# Supplement S2: Complete Derivations (Dimensionless)

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## Complete Mathematical Derivations for All 33 Type I Predictions

*This supplement provides mathematical derivations for the 33 Type I dimensionless predictions in the GIFT framework (Relations #1–#33, all Lean-certified). Each derivation proceeds from topological definitions to numerical predictions. Two additional BH remnant topological predictions (Relations #34–#35 in §24.3b) are included as an appendix; these are classified as Type IV structural diagnostics in the main 95-observable dataset.*

**Status**: All 33 Type I observables Lean-certified. 213 certificate conjuncts, 4 axioms, 0 sorry.

**Note on the 95-observable dataset**: The main paper presents 95 observables across 4 types (33 I + 19 II + 21 III + 22 IV). This supplement covers the 33 Type I predictions (direct algebraic from topology). Type II extensions (19 one-step physical extractions), Type III dynamical results (21 multi-step chains), and Type IV structural diagnostics (22 internal consistency checks, including BH remnant M_res and N_QNM from Pinčák et al. 2026 [46]) are documented in Supplement S3 and the main text §4–§6.

The topological constants that determine these relations are described in S1.

---

## Table of Contents

- [Part I: Foundations](#part-i-foundations)
- [Part II: Foundational Theorems (4 relations)](#part-ii-foundational-theorems)
- [Part III: Gauge Sector (2 relations)](#part-iii-gauge-sector)
- [Part IV: Lepton Sector (3 relations)](#part-iv-lepton-sector)
- [Part V: Quark Sector (1 relation)](#part-v-quark-sector)
- [Part VI: Neutrino Sector (4 relations)](#part-vi-neutrino-sector)
- [Part VII: Higgs & Cosmology (3 relations)](#part-vii-higgs--cosmology)
- [Part VIII: Summary Table](#part-viii-summary-table)
- [Part IX: Observable Catalog](#part-ix-observable-catalog)

---

# Part 0: Derivation Philosophy

## 0. What "Derivation" Means in GIFT

Before presenting derivations, we clarify the logical structure:

### 0.1 Inputs vs Outputs

**Inputs** (taken as given):
- The octonion algebra 𝕆 and its automorphism group G₂ = Aut(𝕆)
- The E₈×E₈ gauge structure
- The K₇ manifold (compact G₂ 7-manifold with b₂ = 21, b₃ = 77; explicit metric certified in [A]; topological construction via Joyce-Karigiannis Z₂³ orbifold route (S1 §8.4); explicit closed-form neck ansatz with Wirtinger certificate (S1 §8.4, [D]))

**Outputs** (derived from inputs):
- The 33 Type I dimensionless predictions

### 0.2 What We Do NOT Claim

- That 𝕆 → G₂ → K₇ is the unique geometry for physics
- That the formulas are uniquely determined by geometric principles
- That the selection rule for specific combinations (b₂/(b₃ + dim_G₂) vs b₂/b₃) is understood

### 0.3 What We Observe

- Given the inputs, the outputs follow by algebra
- The outputs match experiment to 0.73% mean deviation (PDG 2024, NuFIT 6.0, 33 Type I core)
- No continuous parameters are fitted

### 0.4 Torsion Independence

**Important**: All 33 Type I predictions use only topological invariants. The torsion T does not appear in any formula. Therefore:
- Predictions depend only on topology, not on the actual torsion value
- The value κ_T = 1/61 is a topological bound, not a prediction ingredient

### 0.5 Results from Analysis of the NK-Certified Metric

Analysis of the NK-certified metric ([B]) computed all invariants directly from the explicit metric. Key results:

- **Topological realization**: The pair (b₂, b₃) = (21, 77) is realized topologically by the Joyce-Karigiannis Z₂³ orbifold route (S1 §8.4); an explicit closed-form neck ansatz with five-layer Wirtinger certificate is established in [D]. Orthogonal TCS is excluded by parity (b₂+b₃=98 even). Any Betti decomposition via Mayer-Vietoris is conditional on the building block identification, which is open.
- **ν̄ = 0**: Lean-certified in TCSConstruction.lean. For rectangular TCS (k₊=k₋=1), ν̄=0 automatically (CGN invariant); this is conditional on TCS realization.
- **New topological formula**: V_min = √(Vol(K₇)/11), where 11 = b₃/n = 77/7. NK numerical: 219.90; formula: 221.24 (0.6% ✓). Identity: V_min² × b₃/n = Vol(K₇): a universal G₂ relation.
- **U(1)² exact symmetry**: The T² directions θ and ψ are degenerate to 2×10⁻⁵ in the metric eigenvalues, propagating to S_θ = S_ψ = 6.1265 in all period integrals (exact to 2.6×10⁻⁸).
- **SM gauge group origin**: NOT from ADE singularities (TCS is smooth). From g₂ ⊂ so(7) spectral structure and so(8) = g₂ ⊕ L ⊕ R triality → N_gen = 3.

---

# Part I: Foundations

## 1. Status Classification

| Status | Criterion |
|--------|-----------|
| **VERIFIED** | Complete mathematical proof, exact result from topology |
| **VERIFIED (Lean 4)** | Verified by Lean 4 kernel with Mathlib (machine-checked) |
| **TOPOLOGICAL** | Direct consequence of manifold structure |

## 2. Notation

| Symbol | Value | Definition |
|--------|-------|------------|
| dim(E₈) | 248 | E₈ Lie algebra dimension |
| rank(E₈) | 8 | E₈ Cartan subalgebra dimension |
| dim(G₂) | 14 | G₂ holonomy group dimension |
| dim(K₇) | 7 | Internal manifold dimension |
| b₂(K₇) | 21 | Second Betti number |
| b₃(K₇) | 77 | Third Betti number |
| H* | 99 | Effective cohomology = b₂ + b₃ + 1 |
| dim(J₃(O)) | 27 | Exceptional Jordan algebra dimension |
| N_gen | 3 | Number of fermion generations |
| p₂ | 2 | Binary duality parameter |
| Weyl | 5 | Weyl factor: (dim(G₂)+1)/N_gen = b₂/N_gen - p₂ = dim(G₂) - rank(E₈) - 1 |

---

# Part II: Foundational Theorems

## 3. Relation #1: Generation Number N_gen = 3

**Statement**: The number of fermion generations is exactly 3.

**Classification**: VERIFIED (three independent derivations)

### Proof Method 1: Fundamental Topological Constraint

*Theorem*: For G₂ holonomy manifold K₇ with E₈ gauge structure:

$$(\text{rank}(E_8) + N_{\text{gen}}) \cdot b_2(K_7) = N_{\text{gen}} \cdot b_3(K_7)$$

*Derivation*:
$$(8 + N_{\text{gen}}) \times 21 = N_{\text{gen}} \times 77$$
$$168 + 21 \cdot N_{\text{gen}} = 77 \cdot N_{\text{gen}}$$
$$168 = 56 \cdot N_{\text{gen}}$$
$$N_{\text{gen}} = \frac{168}{56} = 3$$

*Verification*:
- LHS: (8 + 3) × 21 = 231
- RHS: 3 × 77 = 231 ✓

### Proof Method 2: Atiyah-Singer Index Theorem

$$\text{Index}(D_A) = \left( 77 - \frac{8}{3} \times 21 \right) \times \frac{1}{7} = 3$$

**Status**: VERIFIED ∎

---

## 4. Relation #2: Hierarchy Parameter τ = 3472/891

**Statement**: The hierarchy parameter is exactly rational.

**Classification**: VERIFIED

### Proof

*Step 1: Definition from topological integers*
$$\tau := \frac{\dim(E_8 \times E_8) \cdot b_2(K_7)}{\dim(J_3(\mathbb{O})) \cdot H^*}$$

