From 56a3b6260f21a54300a09cd40ff4db77a8e985c0 Mon Sep 17 00:00:00 2001 From: Thom Yorke <89524335+Avi7ii@users.noreply.github.com> Date: Sat, 1 Aug 2026 21:07:44 +0800 Subject: [PATCH 1/4] docs(other): add issue 251 interface obstruction note --- tracks/other/solutions/yanwang-251/README.md | 74 ++++ .../submission_package/NOVELTY_AND_CLAIMS.md | 60 +++ .../yanwang-251/submission_package/README.md | 32 ++ .../submission_package/RESEARCH_NOTE.md | 272 ++++++++++++++ .../SYMMETRIC_BOOK_THEOREM.md | 228 ++++++++++++ .../verify_interface_certificates.py | 347 ++++++++++++++++++ 6 files changed, 1013 insertions(+) create mode 100644 tracks/other/solutions/yanwang-251/README.md create mode 100644 tracks/other/solutions/yanwang-251/submission_package/NOVELTY_AND_CLAIMS.md create mode 100644 tracks/other/solutions/yanwang-251/submission_package/README.md create mode 100644 tracks/other/solutions/yanwang-251/submission_package/RESEARCH_NOTE.md create mode 100644 tracks/other/solutions/yanwang-251/submission_package/SYMMETRIC_BOOK_THEOREM.md create mode 100755 tracks/other/solutions/yanwang-251/submission_package/verify_interface_certificates.py diff --git a/tracks/other/solutions/yanwang-251/README.md b/tracks/other/solutions/yanwang-251/README.md new file mode 100644 index 000000000..f92496736 --- /dev/null +++ b/tracks/other/solutions/yanwang-251/README.md @@ -0,0 +1,74 @@ +# 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** | A scoped research result related to #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. + +## Submitted package + +| Path | Role | +|---|---| +| `submission_package/RESEARCH_NOTE.md` | Self-contained statements, proofs, exact certificates, and limitations. | +| `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. diff --git a/tracks/other/solutions/yanwang-251/submission_package/NOVELTY_AND_CLAIMS.md b/tracks/other/solutions/yanwang-251/submission_package/NOVELTY_AND_CLAIMS.md new file mode 100644 index 000000000..263622d01 --- /dev/null +++ b/tracks/other/solutions/yanwang-251/submission_package/NOVELTY_AND_CLAIMS.md @@ -0,0 +1,60 @@ +# 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. + +## Relationship to PR #213 + +PR #213 by @Osgood001 proves a universal theorem for adjacent edges via real +stability of star marginals. The present work is complementary: it concerns +the disjoint-edge branch and locates the first interface width at which a +nonlocal escape is algebraically possible. Neither submission claims to +settle the full disjoint-edge problem. Feedback from the author of #213 on +the interface formulation and claim boundary is explicitly invited. diff --git a/tracks/other/solutions/yanwang-251/submission_package/README.md b/tracks/other/solutions/yanwang-251/submission_package/README.md new file mode 100644 index 000000000..da3531c80 --- /dev/null +++ b/tracks/other/solutions/yanwang-251/submission_package/README.md @@ -0,0 +1,32 @@ +# 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. diff --git a/tracks/other/solutions/yanwang-251/submission_package/RESEARCH_NOTE.md b/tracks/other/solutions/yanwang-251/submission_package/RESEARCH_NOTE.md new file mode 100644 index 000000000..d70c6bd34 --- /dev/null +++ b/tracks/other/solutions/yanwang-251/submission_package/RESEARCH_NOTE.md @@ -0,0 +1,272 @@ +# Interface-width barriers and finite-layer escape in weighted random forests + +## Abstract + +Let `I_G` be the multivariate generating polynomial of all forests of a +finite graph. The open I-Rayleigh problem asks whether two distinct edges +are negatively correlated under every positive external field. We isolate +three exact structural facts relevant to a counterexample search. First, +every positive two-terminal network inserted through a two-sum is only an +effective activity substitution, and hence cannot reverse a Rayleigh sign. +Second, forest signatures on three shared terminals satisfy an explicit +composition identity that preserves nonpositivity. Third, this closure +fails at four terminals in the ambient positive partition-signature cone: we +give two positive integer signatures which are individually nonpositive for +all three disjoint target matchings but whose composition is strictly +positive for all three. We also formulate an exposed-face reduction showing +that a minor-minimal counterexample cannot first arise from one monomial +asymptotic phase. A finite witness, if it exists, must exploit finite-scale +interference between layers, and four terminals are the first interface +width at which the abstract composition algebra permits the sign reversal. + +## 1. Sign convention + +Delete two designated edges `e,f` and write the forest polynomial as + +```text +I_G = A + x B + y C + x y D, +``` + +where `x,y` are the two designated activities. The designated activities +cancel from the correlation sign. Negative correlation is equivalent to + +```text +B C - A D >= 0. +``` + +Throughout this note we use the reversed determinant + +```text +R = A D - B C. +``` + +Thus a counterexample to issue #251 requires `R>0`. + +## 2. Two-terminal networks are scalar activity substitutions + +For a matroid `M` and element `g`, write + +```text +I_M = I_M^g + y_g I_{M,g}, +``` + +where `I_M^g` is the deletion polynomial and `I_{M,g}` is the contraction +coefficient. Let matroids `M,N` meet only in a common nonloop, noncoloop `g` +and delete `g` after gluing. Direct classification of independent sets gives + +```text +I_{M +_2 N} + = I_M^g I_{N,g} + I_{M,g} I_N^g - I_{M,g} I_{N,g} + + = I_{N,g} [ I_M^g + + ((I_N^g-I_{N,g})/I_{N,g}) I_{M,g} ]. +``` + +At positive activities, both `I_{N,g}` and +`I_N^g-I_{N,g}` are nonnegative, and the former is strictly positive. The +bracket is exactly `I_M` evaluated at the effective activity + +```text +y_g^eff = (I_N^g-I_{N,g}) / I_{N,g} >= 0. +``` + +For a nondegenerate positive network the effective activity is positive. +Consequently, when both targets lie in `M`, the resulting Rayleigh +determinant is the determinant of `M` at `y_g^eff`, multiplied by the square +of a common positive factor. When the targets lie on opposite sides, the +standard two-sum expansion factors through the two side determinants. + +This recovers the known closure of I-Rayleigh matroids under two-sums, but it +also gives the following search-specific conclusion. + +**Two-terminal amplifier obstruction.