

---

## Spectral Exclusion for RH (Part XXXV)

| Theorem | Line | Statement |
|---------|------|-----------|
| `cross_term_decay` | 1779 | Cross-terms decay for σ > 1/2 |
| `spectral_exclusion_main` | 1787 | **σ > 1/2 → spectral gap exists** |
| `rh_spectral_equivalence` | 1793 | RH ⟺ spectral condition |

---

## Yang-Mills from CFP (Part XXXVI)

| Theorem | Line | Statement |
|---------|------|-----------|
| `gauge_asymptotic_freedom` | 1825 | g_{k+1} ≤ g_k (asymptotic freedom) |
| `bianchi_identity` | 1838 | dF + [A, F] = 0 |
| `beta_negative` | 1842 | Beta function is negative |
| `yang_mills_equation` | 1850 | [D, F] = 0 |

---

## Mass Gap from CFP (Part XXXVII)

| Theorem | Line | Statement |
|---------|------|-----------|
| `mass_gap_positive` | 1876 | Mass gap ≥ 0 |
| `mass_from_imbalance` | 1884 | Mass arises from λ_k ≠ 1 |
| `no_laplacian_needed` | 1888 | M(X) = [D, [D, X]], not Δ |

---

## Gravity from Lifting (Part XXXVIII)

| Theorem | Line | Statement |
|---------|------|-----------|
| `gravity_gauge_scaling` | 1924 | g_grav ~ 1/log k |
| `asymptotic_flatness` | 1928 | R_k → 0 as k → ∞ |
| `no_spacetime_primitive` | 1932 | Geometry emerges from operators |

---

## Ramanujan Projector (Part XXXIX)

| Theorem | Line | Statement |
|---------|------|-----------|
| `ramanujan_selects_admissible` | 1970 | ℛ_k selects admissible modes |
| `sm_is_ramanujan_projected` | 1978 | SM = Ramanujan-projected CFP |
| `ramanujan_multiplicative` | 1992 | c_{q₁q₂}(n) = c_{q₁}(n) · c_{q₂}(n) |

---

## Black Hole Physics (Part XLI)

| Theorem | Line | Statement |
|---------|------|-----------|
| `information_conservation` | 2111 | I_total = I_visible + I_complement |
| `area_monotonicity` | 2115 | **Second law**: area never decreases |
| `zeroth_law_uniform_balance` | 2124 | Uniform balance on horizon |
| `third_law_extremal` | 2127 | T → 0 as M → ∞ |

---

## Multi-Channel Projection (Part XLIII)

| Theorem | Line | Statement |
|---------|------|-----------|
| `channel_separation_preserves_balance` | 2258 | Channel separation preserves balance |
| `asymmetry_antisymmetric` | 2265 | A_{kl} = -A_{lk} |
| `fusion_symmetric` | 2276 | C_{kl} = C_{lk} |

---

## Summary

| Category | Count |
|----------|-------|
| **Core CFP** | 4 |
| **Number Theory** | 2 |
| **κ-Spectrum Category** | 4 |
| **Formalization Tasks** | 4 |
| **Physics Critical** | 5 |
| **k-Color Darboux** | 7 |
| **L-Functions** | 3 |
| **Transcendental** | 5 |
| **Ribbon Surfaces** | 2 |
| **Regime Transition** | 3 |
| **Universal Darboux** | 3 |
| **Spectral Gap** | 2 |
| **Dark Matter** | 2 |
| **Arithmetic Operators** | 3 |
| **Prime/Composite** | 5 |
| **Spectral Exclusion RH** | 3 |
| **Yang-Mills** | 4 |
| **Mass Gap** | 3 |
| **Gravity** | 3 |
| **Ramanujan** | 3 |
| **Black Holes** | 4 |
| **Multi-Channel** | 3 |
| **TOTAL** | **77+ key theorems** |

---

## Verification Commands

```bash
# Build the Lean 4 project
cd /home/alfons/CascadeProjects/mr-nec-platform/lean4
lake build

# Check for sorry statements
grep -c "sorry" CFP_MCFP_Complete_Canonical.lean
# Expected: 0

# Count theorems
grep -c "^theorem" CFP_MCFP_Complete_Canonical.lean
# Expected: 200+
```

---

## Repository Structure

```
lean4/
├── CFP_MCFP_Complete_Canonical.lean  -- Main canonical file (3525 lines)
├── lakefile.lean                      -- Build configuration
└── archive/                           -- Historical versions
    └── 2026_03_20_complete_versions/
        ├── CFP_MCFP_Complete_v7.lean
        ├── CFP_MCFP_Complete_v10.lean
        └── ... (24 versions)
```