*Step 2: Substitute values*
$$\tau = \frac{496 \times 21}{27 \times 99} = \frac{10416}{2673}$$

*Step 3: Reduce*
$$\gcd(10416, 2673) = 3$$
$$\tau = \frac{3472}{891}$$

*Step 4: Prime factorization*
$$\tau = \frac{2^4 \times 7 \times 31}{3^4 \times 11}$$

*Step 5: Numerical value*
$$\tau = 3.8967452300785634...$$

**Status**: VERIFIED ∎

---

## 5. Relation #3: Torsion Capacity κ_T = 1/61

**Statement**: The topological torsion capacity equals exactly 1/61.

**Classification**: TOPOLOGICAL (structural parameter, not physical prediction)

### Proof

*Step 1: Define from cohomology*
$$61 = b_3(K_7) - \dim(G_2) - p_2 = 77 - 14 - 2 = 61$$

*Step 2: Formula*
$$\kappa_T = \frac{1}{b_3 - \dim(G_2) - p_2} = \frac{1}{61}$$

*Step 3: Geometric interpretation*
- 61 = effective degrees of freedom available for torsional deformation
- 61 = dim(F₄) + N_gen² = 52 + 9

### Clarification

| Quantity | Definition | Value |
|----------|------------|-------|
| κ_T | Topological capacity | 1/61 (fixed) |
| T_base | Torsion for torsion-free metric (Joyce) | **0** (by theorem) |
| T_physical | Effective torsion for interactions | **Open question** |

**Role in predictions**: κ_T appears in only one formula (α⁻¹, as a small correction term det(g)×κ_T ≈ 0.033). The other 32 Type I predictions are independent of torsion capacity. It is primarily a structural parameter characterizing K₇, not a directly measured observable.

**Joyce's theorem**: For compact G₂ manifolds satisfying perturbation bounds, guarantees existence of a torsion-free metric. The Newton-Kantorovich certificate (Paper I) establishes these bounds for K₇.

**Status**: TOPOLOGICAL (structural, not predictive) ∎

---

## 6. Relation #4: Metric Determinant det(g) = 65/32

**Statement**: The K₇ metric determinant is exactly 65/32.

**Classification**: METRIC NORMALIZATION (not TOPOLOGICAL, see S1 §10.3)

### Proof

*Step 1: Define from structural normalization integers*
$$\det(g) = p_2 + \frac{1}{b_2 + \dim(G_2) - N_{gen}}$$

*Step 2: Compute denominator*
$$b_2 + \dim(G_2) - N_{gen} = 21 + 14 - 3 = 32$$

*Step 3: Compute determinant*
$$\det(g) = 2 + \frac{1}{32} = \frac{65}{32}$$

*Step 4: Alternative derivation*
$$\det(g) = \frac{\text{Weyl} \times (\text{rank}(E_8) + \text{Weyl})}{2^5} = \frac{5 \times 13}{32} = \frac{65}{32}$$

**Verification**: The analytical metric g = (65/32)^{1/7} × I₇ has det(g) = [(65/32)^{1/7}]^7 = 65/32 exactly, consistent with the topological formula.

**Epistemic note**: Three independent algebraic paths converge to 65/32, which is suggestive but does not constitute a derivation from topology: the metric optimization was constrained to this normalization target, and the formulas were identified post-hoc. See S1 §10.3 for full discussion. Six observables depending on det(g)_num or det(g)_den carry this normalization dependence.

**Status**: STRUCTURAL (metric normalization, algebraically exact; see S1 §10.3) ∎

---

# Part III: Gauge Sector

## 7. Relation #5: Weinberg Angle sin²θ_W = 3/13

**Statement**: The weak mixing angle has exact rational form 3/13.

**Classification**: VERIFIED

### Proof

*Step 1: Define ratio from Betti numbers*
$$\sin^2\theta_W = \frac{b_2(K_7)}{b_3(K_7) + \dim(G_2)} = \frac{21}{77 + 14} = \frac{21}{91}$$

*Step 2: Simplify*
$$\gcd(21, 91) = 7$$
$$\sin^2\theta_W = \frac{3}{13} = 0.230769...$$

*Step 3: Experimental comparison*

| Quantity | Value |
|----------|-------|
| Experimental (PDG 2024) | 0.23122 ± 0.00004 |
| GIFT prediction | 0.230769 |
| Deviation | 0.195% |

**Status**: VERIFIED ∎

---

## 8. Relation #6: Strong Coupling α_s = √2/12

**Statement**: The strong coupling at M_Z scale.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\alpha_s(M_Z) = \frac{\sqrt{2}}{\dim(G_2) - p_2} = \frac{\sqrt{2}}{14 - 2} = \frac{\sqrt{2}}{12}$$

*Components*:
- √2: E₈ root length
- 12 = dim(G₂) - p₂: Effective gauge degrees of freedom

*Numerical value*: α_s = 0.117851

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 0.1180 ± 0.0009 |
| GIFT prediction | 0.11785 |
| Deviation | 0.127% |

**Status**: TOPOLOGICAL ∎

---

# Part IV: Lepton Sector

## 9. Relation #7: Koide Parameter Q = 2/3

**Statement**: The Koide parameter equals exactly 2/3.

**Classification**: VERIFIED

### Proof

*Formula*:
$$Q_{\text{Koide}} = \frac{\dim(G_2)}{b_2(K_7)} = \frac{14}{21} = \frac{2}{3}$$

*Physical definition*:
$$Q = \frac{m_e + m_\mu + m_\tau}{(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau})^2}$$

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 0.666661 ± 0.000007 |
| GIFT prediction | 0.666667 |
| Deviation | 0.0009% |

**Status**: VERIFIED ∎

---

## 10. Relation #8: Tau-Electron Mass Ratio m_τ/m_e = 3477

**Statement**: The tau-electron mass ratio is exactly 3477.

**Classification**: VERIFIED

### Proof

*Formula*:
$$\frac{m_\tau}{m_e} = \dim(K_7) + 10 \cdot \dim(E_8) + 10 \cdot H^*$$
$$= 7 + 10 \times 248 + 10 \times 99 = 7 + 2480 + 990 = 3477$$

*Prime factorization*:
$$3477 = 3 \times 19 \times 61 = N_{gen} \times prime(8) \times \kappa_T^{-1}$$

*Period integral cross-confirmation*: Analysis of the NK-certified metric ([B], §5) extracts associative 3-cycle volumes directly. The SD associative volumes yield ln(m_τ/m_e) = 8.154 from the Fano-plane volume hierarchy, giving e^8.154 = 3477 ✓. This provides an independent geometric confirmation of the algebraic prediction.

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 3477.15 ± 0.05 |
| GIFT prediction | 3477 (exact) |
| Deviation | 0.0043% |

**Status**: VERIFIED ∎

---

## 11. Relation #9: Muon-Electron Mass Ratio

**Statement**: m_μ/m_e = 27^φ

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\frac{m_\mu}{m_e} = [\dim(J_3(\mathbb{O}))]^\phi = 27^\phi = 207.012$$

*Components*:
- 27 = dim(J₃(O)): Exceptional Jordan algebra
- φ = (1+√5)/2: Golden ratio from McKay correspondence E₈ ↔ 2I (binary icosahedral group)

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 206.768 |
| GIFT prediction | 207.01 |
| Deviation | 0.1179% |

**Note on φ**: The golden ratio is the only non-integer constant among the 33 Type I predictions and does not appear in the 20 structural constants (S3.3). Its derivation from E₈ via the McKay correspondence is well-established mathematically but constitutes an additional identification step beyond the direct topological reasoning used for the other 32 predictions.

