# Supplement S3: Observable Dataset

## S3.1 Introduction

This supplement provides the complete GIFT observable dataset: 95 observables derived from a single G₂ metric (169 Chebyshev-Cholesky geometric parameters, zero freely adjustable physical parameters), organized by derivation type and cross-referenced with Lean formal verification. The dataset includes 2 BH remnant predictions from Pinčák et al. 2026 [46] (classified Type IV) and 6 new Type IV metric block eigenvalues (v3.4.13).

The 95 observables decompose as **33(I) + 19(II) + 21(III) + 22(IV)** across four types of increasing derivation complexity. All predictions trace to a single compact G₂ manifold K₇ with (b₂, b₃) = (21, 77) and E₈×E₈ gauge structure, 169 metric parameters, zero continuously adjustable physical parameters. The pair (b₂, b₃) = (21, 77) does not appear among the catalogued compact G₂ manifolds; a complete geometric construction remains an open problem. The certified metric provides computational evidence for a new G₂ manifold.

The dataset serves as the canonical reference for all GIFT predictions, superseding scattered results across supplements S6–S25. Every entry traces back to a specific computation, a JSON artifact, and (where applicable) a Lean theorem.

---

## S3.2 Type Classification

Observables are classified into four types by derivation directness. The type assignment reflects the number of physical identification steps between the G₂ metric and the final prediction:

| Type | Count | Description | Derivation chain | Lean coverage |
|------|-------|-------------|------------------|---------------|
| I    | 33    | Direct algebraic from structural constants | topology → formula → ratio | 33/33 (100%) |
| II   | 19    | One physical identification step | formula + VEV or scale → dimensional quantity | 0/19 (0%) |
| III  | 21    | Multi-step dynamical chains | metric → mechanism → observable | 14/21 (67%) |
| IV   | 22    | Structural/internal quantities | metric → diagnostic | 8/22 (36%) |

**Type I** (33 observables): Direct from metric geometry and topological invariants (b₂, b₃, dim(G₂), dim(E₈), etc.). These are dimensionless ratios expressed as algebraic combinations of integers. Examples: sin²θ_W = 3/13, Q_Koide = 2/3, m_τ/m_e = 3477. All 33 are Lean-certified. The two BH remnant topological predictions are classified as Type IV structural diagnostics (not part of the 33 Type I Lean-certified core).

**Type II** (19 observables): One physical identification step beyond Type I. Absolute masses from ratios × VEV (e.g., m_u = (m_u/m_d) × m_d_exp), CKM magnitudes from Wolfenstein parametrization, extended quark ratios. From the `gift_observables.csv` pipeline.

**Type III** (21 observables): Multi-step chains involving dynamical mechanisms:
- wilson_line non-adiabatic eigenvalue splitting (3 observables: raw lepton ratios)
- S11 RGE gauge coupling running (4 observables: sin²θ_W, α_em⁻¹, α_s at M_Z, M_GUT)
- spectral 7D spectral geometry (5 observables: Weyl exponent, KK states, fiber channels)
- gauge_bundle gauge bundle data (4 observables: f_IJ conditioning, α_ratio, Yukawa rank)
- instanton instanton volumes + combined wilson_line+instanton (5 observables: ΔV's, combined ratios)

**Type IV** (22 observables): Structural and internal consistency quantities with no direct experimental comparison. These include topology diagnostics (b₂, b₃, χ(K₇)), Newton-Kantorovich certification values (h, distance, margin), Gram matrix conditioning (cond(G_K3), cond(G_77)), spectral counts, metric block eigenvalues (g_ss = 19/6, g_{T²} = 7/6, g_K3 ≈ 64/77), and Pinčák et al. 2026 [46] topological diagnostics (N_QNM = 98, M_res, instanton suppression ×77). M_res and N_QNM are the two BH-remnant-related entries; they are classified here as structural diagnostics, not as Type I or Type III, because they require a physical scale identification (τ₀ = v_EW) not derivable from topology alone. These quantities verify internal consistency rather than predict measurements directly.

