A Dichotomy for Complex Boolean Holant with Binary Disequality

Code Availability

• The exact-verification package for Appendix A’s finite computer-assisted statements is provided in the Holant-X-Dichotomy repository’s certificates/ directory.The paper gives the mathematical certificate statements and all noncomputational interfaces.Source: Code Availability · p. 1
• Run the complete verification suite from the repository root with the Python 3 entry point certificates/run_all_exact.py.Source: Code Availability · p. 1
• The coverage map links manuscript anchors to verifier entry points, while the README documents requirements, payload semantics, and the package-wide computational trust boundary.The files are certificates/theorem-to-script-map.md and certificates/README.md.Source: Code Availability · p. 1

Binary disequality and K-Holant

• Binary disequality is not interchangeable with unary pinnings; its K-Holant formulation exposes structure and supports signature-preserving equivalence between Holant(F, X) and K-Holant(F, I).Degree-two X-components reduce to I or X according to path parity, while cycles contribute an exactly computable scalar.Source: §1 · pp. 5–7

Main dichotomy

• Holant(F, X) is polynomial-time computable when TractX(K(F ∪ {I})) holds and #P-hard otherwise; the criterion is decidable from exact algebraic input.TractX retains four cases: arity-two tensor closure and common affine, product, or local-affine transformability.Source: §1 · pp. 7–8

Equality-free quaternary dichotomy

• Without assuming equality is available, an endpoint-nondegenerate tensor-prime quaternary signature yields a K-Holant dichotomy: Tract decides polynomial time versus #P-hardness.This boundary is decidable from exact algebraic input and removes the factorization branch of the endpoint-nondegenerate eight-vertex interface.Source: §1 · pp. 7–8

Proof strategy

• The proof first recognizes terminal cases, then stabilizes a minimum-arity tensor-prime signature with its complete safe-binary group and matching deck before analyzing group-specific successors.Prime arity can decrease only finitely often, and strict group enlargement is also finite, so each stable pair must reach a terminal, a smaller prime, a larger group, or resolution.Source: §1 · pp. 9–10

Marked dihedral groups

• Marked dihedral analysis combines root-of-unity phase pencils, matching rigidity, and code geometry; an exact six-port support atlas resolves the remaining finite case.The RM(1, 3) exceptional support is eliminated by a 1−t2 minor, and the standard calculation transfers to the other three marked forms.Source: §1 · p. 10

Proper cyclic groups

• Proper-cyclic analysis uses phase pencils and self-dual-code geometry to identify RM(1, 3) as the sole nonproduct core, force the safe group to C2, and obtain a common flat-Lagrangian presentation.At arity four, the analysis reaches an eight-vertex terminal or a reflection that enlarges the group to a dihedral successor.Source: §1 · p. 10

Klein-four groups

• The two marked Klein-four forms use distinct physical calculi: Bell and stabilizer-code methods localize standard-form obstructions, while Frobenius-dual recoupling and finite atlases resolve the exotic form.The standard form reduces low-arity obstructions to H6 and RM8; the exotic-form analysis exposes a proper factor in every remaining nonaffine survivor.Source: §1 · p. 11

Platonic groups

• Platonic-group analysis uses D4, F4, and H4 projective root configurations and exact finite verification to force tensor factors, six-port bridges, or terminal signatures.The tetrahedral triality split ends in an eight-vertex terminal or a common affine presentation; octahedral and icosahedral cases produce an eight-vertex signature, generalized equality, or a binary outside the current group.Source: §1 · p. 11

2.1.1 Sets, maps, and exact fields

• An effective number field representation supports exact field operations and equality tests, while the preliminaries fix notation for sets, maps, groups, rings, fields, and algebraic numbers.The representation may specify a field as Q(θ) using a minimal polynomial and isolating data.Source: §2.1.1 · pp. 11–12

2.1.2 Linear, polynomial, and tensor conventions

• Linear, polynomial, matrix, and tensor conventions include multilinearity, tensor products, and flattening rank for grouping tensor coordinates into a matrix.Matrix rank is identified with linear-map rank.Source: §2.1.2 · p. 13

2.1.3 Groups and projective spaces

• The group conventions cover actions and representations, while projective space identifies nonzero vectors up to scalar and invertible linear maps induce projective automorphisms.Scalar multiples induce the same projective automorphism.Source: §2.1.3 · pp. 13–14

2.2 Complexity conventions

• Complexity is measured in the encoded network size; FP denotes polynomial-time functions, #P counts accepting branches, and #P-hardness uses exact algebraic Turing reductions.Signature sets, their number fields, and fixed gadgets are nonuniform, while the network is the uniform encoded input.Source: §2.2 · p. 14

