A Dichotomy for Complex Boolean Holant with Binary Disequality

Chenghua Liu; Boning Meng

Source-linked AI summary

A Dichotomy for Complex Boolean Holant with Binary Disequality

Chenghua Liu, Boning Meng

arXiv:2609.00219v1Aug 31, 2026cs.CC

TL;DR

The paper asks for the complexity boundary of Boolean Holant problems with arbitrary finite algebraic complex-valued signatures when binary disequality is supplied. It combines terminal and lifting interfaces with stabilized safe-binary transfer-group analysis to obtain an explicit criterion. The result is a decidable FP-versus-#P-hard dichotomy, with a quaternary equality-free variant under an endpoint-nondegenerate tensor-prime hypothesis.

Problem
The paper addresses the missing dichotomy for arbitrary finite algebraic complex-valued Boolean signature sets with only binary disequality supplied, beyond settings with prescribed pinnings or other auxiliary resources.
Method
The proof combines terminal identification, stable-pair recording, gadget-provenance-preserving lifting interfaces, and group-specific analysis of finite safe binary transfer groups.
Results
The main dichotomy is decidable: Holant(F, X) is in FP when TractX(K(F ∪{I})) holds and is #P-hard otherwise.
Takeaways & Limitations
Binary disequality collapses the general tractability boundary to four alternatives: arity-two closure and common affine, product, or local-affine transformability.
Takeaways & Limitations
A later branch analysis has occurrence-level decreases rather than an asserted decrease of the active tuple’s Ω, and bounded outputs must be physical current-coordinate q4 or q6 signatures.

Abstract

from arXiv · show

We prove a complexity dichotomy for Boolean Holant problems defined by arbitrary finite sets of algebraic complex-valued signatures when binary disequality is available. The tractable cases are characterized by an explicit, decidable criterion.

Code Availability

The paper provides an exact-verification package for its finite computer-assisted claims in the repository’s certificates/ directory.

Code Availability: The certificates/ directory contains verification support for finite computer-assisted statements in Appendix A.The package includes a complete-suite entry point and documentation of the computational trust boundary.
Code Availability: The certificate index covers residue, matching-deck, Klein-four, and Platonic-form claims.The listed certificate sections include Sections 5, 6, 8, and the Platonic classifications.

1 Introduction

The paper studies arbitrary finite algebraic complex-valued Boolean Holant signature sets with binary disequality available, and proves a decidable FP-versus-#P-hard dichotomy. Its proof combines established tractability boundaries, terminal interfaces, stabilized safe-binary transfer groups, and group-specific rigidity analyses.

Problem setting: Boolean Holant evaluates tensor-network partition functions whose vertices carry complex-valued Boolean signatures and whose edges contract incident bits.The framework includes counting problems and basis changes that are not visible in standard constraint-language presentations.
Problem setting: The K-Holant presentation makes binary disequality the native edge kernel while equality is supplied as an available binary signature.This is holographically equivalent to ordinary Holant and exposes tractable structure over complex numbers.
Problem setting: The paper addresses the gap left by Holantc dichotomies, because hardness after adjoining pinnings need not persist when those pinnings are removed.It targets arbitrary finite algebraic complex-valued Boolean signature sets with only binary disequality supplied.
Contributions: The main contribution is a decidable dichotomy for the binary-disequality model over every finite set of algebraic complex-valued Boolean signatures.The criterion packages the broadest currently known general complex-valued Boolean Holant tractability boundary.
Contributions: The binary anchor collapses the general seven-alternative boundary to four TractX cases: arity-two closure and common affine, product, or local-affine transformability.The anchor rules out the one-sided, single-Hamming-layer, and matching alternatives.
Contributions: An additional equality-free quaternary dichotomy applies when the signature set contains an endpoint-nondegenerate tensor-prime quaternary signature.Under that hypothesis, K-Holant is polynomial-time computable when Tract holds and #P-hard otherwise, with a decidable criterion.
Proof strategy: The proof stabilizes a minimum-arity tensor-prime signature, its safe projective binary transfer group, and a regenerated dressed matching deck before group-specific analysis.The analyses cover marked cyclic, dihedral, Klein-four, and Platonic possibilities and use rigidity mechanisms such as root-of-unity geometry and Reed–Muller structure.

2 Preliminaries and the Tractable Boundary

The preliminaries define the algebraic and tensor vocabulary, establish transported-normalization and factor-saturation machinery, and characterize the tractable boundary for Boolean Holant with binary disequality.