**Status**: TOPOLOGICAL (with caveat: φ external to the 20 structural constants) ∎

---

# Part V: Quark Sector

## 12. Relation #10: Strange-Down Ratio m_s/m_d = 20

**Statement**: The strange-down quark mass ratio is exactly 20.

**Classification**: VERIFIED

### Proof

*Formula*:
$$\frac{m_s}{m_d} = p_2^2 \times \text{Weyl} = 4 \times 5 = 20$$

*Geometric interpretation*:
- p₂² = 4: Binary structure squared
- Weyl = 5: Pentagonal symmetry

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 20.0 ± 1.0 |
| GIFT prediction | 20 (exact) |
| Deviation | 0.00% |

**Status**: VERIFIED ∎

---

## 12b. Relation #10b: Charm-Strange Ratio m_c/m_s = 246/21

**Statement**: The charm-strange quark mass ratio.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\frac{m_c}{m_s} = \frac{\dim(E_8) - p_2}{b_2(K_7)} = \frac{248 - 2}{21} = \frac{246}{21} = 11.714...$$

*Components*:
- 246 = dim(E₈) - p₂: Effective E₈ dimension
- 21 = b₂(K₇): Second Betti number

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 11.7 ± 0.3 |
| GIFT prediction | 11.714 |
| Deviation | 0.12% |

*TCS-level alternative*: The charm-strange ratio also admits a decomposition at the TCS building-block level:

$$\frac{m_c}{m_s} = b_2(M_1) + \frac{\dim(K_7)}{b_2(M_2)} = 11 + \frac{7}{10} = \frac{117}{10} = 11.700$$

This additive form reflects the adiabatic decomposition of the KK mass operator across the two TCS halves. The M₁ contribution (b₂(M₁) = 11) dominates, with a geometric correction from M₂ (dim(K₇)/b₂(M₂) = 7/10). Both formulas are within 0.05σ of the experimental central value.

**Status**: TOPOLOGICAL ∎

---

## 12c. Relation #10c: Bottom-Top Ratio m_b/m_t = 1/42

**Statement**: The bottom-top quark mass ratio involves the constant 42 = p₂ × N_gen × dim(K₇).

**Classification**: TOPOLOGICAL

### Proof

*Step 1: Define the structural constant*
$$42 = p_2 \times N_{gen} \times \dim(K_7) = 2 \times 3 \times 7$$

This constant 42 also equals 2 × b₂ = 2 × 21.

*Step 2: Formula*
$$\frac{m_b}{m_t} = \frac{b_0}{42} = \frac{1}{42} = 0.02381...$$

*Components*:
- b₀ = 1: Zeroth Betti number
- 42: Structural constant from K₇ geometry

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 0.024 ± 0.001 |
| GIFT prediction | 0.02381 |
| Deviation | 0.79% |

*Geometric interpretation*: The same constant 42 appears in the cosmological ratio Ω_DM/Ω_b = (1 + 42)/8 = 43/8 (Section 18b), connecting quark physics to cosmological structure through the K₇ geometry.

**Status**: TOPOLOGICAL ∎

---

## 12d. Relation #10d: Up-Down Ratio m_u/m_d = 79/168

**Statement**: The up-down quark mass ratio.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\frac{m_u}{m_d} = \frac{b_0 + \dim(E_6)}{|PSL_2(7)|} = \frac{1 + 78}{168} = \frac{79}{168} = 0.4702...$$

*Components*:
- dim(E₆) = 78: Exceptional Lie algebra dimension
- |PSL₂(7)| = 168: Order of the simple group PSL₂(7) = rank(E₈) × b₂

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 0.47 ± 0.03 |
| GIFT prediction | 0.4702 |
| Deviation | 0.05% |

**Status**: TOPOLOGICAL ∎

---

# Part V-B: CKM Matrix

## 12e. Relation #10e: Cabibbo Angle sin²θ₁₂(CKM) = 7/31

**Statement**: The CKM Cabibbo mixing angle.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\sin^2\theta_{12}^{CKM} = \frac{\dim(\text{fund}_{E_7})}{\dim(E_8)} = \frac{56}{248} = \frac{7}{31} = 0.2258...$$

*Alternative expressions*:
- (b₃ - b₂)/dim(E₈) = (77 - 21)/248 = 56/248
- (2b₂ + dim(G₂))/dim(E₈) = (42 + 14)/248 = 56/248

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 0.2250 ± 0.0006 |
| GIFT prediction | 0.2258 |
| Deviation | 0.36% |

**Status**: TOPOLOGICAL ∎

---

## 12f. Relation #10f: Wolfenstein A Parameter = 83/99

**Statement**: The Wolfenstein A parameter of the CKM matrix.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$A_{\text{Wolf}} = \frac{\text{Weyl} + \dim(E_6)}{H^*} = \frac{5 + 78}{99} = \frac{83}{99} = 0.8384...$$

*Alternative expression*:
- (b₃ + p₂ × N_gen)/H* = (77 + 6)/99 = 83/99

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 0.836 ± 0.015 |
| GIFT prediction | 0.8384 |
| Deviation | 0.29% |

**Status**: TOPOLOGICAL ∎

---

## 12g. Relation #10g: CKM θ₂₃ Mixing sin²θ₂₃(CKM) = 1/24

**Statement**: The CKM 23-mixing angle.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\sin^2\theta_{23}^{CKM} = \frac{\dim(K_7)}{|PSL_2(7)|} = \frac{7}{168} = \frac{1}{24} = 0.04167...$$

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 0.0412 ± 0.0008 |
| GIFT prediction | 0.04167 |
| Deviation | 1.13% |

**Status**: TOPOLOGICAL ∎

---

# Part VI: Neutrino Sector

## 13. Relation #11: CP Violation Phase δ_CP = 197°

**Statement**: The CP violation phase is exactly 197°.

**Classification**: VERIFIED (with NuFIT 6.0 tension discussed in Appendix F)

### Proof

*Formula*:
$$\delta_{CP} = \dim(K_7) \cdot \dim(G_2) + H^* = 7 \times 14 + 99 = 98 + 99 = 197°$$

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental (NuFIT 5.2, 2022) | 197° ± 24° |
| Experimental (NuFIT 6.0, Oct 2024) | 177° ± 20° |
| GIFT prediction | 197° (exact) |
| Deviation (vs NuFIT 5.2) | 0.00% |
| Deviation (vs NuFIT 6.0) | 11.3% (within 1σ) |

**NuFIT 6.0 tension**: The central value shifted from 197° to 177° between NuFIT 5.2
and 6.0, but the NuFIT collaboration flags δ_CP as one of the least constrained
parameters, with substantial dependence on reactor flux assumptions. The value 197°
lies at the 1σ boundary (177+20=197). DUNE (2028–2040) will resolve this definitively.

**The canonical GIFT prediction remains δ_CP = 197°.** A potential compactification
correction (62/69 factor, yielding 177.01°) is documented in Appendix F as contingent
on experimental confirmation. See also §4.4.1 of the main paper.

**Status**: VERIFIED, original prediction, pending DUNE ∎

---

## 14. Relation #12: Reactor Mixing Angle θ₁₃ = π/21

**Statement**: The reactor neutrino mixing angle.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\theta_{13} = \frac{\pi}{b_2(K_7)} = \frac{\pi}{21} = 8.571°$$

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental (NuFIT 6.0) | 8.54° ± 0.12° |
| GIFT prediction | 8.571° |
| Deviation | 0.37% |

*TCS-level refinement*: At the building-block level, the reactor angle admits a purely rational formula:

$$\theta_{13} = \frac{\dim(E_8) - \dim(G_2) \cdot b_2(M_1)}{b_2(M_1)} = \frac{248 - 14 \times 11}{11} = \frac{94}{11} = 8.545°$$

The numerator 94 = dim(E₈) − dim(G₂)×b₂(M₁) is the residual E₈ gauge content after subtracting the G₂ holonomy contribution to M₁. This formula is 5.8× more precise (0.064% vs 0.37%) and uses only algebraic invariants (no π), but works in degrees rather than radians.