2.3 Signatures, tensors, and Holant instances

• A Boolean signature is an ordered-port function on {0, 1}^r represented as a tensor, with tensor-prime factorization unique up to port permutation, nonzero scalar, and factor order.The tensor closure includes zero and the arity-zero empty product.Source: §2.3 · pp. 14–15
• Ordinary Holant uses closed networks with equality edges, while native K-Holant uses disequality edges and is Turing-equivalent to Holant(X | F).Ordinary edges identify endpoint bits; native X-edges retain separate endpoint bits, and subdividing them by X-vertices gives the bipartite presentation.Source: §2.3 · pp. 16–17

2.4 Holographic transformations and the K-basis

• The holographic-transformation setup fixes i with i^2 = −1 and a K-coordinate matrix.Exact algebraic postprocessing, including known nonzero rescaling, is included in reductions.Source: §2.4 · p. 17
• Group-theoretic conjugacy alone does not establish a legal global K-Holant normalization or an actual binary gadget; later applications retain the distinguished classes [X] and [Z].In K-coordinates, hats record transformed signatures, while Walsh coordinates are analytic bookkeeping rather than another legal global transformation.Source: §2.4 · p. 21

2.5 Gadgets, interpolation, and factor reductions

• A direct gadget realizing a known nonzero scalar multiple of h implements h, whereas interpolation or tensor-factor extraction is only Turing-exposed absent a separate direct gadget.Analytic identities and linear combinations do not themselves add signature resources; gadget provenance and discarded scalars must be tracked.Source: §2.5 · pp. 21–22

2.5.1 Two-port fibres and observable directions

• Contractions against a kernel space determine a two-port fibre modulo its Frobenius-orthogonal complement: kernel directions are observable, and orthogonal directions are invisible.Source: §2.5.1 · p. 22
• A retained structural state may satisfy AΓ⋆⊆Λ for A∈GO(X), with K-Holant(Λ) ≡T K-Holant(Γ⋆); continuing coordinate changes preserve native X and literal I projectively.The interface pulls states back to caller coordinates before passing a continuing branch onward.Source: §2.5.1 · p. 23

2.6 Zero and nullary preprocessing

• Preprocessing deletes zero and nullary signatures before structural classification, returns zero for instances containing a zero signature, and restores nullary scalar contributions in the root evaluator.The tractable predicates apply to the remaining nonzero positive-arity signature set.Source: §2.6 · p. 24

2.7 Charge and Hamming layers

• The charge framework distinguishes doubled charge from integer half-charge for even arity and organizes support by Hamming layers, parity, and Hamming half-spaces.An exact support mask specifies precisely which inputs have nonzero values.Source: §2.7 · p. 24
• Closed networks of native X-contractions preserve total charge in every contributing assignment.For open networks, the boundary word determines the charge contributions on unpaired boundary half-edges.Source: §2.7 · p. 25

2.8.1 Boolean and affine tools

• The Boolean and affine toolkit includes a difference identity characterizing affine functions and a Möbius expansion over the stated coefficient ring.The identities Δ_aΔ_bF = 0 for all a, b hold exactly when F is a constant plus an additive map; the Möbius expansion is unique, including over F2 and Z4.Source: §2.8.1 · p. 26

2.8.2 Symmetric binary spaces

• The binary-space framework uses orthogonal complements, radicals, and standard characteristic-two Witt facts about nondegenerate symmetric spaces.In particular, even-dimensional nondegenerate spaces have self-dual subspaces, while odd-dimensional ones are nonalternating and contain an anisotropic vector.Source: §2.8.2 · pp. 26–27

2.9 Bell signatures and the two affine cores

• A retained six-port core cH6 exposes Z and Y by Turing reductions and yields an actual four-port product and binary signature.The direct gadget produces 2c Z ⊗Y, not either factor separately; a later circuit gives H6 → RM8, with no inverse gadget used.Source: §2.9 · p. 29

2.10.1 The standard signature classes

• A nowhere-zero signature belongs to the standard affine class exactly when its multiplicative third-difference ratio is identically 1.The criterion allows repeated directions.Source: §2.10.1 · pp. 29–30
• The standard classes A, P, L, flat-Lagrangian, and character-flat use common K-presentations with one basis for the edge and the entire signature set.Source: §2.10.1 · pp. 31–32

2.10.2 The Eulerian-orientation predicate

• EOTract requires one shared sign and one shared class A or P to satisfy a support condition and place every oriented-pairing restriction in that class.This oriented predicate is equivalent to the source papers’ unoriented-pair predicate, with the same exact polynomial-time tests [MWXZ25] [GSS26].Source: §2.10.2 · pp. 32–33

