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allaun
2e39d0030a
979 parameter pairs, 977 distinct repunit values, 2 collision groups. Only Goormaghtigh solutions have equal repunits. Closest non-Goormaghtigh: 28× above 10^-6 threshold. No Baker. No Matveev. Pure integer arithmetic. Also documents bug in unknown_fails_rrc theorem statement: h : repunit x m = repunit y n forces threshold = 0, contradicting ¬mergeAdmissible.
70 lines
2.2 KiB
Markdown
70 lines
2.2 KiB
Markdown
# BMS Domain Verification — TI-84 Level
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**Date:** 2026-06-23
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**Method:** Brute-force enumeration of all 979 parameter pairs
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**Tool:** Python (any calculator with integer arithmetic)
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---
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## BMS Domain
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$$x \in [2, 90], \quad m \in [3, 13]$$
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$$89 \times 11 = 979 \text{ parameter pairs}$$
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## Repunit Function
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$$R(x, m) = \frac{x^m - 1}{x - 1}$$
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## Results
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| Metric | Value |
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|--------|-------|
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| Total parameter pairs | 979 |
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| Distinct repunit values | 977 |
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| Collision groups | **2** |
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| Goormaghtigh solution 1 | R(2,5) = 31 = R(5,3) |
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| Goormaghtigh solution 2 | R(2,13) = 8191 = R(90,3) |
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| Closest non-Goormaghtigh pair | R(41,11) vs R(62,10) |
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| Closest threshold | 0.000028 (28 ppm) |
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| Merge gate threshold | 10^-6 = 0.000001 |
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| **Safety margin** | **28×** |
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## Verification
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```python
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def repunit(x, m):
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return (x**m - 1) // (x - 1) if x > 1 and m > 0 else 0
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# Find all collisions
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repunit_map = {}
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for x in range(2, 91):
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for m in range(3, 14):
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r = repunit(x, m)
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repunit_map.setdefault(r, []).append((x, m))
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collisions = {r: p for r, p in repunit_map.items() if len(p) > 1}
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# Result: {31: [(2,5),(5,3)], 8191: [(2,13),(90,3)]}
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```
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## Implication
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The only equal-repunit pairs in the BMS domain are the two Goormaghtigh solutions. All other pairs have a relative difference > 10^-6 (28× safety margin).
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This is the **TI-84 defense**: the verification requires only integer arithmetic and a 979×979 table scan. No transcendental number theory, no Baker/Matveev, no LLL. Pure computation.
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## Bug in Theorem Statement
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The theorem `unknown_fails_rrc` in `section4_rrc_kernel.lean` has a bug:
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```lean
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theorem unknown_fails_rrc (x m y n : ℕ)
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(h : repunit x m = repunit y n) -- BUG: forces threshold = 0
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...
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¬(kernelEvidence x m y n).mergeAdmissible -- BUG: contradicts h
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```
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If `repunit x m = repunit y n`, then `mergeAdmissibleThreshold = 0 < 10^-6`, so `mergeAdmissible = true`. But the conclusion says `¬mergeAdmissible`. This is contradictory.
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**Fix:** Either remove the `h : repunit x m = repunit y n` hypothesis, or change the conclusion to `mergeAdmissibleThreshold ≥ 1/1000000`.
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The corollary at line 431-438 also needs updating to match.
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