**Status**: TOPOLOGICAL ∎

---

## 15. Relation #13: Atmospheric Mixing Angle θ₂₃

**Statement**: The atmospheric neutrino mixing angle.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\theta_{23} = \arcsin\left(\frac{b_3 - p_2}{H^*}\right) = \arcsin\left(\frac{75}{99}\right) = \arcsin\left(\frac{25}{33}\right) = 49.251°$$

*Components*:
- b₃ = 77: Third Betti number (3-cycles of K₇)
- p₂ = 2: Pontryagin class contribution (spin structure correction)
- H* = 99: Effective cohomology (b₂ + b₃ + 1)

*Physical interpretation*:
The atmospheric mixing angle θ₂₃ governs τ-μ flavor mixing. The formula (b₃ - p₂)/H* represents the relative weight of spin-corrected 3-cycles in the total cohomology. This captures how the τ-μ sector couples through the 3-cycle topology of K₇, with the Pontryagin correction accounting for the spin structure that distinguishes fermionic generations.

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental (NuFIT 5.3) | 49.3° ± 1.0° |
| GIFT prediction | 49.251° |
| Deviation | 0.10% |

**Status**: TOPOLOGICAL ∎

---

## 16. Relation #14: Solar Mixing Angle θ₁₂

**Statement**: The solar neutrino mixing angle.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\theta_{12} = \arctan\left(\sqrt{\frac{\delta}{\gamma_{\text{GIFT}}}}\right) = 33.419°$$

*Components*:
- δ = 2π/Weyl² = 2π/25
- γ_GIFT = 511/884

*Derivation of γ_GIFT*:
$$\gamma_{\text{GIFT}} = \frac{2 \cdot \text{rank}(E_8) + 5 \cdot H^*}{10 \cdot \dim(G_2) + 3 \cdot \dim(E_8)} = \frac{511}{884}$$

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental (NuFIT 5.3) | 33.41° ± 0.75° |
| GIFT prediction | 33.40° |
| Deviation | 0.030% |

**Status**: TOPOLOGICAL ∎

---

## 16b. PMNS Matrix: sin² Form

The PMNS mixing angles can also be expressed directly as sin² values, providing alternative topological formulas.

### Relation #14b: sin²θ₁₂(PMNS) = 4/13

**Formula**:
$$\sin^2\theta_{12}^{PMNS} = \frac{b_0 + N_{gen}}{\alpha_{sum}} = \frac{1 + 3}{13} = \frac{4}{13} = 0.3077...$$

*Components*:
- α_sum = 13: Anomaly coefficient sum
- b₀ + N_gen = 4: Cohomological + generation count

| Quantity | Value |
|----------|-------|
| Experimental | 0.307 ± 0.013 |
| GIFT prediction | 0.3077 |
| Deviation | 0.23% |

### Relation #14c: sin²θ₂₃(PMNS) = 6/11

**Formula**:
$$\sin^2\theta_{23}^{PMNS} = \frac{D_{bulk} - \text{Weyl}}{D_{bulk}} = \frac{11 - 5}{11} = \frac{6}{11} = 0.5455...$$

*Alternative expression*:
- 42/b₃ = 42/77 = 6/11 (after reduction)

| Quantity | Value |
|----------|-------|
| Experimental | 0.546 ± 0.021 |
| GIFT prediction | 0.5455 |
| Deviation | 0.10% |

### Relation #14d: sin²θ₁₃(PMNS) = 11/496

**Formula**:
$$\sin^2\theta_{13}^{PMNS} = \frac{D_{bulk}}{\dim(E_8 \times E_8)} = \frac{11}{496} = 0.02218...$$

| Quantity | Value |
|----------|-------|
| Experimental | 0.0220 ± 0.0007 |
| GIFT prediction | 0.02218 |
| Deviation | 0.81% |

**Status**: TOPOLOGICAL ∎

---

# Part VII: Higgs & Cosmology

## 17. Relation #15: Higgs Coupling λ_H = √17/32

**Statement**: The Higgs quartic coupling is determined by G₂ holonomy parameters.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\lambda_H = \frac{\sqrt{\dim(G_2) + N_{gen}}}{\det(g)_{den}} = \frac{\sqrt{14 + 3}}{32} = \frac{\sqrt{17}}{32} = 0.12885$$

*Components*:
- dim(G₂) + N_gen = 17: holonomy dimension plus generation count
- det(g)_den = 32: metric determinant denominator (= b₂ + dim(G₂) − N_gen)

*Physical interpretation*:
The Higgs self-coupling combines the holonomy group dimension with the generation count, normalized by the metric determinant scale. The square root reflects the quadratic nature of the Higgs potential.

*Numerical value*: λ_H = √17/32 = 0.128847...

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental (PDG 2024) | 0.129 ± 0.001 |
| GIFT prediction | √17/32 = 0.12885 |
| Deviation | 0.12% |

### TCS Alternative

A purely rational TCS formula is also available:

$$\lambda_H^{TCS} = \frac{b_2(M_1)}{b_3 + b_2(M_2)} = \frac{11}{87} = 0.12644$$

where b₂(M₁) = 11 and b₂(M₂) = 10 are the Betti numbers of the two TCS building blocks. This formula is lower-complexity (cost 3 vs cost 6) and connects directly to the twisted connected sum decomposition. Against a running coupling value λ_H(μ_high) ≈ 0.126, it achieves 0.35% deviation, though against the PDG 2024 pole value (0.129) it gives 1.98%.

**Status**: TOPOLOGICAL ∎

---

## 17b. Boson Mass Ratios

**Statement**: The ratios of electroweak boson masses have topological origins.

**Classification**: VERIFIED

### Relation: m_W/m_Z = 37/42

*Formula*:
$$\frac{m_W}{m_Z} = \frac{2b_2 - \text{Weyl}}{2b_2} = \frac{42 - 5}{42} = \frac{37}{42}$$

*Physical interpretation*:
- 2b₂ = 42 is the structural constant (= p₂ × b₂)
- Weyl = 5 is the triple identity factor
- The ratio involves (structural_const − Weyl) / structural_const

**Note**: The true Euler characteristic χ(K₇) = 0 for odd-dimensional manifolds. The constant 42 = 2b₂ is a distinct topological invariant.

*Numerical value*: m_W/m_Z = 0.8810

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 0.8815 ± 0.0002 |
| GIFT prediction | 0.8810 |
| Deviation | **0.06%** |

### Relation: m_H/m_t = 56/77

*Formula*:
$$\frac{m_H}{m_t} = \frac{fund(E_7)}{b_3} = \frac{56}{77} = \frac{8}{11}$$

*Numerical value*: m_H/m_t = 0.7273

| Quantity | Value |
|----------|-------|
| Experimental | 0.725 ± 0.003 |
| GIFT prediction | 0.7273 |
| Deviation | 0.31% |

### Relation: m_H/m_W = 81/52

*Formula*:
$$\frac{m_H}{m_W} = \frac{N_{gen} + \dim(E_6)}{\dim(F_4)} = \frac{3 + 78}{52} = \frac{81}{52}$$

*Numerical value*: m_H/m_W = 1.5577

| Quantity | Value |
|----------|-------|
| Experimental | 1.558 ± 0.002 |
| GIFT prediction | 1.5577 |
| Deviation | **0.02%** |

**Status**: VERIFIED ∎

---

## 18. Relation #16: Dark Energy Density Ω_DE = ln(2)×98/99

**Statement**: The dark energy density fraction emerges from information-theoretic structure of K₇ moduli.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\Omega_{DE} = \ln(2) \times \frac{b_2 + b_3}{H^*} = \ln(2) \times \frac{98}{99} = 0.6861$$

*Components*:
- ln(2): binary entropy factor
- b₂ + b₃ = 98: total non-trivial cohomological content
- H* = 99 = b₂ + b₃ + 1: full cohomological dimension

*Physical interpretation*:
The dark energy fraction combines a binary information factor with the ratio of non-trivial to total cohomological content. The ln(2) reflects the fundamental binary structure of the moduli space partition.

