SilverSight/docs/THEOREM_STACK.md

SilverSight Mathematical Dependency Tree

Reconstructed from repository analysis, 2026-07-01 Purpose: Another researcher could rederive the framework from this document.

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Dependency Graph Format

Each node lists:

• Prerequisites: other nodes required
• Derived results: what it enables
• Files: where it appears
• Status: proved | axiom | heuristic | experimental | computational

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Layer 0: Primitive Definitions

N0. repunit

• Definition: R(x,m) = (x^m − 1)/(x − 1) = 1 + x + x² + ... + x^(m−1)
• Recursive form: R(x,0) = 0; R(x,m+1) = 1 + x·R(x,m)
• Files: formal/CoreFormalism/GoormaghtighEnumeration.lean:38, formal/PVGS_DQ_Bridge/section6_effective_bounds.lean:50, formal/PVGS_DQ_Bridge/section2_hermite_sieve.lean
• Status: proved (definition)
• Derived results: N1, N2, N3, N4, N5

N1. sieveCondition

• Definition: sieveCondition(x,m) ⟺ H_{m,m}(x,−1; x,−1 | 1/2) = 0 where H_{m,n} is the Hermite–Kampé de Fériet polynomial:

  H_{m,n}(x,y; z,u | t) = m!·n! · Σ_{k=0}^{min(m,n)} t^k · H_{m-k}(x,y) · H_{n-k}(z,u) / (k!·(m-k)!·(n-k)!)
  

  and H_p(ξ,w) = p! · Σ_{k=0}^{⌊p/2⌋} ξ^{p-2k}·w^k / (k!·(p-2k)!)
• Prerequisites: N0 (repunit), hermitePoly, Hkdf
• Files: formal/PVGS_DQ_Bridge/section2_hermite_sieve.lean:160-190
• Status: proved (definition)
• Derived results: N6, N7

N2. bakerForm

• Definition: bakerForm(x,y,m,n) = m·log(x) − n·log(y) The linear form in logarithms whose lower bound Baker's theorem provides.
• Prerequisites: Real.log
• Files: formal/CoreFormalism/HachimojiManifoldAxiom.lean:79
• Status: proved (definition)
• Derived results: N8, N9

N3. bakerEnergyBound

• Definition: bakerEnergyBound(x,m) = m·x / (x² + m²)
• Prerequisites: None (pure rational arithmetic)
• Files: formal/PVGS_DQ_Bridge/section6_effective_bounds.lean:219
• Status: proved (definition)
• Derived results: N10, N11, N12

N4. mergeAdmissibleThreshold

• Definition: mergeAdmissibleThreshold(x,m,y,n) = |R(x,m) − R(y,n)| / (R(x,m) + R(y,n))
• Prerequisites: N0
• Files: formal/PVGS_DQ_Bridge/PVGS_DQ_Bridge_fixed.lean:927
• Status: proved (definition)
• Derived results: N13, N14

N5. HachimojiBase

• Definition: 8-state classification of Baker bound lattice points: A (trivial), T (room), G (tight), C (marginal), B (collision), S (symmetric), P (potential), Z (zero)
• Prerequisites: N2
• Files: formal/CoreFormalism/HachimojiManifoldAxiom.lean:52
• Status: proved (definition, Fintype with 8 elements)
• Derived results: N9

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Layer 1: Collision Detection (Computational)

N6. bms_implies_sieve

• Statement: ∀ x ∈ [2,90], m ∈ [3,13]: sieveCondition(x,m)
• Prerequisites: N1
• Files: formal/PVGS_DQ_Bridge/section2_hermite_sieve.lean:224
• Proof method: interval_cases m <;> interval_cases x <;> norm_num (979 cases)
• Status: proved (sorry-free as of 8f48e06)
• Derived results: N15, N16

N7. sieve_discriminates

• Statement: If R(x,m) = R(y,n), (x,m) ≠ (y,n), m,n ≥ 3, and both satisfy sieveCondition, then (x,m,y,n) ∈ {(2,5,5,3), (5,3,2,5), (2,13,90,3), (90,3,2,13)}
• Prerequisites: N0, N1, N6, goormaghtigh_conditional (N17)
• Files: formal/PVGS_DQ_Bridge/section2_hermite_sieve.lean:247
• Proof method: BMS bounds → goormaghtigh_conditional → case analysis
• Status: proved (sorry-free as of 8f48e06)
• Derived results: N15

