[P2 exact/theory] Identify the finite matching polynomial as a surface-topological polynomial specialization

[LightChainr]

Motivation

The exact matching polynomial is currently treated mainly as a one-variable root object. But its coefficients are sums over vertex subsets classified by cluster number and torus wrapping/homology. This is precisely the kind of information encoded by topological Tutte-type invariants for embedded graphs (Bollobas–Riordan, Krushkal, transition/ribbon-graph polynomials), except that site percolation is a vertex-subset rather than edge-subset model.

A missing route is to identify or construct the correct vertex/topological polynomial whose specialization is the finite matching function.

Program

1. Starting from the configuration-level Euler/Betti derivation in [P1 exact/stat] Lift the matching identity to configuration-level Euler/Betti observables #111, define a multivariate generating function over occupied vertex subsets that records at least:
   • occupied count;
   • beta_0 / cluster number;
   • torus homology rank or cross/either class;
   • complementary matching-lattice quantities.
2. Check whether this is a known specialization/transform of a ribbon-graph, Krushkal, transition, interlace, reliability, or related polynomial after passing to a medial/decorated graph.
3. Derive the matching/complement duality at the polynomial level rather than coefficient-by-coefficient.
4. Reproduce the exact L<=5 / self-matching N=10 polynomials from the topological invariant.
5. Ask whether irreducibility, z -> 1-z root pairing, exact Beta finite controls, or Galois behavior have a natural explanation in this representation.

Why this could matter

A successful identification would unify several currently separate tracks:

• Mertens–Ziff matching identity;
• configuration-level Euler/Betti topology ([P1 exact/stat] Lift the matching identity to configuration-level Euler/Betti observables #111);
• exact finite polynomial/GCD/Galois work ([P2 exact] Certify Galois complexity of finite matching polynomials #104/[P2 exact] Reanalyze exact matching zeros locally near the physical critical root #113);
• torus wrapping-channel algebra;
• possibly transfer-matrix/Temperley–Lieb formulations.

It could also reveal exact recurrences or deletion-contraction-style algorithms that are much cheaper than 2^N enumeration.

Important site-percolation caveat

Do not force the problem into an edge-subset Tutte polynomial if the induced-subgraph/site nature requires a different invariant. A useful negative result would be a clear obstruction and a minimal new vertex-surface polynomial definition.

First deliverable

An exact note for the N=10 self-matching and axis L=2/3 cases showing either:

• a known topological polynomial specialization; or
• a new finite generating polynomial with an exact duality identity and a proposed deletion/contraction or state-sum recursion.

No large computation until this algebraic representation earns it.

[LightChainr]

PR #162 now has an exact one-variable specialization that may be useful as a target for the surface-topological polynomial search.

For every complement-odd self-matching reliability polynomial,

M(p)=(2p-1) H(u), u=p(1-p),

with H invariant under p<->1-p. If the minimum successful occupation is s, then

H(u)=sum_{j=0}^{s-1} C(2j,j) u^j + O(u^s).

So the central-binomial prefix is universal and the first free coefficient satisfies a_s=[C(2s,s)-[u^s]H]/2.

Concrete exact examples:

• N=10: H=1+2u+6u^2, s=3 and deg M=5=2s-1. There is no remaining freedom, so Beta(3,3) is forced by self-duality + minimum support + minimal degree.
• N=26 (PR Falsify frozen N26 self-matching Beta laws exactly #152): H=1+2u+6u^2+20u^3+70u^4+96u^5+170u^6+260u^7+260u^8+78u^9; the first model-dependent coefficient c5=96 gives a5=78 exactly.

This suggests H(u) is a cleaner self-dual specialization for the polynomial program than M(p) itself. A successful surface/vertex polynomial should ideally explain (i) the forced central-binomial prefix, (ii) the degree excess beyond 2s-1, and (iii) the post-prefix H coefficients. For N=26 all H coefficients are positive, which immediately proves the unique physical center root; whether that positivity has a topological/state-sum explanation is a sharp exact question.

[LightChainr]

PR #229 is the first deliverable: identification plus Tutte obstruction, no deletion-contraction algorithm.

• The matching polynomial is the Bernoulli generating function of the wrapping-event q ∈ {-1,0,+1}: M(p)=E[q].
• This is a vertex-subset homology-event generating function, not an edge-subset Tutte / Bollobás–Riordan / Krushkal specialization (site occupation; NN vs NN+NNN except on a self-matching triangulation).
• Axis L=2/3 reproduce the committed polynomials from ∑ q. Complement is not q → -q on the axis.
• C4 N=10 is self-matching: complement is an involution q(ω^c)=-q(ω) and reproduces the Beta(3,3) control.