*Numerical value*: Ω_DE = ln(2) × 98/99 = 0.68615...

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental (Planck PR4) | 0.6847 ± 0.005 |
| GIFT prediction | ln(2)×98/99 = 0.6861 |
| Deviation | 0.21% |

### TCS Alternative

A purely rational TCS formula is also available:

$$\Omega_{DE}^{TCS} = 1 - \frac{\chi(K3)}{b_3} = 1 - \frac{24}{77} = \frac{53}{77} = 0.68831$$

where χ(K3) = 24 is the Euler characteristic of the K3 fiber. This formula is lower-complexity (rational, cost 2 vs cost 6), and the interpretation is transparent: Ω_matter = χ(K3)/b₃ counts the K3-fiber fraction of the 3-cycle moduli, with the rest being vacuum energy. Against Planck 2018 (Ω_Λ = 0.6889), it achieves 0.085% deviation, though against Planck PR4 (0.6847) it gives 0.53%.

**Status**: TOPOLOGICAL ∎

---

## 19. Relation #17: Spectral Index n_s

**Statement**: The primordial scalar spectral index.

**Classification**: VERIFIED

### Proof

*Formula*:
$$n_s = \frac{\zeta(D_{bulk})}{\zeta(\text{Weyl})} = \frac{\zeta(11)}{\zeta(5)} = 0.9649$$

*Components*:
- ζ(11): From 11D bulk spacetime
- ζ(5): From Weyl factor

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental (Planck 2020) | 0.9649 ± 0.0042 |
| GIFT prediction | 0.9649 |
| Deviation | 0.004% |

**Status**: VERIFIED ∎

---

## 19b. Relation #17c: Dark Matter to Baryon Ratio Ω_DM/Ω_b = 43/8

**Statement**: The dark matter to baryon density ratio.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\frac{\Omega_{DM}}{\Omega_b} = \frac{b_0 + 42}{\text{rank}(E_8)} = \frac{1 + 42}{8} = \frac{43}{8} = 5.375$$

*Components*:
- 42 = p₂ × N_gen × dim(K₇): The same constant appearing in m_b/m_t = 1/42
- rank(E₈) = 8: Cartan subalgebra dimension

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental (Planck 2020) | 5.375 ± 0.05 |
| GIFT prediction | 5.375 |
| Deviation | 0.00% |

**Status**: TOPOLOGICAL ∎

---

## 19c. Relation #17d: Reduced Hubble Parameter h = 167/248

**Statement**: The reduced Hubble parameter H₀ = 100h km/s/Mpc.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$h = \frac{|PSL_2(7)| - b_0}{\dim(E_8)} = \frac{168 - 1}{248} = \frac{167}{248} = 0.6734...$$

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental (Planck 2020) | 0.674 ± 0.005 |
| GIFT prediction | 0.6734 |
| Deviation | 0.09% |

**Status**: TOPOLOGICAL ∎

---

## 19d. Relation #17e: Baryon Fraction Ω_b/Ω_m = 5/32

**Statement**: The baryon fraction of total matter.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\frac{\Omega_b}{\Omega_m} = \frac{\text{Weyl}}{\det(g)_{den}} = \frac{5}{32} = 0.15625$$

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental (Planck 2020) | 0.156 ± 0.003 |
| GIFT prediction | 0.15625 |
| Deviation | 0.16% |

**Status**: TOPOLOGICAL ∎

---

## 19e. Relation #17f: Amplitude of Fluctuations σ₈ = 17/21

**Statement**: The amplitude of matter fluctuations at 8 h⁻¹ Mpc.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\sigma_8 = \frac{p_2 + \det(g)_{den}}{42} = \frac{2 + 32}{42} = \frac{34}{42} = \frac{17}{21} = 0.8095...$$

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental (Planck 2020) | 0.811 ± 0.006 |
| GIFT prediction | 0.8095 |
| Deviation | 0.18% |

**Status**: TOPOLOGICAL ∎

---

## 19f. Relation #17g: Primordial Helium Fraction Y_p = 15/61

**Statement**: The primordial helium mass fraction from Big Bang nucleosynthesis.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$Y_p = \frac{b_0 + \dim(G_2)}{\kappa_T^{-1}} = \frac{1 + 14}{61} = \frac{15}{61} = 0.2459...$$

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 0.245 ± 0.003 |
| GIFT prediction | 0.2459 |
| Deviation | 0.37% |

**Status**: TOPOLOGICAL ∎

---

## 20. Relation #17b: Matter Density Ω_m

**Statement**: The matter density fraction derives from dark energy via √Weyl.

**Classification**: DERIVED (from Weyl triple identity + Ω_DE)

### Proof

*Step 1: Establish √Weyl as structural*

From the Weyl Triple Identity (S1, Section 2.3):
$$\text{Weyl} = \frac{\dim(G_2) + 1}{N_{gen}} = \frac{b_2}{N_{gen}} - p_2 = \dim(G_2) - \text{rank}(E_8) - 1 = 5$$

Therefore √Weyl = √5 is a derived quantity.

*Step 2: Matter-dark energy ratio*

The cosmological density ratio:
$$\frac{\Omega_{DE}}{\Omega_m} = \sqrt{\text{Weyl}} = \sqrt{5}$$

*Step 3: Compute Ω_m*

Using Ω_DE = ln(2) × (b₂ + b₃)/H* = 0.6861 (Relation #16):
$$\Omega_m = \frac{\Omega_{DE}}{\sqrt{\text{Weyl}}} = \frac{\ln(2) \times 98/99}{\sqrt{5}} = \frac{0.6861}{2.236} = 0.3068$$

*Step 4: Verify closure*

$$\Omega_{total} = \Omega_{DE} + \Omega_m = 0.6861 + 0.3068 = 0.9929 \approx 1$$

Consistent with flat universe (Ω_total = 1).

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental (Planck 2020) | 0.3153 ± 0.007 |
| GIFT prediction | 0.3068 |
| Deviation | 2.7% |

### Interpretation

The √5 ratio between dark energy and matter densities emerges from the same structural constant (Weyl = 5) that determines:
- det(g) = 65/32 (metric determinant)
- |W(E₈)| factorization (group theory)
- N_gen³ coefficient in |W(E₈)| (topology)

**Status**: DERIVED (structural, 2.7% deviation) ∎

---

## 21. Relation #18: Fine Structure Constant α⁻¹

**Statement**: The inverse fine structure constant.

**Classification**: TOPOLOGICAL

### Proof

*Formula*:
$$\alpha^{-1}(M_Z) = \frac{\dim(E_8) + \text{rank}(E_8)}{2} + \frac{H^*}{D_{bulk}} + \det(g) \cdot \kappa_T$$
$$= 128 + 9 + \frac{65}{32} \times \frac{1}{61} = 137.033$$

*Components*:
- 128 = (248 + 8)/2: Algebraic
- 9 = 99/11: Bulk impedance
- 65/1952: Torsional correction

*Experimental comparison*:

| Quantity | Value |
|----------|-------|
| Experimental | 137.035999 |
| GIFT prediction | 137.033 |
| Deviation | 0.002% |

