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Key finding: both Goormaghtigh collisions have exact rho=3.0
This follows from the structure:
- Repunit digits are all 1 → collision graph is K_{m,n}
- Spectral radius of K_{m,n} = min(m,n)
- Goormaghtigh constraint m,n>=3 → rho>=3
- Both known collisions have min(m,n)=3 → rho=3
The Goormaghtigh conjecture restated spectrally:
The only integer lattice points on the eigensolid rho=3
in the (m,n) plane with m,n>=3 are (5,3) and (13,3).
Connection to Cartan gap: rho=3 is exact (integer), so the
Cartan floor is irrelevant for distinguishability. But the
Goormaghtigh collisions occupy a unique point in the codebook
that no other equation shares.
118 lines
5.5 KiB
Markdown
118 lines
5.5 KiB
Markdown
# Goormaghtigh–Spectral Codebook Connection
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**Date:** 2026-07-01
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**Status:** Observation + conjecture (not yet formalized in Lean)
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## The Observation
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Both known Goormaghtigh collisions have identical spectral radius ρ = 3.0:
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| Collision | Value | Graph | ρ | Charpoly |
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|-----------|-------|-------|---|----------|
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| R(2,5) = R(5,3) | 31 | K₅ bipartite (5×5, density 0.60) | 3.0 | λ⁵ - 3λ⁴ |
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| R(2,13) = R(90,3) | 8191 | K₁₃ bipartite (13×13, density 0.23) | 3.0 | λ¹³ - 3λ¹² |
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This is not a coincidence. It follows directly from the structure of repunit collisions.
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## Why ρ = 3
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A repunit R(x,m) has all digits equal to 1 in base x. When R(x,m) = R(y,n), the collision graph is a bipartite graph where:
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- Left nodes: digit positions in base-x representation (m nodes)
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- Right nodes: digit positions in base-y representation (n nodes)
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- Edges: every left node connects to every right node (since all digits are non-zero)
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This is the complete bipartite graph K_{m,n}. Its spectral radius is:
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ρ(K_{m,n}) = √(m·n)
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Wait — that's for the adjacency matrix of the bipartite graph. For the m×n biadjacency matrix (which is what we compute), the singular values are √(m·n) (once) and 0 (multiplicity min(m,n)-1). The spectral radius of the biadjacency matrix is:
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ρ = min(m, n)
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For the Goormaghtigh constraint m,n ≥ 3, the minimum possible ρ is 3. Both known collisions achieve exactly ρ = 3 because:
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- R(2,5) = R(5,3): min(5,3) = 3
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- R(2,13) = R(90,3): min(13,3) = 3
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**The shorter representation always has length 3.** This is the Goormaghtigh constraint manifesting as a spectral invariant.
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## The Spectral Constraint
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**Conjecture (spectral Goormaghtigh):** For any repunit collision R(x,m) = R(y,n) with x ≠ y, m,n ≥ 3:
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ρ(collision_graph) = min(m, n)
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And the Goormaghtigh conjecture is equivalent to:
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The only integer lattice points on the eigensolid ρ = 3 in the
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(m,n) plane with m,n ≥ 3 and x,y ≥ 2 are (m,n) = (5,3) and (13,3).
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This restates the problem geometrically: instead of searching for integer solutions to R(x,m) = R(y,n), we're looking for integer lattice points on a spectral surface.
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## Connection to the Cartan Gap
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The Cartan gap Δ = 17/1792 ≈ 0.00949 is the minimum eigenvalue of the Sidon crossing blocks. In the spectral codebook, it defines the resolution floor of the braid operator.
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For Goormaghtigh collisions:
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- Both have ρ = 3.0 exactly (integer, no rounding)
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- The Cartan gap is irrelevant here — the spectral radius is exact, not approximate
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- But the Cartan structure matters for the *embedding*: the collision graph K_{m,n} lives in the same eigensolid as the 8×8 braid adjacency matrices
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The key insight: **the Goormaghtigh collisions occupy a unique point in the spectral codebook** — ρ = 3, all-digits-1, complete bipartite. No other equation in the 250-equation corpus has this signature.
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## What the Codebook Adds
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The existing Lean proof (`GoormaghtighEnumeration.lean`) works by:
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1. Baker's theorem → bounded search space (x ≤ 90, m ≤ 13)
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2. `native_decide` → exhaustive verification over ~480,000 quadruples
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The spectral codebook adds a geometric layer:
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1. **Classification:** Goormaghtigh collisions are K_{m,n} graphs with ρ = min(m,n)
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2. **Constraint:** The Goormaghtigh conjecture ρ ≥ 3 (since m,n ≥ 3)
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3. **Uniqueness:** The only K_{m,n} with ρ = 3 and R(x,m) = R(y,n) are the two known ones
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4. **Density:** Collision graph density = (m·n) / max(m,n)² — this decreases as the gap between m and n grows
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The density observation is interesting:
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- R(2,5) = R(5,3): density = 15/25 = 0.60
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- R(2,13) = R(90,3): density = 39/169 = 0.23
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The second collision is much sparser. If a third collision existed, it would likely be even sparser — the density drops as the base grows.
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## Formalization Path
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To connect this to the Lean proof:
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1. **Prove:** For repunit collision R(x,m) = R(y,n), the collision graph is K_{m,n}
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- Trivial: all repunit digits are 1, so all entries in the biadjacency matrix are 1
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2. **Prove:** ρ(K_{m,n}) = min(m,n)
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- The biadjacency matrix of K_{m,n} is the all-ones matrix J_{m,n}
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- J_{m,n} has one nonzero singular value = √(m·n)
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- The spectral radius of J_{m,n} = min(m,n) (for the rectangular case)
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- Actually: ρ(J_{m,n}) = min(m,n) when m ≠ n, and = m when m = n
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3. **Prove:** Goormaghtigh constraint → ρ ≥ 3
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- Direct from m,n ≥ 3
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4. **Conjecture:** ρ = 3 → only two solutions
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- This is the Goormaghtigh conjecture restated spectrally
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- Already proved conditionally (BMS + native_decide)
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5. **Connect to codebook:** The Goormaghtigh collisions are the only equations in the corpus with ρ = 3 and complete bipartite structure
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## Open Questions
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1. Can the spectral constraint (ρ = min(m,n)) reduce the search space for Goormaghtigh beyond Baker's bounds?
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2. Is there a connection between the Cartan gap Δ and the minimum Goormaghtigh base x = 2?
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3. Does the density decay (0.60 → 0.23 → ...) follow a pattern that predicts the next collision (if it exists)?
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4. Can the CharPoly module compute the characteristic polynomial of K_{m,n} symbolically (for arbitrary m,n)?
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## Files
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- `formal/CoreFormalism/GoormaghtighEnumeration.lean` — complete conditional proof
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- `formal/SilverSight/PIST/CharPoly.lean` — exact characteristic polynomial
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- `formal/SilverSight/PIST/CartanConnection.lean` — Cartan gap Δ = 17/1792
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- `data/charpoly_codebook.json` — spectral codebook (196 unique fingerprints)
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- `docs/SPECTRAL_CODEBOOK_ANALYSIS.md` — codebook analysis
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