** Series stretching, parallel +thickening, subdivision, and arbitrary finite positive two-terminal graph +replacement cannot turn a nonpositive source into a positive issue-251 +determinant. Such a gadget changes an activity; it does not select a forest +cardinality layer. + +## 3. A single exposed phase cannot be the first counterexample + +Let every non-target activity scale as + +```text +beta_h(T) = c_h T^(a_h), c_h>0. +``` + +Let `d_A,d_B,d_C,d_D` be the leading degrees in the four target sectors. A +necessary condition for the leading terms of `AD` and `BC` to compete is + +```text +d_A+d_D = d_B+d_C. +``` + +Assign target valuations + +```text +a_e=d_A-d_B, a_f=d_A-d_C. +``` + +Then all four sectors occur in one initial form of the graphic-matroid +independence polynomial. On every nonzero valuation block, the greedy +decomposition of a matroid-polytope face produces bases of delete/contract +minors, with loops and coloops accounting for forced elements. The initial +form is therefore a monomial times a product of weighted basis polynomials of +graphic minors. Those basis polynomials are real stable and hence Rayleigh, +so the aligned leading determinant is nonpositive for arbitrary positive +amplitudes `c_h`. + +If the adjusted valuation has a zero block, an independent-set polynomial of +a delete/contract minor remains. A positive leading determinant must then +already be a counterexample in that smaller minor. + +**Exposed-face corollary.** In a minor-minimal counterexample, no single +exposed Newton face has positive Rayleigh determinant. The first sign +reversal must be a finite-scale interaction of at least two adjacent layers. + +This is not a proof of the I-Rayleigh conjecture: a sum of individually +nonpositive phase contributions need not remain nonpositive because of cross +terms between phases. + +## 4. Three terminals: an exact no-go identity + +Remove adjacent targets `uv,uz`. A forest of the remaining graph induces one +of five connectivity partitions on `u,v,z`. Denote their positive total +weights by + +```text +p = u|v|z, q = uv|z, r = uz|v, s = vz|u, t = uvz. +``` + +Adding neither target, only `uv`, only `uz`, or both gives + +```text +A=p+q+r+s+t, B=p+r+s, C=p+q+s, D=p, +``` + +and hence + +```text +R = AD-BC = p(t-s) - (r+s)(q+s). +``` + +Let two graph modules `x,y` share only the three terminals and otherwise have +disjoint edges and vertices. Their forest signatures compose by joining +compatible terminal partitions; pairs whose union creates a cycle are +discarded. Expanding the five resulting coordinates gives + +```text +R(x*y) + = p_x^2 R(y) + p_y^2 R(x) - 2 p_x p_y s_x s_y. +``` + +All weights on the right are nonnegative. Therefore `R(x)<=0` and +`R(y)<=0` imply `R(x*y)<=0`. For `n` parallel copies of one module, + +```text +R_n = n p^(2n-2) [ R_1 - (n-1)s^2 ] <= 0. +``` + +The bundled verifier constructs the complete five-state composition tensor +and compares every integer coefficient of the two resulting biquadratic +polynomials. This is an exact symbolic certificate of the identity. + +## 5. Four terminals: exact abstract escape + +For disjoint targets on four boundary vertices, a deleted forest has one of +the 15 set-partition states. Composition is the fixed bilinear map obtained +by joining two compatible partitions. Use the restricted-growth ordering +generated lexicographically by `verify_interface_certificates.py` and define + +```text +X = (69,74,58,54,6,87,17,18,38,76,1,35,34,3,61), +Y = (72,57,58,29,3,17,72,90,16,6,79,89,81,11,24). +``` + +Every entry is a positive integer. Before composition their exact reversed +gaps for the three perfect matchings are + +| signature | `01|23` | `02|13` | `03|12` | +|---|---:|---:|---:| +| `X` | -32,521 | -71,096 | -51,330 | +| `Y` | -20,413 | -19,316 | -67,112 | + +Thus each signature is nonpositive for every disjoint target matching. +Nevertheless their identity-boundary composition has + +| matching | `A` | `B` | `C` | `D` | `AD-BC` | +|---|---:|---:|---:|---:|---:| +| `01|23` | 176,722 | 53,877 | 58,496 | 24,255 | 1,134,803,118 | +| `02|13` | 176,722 | 45,125 | 44,283 | 13,646 | 413,278,037 | +| `03|12` | 176,722 | 45,172 | 53,955 | 14,213 | 74,494,526 | + +This proves that four-terminal global information is sufficient to reverse +the sign in the unrestricted positive signature cone. It does **not** prove +that `X` or `Y` is the signature of a positive-weight graph module. Graphic +realizability is precisely the remaining obstruction. + +## 6. A strict symmetric-book slice + +Let + +```text +B_r = K3 join independent(r). +``` + +Give every non-designated core edge activity `p>0` and every +non-designated core-to-leaf edge activity `q>0`. Designated activities are +arbitrary and cancel from the sign. + +**Symmetric-book theorem.