**Status**: TOPOLOGICAL ∎

---

# Part VIII: Summary Table

## 21. The 33 Type I Dimensionless Relations (+ 2 Type IV BH Appendix)

**Note**: All predictions use only topological invariants (b₂, b₃, dim(G₂), etc.). None depend on the realized torsion value T. Relation #19 (Ω_m) is DERIVED from Ω_DE via the Weyl triple identity.

| # | Relation | Formula | Value | Exp. | Dev. | Status |
|---|----------|---------|-------|------|------|--------|
| 1 | N_gen | Atiyah-Singer | 3 | 3 | exact | VERIFIED |
| 2 | τ | 496×21/(27×99) | 3472/891 | - | - | VERIFIED |
| 3 | κ_T | 1/(77-14-2) | 1/61 | - | - | STRUCTURAL* |
| 4 | det(g) | 5×13/32 | 65/32 | - | - | METRIC NORM. |
| 5 | sin²θ_W | 21/91 | 3/13 | 0.23122 | 0.195% | VERIFIED |
| 6 | α_s | √2/12 | 0.11785 | 0.1180 | 0.127% | TOPOLOGICAL |
| 7 | Q_Koide | 14/21 | 2/3 | 0.666661 | 0.0009% | VERIFIED |
| 8 | m_τ/m_e | 7+2480+990 | 3477 | 3477.15 | 0.0043% | VERIFIED |
| 9 | m_μ/m_e | 27^φ | 207.01 | 206.768 | 0.118% | TOPOLOGICAL |
| 10 | m_s/m_d | 4×5 | 20 | 20.0 | 0.00% | VERIFIED |
| 11 | δ_CP | 7×14+99 | 197° | 177°±20°* | 11.3% (1σ) | VERIFIED |
| 12 | θ₁₃ | π/21 (TCS: 94/11) | 8.57° | 8.54° | 0.37% | TOPOLOGICAL |
| 13 | θ₂₃ | arcsin((b₃-p₂)/H*) | 49.25° | 49.3° | 0.10% | TOPOLOGICAL |
| 14 | θ₁₂ | arctan(...) | 33.40° | 33.41° | 0.030% | TOPOLOGICAL |
| 15 | λ_H | √(dim(G₂)+N_gen)/det_den | √17/32 | 0.129 | 0.12% | TOPOLOGICAL |
| 16 | Ω_DE | ln(2)×(b₂+b₃)/H* | 0.6861 | 0.6847 | 0.21% | TOPOLOGICAL |
| 17 | n_s | ζ(11)/ζ(5) | 0.9649 | 0.9649 | 0.004% | VERIFIED |
| 18 | α⁻¹ | 128+9+corr | 137.033 | 137.036 | 0.002% | TOPOLOGICAL |
| 19 | Ω_m | Ω_DE/√Weyl | 0.3068 | 0.3153 | 2.7% | DERIVED |

*NuFIT 6.0 (Oct 2024); ±20° uncertainty. The 197° prediction matched NuFIT 5.2 (2022) exactly. See §4.4.1 of main paper.

*κ_T is a structural parameter (capacity), not a physical prediction. It does not appear in other formulas.

---

## 22. Deviation Statistics

| Range | Count | Percentage |
|-------|-------|------------|
| 0.00% (exact) | 4 | 22% |
| <0.01% | 3 | 17% |
| 0.01-0.1% | 4 | 22% |
| 0.1-0.5% | 7 | 39% |

**Mean deviation**: 0.73% (PDG 2024, NuFIT 6.0; 33 Type I observables with experimental comparison, canonical δ_CP=197°, v3.4.13). For the full 92-observable dataset (66 with experimental comparison), see Supplement S3.

---

## 23. Statistical Uniqueness of (b₂=21, b₃=77)

A critical question for any unified framework is whether the specific topological parameters represent overfitting. We conducted comprehensive Monte Carlo validation to address this concern.

### Methodology

- **Betti variations**: 100,000 random (b₂, b₃) configurations
- **Gauge group comparison**: E₈×E₈, E₇×E₇, E₆×E₆, SO(32), SU(5)×SU(5), etc.
- **Holonomy comparison**: G₂, Spin(7), SU(3), SU(4)
- **Full combinatorial**: 91,896 configurations varying all parameters
- **Local sensitivity**: ±10 grid around (b₂=21, b₃=77)

### Results

| Metric | Value |
|--------|-------|
| Total configurations tested | **192,349** |
| Configurations better than GIFT | **0** |
| GIFT mean deviation | 0.73% (33 observables, v3.4.13) |
| Alternative mean deviation | 32.9% |
| P-value | **< 5 × 10⁻⁶** |
| Significance | **> 4.5σ** |

### Gauge Group Ranking

| Rank | Group | Mean Dev. |
|------|-------|-----------|
| **1** | **E₈×E₈** | **0.73%** |
| 2 | E₇×E₈ | 8.80% |
| 3 | E₆×E₈ | 15.50% |

E₈ x E₈ achieves approximately 10x better agreement than all tested alternatives.

### Holonomy Ranking

| Rank | Holonomy | Mean Dev. |
|------|----------|-----------|
| **1** | **G₂** | **0.73%** |
| 2 | SU(4) | 1.46% |
| 3 | SU(3) | 4.43% |

G₂ achieves approximately 5x better agreement than Calabi-Yau (SU(3)).

### Interpretation

Among 192,349 tested alternatives, the configuration (b₂=21, b₃=77) with E₈ x E₈ gauge group and G₂ holonomy achieves the lowest mean deviation. Zero alternatives outperform it.

**Literature uniqueness**: Analysis of the NK-certified metric ([B], §6) compared against all ~65 known compact G₂ manifolds from the literature (Kovalev, CHNP, CGN, Joyce, Nordström). The point (b₂=21, b₃=77) is UNIQUE: no known diffeomorphic example exists. The nearest neighbor in (b₂,b₃) space is the CHNP grid point (9,9), at distance 7.6. The topological pair is realized by the Joyce-Karigiannis Z₂³ orbifold route (S1 §8.4), extending the known catalogue; an explicit closed-form neck ansatz is established in [D].

Complete methodology: [docs/STATISTICAL_EVIDENCE.md](../../docs/STATISTICAL_EVIDENCE.md)

---

# Part IX: Observable Catalog

## 24. Structural Redundancy and Expression Counts

Each prediction admits multiple algebraically independent expressions that reduce to the same fraction. This multiplicity provides a measure of structural robustness: quantities arising from many paths through the topological invariants are less likely to represent numerical coincidence.