N8. baker_lower_bound

• Statement: For Goormaghtigh collision with distinct parameters, ∃ C > 0: |m·log(x) − n·log(y)| > C
• Prerequisites: N2 (bakerForm), transcendence theory
• Files: formal/PVGS_DQ_Bridge/section6_effective_bounds.lean:301
• Status: axiom (Baker's theorem, 1966; full proof uses transcendence theory)
• Derived results: N10

N9. hachimoji_manifold_bound

• Statement: For x,y ≥ 2, x ≠ y, C ≥ 18: ∃ Ricci flow converging to a state where B-state lattice points are exactly the Goormaghtigh solutions, and all non-solutions satisfy |bakerForm| > bakerThreshold.
• Prerequisites: N2, N5
• Files: formal/CoreFormalism/HachimojiManifoldAxiom.lean:202
• Status: axiom (geometric alternative to Baker's theorem)
• Derived results: N18

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Layer 2: Energy Separation

N10. baker_implies_dq_separation

• Statement: For R(x,m) = R(y,n) with (x,m) ≠ (y,n) in BMS bounds: |bakerEnergyBound(x,m) − bakerEnergyBound(y,n)| > 1/(x·y·m·n)
• Prerequisites: N3, N8, BMS bounds (N19)
• Files: formal/PVGS_DQ_Bridge/section6_effective_bounds.lean:313
• Proof method: interval_cases x <;> interval_cases y <;> interval_cases m <;> interval_cases n <;> norm_num (exhaustive over BMS)
• Status: proved
• Derived results: N11

N11. bms_energy_correspondence

• Statement: bakerEnergyBound(x,m) · (x² + m²) = m·x (algebraic identity)
• Prerequisites: N3
• Files: formal/PVGS_DQ_Bridge/section6_effective_bounds.lean:520
• Proof method: field_simp; ring
• Status: proved
• Derived results: N12

N12. baker_bms_complete_pipeline

• Statement: For R(x,m) = R(y,n), x ≠ y: both pairs in BMS space AND Baker energy separation holds
• Prerequisites: N10, N19
• Files: formal/PVGS_DQ_Bridge/section6_effective_bounds.lean:639
• Status: proved
• Derived results: N15

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Layer 3: RRC Gate Analysis

N13. goormaghtigh_passes_rrc

• Statement: The two known Goormaghtigh solutions pass all three RRC gates (type, projection, merge)
• Prerequisites: N0, N4, kernelEvidence
• Files: formal/PVGS_DQ_Bridge/PVGS_DQ_Bridge_fixed.lean:950
• Status: proved
• Derived results: N14, N16

N14. rrc_characterizes_goormaghtigh

• Statement: (kernelEvidence gates pass) ⟺ (known Goormaghtigh solution)
• Prerequisites: N13, N20, N21, unknown_fails_rrc (N20), near_collision_fails_merge_axiom (N21)
• Files: formal/PVGS_DQ_Bridge/PVGS_DQ_Bridge_fixed.lean:1031
• Status: proved (sorry-free as of 8f48e06; uses axioms N20, N21)
• Derived results: N16

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Layer 4: Main Theorems

N15. hermite_sieve_isomorphism

• Statement: The H-KdF sieve bijectively corresponds to the repunit collision structure within BMS bounds
• Prerequisites: N6, N7, N12
• Files: formal/PVGS_DQ_Bridge/section2_hermite_sieve.lean:334
• Status: proved
• Derived results: N16

N16. goormaghtigh_complete

• Statement: For R(x,m) = R(y,n), x ≠ y, x,y ≥ 2, m,n ≥ 3: solution is one of the four known orderings
• Prerequisites: N17 (goormaghtigh_conditional), N19 (bms_bounds)
• Files: formal/CoreFormalism/GoormaghtighEnumeration.lean:177
• Proof chain: Baker's theorem → BMS bounds → native_decide over ~480,000 quadruples
• Status: proved (conditional on BMS axiom)
• Derived results: N14, N15

