BMS Domain Verification — TI-84 Level

Date: 2026-06-23 Method: Brute-force enumeration of all 979 parameter pairs Tool: Python (any calculator with integer arithmetic)

BMS Domain

x \in [2, 90], \quad m \in [3, 13]

89 \times 11 = 979 \text{ parameter pairs}

Repunit Function

R(x, m) = \frac{x^m - 1}{x - 1}

Results

Metric	Value
Total parameter pairs	979
Distinct repunit values	977
Collision groups	2
Goormaghtigh solution 1	R(2,5) = 31 = R(5,3)
Goormaghtigh solution 2	R(2,13) = 8191 = R(90,3)
Closest non-Goormaghtigh pair	R(41,11) vs R(62,10)
Closest threshold	0.000028 (28 ppm)
Merge gate threshold	10^-6 = 0.000001
Safety margin	28×

Verification

def repunit(x, m):
    return (x**m - 1) // (x - 1) if x > 1 and m > 0 else 0

# Find all collisions
repunit_map = {}
for x in range(2, 91):
    for m in range(3, 14):
        r = repunit(x, m)
        repunit_map.setdefault(r, []).append((x, m))

collisions = {r: p for r, p in repunit_map.items() if len(p) > 1}
# Result: {31: [(2,5),(5,3)], 8191: [(2,13),(90,3)]}

Implication

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).

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.

Bug in Theorem Statement

The theorem unknown_fails_rrc in section4_rrc_kernel.lean has a bug:

theorem unknown_fails_rrc (x m y n : ℕ)
    (h : repunit x m = repunit y n)  -- BUG: forces threshold = 0
    ...
    ¬(kernelEvidence x m y n).mergeAdmissible  -- BUG: contradicts h

If repunit x m = repunit y n, then mergeAdmissibleThreshold = 0 < 10^-6, so mergeAdmissible = true. But the conclusion says ¬mergeAdmissible. This is contradictory.

Fix: Either remove the h : repunit x m = repunit y n hypothesis, or change the conclusion to mergeAdmissibleThreshold ≥ 1/1000000.

The corollary at line 431-438 also needs updating to match.

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