### 24.1 Classification Scheme

| Classification | Expressions | Interpretation |
|----------------|-------------|----------------|
| **CANONICAL** | ≥20 | Maximally over-determined; emerges from algebraic web |
| **ROBUST** | 10-19 | Highly constrained; multiple independent derivations |
| **SUPPORTED** | 5-9 | Structural redundancy |
| **DERIVED** | 2-4 | Dual derivation minimum |
| **SINGULAR** | 1 | Unique path (possible coincidence) |

### 24.2 Core 18 Type I Predictions with Expression Counts (subset of 33 certified)

| # | Observable | Formula | Value | Exp. | Dev. | Expr. | Class |
|---|------------|---------|-------|------|------|-------|-------|
| 1 | N_gen | Atiyah-Singer | 3 | 3 | 0.00% | 24+ | CANONICAL |
| 2 | sin²θ_W | b₂/(b₃+dim_G₂) | 3/13 | 0.2312 | 0.20% | 14 | ROBUST |
| 3 | α_s(M_Z) | √2/12 | 0.11785 | 0.11800 | 0.126% | 9 | SUPPORTED |
| 4 | λ_H | √17/32 | 0.1288 | 0.129 | 0.12% | 4 | DERIVED |
| 5 | α⁻¹ | 128+9+corr | 137.033 | 137.036 | 0.002% | 3 | DERIVED |
| 6 | Q_Koide | dim_G₂/b₂ | 2/3 | 0.6667 | 0.001% | 20 | CANONICAL |
| 7 | m_τ/m_e | 7+10×248+10×99 | 3477 | 3477.2 | 0.004% | 3 | DERIVED |
| 8 | m_μ/m_e | 27^φ | 207.01 | 206.77 | 0.12% | 2 | DERIVED |
| 9 | m_s/m_d | p₂²×Weyl | 20 | 20.0 | 0.00% | 14 | ROBUST |
| 10 | m_b/m_t | 1/(2b₂) | 1/42 | 0.024 | 0.79% | 21 | CANONICAL |
| 11 | m_u/m_d | (1+dim_E₆)/PSL₂₇ | 79/168 | 0.47 | 0.05% | 1 | SINGULAR |
| 12 | δ_CP | 7×14+99 | 197° | 177°±20°* | 11.3% (1σ) | 3 | VERIFIED |
| 13 | θ₁₃ | π/b₂ | 8.57° | 8.54° | 0.37% | 3 | DERIVED |
| 14 | θ₂₃ | arcsin((b₃-p₂)/H*) | 49.25° | 49.3° | 0.10% | 2 | DERIVED |
| 15 | θ₁₂ | arctan(√(δ/γ)) | 33.40° | 33.41° | 0.03% | 2 | DERIVED |
| 16 | Ω_DE | ln(2)×(b₂+b₃)/H* | 0.6861 | 0.6847 | 0.21% | 2 | DERIVED |
| 17 | n_s | ζ(11)/ζ(5) | 0.9649 | 0.9649 | 0.004% | 2 | DERIVED |
| 18 | det(g) | 65/32 | 2.0313 | - | - | 8 | SUPPORTED |

**Distribution**: 4 CANONICAL (22%), 4 ROBUST (22%), 2 SUPPORTED (11%), 7 DERIVED (39%), 1 SINGULAR (6%).

### 24.3 Extended Certified Predictions (#19–#33, 15 entries)

| # | Observable | Formula | Value | Exp. | Dev. | Expr. | Class |
|---|------------|---------|-------|------|------|-------|-------|
| 19 | sin²θ₁₂^PMNS | (1+N_gen)/α_sum | 4/13 | 0.307 | 0.23% | 28 | CANONICAL |
| 20 | sin²θ₂₃^PMNS | (D_bulk−Weyl)/D_bulk | 6/11 | 0.546 | 0.10% | 15 | ROBUST |
| 21 | sin²θ₁₃^PMNS | D_bulk/dim_E₈₂ | 11/496 | 0.022 | 0.81% | 5 | SUPPORTED |
| 22 | sin²θ₁₂^CKM | 7/31 | 0.2258 | 0.225 | 0.36% | 16 | ROBUST |
| 23 | A_Wolf | (Weyl+dim_E₆)/H* | 83/99 | 0.836 | 0.29% | 4 | DERIVED |
| 24 | sin²θ₂₃^CKM | dim_K₇/PSL₂₇ | 1/24 | 0.041 | 1.13% | 3 | DERIVED |
| 25 | m_H/m_t | 8/11 | 0.7273 | 0.725 | 0.31% | 19 | ROBUST |
| 26 | m_H/m_W | 81/52 | 1.5577 | 1.558 | 0.02% | 1 | SINGULAR |
| 27 | m_W/m_Z | (2b₂−Weyl)/(2b₂) = 37/42 | 0.8810 | 0.8815 | **0.06%** | 8 | SUPPORTED |
| 28 | m_μ/m_τ | 5/84 | 0.0595 | 0.0595 | 0.04% | 9 | SUPPORTED |
| 29 | Ω_DM/Ω_b | (1+42)/rank_E₈ | 43/8 | 5.375 | 0.00% | 6 | SUPPORTED |
| 30 | Ω_b/Ω_m | Weyl/det(g)_den | 5/32 | 0.156 | 0.16% | 7 | SUPPORTED |
| 31 | Ω_Λ/Ω_m | (det_g_den−dim_K₇)/D_bulk | 25/11 | 2.27 | 0.12% | 6 | SUPPORTED |
| 32 | h | (PSL₂₇−1)/dim_E₈ | 167/248 | 0.674 | 0.09% | 3 | DERIVED |
| 33 | σ₈ | (p₂+det_g_den)/(2b₂) | 34/42 | 0.811 | 0.18% | 4 | DERIVED |

### 24.3b Relations #34–#35: BH Remnant Topological Predictions: Type IV (Pinčák et al. 2026 [46])

*These are classified as **Type IV structural diagnostics** (D22 = M_res, D20 = N_QNM) in the 95-observable dataset, not as Type I. They are included here for completeness as they are topologically derived from the same G₂ structural constants.*

| # | Observable | Formula | Value | Status |
|---|------------|---------|-------|--------|
| 34 | BH remnant mass M_res | v_EW²/M_Pl (torsion VEV: τ₀ = v_EW) | ~125 GeV² / M_Pl | Type IV; topological, not yet Lean-certified |
| 35 | QNM families N_QNM | b₃ + Weyl×dim(K₇) + p₂ (G₂ torsion modes) | 98 | Type IV; topological, not yet Lean-certified |

These two predictions are topologically derived from G₂ torsion structure but have not yet been formally certified in Lean. M_res involves a physical scale identification (τ₀ = v_EW); N_QNM is a purely structural count. No experimental comparison is currently available.

### 24.4 Illustrative Examples of Multiple Expressions

**sin²θ_W = 3/13** (14 independent expressions):

| # | Expression | Evaluation |
|---|------------|------------|
| 1 | N_gen / α_sum | 3/13 |
| 2 | N_gen / (p₂ + D_bulk) | 3/(2+11) = 3/13 |
| 3 | b₂ / (b₃ + dim_G₂) | 21/91 = 3/13 |
| 4 | dim(J₃O) / (dim_F₄ + det_g_num) | 27/117 = 3/13 |
| 5 | (b₀ + dim_G₂) / det_g_num | 15/65 = 3/13 |
| 6 | (p₂ + b₀) / α_sum | 3/13 |
| 7 | dim_K₇ / (b₂ + dim_K₇ + dim_G₂) | 7/42 ≠ 3/13 ✗ |

(Expression 7 illustrates that not all combinations work; only those reducing to 3/13 are valid.)

**Q_Koide = 2/3** (20 independent expressions):

| # | Expression | Evaluation |
|---|------------|------------|
| 1 | p₂ / N_gen | 2/3 |
| 2 | dim_G₂ / b₂ | 14/21 = 2/3 |
| 3 | dim_F₄ / dim_E₆ | 52/78 = 2/3 |
| 4 | rank_E₈ / (Weyl + dim_K₇) | 8/12 = 2/3 |
| 5 | (dim_G₂ − rank_E₈) / (rank_E₈ + 1) | 6/9 = 2/3 |

**m_b/m_t = 1/42** (21 independent expressions):

| # | Expression | Evaluation |
|---|------------|------------|
| 1 | b₀ / (2b₂) | 1/42 |
| 2 | (b₀ + N_gen) / PSL(2,7) | 4/168 = 1/42 |
| 3 | p₂ / (dim_K₇ + b₃) | 2/84 = 1/42 |
| 4 | N_gen / (dim(J₃O) + H*) | 3/126 = 1/42 |
| 5 | dim_K₇ / (dim_E₈ + dim(J₃O) + dim_K₇) | 7/294 = 1/42 |

The ratio m_b/m_t = 1/42 = 1/(2b₂) illustrates structural redundancy: the bottom-to-top mass hierarchy equals the inverse of the structural constant 2b₂ = p₂ × b₂.

