# EP-278 verification report

Audit date: 2026-08-29.

Current verdict: **mathematically complete internal preprint candidate**.
This means the written implication chain is complete and all implemented
falsification tests pass. It does not mean peer reviewed or independently
certified.

## Proof-obligation audit

| Obligation | Resolution | Status |
|---|---|---|
| Density exists | Every chosen union is periodic modulo the common lcm. | PASS |
| Higher intersection criterion | Generalized CRT gives simultaneous solubility iff every pair agrees modulo the pairwise gcd. | PASS |
| Density depends only on graph | Finite inclusion–exclusion retains exactly the clique subsets. | PASS |
| Forced-edge count | It is the kernel size of a well-defined homomorphism from (prod_i\mathbb Z/n_i\mathbb Z) to (prod_e\mathbb Z/d_e\mathbb Z). | PASS |
| Lattice index direction | The cokernel has order ([\mathbb Z^q:D_T\mathbb Z^q+B_T\mathbb Z^r]); the first isomorphism theorem gives the stated numerator/denominator. | PASS |
| Exact graph count | Superset zeta relation followed by Boolean Möbius inversion has the stated sign and direction. | PASS |
| Extremal equality | Positive exact count is equivalent to realizability; graph-only density then makes the two maxima identical. | PASS |
| Prime-layer formula | Base-(p) digits are independent; each active connected component contributes one free digit. | PASS |
| Factoring-free complexity | gcd/lcm and polynomial-bit-complexity Smith normal form suffice; there are (2^{\binom r2}) edge states. | PASS |
| Gcd-kernel reduction | Every incident constraint reads coordinate (i) only modulo the lcm of its pairwise gcds. | PASS |
| Component factorization | Component lcms are pairwise coprime; CRT makes missed densities and their minima multiply. | PASS |

## Computational audit

Main checker:

- 59 primary modulus sets;
- 4,228 prime-layer forced-edge instances;
- 221 realizable graphs;
- 65,536 six-vertex forced-edge stress instances;
- 1,099 structural graph reductions;
- 127 nonempty small subsets;
- exact recovery of (M(\{2,3,\ldots,10\})=69/70).

Independent checker:

- no import from the main project checker;
- 101 modulus sets;
- 24,745 literal residue tuples;
- 3,665 exact graph-count comparisons;
- direct period-union maxima equal graph-formula maxima in every case;
- status PASS.

## Publication-risk audit

- Direct same formula located in checked literature: NO.
- Proof that no equivalent prior formula exists: NO; impossible to guarantee
  from search alone.
- Core Smith/arrangement machinery is prior art: YES and explicitly credited.
- External mathematical review completed: NO.
- Maintained problem tracker acceptance obtained: NO.
- Author identity and responsibility supplied: NO.
- Actual arXiv upload authorized: NO.

The correct label is therefore “unrefereed preprint candidate,” with the
result stated as an exact effective characterization. The package should not
claim community acceptance, peer review, or absolute priority.

## Public release and license

The author approved public release on 2026-09-02. This paper, its source files, and this verification report are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/