**Derivation depth**: The type assignment correlates with derivation chain length. Type I observables require 1 algebraic step; Type II requires 2 steps (algebra + scale identification); Type III requires 3–5 steps (algebra + dynamics + calibration); Type IV involves diagnostic computation chains of variable length.

---

## S3.3 Methodology

### Experimental Sources

All experimental reference values are drawn from four standard compilations:

| Source | Coverage | Date |
|--------|----------|------|
| **PDG 2024** [1] | Particle masses, gauge couplings, CKM elements | 2025 |
| **NuFIT 6.0** [2] | Neutrino oscillation parameters (NO w/o SK) | Oct 2024 |
| **Planck PR4** (Tristram+ 2024) [3] | Cosmological parameters: h, σ₈, Ω_Λ | 2024 |
| **CODATA 2022** [4] | Lepton mass ratios | 2022 |

Selected NuFIT 6.0 values: sin²θ₂₃ = 0.561, δ_CP = 177°. Selected Planck PR4 values: h = 0.6764, σ₈ = 0.807, Ω_Λ = 0.6847.

### Structural Constants

The 20 structural constants from which all 95 observables derive:

| # | Constant | Value | Origin |
|---|----------|-------|--------|
| 1 | dim(E₈) | 248 | E₈ Lie algebra |
| 2 | rank(E₈) | 8 | Cartan subalgebra |
| 3 | dim(G₂) | 14 | Aut(𝕆) |
| 4 | dim(K₇) | 7 | Im(𝕆) |
| 5 | b₂(K₇) | 21 | Input; confirmed by spectral analysis (Paper B [B], §2.3). Any Mayer-Vietoris decomposition is conditional on building block identification (open problem). |
| 6 | b₃(K₇) | 77 | Input; confirmed by spectral analysis (Paper B [B], §2.3). Any Mayer-Vietoris decomposition is conditional on building block identification (open problem). |
| 7 | N_gen | 3 | Index theorem |
| 8 | p₂ | 2 | dim(G₂)/dim(K₇) |
| 9 | Weyl | 5 | Triple identity |
| 10 | H* | 99 | b₂+b₃+1 |
| 11 | dim(J₃(𝕆)) | 27 | Jordan algebra |
| 12 | dim(E₆) | 78 | E₆ Lie algebra |
| 13 | dim(E₇) | 133 | E₇ Lie algebra |
| 14 | dim(F₄) | 52 | Aut(J₃(𝕆)) |
| 15 | fund(E₇) | 56 | E₇ fundamental |
| 16 | \|PSL(2,7)\| | 168 | Fano automorphisms |
| 17 | D_bulk | 11 | 4+7 |
| 18 | α_sum | 13 | rank(E₈)+Weyl |
| 19 | det(g)_den | 32 | b₂+dim(G₂)−N_gen |
| 20 | det(g)_num | 65 | Weyl×α_sum |

### Consolidation Pipeline

The master dataset (`gift_observables.json`) is generated by `run_observable_dataset.py` (observable_dataset), which collects all individual JSON results from S6–S25 and performs 5 verification checks:

1. **Completeness**: All 95 entries present across 4 types
2. **Uniqueness**: No duplicate observable IDs
3. **Consistency**: GIFT values match source JSONs to machine precision
4. **Source tracing**: Every entry maps to a computation step (S6–S25)
5. **Format validation**: JSON, CSV, and LaTeX outputs agree

Each observable record contains: GIFT prediction (float64), GIFT fraction (exact symbolic), experimental value and uncertainty, experimental source, relative deviation (%), Lean theorem name (if certified), and provenance (computation step).

---

## S3.4 Complete Observable Table

<!-- INSERT: gift_observables_latex.tex -->

**Table S3.1**: Complete GIFT observable dataset (95 entries, v3.4.13). Columns: name, GIFT prediction, experimental value, relative deviation (%), type (I/II/III/IV), source computation (S6–S25), Lean verification status.

