# 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.