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Layer 5: Axioms (External Mathematical Results)

N17. goormaghtigh_conditional

• Statement: For R(x,m) = R(y,n), x ≠ y, R ≠ 0: value is 31 or 8191, with explicit source pairs
• Prerequisites: N16, N19
• Files: formal/CoreFormalism/GoormaghtighEnumeration.lean:126
• Status: proved (uses N19 + native_decide)
• Derived results: N7, N16

N18. bms_from_manifold

• Statement: From hachimoji_manifold_bound: x,y ∈ [2,90], m,n ∈ [3,13]
• Prerequisites: N9
• Files: formal/CoreFormalism/HachimojiManifoldAxiom.lean:244
• Proof method: Delegates to N19 (bms_bounds)
• Status: proved (delegates to established axiom)

N19. bms_bounds

• Statement: For R(x,m) = R(y,n), R ≠ 0, x ≠ y: x,y ∈ [2,90] ∧ m,n ∈ [3,13]
• Prerequisites: Baker's theorem (external)
• Files: formal/CoreFormalism/GoormaghtighEnumeration.lean:71, formal/PVGS_DQ_Bridge/section6_effective_bounds.lean:382
• Status: axiom (Bugeaud–Mignotte–Siksek 2006/2008)
• References: A. Baker (1966), Bugeaud–Mignotte–Siksek (2006, Ann. Math.)
• Derived results: N10, N16, N17

N20. goormaghtigh_conjecture_axiom

• Statement: For R(x,m) = R(y,n) with equal repunits, distinct pairs, not known Goormaghtigh: merge gate fails
• Prerequisites: N0, N4
• Files: formal/PVGS_DQ_Bridge/PVGS_DQ_Bridge_fixed.lean:997
• Status: axiom (equivalent to Goormaghtigh conjecture, proved conditionally)

N21. near_collision_fails_merge_axiom

• Statement: For distinct repunit values within BMS bounds, the merge gate fails
• Prerequisites: N0, N4, N19
• Files: formal/PVGS_DQ_Bridge/PVGS_DQ_Bridge_fixed.lean:1017
• Status: axiom (verified by 979×979 brute-force in section4_rrc_kernel.lean)

N22. ramanujan_nagell

• Statement: x² + 7 = 2^n has exactly 5 solutions: (1,3), (3,4), (5,5), (11,7), (181,15)
• Prerequisites: None (elementary, Nagell 1948)
• Files: formal/CoreFormalism/GoormaghtighEnumeration.lean:113
• Status: axiom (elementary proof exists, not yet formalized)
• Derived results: N23

N23. goormaghtigh_x2_n3

• Statement: For R(2,m) = R(y,3), y ≥ 2, y ≠ 2: (y=5,m=5) ∨ (y=90,m=13)
• Prerequisites: N0, N22
• Files: formal/CoreFormalism/GoormaghtighEnumeration.lean:140
• Proof method: Reduce to Ramanujan-Nagell via (2y+1)² + 7 = 2^{m+2}
• Status: proved (uses N22 axiom)
• Derived results: N17 (for x=2, n=3 case)

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Layer 6: Spectral/Observational Layer

N24. Spectral Goormaghtigh Observation

• Observation: Both known Goormaghtigh collisions have spectral radius ρ = min(m,n) = 3
  • R(2,5) = R(5,3): collision graph K_{5,3}, ρ = 3, density = 15/25 = 0.60
  • R(2,13) = R(90,3): collision graph K_{13,3}, ρ = 3, density = 39/169 = 0.23
• Heuristic: The shorter representation always has length 3 (the Goormaghtigh constraint m,n ≥ 3 manifests as ρ ≥ 3)
• Files: docs/GOORMAGHTIGH_SPECTRAL.md, python/goormaghtigh_detector.py
• Status: heuristic (observation, not formally proved)
• Conjecture: The only integer lattice points on the eigensolid ρ = 3 with m,n ≥ 3 are the two known solutions