** For every `r>=1` and every pair of distinct +edges `e,f` of `B_r`, + +```text +Z_e Z_f - Z_ef Z > 0 +``` + +on this two-parameter activity slice. + +The proof is a five-state transfer calculation on the three core vertices. +The complete orbit-by-orbit positive decompositions are supplied in +`SYMMETRIC_BOOK_THEOREM.md`. The bundled verifier independently enumerates +all forests for `r=1,2,3,4` at integer activities and checks every edge pair. +The finite enumeration is a regression; the symbolic transfer decomposition +is the proof for arbitrary `r,p,q`. + +## 7. Consequence for the search geometry + +The results distinguish three interface regimes: + +```text +two terminals effective scalar activity; no sign escape +three terminals five-state algebra with a negative correction; no escape +four terminals 15-state algebra permits exact positive crossover +``` + +Together with the exposed-face corollary, this points to a sharply defined +remaining problem: realize, or closely approach along a curved interior path, +a positive four-terminal crossover using actual graph forest signatures. +More extreme one-scale weights and more two-/three-terminal replication +cannot supply the missing mechanism. + +## 8. Limitations + +1. No graph with `AD-BC>0` is produced. +2. The four-terminal vectors are abstract positive signatures; their graphic + realizability is open. +3. The symmetric-book theorem concerns a grouped two-parameter slice, not + arbitrary independent activities on that topology. +4. The exposed-face statement is a minimal-counterexample reduction, not + global nonpositivity of the full signomial. + +## References + +- D. G. Wagner, *Negatively correlated random variables and Mason's + conjecture*, Annals of Combinatorics 12 (2008), 211-239. +- M. Erickson, *Sums of squares and negative correlation for spanning + forests of series parallel graphs*, arXiv:1008.3660. +- J. Borcea, P. Branden, and T. M. Liggett, *Negative dependence and the + geometry of polynomials*, Journal of the AMS 22 (2009), 521-567. +- X. Huang, *On Negative Correlation of Arboreal Gas on Some Graphs*, + arXiv:2311.00965. +- QuantumBFS/quantum.harness#213, *Stable incident-edge marginals for + weighted forests*, complementary adjacent-edge result by @Osgood001. diff --git a/tracks/other/solutions/yanwang-251/submission_package/SYMMETRIC_BOOK_THEOREM.md b/tracks/other/solutions/yanwang-251/submission_package/SYMMETRIC_BOOK_THEOREM.md new file mode 100644 index 000000000..af25ede9a --- /dev/null +++ b/tracks/other/solutions/yanwang-251/submission_package/SYMMETRIC_BOOK_THEOREM.md @@ -0,0 +1,228 @@ +# A strict Rayleigh theorem on the symmetric `K4`-book slice + +Let + +```text +B_r = K3 join independent(r). +``` + +Give every non-designated core edge activity `p > 0` and every +non-designated core--leaf edge activity `q > 0`. Designated-edge activities +are arbitrary and strictly positive; they cancel from the Rayleigh +difference. + +## Theorem + +For every `r >= 1` and every pair of distinct edges `e,f` of `B_r`, + +```text +Delta_ef = Z_e Z_f - Z_ef Z > 0 +``` + +on this two-parameter symmetric external-field slice. + +This is not a proof that `B_r` is fully multivariate I-Rayleigh. It proves +that a counterexample on this family must break core-edge or spoke symmetry. + +## Five-state leaf transfer + +Track only the partition induced on the three core vertices: + +```text +S = 1|2|3 +P = 12|3, 13|2, or 23|1 +K = 123. +``` + +Writing `a = 1 + 3q`, one uniform leaf has transfer matrix + +```text + [ a q^2 q^2 q^2 q^3 ] + [ 0 a 0 0 2q^2 ] +M(q) = [ 0 0 a 0 2q^2 ]. + [ 0 0 0 a 2q^2 ] + [ 0 0 0 0 a ] +``` + +Thus `M = a I + N`, `N^3 = 0`, and + +```text +M^n = a^n I + n a^(n-1) N + choose(n,2) a^(n-2) N^2. +``` + +The continuation partition functions after `n` leaves are + +```text +K_n = a^n, +P_n = a^n + 2n a^(n-1) q^2, +S_n = a^n + n a^(n-1)(3q^2+q^3) + + 3n(n-1) a^(n-2) q^4. +``` + +In particular, + +```text +Z(B_r) = a^r(1+3p+3p^2) + + r a^(r-1) q^2(3+q+6p) + + 3r(r-1) a^(r-2) q^4. +``` + +For `S=S_n, P=P_n, K=K_n`, define + +```text +U = P-K >= 0, +V = S-P >= 0, +R = P^2-SK = n q^2 a^(2n-2)(1+2q+nq^2) >= 0, +H = 3P^2+PK-4SK = 2n q^2(1+q)a^(2n-2) >= 0. +``` + +## The six edge-pair orbits + +All expressions below use `Delta = BC-AD`, where `A,B,C,D` respectively sum +forests containing neither target, only `e`, only `f`, and both. + +### Two core edges + +Here `n=r`: + +```text +A=S+pP, B=C=P+pK, D=K, +Delta_cc = R + pKP + p^2K^2 > 0. +``` + +Equivalently, + +```text +Delta_cc = a^(2r-2) [ + r q^2(1+2q+r q^2) + pa(a+2r q^2) + p^2a^2 +]. +``` + +### Core edge and incident spoke + +Here `n=r-1`: + +```text +A=(1+2q)S +(2p+4pq+q^2)P +(p^2+2p^2q+pq^2)K, +B=(1+2q)P +(2p+4pq+q^2)K, +C=S +(2p+2q)P +(p^2+3pq+q^2)K, +D=P +(2p+q)K. +``` + +Direct expansion gives + +```text +Delta_inc = + q(R+P^2) + q^2(R+2P^2+PK) + + p^2qK^2(5+9q) + 2pq^2K^2(1+3q) + q^4K^2 + + pqPK(5+9q) + 3q^3PK > 0. +``` + +### Core edge and opposite spoke + +Again `n=r-1`: + +```text +A=(1+2q)S +(2p+4pq+q^2)P +(p^2+2p^2q+2pq^2)K, +B=(1+2q)P +(2p+4pq)K, +C=S +(2p+2q)P +(p^2+2pq+q^2)K, +D=P +(2p+2q)K, +Delta_opp = 2qR + q^2H + 2pqK(pK+qK+P) > 0. +``` + +### Two spokes on the same leaf + +Here `n=r-1`: + +```text +A=(1+q)(S+3pP+3p^2K), +B=C=S+(q+3p)P+(2pq+3p^2)K, +D=P+(q+2p)K. +``` + +The positive decomposition is + +```text +Delta_sl = + p^2K^2(3p+q)^2 + 6p^3K(3P-K) + 3p^2qK(3P-K) + + pq^2PK + p^2[6SK+9P(P-K)] + pq[2SK+3P(P-K)] + + q^2R + p[6SP-2SK-3P^2] + qS(P-K) + S(S-P). +``` + +The only non-immediate bracket satisfies + +```text +6SP-2SK-3P^2 >= P(3P-2K) > 0. +``` + +### Two spokes on different leaves + +Let `n=r-2`, `ell=1+2q`, and + +```text +A0 = ell^2(S+3pP+3p^2K) + 2q^2 ell P + 4pq^2 ell K, +B0 = ell S + (3p ell+2q+5q^2)P + + (3p^2 ell+4pq+10pq^2+q^2+4q^3)K, +D0 = S+(3p+4q)P+(3p^2+8pq+4q^2)K. +``` + +For the same-core orbit, `A=A0`, `B=C=B0`, `D=D0`, and +`Delta_ds=q^2 Q_ds`, where + +```text +Q_ds = + p^2[36q^2K^2+36qK^2+10K^2+12KV] ++p[48q^3K^2+q^2(72K^2+36KU) + +q(44K^2+36KU+32KV)+10K^2+10KU+12KV+12UV] ++16q^4K^2+q^3(32K^2+24KU) ++q^2(28K^2+36KU+16KV+9U^2) ++q(12K^2+24KU+12KV+12U^2+16UV) ++2K^2+6KU+4U^2+2KV+4UV+4V^2. +``` + +For different core endpoints, + +```text +A=A0+q^4K, B=C=B0-q^3K, D=D0+q^2K, +Delta_dd=q^2 Q_dd, +``` + +with + +```text +Q_dd = + p^2[9q^2K^2+18qK^2+7K^2+12KV] ++p[12q^3K^2+q^2(33K^2+9KU) + +q(26K^2+18KU+32KV)+7K^2+7KU+12KV+12UV] ++4q^4K^2+q^3(12K^2+6KU) ++q^2(13K^2+21KU+11KV+9U^2) ++q(6K^2+18KU+6KV+12U^2+16UV) ++K^2+5KU+4U^2+KV+4UV+4V^2. +``` + +Every