Notation and algebraic setting: The preliminaries define finite-set, partition, field, matrix, algebraic-number, and Boolean-field notation used throughout the paper.They also specify cardinality, complements, fibers, lexicographic order, and indicator notation.
Holographic transformations: The paper distinguishes group-theoretic conjugacy from legal K-Holant normalization because a conjugating matrix need not lie in GO(X).Consequently, conjugacy alone does not provide a legal global normalization or an actual binary gadget.
Normalization: A transported inclusion AΓ⋆⊆Λ preserves polynomial-time Turing equivalence when A∈GO(X).Literal augmentation is the special case A=I, while the normalization ledger tracks coefficient-field and group data.
Tractable boundary: The general Tract predicate combines seven known tractable alternatives with polynomial-time algorithms for its two support-defined alternatives.Tract is explicit and decidable, and its truth implies Holant(F)∈FP.
TractX: With binary disequality available, the tractable boundary collapses to four alternatives because the anchor excludes one-sided, single-weight, and matching cases.The surviving alternatives are arity-two closure and common affine, product, or local-affine transformability.
TractX: The local-affine alternative is essential because a finite signature set containing X can satisfy it while satisfying none of the other retained alternatives.A separate comparison shows that endpoint-nondegenerate eight-vertex instances have a more rigid local-affine boundary.
Effective recognition: The paper establishes effective algebraic transformability: rational witnesses suffice, and witness existence is decidable for exact algebraic input.The result applies to the stated transformability classes.
Imported boundaries: The odd-arity dichotomy gives an FP-versus-#P-hard classification under Tract, while one-hot support admits linear-time evaluation in the stated setting.The bound is O(|V(Γ)| + |Eint(Γ)|).

3 Shared Terminal and Lifting Interfaces

The shared interfaces reduce exposed signatures to terminal outcomes and provide controlled lifting, interpolation, replication, and unary-access mechanisms without treating analytic constructions as free signature resources.

Terminal framework: The interface layer resolves exposed factors and gadgets through terminal identification, stable-pair recording, and retained provenance.Continuing states record a prime signature, safe binary transfers, retained factors, and a regenerated matching deck.
Binary interpolation: Binary interpolation uses a directly realizable ordered binary transfer to expose rank-one signatures when the transfer is singular or has infinite projective order.The construction handles diagonalizable and Jordan cases through polynomial interpolation.
Binary interpolation: For m designated occurrences, interpolation requires m+1 oracle queries and O(m^3) added vertices overall.Each query uses O(m^2) added vertices.
Unary exits: The method stops when it exposes any nonzero unary, while stronger unary outputs are recorded only when later arguments require them.Higher-order equality access can expose both unary point masses, and one nonzero unary suffices for the common-unary exit.
One-pair reduction: Universal one-pair reduction reconstructs a deleted equality contribution using ordered boundary orientations and a constant number of oracle queries.The ordered ports matter in the subtraction, and the witness network has constant size.
One-pair reduction: The reduction distinguishes symmetric binary behavior from an asymmetric realized binary witness and uses the corresponding oracle identities.The asymmetric case uses at most three oracle queries.
Replication: Single-occurrence replication nests a carrier-preserving context to create multiple designated copies of h while retaining exactly one occurrence of g.The k-fold construction has size O(k) and a known nonzero scalar that can be divided out.
Unary exits: A common unary exit reduces access to a nonzero unary to the original problem unless the original tractability predicate holds.The argument applies in both ordinary Holant and K-Holant.

4 Endpoint-Nondegenerate Eight-Vertex Signatures

The section classifies endpoint-nondegenerate eight-vertex signatures through product-transformable, local-affine, and affine cases, building operational interfaces for retained signature sets. These interfaces reduce surviving branches to tractable, hard, or weighted-equality-factor outcomes while preserving legal normalizations and factorization data.