[The full LaTeX table is provided in `data/gift_observables_latex.tex` and will be included at typesetting.]

### Sector-by-Sector Performance

The 66 observables with experimental comparison (Types I+II+III) organize into 11 sectors with varying precision:

**Best-performing sectors** (mean dev < 0.5%):
- **Boson** (3 obs): mean 0.13%, m_H/m_W (0.02%), m_W/m_Z (0.06%), m_H/m_t (0.31%)
- **Quark** (4 obs): mean 0.21%, m_s/m_d (0.00%), m_u/m_d (0.05%), m_c/m_s (0.12%), m_b/m_t (0.79%)
- **CKM** (3 obs): mean 0.59%: A_Wolf (0.29%), sin²θ₁₂ (0.36%), sin²θ₂₃ (1.13%)
- **Cosmology** (7 obs): mean 0.15%, n_s (0.004%), h (0.09%), Ω_b/Ω_m (0.16%), σ₈ (0.18%), Ω_DE (0.21%), Y_p (0.37%), Ω_DM/Ω_b (0.00%)

**Good sectors** (mean dev 0.5–3%):
- **Electroweak** (3 obs): mean 0.11%, α⁻¹ (0.002%), α_s (0.126%), sin²θ_W (0.20%)
- **Lepton** (3 obs): mean 0.04%: Q_Koide (0.001%), m_τ/m_e (0.004%), m_μ/m_e (0.12%)
- **PMNS** (4 obs): mean 0.29%, θ₁₂ (0.03%), θ₂₃ (0.10%), sin²θ₁₂ (0.23%), θ₁₃ (0.37%)

**Challenging sectors** (mean dev > 3%):
- **Gauge running** (4 obs): mean 2.3%, sin²θ_W(RGE) (2.78%), α_em⁻¹(RGE) (2.53%), α_s(RGE) (3.7%), M_GUT (0.00%)
- **Instanton** (3 obs): mean 7.4%: ΔV(e-τ) (5.9%), ΔV(e-μ) (15.9%), κ(gauge) (4.7%)

**B-test consistency** (derived, not counted as independent observable): The MSSM B parameter B = (α₁⁻¹−α₂⁻¹)/(α₂⁻¹−α₃⁻¹) evaluates to 1.4033 using GIFT couplings (0.23% from the theoretical 7/5), closer than the purely experimental B = 1.3948 (0.37%). The identity B = 7/5 is equivalent to α_em⁻¹(M_Z) = 91√2 = dim(Λ²𝔤₂)·√2 and implies the holonomy sequence α₁⁻¹:α₂⁻¹:α₃⁻¹ = dim(G₂):dim(K₇):p₂ = 14:7:2. See §5.4 of the main text.

### Exact Matches

11 observables achieve deviations below 0.01% (effectively exact):

| Observable | GIFT | Exp. | Dev. |
|-----------|------|------|------|
| α⁻¹ | 137.033 | 137.036 | 0.002% |
| Q_Koide | 2/3 | 0.666661 | 0.001% |
| m_τ/m_e | 3477 | 3477.15 | 0.004% |
| m_s/m_d | 20 | 20.0 | 0.000% |
| n_s | 0.9649 | 0.9649 | 0.004% |
| m_u | 2.16 | 2.16 | 0.00% |
| \|V_tb\| | 0.999 | 0.999 | 0.00% |
| m_c/m_d | 234.3 | 234.0 | 0.004% |
| M_GUT | 2×10¹⁶ | 2×10¹⁶ | 0.00% |
| α_GUT⁻¹ | 25.3 | 25.3 | 0.00% |
| rank(Y) | 3 | 3 | 0.00% |

Note: δ_CP canonical prediction is 197° (11.3% from NuFIT 6.0 central); a compactification factor 62/69 brings it to 177.014° (0.008%) but is not adopted, see §4.4.1. m_c/m_s deviation is 0.12%, above the 0.01% threshold.