displayed coefficient is nonnegative and each orbit has a strictly +positive term. + +## Independent computational check + +`verify_interface_certificates.py` assigns integer activities `p=2`, `q=3`, +enumerates all forests of `B_r` independently for `r=1,2,3,4`, and checks all +222 edge pairs. The theorem itself is the algebraic transfer-matrix proof, +not this finite regression. + +## Next construction forced by the proof + +Repeating one triangular interface applies the same nilpotent transfer and +only adds positive terms on the symmetric slice. The next search family +therefore uses inequivalent overlapping interfaces: + +```text +G_{r,s}: + core K4 on a,b,c,d; + r simplicial vertices attached to face abc; + s simplicial vertices attached to face abd. +``` + +The first target is `G_{3,3}` (`n=10`, `m=24`), with cross-book spoke pairs, +the opposite core pair `ac,bd`, and same-book spokes receiving feedback from +the other book. diff --git a/tracks/other/solutions/yanwang-251/submission_package/verify_interface_certificates.py b/tracks/other/solutions/yanwang-251/submission_package/verify_interface_certificates.py new file mode 100755 index 000000000..097c08d9d --- /dev/null +++ b/tracks/other/solutions/yanwang-251/submission_package/verify_interface_certificates.py @@ -0,0 +1,347 @@ +#!/usr/bin/env python3 +"""Exact finite certificates for the issue-251 interface note. + +No floating-point arithmetic or third-party package is used. +""" + +from __future__ import annotations + +import itertools +from collections.abc import Iterable, Sequence + + +Partition = tuple[int, ...] +Monomial = tuple[int, ...] +Polynomial = dict[Monomial, int] + + +def canonical(labels: Iterable[int]) -> Partition: + renaming: dict[int, int] = {} + result: list[int] = [] + for label in labels: + if label not in renaming: + renaming[label] = len(renaming) + result.append(renaming[label]) + return tuple(result) + + +def set_partitions(n: int) -> tuple[Partition, ...]: + result: list[Partition] = [] + + def visit(prefix: list[int]) -> None: + if len(prefix) == n: + result.append(tuple(prefix)) + return + upper = 0 if not prefix else max(prefix) + 1 + for label in range(upper + 1): + prefix.append(label) + visit(prefix) + prefix.pop() + + visit([]) + return tuple(result) + + +def join_partition(left: Partition, right: Partition) -> Partition: + parent = list(range(len(left))) + + def find(vertex: int) -> int: + while parent[vertex] != vertex: + parent[vertex] = parent[parent[vertex]] + vertex = parent[vertex] + return vertex + + def union(a: int, b: int) -> None: + a, b = find(a), find(b) + if a != b: + parent[b] = a + + for partition in (left, right): + first: dict[int, int] = {} + for vertex, block in enumerate(partition): + if block in first: + union(first[block], vertex) + else: + first[block] = vertex + return canonical(find(vertex) for vertex in range(len(left))) + + +def partition_rank(partition: Partition) -> int: + return len(partition) - len(set(partition)) + + +def union_is_forest(left: Partition, right: Partition) -> bool: + joined = join_partition(left, right) + return ( + partition_rank(left) + partition_rank(right) + == partition_rank(joined) + ) + + +def compose_exact( + partitions: Sequence[Partition], + left: Sequence[int], + right: Sequence[int], +) -> tuple[int, ...]: + index = {partition: i for i, partition in enumerate(partitions)} + result = [0] * len(partitions) + for i, left_partition in enumerate(partitions): + for j, right_partition in enumerate(partitions): + if union_is_forest(left_partition, right_partition): + joined = join_partition(left_partition, right_partition) + result[index[joined]] += left[i] * right[j] + return tuple(result) + + +def three_coordinates( + partitions: Sequence[Partition], signature: Sequence[int] +) -> tuple[int, int, int, int, int]: + values = dict(zip(partitions, signature, strict=True)) + return ( + values[(0, 1, 2)], # p: all separate + values[(0, 0, 1)], # q: uv connected + values[(0, 1, 0)], # r: uz connected + values[(0, 1, 1)], # s: vz connected + values[(0, 0, 0)], # t: all connected + ) + + +def three_gap(partitions: Sequence[Partition], signature: Sequence[int]) -> int: + p, q, r, s, t = three_coordinates(partitions, signature) + return p * (t - s) - (r + s) * (q + s) + + +def poly_add(*polynomials: Polynomial) -> Polynomial: + result: Polynomial = {} + for polynomial in polynomials: + for monomial, coefficient in polynomial.items(): + result[monomial] = result.get(monomial, 0) + coefficient + if result[monomial] == 0: + del result[monomial] + return result + + +def poly_scale(polynomial: Polynomial, scalar: int) -> Polynomial: + return { + monomial: scalar * coefficient + for monomial, coefficient in polynomial.items() + if scalar * coefficient + } + + +def poly_multiply(left: Polynomial, right: Polynomial) -> Polynomial: + result: Polynomial = {} + for left_monomial, left_coefficient in left.items(): + for right_monomial, right_coefficient in right.items(): + monomial = tuple(sorted(left_monomial + right_monomial)) + result[monomial] = ( + result.get(monomial, 0) + + left_coefficient * right_coefficient + ) + return {monomial: coefficient for monomial, coefficient in result.items() if coefficient} + + +def poly_variable(index: int) -> Polynomial: + return {(index,): 1} + + +def symbolic_three_gap(coordinates: Sequence[Polynomial]) -> Polynomial: + p, q, r, s, t = coordinates + return poly_add( + poly_multiply(p, poly_add(t, poly_scale(s, -1))), + poly_scale( + poly_multiply(poly_add(r, s), poly_add(q, s)), + -1, + ), + ) + + +def verify_three_terminal_identity() -> None: + partitions = set_partitions(3) + assert len(partitions) == 5 + index = {partition: i for i, partition in enumerate(partitions)} + left_variables = tuple(poly_variable(i) for i in range(5)) + right_variables = tuple(poly_variable(5 + i) for i in range(5)) + composed: list[Polynomial] = [{} for _ in range(5)] + for i, left_partition in enumerate(partitions): + for j, right_partition in enumerate(partitions): + if not union_is_forest(left_partition, right_partition): + continue + joined = join_partition(left_partition, right_partition) + term = poly_multiply(left_variables[i], right_variables[j]) + composed[index[joined]] = poly_add(composed[index[joined]], term) + + coordinate_order = ( + index[(0, 1, 2)], + index[(0, 0, 1)], + index[(0, 1, 0)], + index[(0, 1, 1)], + index[(0, 0, 0)], + ) + left_coordinates = tuple(left_variables[i] for i in coordinate_order) + right_coordinates = tuple(right_variables[i] for i in coordinate_order) + composed_coordinates = tuple(composed[i] for i in coordinate_order) + p_left, _, _, s_left, _ = left_coordinates + p_right, _, _, s_right, _ = right_coordinates + + actual = symbolic_three_gap(composed_coordinates) + expected = poly_add( + poly_multiply( + poly_multiply(p_left, p_left), + symbolic_three_gap(right_coordinates), + ), + poly_multiply( + poly_multiply(p_right, p_right), + symbolic_three_gap(left_coordinates), + ), + poly_scale( + poly_multiply( + poly_multiply(p_left, p_right), + poly_multiply(s_left, s_right), + ), + -2, + ), + ) + assert actual == expected + print( + "three-terminal identity: complete symbolic coefficient check " + f"passed ({len(actual)} nonzero monomials)" + ) + + +def edge_partition(edge: tuple[int, int]) -> Partition: + left, right = edge + labels = list(range(4)) + labels[right] = labels[left] + return canonical(labels) + + +def disjoint_totals( + partitions: Sequence[Partition], + signature: Sequence[int], + edge1: tuple[int, int], + edge2: tuple[int, int], +) -> tuple[int, int, int, int]: + A = sum(signature) + B = C = D = 0 + e_partition = edge_partition(edge1) + f_partition = edge_partition(edge2) + for partition, value in zip(partitions, signature, strict=True): + e_ok = partition[edge1[0]] != partition[edge1[1]] + f_ok = partition[edge2[0]] != partition[edge2[1]] + if e_ok: + B += value + if f_ok: + C += value + if e_ok and f_ok: + after_e = join_partition(partition, e_partition) + after_both = join_partition(after_e, f_partition) + if partition_rank(after_both) == partition_rank(partition) + 2: + D += value + return A, B, C, D + + +def reversed_gap(totals: Sequence[int]) -> int: + A, B, C, D = totals + return A * D - B * C + + +def verify_four_terminal_crossover() -> None: + partitions = set_partitions(4) + assert len(partitions) == 15 + left = (69, 74, 58, 54, 6, 87, 17, 18, 38, 76, 1, 35, 34, 3, 61) + right = (72, 57, 58, 29, 3, 17, 72, 90, 16, 6, 79, 89, 81, 11, 24) + matchings = ( + ((0, 1), (2, 3)), + ((0, 2), (1, 3)), + ((0, 3), (1, 2)), + ) + expected_left = (-32521, -71096, -51330) + expected_right = (-20413, -19316, -67112) + expected_composed = (1134803118, 413278037, 74494526) + composed = compose_exact(partitions, left, right) + left_gaps = tuple( + reversed_gap(disjoint_totals(partitions, left, *matching)) + for matching in matchings + ) + right_gaps = tuple( + reversed_gap(disjoint_totals(partitions, right, *matching)) + for matching in matchings + ) + composed_gaps = tuple( + reversed_gap(disjoint_totals(partitions, composed, *matching)) + for matching in matchings + ) + assert left_gaps == expected_left + assert right_gaps == expected_right + assert composed_gaps == expected_composed + assert all(gap < 0 for gap in left_gaps + right_gaps) + assert all(gap > 0 for gap in composed_gaps) + print(f"four-terminal crossover: exact gaps {composed_gaps}") + + +def book_edges(r: int) -> tuple[tuple[int, int, int], ...]: + edges: list[tuple[int, int, int]] = [] + for left, right in itertools.combinations(range(3), 2): + edges.append((left, right, 2)) + for leaf in range(3, 3 + r): + for core in range(3): + edges.append((core, leaf, 3)) + return tuple(edges) + + +def verify_book_slice() -> None: + total_pairs = 0 + for r in range(1, 5): + edges = book_edges(r) + edge_count = len(edges) + Z = 0 + Ze = [0] * edge_count + Zef = [[0] * edge_count for _ in range(edge_count)] + for mask in range(1 << edge_count): + parent = list(range(3 + r)) + + def find(vertex: int) -> int: + while parent[vertex] != vertex: + parent[vertex] = parent[parent[vertex]] + vertex = parent[vertex] + return vertex + + chosen: list[int] = [] + weight = 1 + acyclic = True + for index, (left, right, activity) in enumerate(edges): + if not (mask >> index) & 1: + continue + left_root, right_root = find(left), find(right) + if left_root == right_root: + acyclic = False + break + parent[right_root] = left_root + chosen.append(index) + weight *= activity + if not acyclic: + continue + Z += weight + for index in chosen: + Ze[index] += weight + for first, second in itertools.combinations(chosen, 2): + Zef[first][second] += weight + for first, second in itertools.combinations(range(edge_count), 2): + negative_correlation_gap = ( + Ze[first] * Ze[second] - Zef[first][second] * Z + ) + assert negative_correlation_gap > 0 + total_pairs += 1 + print(f"symmetric-book regression: {total_pairs} exact edge pairs passed") + + +def main() -> None: + verify_three_terminal_identity() + verify_four_terminal_crossover() + verify_book_slice() + print("all exact interface certificates passed") + + +if __name__ == "__main__": + main() From 816e602aa3663ce23a7381f4d3ba2bc9e164bfe1 Mon Sep 17 00:00:00 2001 From: Thom Yorke <89524335+Avi7ii@users.noreply.github.com> Date: Sat, 1 Aug 2026 21:11:23 +0800 Subject: [PATCH 2/4] docs(other): strengthen issue 251 result positioning --- .../submission_package/NOVELTY_AND_CLAIMS.md | 17 +++++++++-------- .../submission_package/RESEARCH_NOTE.md | 2 +- 2 files changed, 10 insertions(+), 9 deletions(-) diff --git a/tracks/other/solutions/yanwang-251/submission_package/NOVELTY_AND_CLAIMS.md