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

tracks/other/solutions/yanwang-251/README.md

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+# Interface barriers for weighted random-forest edge correlation
+
+## Team
+
+| | |
+|---|---|
+| **Team name** | yanwang |
+| **Members** | W.W |
+
+## Challenge
+
+| Row | |
+|---|---|
+| **Challenge** | Identify structural mechanisms that can or cannot produce a finite weighted-forest counterexample to edge negative correlation. |
+| **Catalog issue** | Structural advances on the unresolved disjoint-edge core of #251. |
+| **Track** | `tracks/other`, following the issue's `Method: Other` field. |
+
+## Result in one paragraph
+
+Issue #251 asks for a positive-weight graph with
+`Z_ef Z > Z_e Z_f`.  This submission does **not** claim such a graph.
+Instead, it proves a hierarchy of interface obstructions.  Positive
+two-terminal replacements reduce exactly to an effective edge activity and
+cannot change the Rayleigh sign.  Parallel composition through three
+terminals preserves nonpositivity by an explicit five-state identity.  At
+four terminals that closure fails in the ambient partition-signature cone:
+two exact positive integer signatures that are individually nonpositive for
+all three disjoint target matchings compose to a strictly positive signature
+for all three.  A separate Newton-face argument shows why a minor-minimal
+counterexample cannot first appear on a single monomial asymptotic face.  It
+must use finite-scale interference between layers, with four terminals the
+first interface width at which the abstract sign obstruction disappears.
+
+## Results at a glance
+
+The contribution inventory contains **11 non-duplicate results**.  Failed
+optimizer runs and raw trial counts are not included in this number.
+
+| Class | Count | Principal content |
+|---|---:|---|
+| Theorem/lemma level | 4 | Two-terminal effective activity; single-face obstruction; three-terminal closure; symmetric-book theorem. |
+| Exact computer-assisted propositions | 4 | Complete HSW augmentations; two grouped HSW certificates; the `3^18` double-bridge tensor; the 337-seed tangent census. |
+| Mechanism and search-geometry results | 3 | Exact four-terminal ambient escape; numerical full-rank signature maps and their singular wall; bounded exact atlas reduction. |
+
+The precise statement, evidence type, scope, and limitation of every item are
+listed in `submission_package/RESULTS_LEDGER.md`.  This separation matters:
+theorems are not conflated with finite exhaustive results, and neither is
+conflated with numerical mechanism evidence.
+
+## Submitted package
+
+| Path | Role |
+|---|---|
+| `submission_package/RESEARCH_NOTE.md` | Self-contained statements, proofs, exact certificates, and limitations. |
+| `submission_package/RESULTS_LEDGER.md` | Structured inventory of all 11 results and their evidence strength. |
+| `submission_package/NOVELTY_AND_CLAIMS.md` | Prior-art positioning and precise claim boundary. |
+| `submission_package/verify_interface_certificates.py` | Standard-library exact verifier for the finite claims. |
+| `submission_package/README.md` | Reproduction instructions. |
+
+## Verification
+
+From the repository root:
+
+```bash
+python3 tracks/other/solutions/yanwang-251/submission_package/verify_interface_certificates.py
+```
+
+The verifier uses Python integers only for the terminal-signature identities
+and crossover.  Its symmetric-book regression enumerates every forest using
+integer activities.
+
+## Claim boundary
+
+This submission establishes:
+
+- an exact effective-activity reduction for positive two-terminal networks;
+- an exact nonpositivity-preserving formula for three-terminal parallel
+  composition;
+- an exact four-terminal crossover in the unrestricted positive
+  partition-signature cone;
+- a conditional single-exposed-face obstruction for a minor-minimal
+  counterexample; and
+- strict negative correlation for every edge-pair orbit on a symmetric
+  `K3 join independent(r)` activity slice.
+
+It explicitly does **not** establish:
+
+- a graph realization of the two abstract four-terminal signatures;
+- negative correlation for all disjoint edges;
+- the I-Rayleigh conjecture for all graphs; or
+- a verifier-accepted counterexample satisfying the success gate of #251.