### Known Outliers

Two observables deviate by more than 5%, each with an identified physical origin:

**δ_CP**: The GIFT prediction is 197° (pure topological: 7 × 14 + 99), deviating 11.3% from NuFIT 6.0 central (177° ± 20°) but at the edge of 1σ. PSLQ analysis identifies a compactification factor 62/69 (§4.4.1 of main paper), documented as a structural observation but not adopted: the canonical prediction remains 197°. The experimental central value shifted from 197° (NuFIT 5.2) to 177° (NuFIT 6.0) and carries ±20° uncertainty. Falsifiable by DUNE (2028–2040).

**α_s(RGE) = +3.7%**: G₂-MSSM split-spectrum matching (MSSM above M_squark = 3165 GeV, SM + gauginos below (b₃ = −5)) gives α_s = 0.1224 (exp 0.1180, dev 3.7%). A naive degenerate-spectrum treatment would give 12.1%. The topological value α_s = √2/12 = 0.11785 (Type I) matches experiment to 0.126%.

**ΔV(e-μ) = +15.9%**: The instanton volume assignment ΔV(e-μ) = 3.271 versus target ln(16.82) = 2.823 reflects the optimal assignment problem for 57 associative cycles. The combined wilson_line+instanton pipeline with α = e^K (§S3.5) reduces this to 0.75%.

**κ(gauge) = +4.7%**: The gauge kinetic function conditioning cond(f_IJ) = 1.047 measures departure from exact universality. The deviation from 1.000 reflects K₇ geometry at the percent level.

### Summary Statistics

| Metric | Value |
|--------|-------|
| Total observables | 95 |
| Type breakdown | 33(I) + 19(II) + 21(III) + 22(IV) |
| With experimental comparison | 66 |
| Exact matches (dev < 0.01%) | 11 |
| Within 1% of experiment | 53 |
| Within 5% of experiment | 63 |
| Lean-certified | 55 |
| Structural constants | 20 |
| Metric parameters | 169 |

---

## S3.5 The Combined wilson_line+instanton Pipeline

The lepton mass hierarchy (combined lepton ratio observables) uses two complementary geometric mechanisms. The complete derivation is in §6 of the main text; results are summarised here for cross-reference.

Raw wilson_line (K3 fiber eigenvalue splitting at c = 0.452): m_τ/m_e = 3403 (2.1%), m_τ/m_μ = 16.54 (1.7%), m_μ/m_e = 205.7 (0.5%). Raw instanton (associative 3-cycle volumes, 57 cycles): ΔV(e-τ) = 8.633 (5.9%), ΔV(e-μ) = 3.271 (15.9%). Combined with α = e^{K₀} = V̂^{−3} = 0.002763 (Kähler potential of K₇, zero free parameters):

- m_τ/m_e = 3485 (exp 3477, **0.24%**)
- m_τ/m_μ = 16.69 (exp 16.82, **0.75%**)
- m_μ/m_e = 208.8 (exp 206.8, **0.97%**)

Certified in Lean: `AssociativeVolumes.lean` (14 conjuncts, 0 sorry). Full mechanism description: main §6.4.

---

## S3.6 Sensitivity Cross-Reference

The complete sensitivity analysis is in §7 of the main text. Key results for cross-reference:

- **Effective DOF**: r_eff = 15.53 (SVD of 20×33 constant-usage Jacobian)
- **Overdetermination**: 33/15.53 = 2.13× (more constraints than free dimensions)
- **Cross-coupling**: 155/528 observable pairs share a structural constant; all form one connected component
- **P(coincidence, uniform)** = 10⁻³⁴⁶ (χ² + Fisher combined, main §7.5)
- **P(coincidence, algebraic)** = 10⁻¹³³ (4.2M random formulas from same 20 constants, main §7.5)

Figures: `fig_sensitivity_heatmap.png`, `fig_constant_usage.png`, `fig_observable_correlations.png`, `fig_mc_per_observable.png`.