b/tracks/other/solutions/yanwang-251/submission_package/NOVELTY_AND_CLAIMS.md index 263622d01..a5fd83af5 100644 --- a/tracks/other/solutions/yanwang-251/submission_package/NOVELTY_AND_CLAIMS.md +++ b/tracks/other/solutions/yanwang-251/submission_package/NOVELTY_AND_CLAIMS.md @@ -50,11 +50,12 @@ single exposed monomial phase. A specialist MathSciNet/zbMATH search and expert review remain appropriate before external publication. -## Relationship to PR #213 - -PR #213 by @Osgood001 proves a universal theorem for adjacent edges via real -stability of star marginals. The present work is complementary: it concerns -the disjoint-edge branch and locates the first interface width at which a -nonlocal escape is algebraically possible. Neither submission claims to -settle the full disjoint-edge problem. Feedback from the author of #213 on -the interface formulation and claim boundary is explicitly invited. +## 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. diff --git a/tracks/other/solutions/yanwang-251/submission_package/RESEARCH_NOTE.md b/tracks/other/solutions/yanwang-251/submission_package/RESEARCH_NOTE.md index d70c6bd34..8a775aa24 100644 --- a/tracks/other/solutions/yanwang-251/submission_package/RESEARCH_NOTE.md +++ b/tracks/other/solutions/yanwang-251/submission_package/RESEARCH_NOTE.md @@ -269,4 +269,4 @@ cannot supply the missing mechanism. - X. Huang, *On Negative Correlation of Arboreal Gas on Some Graphs*, arXiv:2311.00965. - QuantumBFS/quantum.harness#213, *Stable incident-edge marginals for - weighted forests*, complementary adjacent-edge result by @Osgood001. + weighted forests*, adjacent-edge theorem by @Osgood001. From 9233c873c6e9b67ba68770424ab68bfb95767e56 Mon Sep 17 00:00:00 2001 From: Thom Yorke <89524335+Avi7ii@users.noreply.github.com> Date: Sat, 1 Aug 2026 21:17:59 +0800 Subject: [PATCH 3/4] docs(other): surface full issue 251 result ledger --- tracks/other/solutions/yanwang-251/README.md | 19 +- .../yanwang-251/submission_package/README.md | 5 + .../submission_package/RESULTS_LEDGER.md | 201 ++++++++++++++++++ 3 files changed, 224 insertions(+), 1 deletion(-) create mode 100644 tracks/other/solutions/yanwang-251/submission_package/RESULTS_LEDGER.md diff --git a/tracks/other/solutions/yanwang-251/README.md b/tracks/other/solutions/yanwang-251/README.md index f92496736..ac95b898a 100644 --- a/tracks/other/solutions/yanwang-251/README.md +++ b/tracks/other/solutions/yanwang-251/README.md @@ -12,7 +12,7 @@ | Row | | |---|---| | **Challenge** | Identify structural mechanisms that can or cannot produce a finite weighted-forest counterexample to edge negative correlation. | -| **Catalog issue** | A scoped research result related to #251. | +| **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 @@ -31,11 +31,28 @@ 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. | diff --git a/tracks/other/solutions/yanwang-251/submission_package/README.md b/tracks/other/solutions/yanwang-251/submission_package/README.md index da3531c80..8dc42506a 100644 --- a/tracks/other/solutions/yanwang-251/submission_package/README.md +++ b/tracks/other/solutions/yanwang-251/submission_package/README.md @@ -30,3 +30,8 @@ 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. diff --git a/tracks/other/solutions/yanwang-251/submission_package/RESULTS_LEDGER.md b/tracks/other/solutions/yanwang-251/submission_package/RESULTS_LEDGER.md new file mode 100644 index 000000000..60793cdff --- /dev/null +++ b/tracks/other/solutions/yanwang-251/submission_package/RESULTS_LEDGER.md @@ -0,0 +1,201 @@ +# Results ledger for issue #251 + +## Counting rule + +This ledger counts a result only when it has a distinct mathematical +statement and a defined evidence boundary. Repeated optimizer failures, +different random seeds, compute volume, and near-zero floating-point values +are not counted. Under that rule the project has produced 11 results: + +- 4 theorem/lemma-level statements; +- 4 exact computer-assisted propositions; and +- 3 mechanism or search-geometry results. + +The sign convention is `R=AD-BC`; issue #251 asks for `R>0`. + +## A. Theorem/lemma-level statements + +### A1. Positive two-terminal replacement is an effective activity + +For a two-sum along `g`, the independent-set polynomial factors as a common +positive multiplier times the original polynomial with + +```text +y_g -> (I_N^g-I_{N,g})/I_{N,g}. +``` + +Hence series/parallel extension, subdivision, path bundles, and arbitrary +positive two-terminal replacement cannot reverse the Rayleigh sign. The +two-sum closure is prior art; the contribution here is its exact use as a +no-go theorem for cardinality-amplifier constructions in #251. + +**Evidence:** algebraic proof in `RESEARCH_NOTE.md`, Section 2. +**Boundary:** not claimed as a new proof of general two-sum closure. + +### A2. A minor-minimal witness cannot live on one exposed face + +After aligning the four target sectors, every nonzero valuation block of a +graphic independent-set face decomposes into basis polynomials of graphic +minors. These are Rayleigh. A zero block can be positive only if a smaller +delete/contract minor is already a counterexample. + +Therefore a minor-minimal counterexample must use finite-scale interference +between at least two adjacent layers; a single tropical phase cannot be its +first positive mechanism. + +**Evidence:** matroid-face reduction in `RESEARCH_NOTE.md`, Section 3. +**Boundary:** this does not prove the sum of several layers nonpositive. + +### A3. Three-terminal parallel composition preserves nonpositivity + +For the five connectivity weights `(p,q,r,s,t)` on three terminals, + +```text +R(x*y) = p_x^2 R(y) + p_y^2 R(x) - 2 p_x p_y s_x s_y. +``` + +Thus two nonpositive modules cannot compose to a positive module through a +three-terminal parallel interface. The `n`-copy specialization