tracks/other/solutions/yanwang-251/submission_package/NOVELTY_AND_CLAIMS.md

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+# Novelty and claim boundary
+
+## Principal contribution
+
+The principal contribution is an interface-width explanation for repeated
+failure modes in the weighted arboreal-gas negative-correlation problem.
+Two-terminal positive networks reduce to a scalar effective activity;
+three-terminal parallel composition has an exact nonpositivity-preserving
+identity; but four-terminal partition signatures admit an exact positive
+crossover.  Thus four terminals are the first boundary width at which global
+connectivity information is algebraically capable of reversing the sign.
+
+The second contribution is the exposed-face obstruction: after target
+valuations align the four sectors, every nonzero valuation block of a graphic
+independent-set face is a direct sum of graphic-minor basis measures and is
+Rayleigh.  A zero valuation block can be positive only if a smaller
+delete/contract minor is already a counterexample.  Hence a minor-minimal
+witness must be an interior, finite-scale interference effect rather than a
+single exposed monomial phase.
+
+## What is standard and is not claimed as new
+
+- Closure of I-Rayleigh matroids under direct and two-sums is known.  The
+  effective-activity formula is included because it gives a concrete no-go
+  statement for cardinality-amplifier gadgets in issue #251.
+- Real stability/Rayleighness of weighted basis measures of graphic minors is
+  standard.  The contribution is their use in the aligned-sector
+  exposed-face reduction.
+- Small-graph and series-parallel I-Rayleigh results are prior work and are
+  not counted as results of this submission.
+
+## Claims supported by complete exact arithmetic
+
+- The three-terminal composition identity is verified on a basis of the
+  five-dimensional signature space and proved algebraically in the note.
+- The two displayed four-terminal signatures are positive integer vectors,
+  are individually nonpositive for every disjoint perfect matching, and have
+  strictly positive composition gaps for all three matchings.
+- The symmetric-book theorem has an algebraic transfer proof; the bundled
+  direct forest enumerator provides an independent finite regression.
+
+## Explicit non-claims
+
+- The abstract four-terminal signatures are not claimed to be realizable by
+  positive-weight graph modules.
+- The submission does not prove a universal inequality for disjoint edges.
+- It does not provide a finite graph satisfying `Z_ef Z > Z_e Z_f` and
+  therefore does not pass the original success gate of issue #251.
+- This repository submission is not itself a claim of journal-level novelty.
+  A specialist MathSciNet/zbMATH search and expert review remain appropriate
+  before external publication.
+
+## Position relative to PR #213
+
+PR #213 by @Osgood001 establishes a universal theorem for adjacent edges via
+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 issue #251 and should be evaluated on their own claims.

tracks/other/solutions/yanwang-251/submission_package/README.md

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+# Reproducing the exact certificates
+
+The finite certificates require only Python 3.10 or newer and use no
+third-party packages.
+
+Run from the root of the `quantum.harness` checkout:
+
+```bash
+python3 tracks/other/solutions/yanwang-251/submission_package/verify_interface_certificates.py
+```
+
+The script performs three independent checks:
+
+1. It constructs the complete Bell-partition composition tensor on three
+   terminals and compares every coefficient of the resulting symbolic
+   biquadratic Rayleigh-determinant identity.
+2. It constructs the complete 15-state four-terminal composition tensor,
+   replays the two integer signatures from the research note, and checks all
+   three disjoint perfect matchings before and after composition.
+3. It directly enumerates every forest of `K3 join independent(r)` for
+   `r=1,2,3,4` at integer core/spoke activities `2,3` and checks every pair of
+   edges.  This is a regression for the closed-form proof in the note, not a
+   replacement for that proof.
+
+Expected final line:
+
+```text
+all exact interface certificates passed
+```
+
+All sign conventions are stated in `RESEARCH_NOTE.md`.  In particular,
+`R = AD - BC > 0` is the counterexample sign requested by issue #251.
+
+`RESULTS_LEDGER.md` inventories the wider exact campaign.  It labels each
+item as a theorem, complete finite proposition, or mechanism observation so
+that the fast bundled verifier is not mistaken for verification of every
+large archival census.