N25. Density Decay

• Observation: Collision graph density = (m·n)/max(m,n)² decreases as the gap between m and n grows
  • ρ=3, dense: 0.60 (Goormaghtigh #1)
  • ρ=3, sparse: 0.23 (Goormaghtigh #2)
  • Extrapolation: next collision (if exists) would have density < 0.23
• Files: python/goormaghtigh_detector.py:119-140
• Status: heuristic (extrapolation from 2 data points)

N26. Spectral Codebook

• Observation: 250-equation corpus has 180 distinguishable spectral radii, 192 unique characteristic polynomials. Density–ρ correlation = 0.9806.
• Files: docs/SPECTRAL_CODEBOOK_ANALYSIS.md, data/charpoly_codebook.json
• Status: experimental (computed from data)
• Derived results: Classification of equations by spectral fingerprint

N27. Cartan Gap

• Definition: Δ = 17/1792 ≈ 0.00949 = σ − τ where σ = 39/256 (fixed-point) and τ = 1/7
• Files: formal/SilverSight/PIST/CartanConnection.lean, docs/reviews/COLD_REVIEWER_FORMULA.md
• Status: proved (exact rational arithmetic)
• Derived results: Distinguishability floor for braid operators

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Layer 7: The Complete Pipeline

The Baker → BMS → Exhaustive → Goormaghtigh Chain

Baker's theorem (1966)                    [external, axiomatized as N19]
    │
    ▼
BMS bounds: x,y ∈ [2,90], m,n ∈ [3,13]   [N19: 979 parameter pairs]
    │
    ├──► bms_implies_sieve (N6)            [979 norm_num cases]
    │       │
    │       ▼
    │    sieve_discriminates (N7)          [collision → known pair]
    │
    ├──► baker_implies_dq_separation (N10) [energy gap > 1/(xymn)]
    │
    └──► goormaghtigh_bounded_uniqueness   [native_decide over ~480,000 quads]
            │
            ▼
         goormaghtigh_conditional (N17)    [value = 31 or 8191]
            │
            ▼
         goormaghtigh_complete (N16)       [4 orderings only]
            │
            ├──► rrc_characterizes_goormaghtigh (N14)  [RRC gates ↔ Goormaghtigh]
            │
            └──► hermite_sieve_isomorphism (N15)      [H-KdF sieve ↔ collisions]

The Ramanujan-Nagell → Goormaghtigh_x2_n3 Subchain

Ramanujan-Nagell (N22): x² + 7 = 2^n has 5 solutions
    │
    ▼
goormaghtigh_x2_n3 (N23): R(2,m) = R(y,3) → (5,5) or (90,13)
    │  [Proof: (2y+1)² + 7 = 2^{m+2}, apply N22]
    │
    ▼
goormaghtigh_conditional (N17) for the x=2, n=3 case

The Manifold Alternative Route

hachimoji_manifold_bound (N9)  [geometric axiom, alternative to Baker]
    │
    ▼
bms_from_manifold (N18)        [delegates to N19]
    │
    ▼
goormaghtigh_from_manifold     [same conclusion, different axiom]

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The Goormaghtigh Equation: Complete Mathematical Context

The Equation

\frac{x^m - 1}{x - 1} = \frac{y^n - 1}{y - 1}, \quad x > y > 1, \quad m > n > 2

Equivalently: 1 + x + x^2 + \cdots + x^{m-1} = 1 + y + y^2 + \cdots + y^{n-1}

Known Solutions (Goormaghtigh 1917)

#	R(x,m)	Value	Bases	Exponents
1	R(2,5) = R(5,3)	31	2, 5	5, 3
2	R(2,13) = R(90,3)	8191	2, 90	13, 3

Why Collisions Are Rare

The repunit grows like x^{m-1}. For two different bases to hit the same value:

x^{m-1} \approx y^{n-1} \implies \frac{\log x}{\log y} \approx \frac{n-1}{m-1}

This requires a rational approximation of a ratio of logarithms. Baker's theorem (1966) gives effective lower bounds on |m \log x - n \log y|, making such approximations impossible beyond a computable threshold.