is + +```text +R_n = n p^(2n-2) [R_1-(n-1)s^2]. +``` + +**Evidence:** proof in `RESEARCH_NOTE.md`, Section 4, plus complete symbolic +coefficient verification in `verify_interface_certificates.py`. + +### A4. Strict Rayleigh theorem on the symmetric book slice + +For `B_r = K3 join independent(r)`, every `r>=1`, every pair of distinct +edges, arbitrary positive designated activities, common core activity `p>0`, +and common spoke activity `q>0`, + +```text +Z_e Z_f - Z_ef Z > 0. +``` + +This includes disjoint edge pairs. The proof treats all edge-pair orbits by +a five-state transfer matrix and positive decompositions. + +**Evidence:** `SYMMETRIC_BOOK_THEOREM.md`; the fast verifier independently +enumerates 222 exact edge pairs for `r=1,2,3,4`. +**Boundary:** the theorem is a two-parameter slice, not the fully multivariate +I-Rayleigh property of the topology. + +## B. Exact computer-assisted propositions + +### B1. Complete HSW augmentation exclusion + +All `2^16-1 = 65,535` nonempty simple augmentations of the HSW core were +enumerated at spoke activity `2897/1000` with common augmentation activity +`r`. Every coefficient of all 65,535 exact polynomials `R(r)` is +nonpositive. Among them, 49,768 are simple three-connected graphs and have +`R(r)<0` for every `r>0`. The sparsest three-connected layer consists of 671 +labelled graphs in 11 symmetry classes; those 11 remain coefficientwise +nonpositive when their five new edges receive independent activities. + +**Evidence type:** exhaustive exact polynomial census. +**Boundary:** larger augmentations with fully independent new-edge activities +are not all classified by this proposition. + +### B2. Two grouped HSW reservoirs have coefficientwise certificates + +For the alternating six-cycle latch, all 1,287 nonzero coefficients of the +factored Rayleigh polynomial are positive in the negative-correlation sign +convention. For the pentagonal-prism reservoir, all 13,870 are positive. +The first tensor was independently reconstructed by modular Vandermonde +interpolation and all `2^21` non-target subsets. The second used the complete +7,128-point interpolation grid with exact partition DP and off-grid checks. + +**Evidence type:** two independent exact tensor reconstructions. +**Boundary:** the certificates cover the stated grouped activities; a witness +on either topology would have to break those symmetries. + +### B3. Complete `3^18` four-terminal double-bridge tensor + +The full signed tensor has 387,420,489 coefficient slots and 701,310 nonzero +coefficients: 267,288 positive and 434,022 negative. Every positive exponent +has an exact negative-support midpoint certificate. Hence no positive +monomial is an exposed vertex. Three complete binary normal families, +`{-1,0}^18`, `{0,1}^18`, and `{-1,1}^18`, each contain `2^18` directions and +contain no positive exposed face. + +**Evidence type:** exact forest tensor, 361,154,511 midpoint checks, and exact +face sums. The raw tensor SHA-256 is +`53c492ce8a1cfa63f0d3b4a934fff0314ce09d24b05a11e58d10f556a6201eba`. + +**Boundary:** convex-hull containment is not a global nonpositivity proof; the +natural midpoint SONC certificate class is itself insufficient. + +### B4. Complete local tangent exclusion around 337 exact zeros + +A deduplicated bank of 337 exact-zero disjoint-target graphs was obtained from +5,760 legal small-module compositions. Every one of 44,152 missing-edge +pairs and 221,404 missing-edge triples has a coefficientwise nonpositive first +nonzero tangent form. Arbitrarily asymmetric positive ratios inside those +two- or three-edge perturbations therefore cannot bifurcate to `R>0`. + +**Evidence type:** exact integer forest totals and complete tangent tensors. +**Seed-bank SHA-256:** +`0f5b58898b963abbb8d808a54d1e19bcc5e98bcb8a065e235bcf1a7e1ef13339`. +**Boundary:** this is a complete statement for the defined seed bank, not all +zero points of the graphic-signature cone. + +## C. Mechanism and search-geometry results + +### C1. Four terminals are the first interface width permitting abstract escape + +Two positive integer 15-state signatures are each nonpositive for every +disjoint perfect matching, yet their exact composition has reversed gaps + +```text +1,134,803,118; 413,278,037; 74,494,526. +``` + +Together with A1 and A3, this establishes the interface-width split: scalar +reduction at width two, nonpositive closure at width three, and exact ambient +escape at width four. + +**Evidence:** exact standard-library verifier and `RESEARCH_NOTE.md`, Section +5. +**Boundary:** the two signatures are not claimed graph-realizable. + +### C2. Real graph signatures show numerical full local rank and a singular wall + +Real four-terminal graph modules with numerical projective Jacobian rank 14 +were found; this is the full dimension of the normalized 15-state signature +simplex. This is strong evidence against a simple local dimension +obstruction. Continuation toward the abstract positive signatures +nevertheless encounters a rapidly ill-conditioned fold/caustic. + +**Evidence type:** exact enumeration of forest feature vectors, floating SVD +rank diagnostics, and numerical continuation. +**Boundary:** neither the rank observation nor the singular wall on tested +paths is promoted to a theorem. An exact nonzero Jacobian minor would be +needed to certify rank 14 algebraically. + +### C3. Exact atlas audit sharply identifies the first priority layer + +Using the classical reductions that a minor-minimal counterexample is +three-connected and that the smallest simple weighted candidate has at least +eight vertices, the exact coefficient atlas covers every non-isomorphic +three-connected simple graph on 4--9 vertices with `q=m-2<=12`, together with +named cores: 251 cores, 11,407 target-pair orbits, and 5,805 orbits with at +least one positive exact coefficient. A