Archive BMS_VERIFICATION.md

archive/2026-07-02/docs/BMS_VERIFICATION.md

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+# 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
+
+```python
+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:
+
+```lean
+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.