Definition and setup: Endpoint-nondegenerate eight-vertex signatures have even-parity support with nonzero values at 0000 and 1111.Their six weight-two entries are central coordinates, and the three matching flattenings determine a rank profile.
Classification strategy: The classification proceeds through product, local-affine, and affine-transformable cases before supplying a quaternary interface for the matching-deck proof.The same classification yields the independent quaternary dichotomy in Theorem 1.3.
Product-transformable cases: P-transformable signatures have exactly one of three structural outcomes: pure generalized equality, two nonsingular weighted equalities, or a full-support dense form with an equality anchor.These outcomes imply support size 2, 4, or 8, up to port permutation.
Affine cases: Affine-transformable signatures admit an algebraic invertible diagonal normalization, and their genuine two-block factorization can expose a nonsingular weighted-equality factor after retained factor saturation.The factor is Turing-exposed rather than necessarily directly realizable, while normalization changes are recorded in the ledger.
Signature-set-wide synthesis: The operational interface reduces each retained branch to #P-hardness, a tractable presentation, or a retained augmentation containing a proper weighted-equality factor.The reduced interface postpones equality-accessible closure, and the common unary exit merges the resulting routes.
Tensor-prime consequence: For a tensor-prime quaternary signature, the continuing factorization outcome contradicts tensor-primality, leaving #P-hardness as the remaining outcome.The contradiction is preserved under invertible portwise dressing.

5 Equality-Accessible Matching Decks

The equality-accessible analysis builds a finite matching-deck procedure that either reaches a stable state or exits through a terminal, tractable conclusion, or successor. Stable tuples are routed through four marked finite-group branches, yielding a decidable tractability test and a common structural presentation for nonhard cases.

Equality-accessible setup: The equality-wire equivalence establishes the K-Holant representation with native kernel X and ordinary equality wires.The conversion uses exact path and cycle identities while preserving the available X kernel.
Finite-group control: Bounded torsion makes the finite projective transfer group finite whenever all directly realizable nonsingular binary gadgets have finite order.This finiteness supports the subsequent complete-group and restart analysis.
Matching-deck construction: The dispatch procedure either resolves a terminal or tractable branch, or produces a retained successor with a minimum-arity nonbinary tensor-prime and a stable matching deck.On the continuing branch, every nonzero card is a matching signature whose factors are retained and projectively drawn from the fixed binary transfer classes.
Stabilization and restart: The stabilization scheme records a deck-stable tuple and uses strict group growth or decreasing measures to make restarts finite and acyclic.Continuing transitions strictly increase the complete group, while frozen-token coverage discharges within-node work or decreases the remaining measure.
Stable-state conclusion: In a deck-stable state, K-Holant is in FP when Tract(K(G)) holds and is #P-hard otherwise, equivalently using TractX(K(G)).Every nonhard rigidity survivor has one common product, affine, or flat-Lagrangian K-presentation, and the same basis places the transformed edge in that class.
Four-way routing: The complete finite safe transfer group is routed into cyclic, dihedral, Klein-four, or Platonic marked forms, with Figure 1 recording necessary strict-growth paths.The marked refinements include the specified dihedral, Klein-four, tetrahedral, octahedral, and icosahedral displays; grouped nodes do not identify distinct forms.

6 Marked Dihedral Matching Decks

The marked-dihedral analysis localizes unresolved matching and coefficient disagreements to bounded quaternary or six-port configurations, then forces the remaining support into one global matching coset. Subsequent coefficient dispatch establishes the corresponding structural alternatives or closes the branch.

Trace-cover localization: Compatible closures preserve nonzero products and protected determinants while shortening alternating cycles and reducing two typed tags to at most one.These operations use only the native X-wire, explicit normalized equality link, and actual group representatives.
Trace-cover localization: Bounded trace-cover localization reduces every minimal matching, parity, or coefficient disagreement to one of three configurations involving a deleted pair and at most two typed structures.The configurations produce either two six-port cards with different matchings, two six-port cards with protected-factor differences, or a quaternary with a protected minor.
Sector pencils: Sector-ruling rigidity leaves each sampled pencil with one fixed matching and at most one varying Segre factor, with all other binary factors fixed projectively.The root-pencil normal form excludes an additional rotation/inversion branch after exact normalization.
Endpoint analysis: The sole nondischarged endpoint survivor is the C/C case with an endpoint parallelogram and one compatible residual matching.Zero or proportional endpoint pencils otherwise produce a concrete resolved outcome.
Coefficient dispatch: Six-port coefficient dispatch reduces the surviving fixed three-edge-matching pattern to zero, a current-group matching product, or rank-one logical flattenings.The analysis closes produced q4 cards and factor batches before completing the parent token.

7 Proper Cyclic Matching Decks and Reed–Muller Rigidity

The proper-cyclic analysis studies sampled root-of-unity phase pencils and their physical port dressings. Its rigidity alternatives constrain endpoint and central Walsh coordinates, providing the phase-structured basis for the quaternary boundary and the remaining Reed–Muller analysis.

