[other] yanwang: interface-width barriers and exact four-terminal signature escape

[Avi7ii]

Team

Team name	yanwang
Members	W.W

Challenge relationship

This research result addresses the structural core of #251. It does not
claim a finite weighted-graph counterexample and does not close the issue.

The submission isolates an interface-width hierarchy for possible escape
from edge negative correlation:

two terminals    exact effective-activity substitution; no sign escape
three terminals  exact five-state composition identity; no sign escape
four terminals   exact positive crossover in the ambient signature cone

It also gives a conditional Newton-face reduction: a minor-minimal
counterexample cannot first appear on one exposed monomial phase, so a real
witness must use finite-scale interference between adjacent layers.

Results at a glance

The submission records 11 non-duplicate results; raw no-hit trial counts
are deliberately excluded.

Four theorem/lemma-level results

1. Positive two-terminal replacement is exactly an effective-activity
   substitution and cannot reverse the Rayleigh sign.
2. A minor-minimal counterexample cannot first occur on one exposed Newton
   face; it must use finite-scale adjacent-layer interference.
3. Three-terminal parallel composition preserves nonpositivity by an exact
   five-state identity.
4. Every edge-pair orbit of K3 join independent(r) is strictly negatively
   correlated on the symmetric two-parameter slice for all r>=1.

Four exact computer-assisted propositions

1. At spoke activity 2897/1000, all 65,535 HSW augmentations have
   coefficientwise nonpositive common-r polynomials; this includes 49,768
   three-connected graphs.
2. Two HSW reservoirs have complete same-sign certificates with 1,287 and
   13,870 nonzero coefficients, independently reconstructed.
3. The complete 3^18 double-bridge tensor has 267,288 positive exponents,
   every one certified inside negative support by an exact midpoint.
4. For 337 exact-zero seeds, all 44,152 missing-edge pairs and 221,404 triples
   have coefficientwise nonpositive leading tangent forms.

Three mechanism/search-geometry results

1. Four terminals are the first interface width permitting exact positive
   crossover in the ambient signature cone.
2. Real graph signatures attain numerical projective Jacobian rank 14; tested
   positive continuations fail at a fold/caustic. This is mechanism evidence,
   not an exact rank theorem.
3. The bounded exact 3-connected atlas (n=4..9, q=m-2<=12, plus named
   cores) contains 251 cores and 11,407 target-pair orbits; the first minimal
   cubic priority layer reduces to four cores and 23 target orbits.

submission_package/RESULTS_LEDGER.md gives the evidence type, exact scope,
and limitation of every item. Theorems, exhaustive finite propositions, and
numerical mechanism evidence are kept explicitly separate.

Position relative to #213

PR #213 by @Osgood001 establishes a universal adjacent-edge theorem through
real stability of star marginals, closing the adjacent-edge branch. This
submission advances the unresolved disjoint-edge branch: it proves structural
no-go results at interface widths two and three, demonstrates exact sign
reversal at width four in the ambient signature cone, and isolates graphic
realizability as the remaining barrier. The two submissions address different
parts of #251 and should be evaluated on their own claims.

What is included

• a self-contained research note with the two-terminal reduction,
  three-terminal identity, exposed-face argument, and exact four-terminal
  crossover;
• a structured ledger of all 11 results with exact counts and claim
  boundaries;
• an orbit-by-orbit algebraic proof for the symmetric
  K3 join independent(r) activity slice;
• a precise novelty/non-claim statement; and
• a standard-library exact verifier.

The two positive integer four-terminal signatures are individually
nonpositive for every disjoint perfect matching. Their composition has exact
reversed gaps

1,134,803,118
  413,278,037
   74,494,526

for the three matchings. The signatures are abstract and are not claimed
to be graph-realizable.

Verification

python3 tracks/other/solutions/yanwang-251/submission_package/verify_interface_certificates.py

Fresh output:

three-terminal identity: complete symbolic coefficient check passed (13 nonzero monomials)
four-terminal crossover: exact gaps (1134803118, 413278037, 74494526)
symmetric-book regression: 222 exact edge pairs passed
all exact interface certificates passed

git diff HEAD^ --check also passes.

Explicit non-claims

• no graph realization of the abstract crossover signatures;
• no universal theorem for disjoint edges;
• no proof of the full I-Rayleigh conjecture; and
• no witness satisfying Z_ef Z > Z_e Z_f for issue [challenge]: Finite weighted-forest counterexample to arboreal-gas edge negative association #251.