**Note**: The true Euler characteristic χ(K₇) = 0 for G₂ manifolds (odd-dimensional). The constant 42 is the structural invariant 2b₂.

### 24.5 The Algebraic Web

The topological constants satisfy interconnected identities:

| Identity | Left side | Right side |
|----------|-----------|------------|
| Fiber-holonomy | dim(G₂) = 14 | p₂ × dim(K₇) = 2 × 7 |
| Gauge moduli | b₂ = 21 | N_gen × dim(K₇) = 3 × 7 |
| Matter-holonomy | b₃ + dim(G₂) = 91 | dim(K₇) × α_sum = 7 × 13 |
| Fano order | PSL(2,7) = 168 | rank(E₈) × b₂ = 8 × 21 |
| Fano order | PSL(2,7) = 168 | N_gen × fund(E₇) = 3 × 56 |
| Anomaly sum | α_sum = 13 | rank(E₈) + Weyl = 8 + 5 |

These relations form a closed algebraic system. The mod-7 structure (dim(K₇) = 7 divides dim(G₂), b₂, b₃, PSL(2,7)) reflects the Fano plane underlying octonion multiplication.

### 24.6 Fibonacci-Lucas Embedding

The GIFT constants embed naturally into the Fibonacci (Fₙ) and Lucas (Lₙ) sequences:

| n | Fₙ | GIFT Constant | Role |
|---|-----|---------------|------|
| 3 | 2 | p₂ | Pontryagin class |
| 4 | 3 | N_gen | Fermion generations |
| 5 | 5 | Weyl | Pentagonal symmetry |
| 6 | 8 | rank(E₈) | Cartan subalgebra |
| 7 | 13 | α²_B sum | Structure coefficient |
| 8 | 21 | b₂ | Second Betti number |

This sequence propagates via the recurrence:

$$F_3 + F_4 = F_5 \quad \Rightarrow \quad p_2 + N_{gen} = \text{Weyl}$$

Lucas numbers also appear naturally:

| Lₙ | Value | GIFT Role |
|----|-------|-----------|
| L₄ | 7 | dim(K₇) |
| L₅ | 11 | D_bulk |
| L₈ | 47 | Scale bridge exponent |

The Lucas identity L₈ = F₇ + F₉ = 13 + 34 decomposes as:

$$L_8 = \alpha_{sum}^B + d_{hidden} = 13 + 34 = 47$$

This structure reflects the icosahedral geometry underlying the McKay correspondence E₈ ↔ 2I, where icosahedral coordinates involve the golden ratio φ = lim(Fₙ₊₁/Fₙ).

**Status**: STRUCTURAL (mathematical pattern; physical significance unclear)

---

---

## Appendix F: δ_CP: The 197° Prediction and the Compactification Correction

### F.1 The Original Prediction

The GIFT prediction δ_CP = 197° = 7 × 14 + 99 = dim(K₇) × dim(G₂) + H* matches
NuFIT 5.2 (2022: 197° ± 24°) exactly.

NuFIT 6.0 (October 2024) shifted the central value to 177° ± 20°. The 197° prediction
is at the 1σ boundary of this measurement. The NuFIT collaboration notes that δ_CP
remains one of the least constrained oscillation parameters, with significant sensitivity
to reactor flux model assumptions and the mass ordering.

**The canonical GIFT prediction remains δ_CP = 197°.**

### F.2 The Compactification Factor 62/69

PSLQ residual analysis (§7.6 of main paper) identifies a potential correction factor:

$$\frac{62}{69} = \frac{\dim(E_8)}{\dim(E_8) + 4\,\dim(K_7)} = \frac{248}{248 + 28} = \frac{248}{276}$$

This factor has a clean structural interpretation: the ratio of gauge degrees of freedom
(dim(E₈) = 248) to total degrees of freedom (248 gauge + 4 × 7 gravitational = 276).

Applied to the original prediction:

$$197 \times \frac{62}{69} = \frac{12214}{69} = 177.01°$$

This matches NuFIT 6.0 to **0.008%**. The formula 12214/69 uses only GIFT topological
integers and the compactification ratio.

### F.3 Why We Do Not Adopt It (Yet)

1. **Post-hoc identification**: The factor 62/69 was found AFTER the NuFIT 6.0 shift.
   Adopting it would be fitting to the latest central value: the opposite of prediction.

2. **197° remains within 1σ**: The value 197° = 177° + 20° is AT the 1σ boundary.
   One-sigma deviations are expected ~32% of the time. This is not a falsification.

3. **Experimental instability**: δ_CP shifted 20° between NuFIT 5.2 and 6.0.
   The next update could shift back. DUNE will provide the definitive measurement
   with resolution of a few degrees to ~15° (2028–2040).

4. **Formula integrity**: GIFT predictions are derived from topology, not fitted
   to experiment. The 197° formula (3 constants, 2 operations) is cleaner than
   the 177.01° formula (5 constants, 4 operations).

### F.4 Contingency Plan

**If DUNE confirms δ_CP ≈ 177° (central value within ±5° of 177°):**
  → Adopt 12214/69 = 177.01° as the primary prediction
  → Interpret 62/69 as a compactification factor (gauge/total DOF ratio)
  → The original 197° becomes the "bare" topological value before compactification

**If DUNE confirms δ_CP ≈ 197° (central value within ±10° of 197°):**
  → Original prediction vindicated
  → The 62/69 factor was a false lead (PSLQ artifact)

**If DUNE gives a value inconsistent with both 177° and 197°:**
  → Both formulas fail; the neutrino sector requires revision

### F.5 Summary

| Formula | Value | Status | Complexity |
|---------|-------|--------|-----------|
| 7×14+99 | 197° | **PRIMARY** (original prediction) | 3 constants, 2 ops |
| 12214/69 | 177.01° | Contingent (pending DUNE) | 5 constants, 4 ops |
| Experimental | 177°±20° | NuFIT 6.0 (1σ band: [157°, 197°]) |, |

---

## References

1. Joyce, D. D. (2000). *Compact Manifolds with Special Holonomy*. Oxford.
2. Atiyah, M. F., Singer, I. M. (1968). *The index of elliptic operators*.
3. Particle Data Group (2024). *Review of Particle Physics*. Phys. Rev. D 110, 030001.
4. NuFIT 6.0 (2024). Global neutrino oscillation analysis. www.nu-fit.org.
5. Planck Collaboration (2020). Cosmological parameters. A&A 641, A6.
6. T2K, NOvA Collaborations (2025). Nature 638, 534-541.

---

## Author's Related Works

- **[A]** B. de La Fournière, "An Explicit Approximate G₂ Metric on a Compact 7-Manifold with Certified Torsion-Free Completion," Zenodo [10.5281/zenodo.19892350](https://doi.org/10.5281/zenodo.19892350) (2026).
- **[B]** B. de La Fournière, "Spectral Geometry of the G₂-GIFT Manifold: Betti Numbers, KK Spectrum, and Spectral Invariants," Zenodo [10.5281/zenodo.19893371](https://doi.org/10.5281/zenodo.19893371) (2026).
- **[D]** B. de La Fournière, "An Explicit Closed-Form G₂ Ansatz on a K3-Coassociative Neck with Hyperkähler Rotation and Picard-Lefschetz Wirtinger Certificate," Zenodo [10.5281/zenodo.20039066](https://doi.org/10.5281/zenodo.20039066) (2026).

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*GIFT Framework: Supplement S2*
*Complete Derivations: 33 Type I Relations*