Definitions and setup: A tensor pencil is a one-parameter linear family, while sampled phase pencils restrict the parameter to a finite root-of-unity group and retain zero and projective information.The analysis uses monomial binary matrices, which in dimension two are diagonal or anti-diagonal.
Definitions and setup: The analytic Walsh transform of a quaternary signature records endpoint coordinates p and t together with central coordinates x_ab on the even-weight sector.These coordinates are defined by p = ê_0000, t = ê_1111, and x_ab = ê_1{a,b}.
Phase-pencil rigidity: Cyclic phase-card rigidity requires each sampled diagonal matrix to be zero or projectively a root-of-unity monomial matrix.The relation is characterized through affine linear forms whose ratio maps the sampled root set into itself.
Physical dressing: The physical cyclic lemmas require fixed actual binary representatives for every root-of-unity class, with orientations and nonzero realization scalars recorded.The Walsh transform is an analytic convention rather than an inserted physical gadget.
Physical dressing: Attaching these representatives through native X-edges creates normalized four-port dressed signatures whose cards are tested across complementary deletions and both residual-port orders.This supplies the physical interface between sampled phase-pencil constraints and the quaternary boundary.
Quaternary phase family: For the quaternary phase family, if p and t are nonzero then t/p lies in µN and each central block is zero or has a root-of-unity ratio; if p = t = 0, nonzero central blocks are anti-diagonal.Exactly one of p and t cannot be nonzero, and the stated alternatives are also sufficient for the zero-or-cyclic property.

Define the map FN : µ4

The quaternary branch either resolves through a tensor-prime endpoint-nondegenerate eight-vertex signature or advances to a strict group successor. The remaining matching-deck analysis reconstructs the only unresolved core as an eight-port Reed–Muller signature and develops a common C2 presentation for the retained signature set.

Quaternary entry: A two-copy gadget using a quaternary signature either exposes an unsafe binary, strictly enlarges the complete group, or realizes a tensor-prime endpoint-nondegenerate eight-vertex signature.This gives the branch a resolved leaf, an outer-deck successor, or the tractable K-Holant route.
Quaternary entry: Invertible X-dressings preserve endpoint-nondegeneracy and tensor-primality, so the resulting signature satisfies the boundary needed for the eight-vertex dichotomy.The dressing is an analytic reconstruction rather than installation of a physical gadget.
Minimum-core reconstruction: The zero-card and matching lemmas eliminate nontrivial zero-card structure and show that any nonempty weight-two matching graph yields either an Even4 anchor or a proper tensor factorization.For minimum cores of arity at least eight, the weight-two graph is empty.
Minimum-core reconstruction: Matching consistency and diagonal-slice integration force the surviving minimum core into the Reed–Muller reconstruction, with a unique weight-four completion implying arity eight and support RM(1, 3).The surviving coefficient structure is then reduced to a linear sign character.
Minimum-core reconstruction: The coefficient atlas and available X-dressings identify the standard core RM8, equivalent to the Shao–Cai signature f8.This closes the core-identification step up to the recorded equivalences.
C2 rigidity and closure: The continuing branch proves that the complete safe transfer group is exactly {[I], [X]} ≅ C2 and derives a common flat-Lagrangian presentation for the entire retained signature set and its edge.The associated calculus supplies later Radon, localization, affine-patching, and recognition arguments, while a shortcut can close the branch by group enlargement.

8 Klein-Four Matching Decks

The Klein-four analysis reduces standard Pauli cores to H6 and RM8, while exotic Klein forms admit no H6 analogue. Higher-arity signatures are localized through actual card contractions, stabilizer methods, and affine closure.

Setup: Theorem 8.1 organizes Klein-four rigidity around stabilized tuples and, in the standard Pauli case, assumes arity at least six.The analysis distinguishes standard V4 from the exotic form Vex and uses complete closure of the relevant branch.
Standard Pauli branch: Six-port standard-Pauli tensors whose actual pair cards are zero or Pauli-binary products are either three-binary products or in the H6 orbit.Every Pauli pair card of H6 is itself a nonzero two-Pauli product.
Standard Pauli branch: The full-V4 fixed-ruling classification leaves H6 and RM8 as the irreducible low-arity cores; all other outcomes factor or become matching products.No irreducible fixed-ruling tensor exists for r ≥5, and the r = 4 case lies in the RM8 orbit.
Standard Pauli branch: The direct four-copy circuit realizes RM8 from H6, after which the normalized closure forces surviving companions into a common character-flat affine class.The resulting alternative is the common affine presentation, with factor and product outcomes removed by closure.
Higher-arity localization: For arity at least ten, five-spread gluing and stabilizer erasure localize any nonaffine signature to an actual nonaffine card, a genuine factor, or arity at most eight.The physical localization corollary makes these the only remaining outcomes under the full-V4 hypotheses.
Exotic Klein branch: In the exotic Klein branch, every six-port tensor with only zero or exotic matching-product cards is a product of three exotic binaries, so no exotic H6 analogue exists.For higher arity, live exotic cards still share one residual perfect matching.