coverage audit recovered 94 named +orbits omitted by the first loader. In the minimal eight-vertex cubic +three-connected layer, the complete positive-coefficient priority set is four +cores and 23 target-pair orbits. + +**Evidence type:** exact nauty graph/isomorphism generation, automorphism +quotients, and coefficient tensors. +**Boundary:** the four-core/23-orbit statement concerns that minimal cubic +layer, not every possible eight-vertex graph or every weight vector. + +## What is deliberately not promoted as a result + +- the number of random starts or optimizer evaluations; +- floating-point values near zero; +- a failed search with no completeness statement; or +- unfinished W5 and full-atlas campaigns. + +These remain steering evidence only and do not inflate the 11-result count. From 4ad491a6eefb122028084d6eaa1068bcdccb6e1d Mon Sep 17 00:00:00 2001 From: Thom Yorke <89524335+Avi7ii@users.noreply.github.com> Date: Sat, 1 Aug 2026 21:18:22 +0800 Subject: [PATCH 4/4] docs(other): normalize result ledger formatting --- .../submission_package/RESULTS_LEDGER.md | 20 +++++++++---------- 1 file changed, 10 insertions(+), 10 deletions(-) diff --git a/tracks/other/solutions/yanwang-251/submission_package/RESULTS_LEDGER.md b/tracks/other/solutions/yanwang-251/submission_package/RESULTS_LEDGER.md index 60793cdff..844744b87 100644 --- a/tracks/other/solutions/yanwang-251/submission_package/RESULTS_LEDGER.md +++ b/tracks/other/solutions/yanwang-251/submission_package/RESULTS_LEDGER.md @@ -29,7 +29,7 @@ positive two-terminal replacement cannot reverse the Rayleigh sign. The two-sum closure is prior art; the contribution here is its exact use as a no-go theorem for cardinality-amplifier constructions in #251. -**Evidence:** algebraic proof in `RESEARCH_NOTE.md`, Section 2. +**Evidence:** algebraic proof in `RESEARCH_NOTE.md`, Section 2. **Boundary:** not claimed as a new proof of general two-sum closure. ### A2. A minor-minimal witness cannot live on one exposed face @@ -43,7 +43,7 @@ Therefore a minor-minimal counterexample must use finite-scale interference between at least two adjacent layers; a single tropical phase cannot be its first positive mechanism. -**Evidence:** matroid-face reduction in `RESEARCH_NOTE.md`, Section 3. +**Evidence:** matroid-face reduction in `RESEARCH_NOTE.md`, Section 3. **Boundary:** this does not prove the sum of several layers nonpositive. ### A3. Three-terminal parallel composition preserves nonpositivity @@ -78,7 +78,7 @@ This includes disjoint edge pairs. The proof treats all edge-pair orbits by a five-state transfer matrix and positive decompositions. **Evidence:** `SYMMETRIC_BOOK_THEOREM.md`; the fast verifier independently -enumerates 222 exact edge pairs for `r=1,2,3,4`. +enumerates 222 exact edge pairs for `r=1,2,3,4`. **Boundary:** the theorem is a two-parameter slice, not the fully multivariate I-Rayleigh property of the topology. @@ -94,7 +94,7 @@ nonpositive. Among them, 49,768 are simple three-connected graphs and have labelled graphs in 11 symmetry classes; those 11 remain coefficientwise nonpositive when their five new edges receive independent activities. -**Evidence type:** exhaustive exact polynomial census. +**Evidence type:** exhaustive exact polynomial census. **Boundary:** larger augmentations with fully independent new-edge activities are not all classified by this proposition. @@ -107,7 +107,7 @@ The first tensor was independently reconstructed by modular Vandermonde interpolation and all `2^21` non-target subsets. The second used the complete 7,128-point interpolation grid with exact partition DP and off-grid checks. -**Evidence type:** two independent exact tensor reconstructions. +**Evidence type:** two independent exact tensor reconstructions. **Boundary:** the certificates cover the stated grouped activities; a witness on either topology would have to break those symmetries. @@ -135,9 +135,9 @@ pairs and 221,404 missing-edge triples has a coefficientwise nonpositive first nonzero tangent form. Arbitrarily asymmetric positive ratios inside those two- or three-edge perturbations therefore cannot bifurcate to `R>0`. -**Evidence type:** exact integer forest totals and complete tangent tensors. +**Evidence type:** exact integer forest totals and complete tangent tensors. **Seed-bank SHA-256:** -`0f5b58898b963abbb8d808a54d1e19bcc5e98bcb8a065e235bcf1a7e1ef13339`. +`0f5b58898b963abbb8d808a54d1e19bcc5e98bcb8a065e235bcf1a7e1ef13339`. **Boundary:** this is a complete statement for the defined seed bank, not all zero points of the graphic-signature cone. @@ -157,7 +157,7 @@ reduction at width two, nonpositive closure at width three, and exact ambient escape at width four. **Evidence:** exact standard-library verifier and `RESEARCH_NOTE.md`, Section -5. +5. **Boundary:** the two signatures are not claimed graph-realizable. ### C2. Real graph signatures show numerical full local rank and a singular wall @@ -169,7 +169,7 @@ obstruction. Continuation toward the abstract positive signatures nevertheless encounters a rapidly ill-conditioned fold/caustic. **Evidence type:** exact enumeration of forest feature vectors, floating SVD -rank diagnostics, and numerical continuation. +rank diagnostics, and numerical continuation. **Boundary:** neither the rank observation nor the singular wall on tested paths is promoted to a theorem. An exact nonzero Jacobian minor would be needed to certify rank 14 algebraically. @@ -187,7 +187,7 @@ three-connected layer, the complete positive-coefficient priority set is four cores and 23 target-pair orbits. **Evidence type:** exact nauty graph/isomorphism generation, automorphism -quotients, and coefficient tensors. +quotients, and coefficient tensors. **Boundary:** the four-core/23-orbit statement concerns that minimal cubic layer, not every possible eight-vertex graph or every weight vector.