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.

---

## 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`

---

## 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

---

## 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

---

## 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

---

## 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

---

## 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

---

## 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)

---

## 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

---

## 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]
```

---

## 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.

---

## 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

---

## 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.