9 Platonic Matching Decks

The Platonic matching-deck analysis classifies marked finite-group branches through physical kernels, card contractions, rank tests, and rigidity arguments. Tetrahedral survivors are affine or factor, while local consumers terminate octahedral and icosahedral branches.

Tetrahedral forms: The two marked tetrahedral forms are classified after a legal fixed-I-preserving normalization, with the physical transfer and reversal marks determining the extension group.One standard case has j ∈G and H = G for the standard tetrahedral group T.
Icosahedral forms: The fixed-equality dictionary transfers standard icosahedral identities to the opposite display while preserving zeros, factors, ranks, matching labels, and endpoint support.It changes physical kernels and scalars but leaves the signature set, active occurrence, and literal equality unchanged.
Quaternary boundary: Endpoint-nondegenerate eight-vertex terminals are decided by rank: K-Holant(Λ) is either #P-hard or satisfies Tract(KΛ).The rank-obstructed terminal excludes the remaining continuing cases after factorization and equality outputs are resolved.
Quaternary boundary: Platonic quaternary cores have no rank-three case; ranks zero, one, and four give zero or factorization, while rank two occurs only for S4.The marked forms then route to equality, terminal, or group-successor outcomes rather than recursion.
Octahedral branches: Marked octahedral branches close through bounded local consumers whose outputs are resolved terminals, lower-arity factors, or strict complete-group successors.Displayed kernels and at most four port dressings realize eight-vertex, generalized-equality, or outside-binary exits.
Internal A4 form: Tetrahedral orientation-preserving tensors are affine, and every tensor-prime eight-port survivor is affine after physical card testing.The alternative is factorization across a 2|6 cut or membership in the standard affine class.
Internal A4 form: Physical tetrahedral localization reduces any nonaffine tensor-prime of arity at least ten to an actual nonaffine card on at most eight ports, unless a factor, terminal, or strict group restart occurs.The construction uses one vertex and O(arity(g)) actual binaries.

10 Synthesis of the Binary-Disequality Dichotomy

The synthesis closes the equality-accessible matching-deck classification and transfers it back to ordinary Holant with binary disequality. A finite decreasing protocol yields an explicit dichotomy whose tractable side is exactly TractX.

Synthesis: The final synthesis first closes the native-X problem with literal equality, then uses equality-wire equivalence to return to the binary-disequality formulation.Endpoint-nondegenerate tensor-prime quaternaries are closed by Theorem 1.3, while Corollary 4.7 handles intermediate factorization routes.
Termination: The protocol represents each proper card and its factor forest by frozen tokens, ensuring same-stratum continuations remain within a finite queue.Multi-copy consumers produce resolved leaves, factors, or group exits rather than untracked recursion.
Termination: Strict group enlargement, lower minimum arity, queue closure, and factor splitting provide decreasing measures at every restart.Safe binaries inside the current group close locally, while unsafe or outside-group binaries terminate or force recomputation.
Final classification: The closure ends every branch in a terminal, a direct TractX leaf, or one common product or affine K-presentation.The flat-Lagrangian survivor is treated as a structured affine subcase rather than a separate tractable alternative.
Final classification: For a finite retained factor-saturated set containing literal equality, TractX(K(Λ0)) gives FP and its failure gives #P-hardness.TractX is equivalent to Tract in this equality-accessible setting because Z = K⊗2I is available.
Final classification: The same closure remains #P-hard after retained factor exposure when TractX fails, with decidability supplied by the classification procedure.Augmented signature sets preserve polynomial-time equivalence and return to fixed-I coordinates after internal normalization.

11 Conclusion

The paper establishes equivalent decidable dichotomies for ordinary Holant and K-Holant with binary disequality. The equality anchor reduces tractability to four explicit alternatives, while a separate theorem handles sets without available equality.