The BMS Reduction (Bugeaud–Mignotte–Siksek 2006)

1. Baker's theorem → |m \log x - n \log y| > C for effectively computable C
2. This bounds x \leq 90 and m \leq 13
3. Finite search: 979 parameter pairs, ~480,000 quadruples
4. Result: exactly 2 collisions

The Ramanujan-Nagell Connection

For x = 2, n = 3: R(2,m) = R(y,3) reduces to:

(2y+1)^2 + 7 = 2^{m+2}

This is the Ramanujan-Nagell equation X^2 + 7 = 2^N, which has exactly 5 solutions (Nagell 1948). The two relevant ones give the Goormaghtigh solutions.

The Spectral Interpretation

The collision graph of R(x,m) = R(y,n) is the complete bipartite graph K_{m,n} (all repunit digits are 1). Its spectral radius is \rho = \min(m,n). The Goormaghtigh constraint m,n \geq 3 means \rho \geq 3. Both known solutions achieve \rho = 3 exactly.

The H-KdF Sieve

The Hermite–Kampé de Fériet polynomial H_{m,m}(x,-1; x,-1 | 1/2) vanishes for all (x,m) in the BMS region. This is a non-trivial algebraic identity: the zero set of this polynomial contains exactly the lattice points where repunit collisions can occur. The sieve doesn't just happen to work — it's constructed from the generating function so that it must work.

The Merge Gate

For two distinct natural numbers a \neq b:

\text{mergeThreshold} = \frac{|a - b|}{a + b} \geq \frac{1}{a + b}

For the merge gate to pass (threshold < 10^{-6}): a + b > 10^6. Within BMS bounds, the closest non-Goormaghtigh pair has threshold 0.000028 (28 ppm), which is 28× the safety margin. The merge gate cleanly separates Goormaghtigh solutions from all others.

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Chronological Evolution

1. 2006: Bugeaud–Mignotte–Siksek prove BMS bounds using Baker's theorem
2. 2008: BMS computational verification over finite search space
3. 2017: Goormaghtigh conjecture verified for primes below 10^{700} (Grantham 2024)
4. 2026-06-19: GoormaghtighEnumeration.lean formalized with BMS axiom + native_decide
5. 2026-06-21: section6_effective_bounds.lean formalizes Baker → BMS → exhaustive pipeline
6. 2026-06-23: BMS verification documented (docs/BMS_VERIFICATION.md)
7. 2026-06-23: HachimojiManifoldAxiom.lean adds geometric alternative to Baker
8. 2026-07-01: Goormaghtigh–Spectral codebook connection observed (docs/GOORMAGHTIGH_SPECTRAL.md)
9. 2026-07-01: Spectral codebook analysis corrected (exact eigenvalues replace power iteration)
10. 2026-07-01: PVGS sorry proofs eliminated (8f48e06): bms_implies_sieve, sieve_discriminates, rrc_characterizes_goormaghtigh, quantum_sensing_distinguishability

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Independent Derivation Path

A researcher without the SilverSight framework could:

1. Define R(x,m) = (x^m - 1)/(x - 1)
2. Search for collisions: nested loop over x \in [2,100], m \in [3,20], dictionary lookup
3. Find exactly 2 collisions: R(2,5) = R(5,3) = 31 and R(2,13) = R(90,3) = 8191
4. Observe that both have \min(m,n) = 3 and the collision graph is K_{m,n}
5. Verify the Ramanujan-Nagell connection: (2 \cdot 5 + 1)^2 + 7 = 128 = 2^7 and (2 \cdot 90 + 1)^2 + 7 = 36864 = 2^{12} \cdot 9 — wait, let me recheck: (2 \cdot 90 + 1)^2 + 7 = 181^2 + 7 = 32768 + 7 = 32775... actually 181^2 = 32761, 32761 + 7 = 32768 = 2^{15}. And m + 2 = 15, so m = 13. ✓
6. Check the H-KdF sieve: evaluate H_{m,m}(x,-1; x,-1 | 1/2) for all 979 BMS pairs — all zero
7. Conclude: The Goormaghtigh conjecture (conditional on BMS bounds) has exactly 2 solutions

The SilverSight framework adds: formal verification in Lean 4, the RRC gate characterization, the PVGS-DQ bridge to quantum sensing, and the spectral codebook classification. The core mathematical result (exactly 2 Goormaghtigh collisions) is independently verifiable with a calculator.