[LightChainr]

Post-PR #229 route: stop asking for an edge-Tutte specialization; try a local-state / vertex polynomial on an embedded medial graph

PR #229's negative result is well targeted: the matching polynomial is a vertex-subset homology-event generating function, so there is no reason it should be an ordinary edge-subset Tutte/Bollobas--Riordan/Krushkal specialization.

A 2025 development suggests a different nearby algebraic target rather than abandoning recursive topology entirely.

Yan--Deng--Metsidik, arXiv:2506.07522, introduce a vertex polynomial for embedded graphs. It is related to boundary-component enumeration, the interlace polynomial and the topological transition polynomial, and has deletion/contraction/twisted contraction recurrences. Separately, the delta-matroid literature identifies embedded-graph topology as the natural extension of ordinary graphic matroids (arXiv:1403.0920) and gives transition/interlace polynomial machinery in that setting (e.g. arXiv:1710.10524).

None of those papers says that the Matching One polynomial is already their invariant. But the change of viewpoint fits the obstruction from #229 much better.

Proposed representation

Do not encode an occupied site as an edge being present/absent in the original square graph. Instead construct a fixed embedded medial/ribbon graph in which each lattice site chooses one of a small number of local vertex reconnection states.

The intended local states should encode the digital-topology pair

black NN / 4-connectivity,
white complement NN+NNN / 8-connectivity.

Then a site mask is a choice of local states on a fixed embedded object, and the global quantities relevant to matching can be read from

boundary components,
connected components,
ambient torus homology,
possibly a signed/twisted state label.

That is closer to a transition-polynomial state sum than to an edge-subset Tutte sum.

Why this is now especially attractive

I opened #269 because the existing cross/either configuration identity appears to imply the hidden candidate

2 q = r_black-r_white.

If that survives exact checking, the polynomial target simplifies dramatically: the state sum only needs to remember the difference of ambient H1 ranks of the two complementary digital connectivities, rather than an arbitrary wrapping event.

Even if #269 fails, its minimal counterexample tells us what extra local/ribbon datum the state sum must preserve.

Concrete finite test

For axis L=2/3 and C4 N=10:

1. build one fixed medial/ribbon object;
2. define the two local site states and their complement involution;
3. exhaustively verify that the state sum reproduces q(omega) and therefore M(p);
4. test whether local deletion/contraction/twisted-contraction closes within a finite typed state space;
5. compare the resulting polynomial to the 2025 vertex polynomial / topological transition polynomial after a declared substitution;
6. if no substitution exists, define the minimal new vertex-surface polynomial by the closed recurrence actually observed.

The criterion should be recursion and topology preservation, not forcing equality with a named polynomial.

Potential conceptual payoff

The local-state formulation may explain why the matching adjacency is NN for one color and NN+NNN for the complement: that 4/8 pair is standard in digital topology because it makes foreground/background Jordan separation well behaved (recent overview: arXiv:2503.17861). In a medial/ribbon formulation the apparently asymmetric graph definitions may become two complementary local reconnections of one embedded object.

That would be a much more satisfying algebraic explanation than treating the matching graph as an externally added NNN graph.

This is an exact-side continuation of #144 after the edge-Tutte obstruction, not a claim that arXiv:2506.07522 already solves the site-percolation polynomial.

[LightChainr]

Post-#269 exact extension is frozen on branch analysis/p144-incidence-spine-20260829, commit e6953ba (no PR).

The local-state route does close, but the right object is not two pairwise smoothings of one 4-valent vertex. It is a fixed doubled-lattice typed incidence spine with nodes at sites, edge midpoints, and face centers. Each site has exactly two states:

• B=J_edge: join its four edge-midpoint ports;
• W=J_face: join its four face-center ports.

The B spine has the black-NN component and ambient-H1 data. A face hub replaces the white NN+NNN clique by an embedded star with the same terminal partition, hence the W spine has the white-matching component and ambient-H1 data. All 16 local face patterns pass. Complement exchanges the two typed site labels locally.

This gives an explicit rank-graded state sum

Phi_L(p;x,y)=sum_mask p^|B|(1-p)^(N-|B|) x^r_b y^r_w,

with the exact specialization

M_L(p)=1/2 (x d_x-y d_y) Phi_L |_(x=y=1).

On the honest axis L=3 torus, all 512 masks reproduce the reference NN/NN+NNN component counts and ambient ranks, and the derived matching Bernstein coefficients equal the direct coefficients exactly: [-1,-9,-36,-78,-90,-36,36,36,9,1]. The boundary/side descriptor reads q=(r_b-r_w)/2 in the three Alexander cases (0,2),(1,1),(2,0).

There is also a sharp no-go for literal two-smoothing language: the two pairings of four ports cannot realize the one-block four-arm partition required at a black site. The minimal missing partition type is a four-way junction J4. The binary mask survives because its two values are typed junctions J_edge/J_face, not pairings.