Main dichotomy: For ordinary Holant with binary disequality, TractX(K(F ∪{I})) characterizes FP, and failure is #P-hard.For finite sets already containing I, the equivalent K-Holant formulation uses TractX(K(F)).
Main dichotomy: The binary anchor collapses the full tractability predicate to arity-two tensor closure and common affine, product, or local-affine transformability.These are the four alternatives of TractX.
Equality-free boundary: Without assuming equality, an endpoint-nondegenerate tensor-prime quaternary still yields an FP-versus-#P-hard dichotomy governed by Tract(K(F)).This theorem generalizes the single-signature eight-vertex dichotomy and supplies the quaternary terminal for the main classification.

A Exact Finite Certificates

This appendix records the exact finite certificate claims supporting the binary-disequality classification and identifies their corresponding verifiers and audit payloads.

The appendix covers endpoint-eight-vertex, equality-accessible, normalized-dihedral, Klein-four, and Platonic finite certificate checks.

A.1 Certificates for Section 4: Endpoint-Nondegenerate Eight-Vertex Signatures

This subsection introduces the two exact finite statements used in Section 4.

Two exact finite statements support the analysis of Section 4.

A.1.1 The fourth-root odd quaternary residue

The fourth-root odd quaternary residue is partitioned by twelve deterministic loop tests into rank-one, infinite-order, and stalled states, with stalled cases handled by further exact constructions and phase tests.

The twelve ordered loop pairs determine a deterministic partition of every normalized full-support fourth-root odd table.
A loop with exactly one zero coordinate yields a rank-one exit, while unequal Gaussian norms certify infinite projective transfer order.
Otherwise the state is stalled, with zero–zero pairs retained in the classification.
Exactly 64 stalled states satisfy the exceptional condition and produce support-six endpoint-nondegenerate eight-vertex signatures through a two-copy construction.
Exactly two phase shifts make each remaining stalled state affine: {0, 2} for 176 states and {1, 3} for 128 states.
The resulting 512-state affine family is split among rank-one native loops and two-copy dense product-transformable constructions.

A.1.2 The affine endpoint-nondegenerate certificate

The affine endpoint certificate reduces normalized affine signatures to finite support types and supplies explicit gadget or factorization witnesses for each relevant case.

There are 36 720 normalized projective affine signatures, and exactly 4200 core-signature pairs satisfy the retention test.
The diagonal-orbit invariant J(g) completely characterizes diagonal orbits of endpoint-nondegenerate eight-vertex signatures.
The 512 full-even references have a disjoint first-witness partition including dense product-transformable forms, rank-one native self-loops, and crossed two-copy gadgets.
Exact replay enumerates affine supports, phase choices, eighteen cores, fourth-root odd states, loop tests, phase shifts, and two-copy contractions over Z[i].
The eighteen-element core set maps normalized affine signatures into complementary endpoints, four-point planes, or full even-parity support.
Support-two references are pure generalized equalities, while support-four references either factor into weighted equalities or yield a pure generalized equality by a two-copy construction.

A.2 Certificates for Section 5: Equality-Accessible Matching Decks

The certificates establish explicit low-arity interfaces that convert equality-accessible matching-deck states into endpoint-nondegenerate eight-vertex signatures, while isolating the exceptional support and phase configurations exactly. These outputs are shown to be tensor-prime and outside the relevant matching-product classes.

Equality-accessible quaternary certificates: Proposition A.7 characterizes equality-accessible quaternary tensor-prime cores whose physical decks use only fixed group representatives and supports the subsequent interface construction.The proposition assumes a deck-stable, factor-saturated retained state containing literal equality and a nonzero quaternary tensor-prime core.
Equality-accessible quaternary certificates: An actual gadget using at most two copies of the quaternary core produces an endpoint-nondegenerate eight-vertex signature.The construction applies across the relevant parity- and complement-structured cases.
Parity and complement structure: Fourier and Walsh-complement conditions force one physical parity and complement closure, after which Pauli X-dressing makes both endpoints live.The resulting signatures are endpoint-nondegenerate eight-vertex signatures.
Safe-locus enumeration: The six-pair safe-locus calculation yields exactly twenty-four maximal two-dimensional linear subspaces, with sixteen one-dimensional solutions contained in them.Actual local exotic dressings and port permutations act transitively on the twenty-four planes.
Normalized-dihedral support lowering: For normalized-dihedral supports, exactly 96 labelled choices are matching cosets; every other choice admits a contraction to a cancellation-free quaternary of support six or eight.The nonmatching output is not a product of two nonsingular monomial binaries.