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## S3.7 Lean Cross-References

Of the 95 observables, **55 are formally verified** in Lean 4 across the following certificate files:

| Lean File | Axioms | Theorems | Conjuncts | Coverage |
|-----------|--------|----------|-----------|----------|
| `Foundations.lean` | 0* | n/a | 38 | Type I: metric, torsion, topology |
| `Predictions.lean` | 0* | n/a | 55 | Types I–III: couplings, masses, mixing |
| `Spectral.lean` | 0* | n/a | 41 | Type II: KK spectrum, Weyl law |
| `MetricEigenvalues.lean` | 0 | n/a | 15 | Metric fractions: g_ss=19/6, g_T²=7/6, γ²=24π²/7 (T² Laplacian) |
| `SpectralInvariants.lean` | 0 | n/a | 10 | Heat kernel, ζ'(0), b₁=0 |
| `CompactificationCorrection.lean` | 0 | n/a | 6 | δ_CP compactification factor |
| `TCSGaugeBreaking.lean` | 0 | 14 | 10 | Type III: gauge breaking chain |
| `GaugeBundleData.lean` | 0 | 12 | 11 | Type III: bundle universality |
| `AssociativeVolumes.lean` | 0 | 19 | 14 | Type III: instanton hierarchy |
| `ComputedWeylLaw.lean` | 0 | 8 | 7 | Type III: 7D spectral geometry |

*4 published axioms. All substantive: standard theorems (Cheeger inequality) + geometric structure (TCS spectral bounds) + physical inputs (literature package: CGN+Joyce). KK_YM_EFT, K7_spectral_data, K7_analysis_data are theorems/defs. G₂ group structure proven by `native_decide`.

**Total certificate**: 213 conjuncts, 0 `sorry`, 134 .lean files (128 core + 6 generated + 12 test/support), 8378 build jobs (Lean 4.29.0).

### Coverage by Type

| Type | Certified | Total | Coverage |
|------|-----------|-------|----------|
| **I** | 33 | 33 | 100% |
| **II** | 0 | 19 | 0% |
| **III** | 14 | 21 | 67% |
| **IV** | 8 | 22 | 36% |
| **Total** | **55** | **95** | **58%** |

**Type II uncertified**: The 19 Type II observables derive from the CSV pipeline (absolute masses from ratio × VEV, CKM magnitudes from Wolfenstein). These involve dimensional quantities and intermediate data that have not yet been axiomatized in Lean. The underlying Type I ratios from which they derive are all certified.

**Type IV partial**: 8 of 22 structural quantities are certified (Betti numbers, holonomy dimension, NK certification bounds). The remaining 14 involve spectral counts, conditioning numbers, metric eigenvalues, and BH remnant estimates that require floating-point Lean infrastructure not yet developed.

---

## S3.8 Data Availability

The complete dataset is available in three formats:

- **JSON**: `gift_observables.json` (machine-readable, full metadata per observable)
- **CSV**: `gift_observables.csv` (tabular, 95 rows × 13 columns, for analysis)
- **LaTeX**: `gift_observables_latex.tex` (publication-ready table)

All artifacts are archived in the Zenodo data deposit (DOI: [10.5281/zenodo.19893371](https://doi.org/10.5281/zenodo.19893371)), which also contains the 9 source JSON files from S6–S25, the 10 computation scripts, and the 10 figures.

---

## References

[1] Particle Data Group, Phys. Rev. D 110, 030001 (2024)
[2] NuFIT 6.0, www.nu-fit.org (Oct 2024)
[3] L. Tristram et al., A&A 682, A37 (2024) (Planck PR4)
[4] CODATA 2022, physics.nist.gov/cuu/Constants

---

*GIFT Framework: Supplement S3: Observable Dataset*
*95 observables, 4 types, 55 Lean-certified*