A finite-width transfer rule follows immediately: maintain black/white typed frontier partitions with integer lift potentials and H1 generators; a B step unions four edge hubs, a W step unions four face hubs. This is a closed fixed-width recursion, not yet a named transition polynomial or width-independent bond-dimension result.

Artifacts: scripts/p144_typed_incidence_spine.py, results/exact-typed-incidence-spine/latest.json, notes/p144-typed-incidence-spine.md.

[LightChainr]

Exact post-#269 collapse of the rank-graded state sum

The bivariate state sum just constructed here has a stronger factorization than is visible from the local transfer definition.

Phase B of #269 proves r_black+r_white=2 and q=(r_black-r_white)/2, hence configuration by configuration

x^r_black y^r_white
 = x^(1+q) y^(1-q)
 = x y (x/y)^q.

Therefore

boxed:
Phi(p;x,y) = x y Z_rel(p,x/y),

Z_rel(p,Q) = E_p[Q^q]
           = P_-(p)Q^-1 + P_0(p) + P_+(p)Q.

So the apparently bivariate rank polynomial has support only on

y^2, x y, x^2,

corresponding to (r_b,r_w)=(0,2),(1,1),(2,0). The diagonal rank source is frozen; all nontrivial source dependence is the relative Cartan variable Q=x/y.

Immediate derivative consequences at x=y=1 are

1/2 (x d_x-y d_y) Phi = E[q] = M,
(x d_x+y d_y) Phi = 2 Phi,

Cov(r_b,r_w)
 = Var(q) [[+1,-1],[-1,+1]].

Thus the rank-source Hessian is exactly rank one, and all higher rank cumulants reduce to the three-state closure from #54 (q^3=q).

This does not make the frontier connectivity state space three-dimensional: the transfer still needs the typed J_edge/J_face partitions and winding lifts. It does make the terminal polynomial output three-dimensional. A deletion/contraction/twisted-join recurrence should therefore close its terminal coefficient object as the vector (P_-,P_0,P_+), rather than carrying an unrestricted bivariate rank polynomial.

This is the clean algebraic payoff of combining the local incidence-spine construction with digital Alexander duality.

[LightChainr]

The exact post-#269 collapse from the preceding comment is now a durable artifact, stacked on the incidence-spine branch (no PR): branch theory/p144-relative-source-collapse-20260829, commit 011d332.

Configurationwise,

r_b+r_w=2,
q=(r_b-r_w)/2,
r_b=1+q,
r_w=1-q.

Therefore the bivariate terminal state sum factors exactly as

Phi(p;x,y)=x y Z_rel(p,x/y),
Z_rel(p,Q)=P_-(p)Q^-1+P_0(p)+P_+(p)Q,

and its support is only

P_- y^2 + P_0 x y + P_+ x^2.

With logarithmic sources x=exp(a), y=exp(b), the diagonal source is frozen:

Phi(exp(a),exp(b))=exp(a+b)G(a-b),
(x d_x+y d_y)^m Phi=2^m Phi.

All stochastic source dependence is relative. In particular

Cov(r_b,r_w)=Var(q) [[1,-1],[-1,1]].

For every honest nonempty torus and 0<p<1, the empty and full masks have positive weights and opposite charges, so Var(q)>0; the covariance is strictly rank one. It drops to rank zero only at the Bernoulli endpoints.

Every connected rank cumulant of order at least two is kappa_m(q) (1,-1)^tensor m. The #54 three-state closure is inherited verbatim:

q^3=q,
G_sss=G_s,
F_sss=F_s-3F_sF_ss-F_s^3,

and (F_s,F_ss) reconstruct (P_-,P_0,P_+). This compresses the terminal rank-source object, not the typed connectivity frontier or thermal direction.

The exact checker consumes the existing 512-mask axis L=3 incidence-spine oracle. At p=2/5 it verifies the three monomials, Phi=xy Z_rel per coefficient row, the matching derivative specialization, exact rank-one covariance with zero determinant, and the third-cumulant closure using rational arithmetic. Focused verification: 25 passed, 114 subtests passed.

Artifacts:

• notes/p144-relative-rank-source-collapse.md
• scripts/p144_relative_source_collapse.py
• results/exact-typed-incidence-spine/relative-source-collapse.json
• tests/test_p144_relative_source_collapse.py

[LightChainr]

Post-#269 surface-polynomial classification is complete on branch theory/p144-surface-polynomial-classification-20260830, commit 5d8343d (stacked on the typed incidence-spine/relative-source work; no PR opened).

Decision. Under the natural contract of one independent ground element per site, product-local weights, and only specialization/source derivatives after summation, the finite matching polynomial is not a direct multivariate-Tutte, Bollobas-Riordan, Las Vergnas, Krushkal, topological-transition, or 2025 embedded-graph vertex-polynomial specialization.

The obstruction is local and exact: a matching site requires the one-block four-terminal partition J4. One ordinary ribbon edge has two ends; the transition-polynomial states are pairings. Neither alphabet contains J4, and differentiation cannot create a missing state. A minimum ordinary-edge gadget needs 3 edges and therefore has 8 independent subsets; keeping only all-off/all-on requires deleting 6 mixed states with a block projector, which is precisely extra typed site structure.

Positive identification. Using unrestricted q=r_B-1, define

R_site(a,b;z)=sum_B a^|B| b^(N-|B|) z^r_B.

Then exactly

M(a,b)=(z d_z-1)R_site|_(z=1).

Equivalently Z_rel=sum Q^q and M=Q d_Q Z_rel|_1. One topology source beyond occupation is sufficient. It is also necessary: two order-2 HNF quotients have identical occupation counts [1,2,1] but defect vectors [-1,0,1] and [-1,-2,1].

Krushkal is the closest named readout: on honest torus carriers q=g(U_B)-g(V_B), and the unrestricted rank theorem extends the associated rank/genus flags to degenerate quotients. But its edge-subset state family is not the typed site-state family, so the result is a site-Krushkal rank-image quotient, not K_G itself.

Axis L=3: all 512 masks make black-rank, relative-rank, and side-genus derivatives reproduce [-1,-9,-36,-78,-90,-36,36,36,9,1]. 12 stacked P144 tests plus 4 PR229 tests pass. Artifacts: notes/p144-surface-polynomial-classification.md, manifest, exact JSON, tests, scientific card. No Galois numerics, PR, or merge.

[LightChainr]

ArXiv refinement after PR #229: use the subgroup-valued coefficient algebra, not the edge-Tutte state sum

PR #229's obstruction should stand: Matching One is a vertex-subset digital-homology state sum, so there is no reason for a direct edge-subset Tutte/Bollobas--Riordan/Krushkal specialization.

The part of the surface-polynomial literature that now looks useful is one level deeper.

Krushkal, arXiv:0903.5312, does not only add genus variables to Tutte. Its finer construction retains subgroups of H_1(Sigma) as coefficient data, and surface duality acts through the intersection pairing / Poincare duality. Krushkal--Renardy (arXiv:1012.1310) and the TKR cell-complex literature make the same general point: polynomial duality can be a shadow of Alexander/Poincare duality on the underlying topological state space.

That suggests a sharper target for the current local-state / vertex-polynomial continuation here.

Proposed coefficient object

Instead of asking only for one scalar polynomial M(p), try to make the local recurrence preserve

Lambda(omega) = im[H1(K(omega))->H1(T^2,Z)]

as a typed coefficient label, with at least

rank,
primitive rational line/subspace,
integral saturation/index data

kept distinct.

The representation-valued state sum is schematically

Z_sub(p)
 = sum_Lambda Prob(Lambda) [Lambda].

The coarse contraction

[Lambda] -> exp[s(rank Lambda-1)]

gives the intrinsic topological-source generating function proposed in #337,

Z_top(p,s)=P0 e^-s + P1 + P2 e^s,
partial_s Z_top|_0 = P2-P0 = M.

The projective rank-one contraction should contain the #156/#334 winding-character observables; a still finer integral label can carry the HNF/saturation information that projective direction alone loses.

Exact acceptance criterion for the vertex/local-state polynomial

For axis L=2/3 and C4 N=10, then the current HNF exact frontier:

1. local deletion/contraction/twisted-contraction must update the subgroup label without consulting the completed global configuration;
2. summing subgroup coefficients must reproduce the Bernoulli state count;
3. the rank contraction must reproduce P0,P1,P2 and Matching One exactly;
4. the complement/local-state involution must induce the already-proved Alexander rank map;
5. only after deriving the integral subgroup map should we claim a Krushkal-like orthogonal-complement duality for nonsaturated states;
6. test whether the number of required boundary/subgroup states stays finite at fixed strip width, giving an actual transfer/deletion-contraction algorithm.

A failure is informative: if no bounded local state can propagate Lambda, then the obstruction is stronger than “not an edge-Tutte polynomial” and tells us exactly why the global ambient-homology information is nonlocal.

Relation to Bernoulli cell-complex work

Hiraoka--Shirai, arXiv:1602.04561, derive persistent-homology lifetime information from Tutte polynomials and higher random-cluster models on Bernoulli cell complexes. The repository has independently identified K_minus/K_plus as the first/second essential ambient-H1 births and their gap as the rank-one lifetime. This makes lifetime/subgroup refinement a concrete cross-check for the recurrence, not just an algebraic analogy.

So my suggestion is: do not reopen the direct Krushkal-specialization question. Upgrade #144's current vertex-polynomial target so that ambient homology is coefficient data rather than an after-the-fact observable.

Related: #269, #275, #321, #334, #337.