2.10.3 The tractability predicates

• Tract packages the broadest currently known general boundary into seven alternatives; after preprocessing, truth marks polynomial-time cases and failure marks #P-hardness [MWXZ25] [GSS26].Source: §2.10.3 · p. 33
• With binary disequality or theorem-coordinate anchor Z, Tract reduces to four clauses: membership in ⟨T⟩ or A-, P-, or L-transformability.The theorem coordinate images contain Z, so their TractX tests equal the full Tract criterion.Source: §2.10.3 · pp. 34–35
• A finite algebraic set containing X can satisfy L-transformability while failing ⟨T⟩, A-transformability, and P-transformability, making the local-affine clause essential.Source: §2.10.3 · p. 35
• The constructed signature g is tensor-prime and cannot be placed in P by an invertible local basis change, showing local-affine transformability cannot be absorbed into the other alternatives.The local-affine alternative remains indispensable for this binary-disequality dichotomy, although at the endpoint-nondegenerate quaternary eight-vertex boundary every local-affine presentation collapses to an affine one.Source: §2.10.3 · p. 36

2.10.4 Effective algebraic geometry

• For the stated transformability classes, a complex witness implies a witness in GL2(Q), and exact algebraic inputs permit deciding witness existence and producing an algebraic witness.The proof enumerates finitely many normal-form types and encodes each as polynomial equalities and inequations; Gröbner-basis methods decide branch emptiness, and exact algebraic sample points yield witnesses.Source: §2.10.4 · pp. 37–39

2.10.5 Imported dichotomies and decomposition

• For any finite preprocessed algebraic signature set F containing a nonzero odd-arity signature, Holant(F) is in FP when Tract(F) holds and #P-hard otherwise [MWXZ25].If supp(hw) ⊆ {0^dw, e1, …, edw}, then ZX(Γ) is computable in O(|V(Γ)| + |Eint(Γ)|).Source: §2.10.5 · pp. 39–40

2.10.6 Decidability and polynomial-time evaluation

• The predicate Tract(F) is decidable for every finite algebraic signature set, and when it holds, Holant(F) is in FP.The proof reduces the alternatives to finite exact tests, then uses the corresponding Eulerian, arity-two, matching, affine, product, or local-affine evaluation procedures.Source: §2.10.6 · p. 41

2.11.1 Projective tensor geometry

• The Segre variety consists of projective classes of decomposable tensors; its ideal is generated by flattening minors, and its projective lines lie in one-factor rulings.CP rank, the least number of simple summands, is distinct from the matrix rank of any individual flattening.Source: §2.11.1 · p. 42

2.11.2 Möbius geometry

• Möbius transformations of the Riemann sphere carry generalized circles—Euclidean circles or affine lines together with infinity—to generalized circles [Bea83, Chapter 3].The sphere is identified with the projective line, with infinity represented by [1 : 0].Source: §2.11.2 · p. 42

2.11.3 Root configurations and incidence

• A reduced finite root system is a finite spanning set of nonzero vectors in a real inner-product space, invariant under its root reflections and meeting each root line only at the root and its negative.The text introduces the associated root-line configuration.Source: §2.11.3 · p. 42
• Projective cross-ratio defines harmonic quadruples by value −1, while D4 triality acts nontrivially on its three outer nodes through the automorphism action.The passages also define finite projective point configurations and incidence structures with automorphisms preserving distinguished incidences.Source: §2.11.3 · pp. 42–43

2.12.1 Laurent polynomials; 2.12.2 Gauges and product weights

• Laurent-polynomial exponents use integer arithmetic, and portwise gauges are analytic rescalings rather than gadgets unless separately realized.Product weights are separable monomial weights; the passage distinguishes them from genuine characters.Source: §2.12.1 · p. 43, §2.12.2 · p. 44

2.13.1 Hilbert-space conventions; 2.13.2 Pauli and stabilizer conventions

• Hilbert-space conventions use Hermitian inner products distinct from Holant’s bilinear pairing; Pauli labels carry a symplectic form, and stabilizer isometries encode logical Paulis.The passages define Lagrangian label spaces and stabilizer encoders with physical representatives of logical operators.Source: §2.13.1 · p. 44, §2.13.2 · p. 45

2.13.3 Stabilizer error correction

• Erasure of a physical output is correctable exactly when the single-site Knill–Laflamme condition holds.In the stabilizer setting, encoder tableau reduction records physical representatives of logical Paulis and residual stabilizer generators.Source: §2.13.3 · p. 45

2.14.1 Orders and termination

• Well-founded orders and decreasing potentials justify termination arguments, while worklists, canonical keys, and finite certificates specify algorithmic bookkeeping.The multiset extension of a well-founded order remains well founded.Source: §2.14.1 · p. 46

2.15.1 Retained factor saturation

• Factor saturation retains Λ∪Π(Λ), preserving witnesses while a finite, order-independent worklist exposes its canonical tensor-prime factors.The worklist terminates by a decreasing potential; the resulting factors support the retained augmentation rather than replacing it.Source: §2.15.1 · pp. 46–47

2.15.2 Downward inheritance and global normalization

• Tractability is inherited by preprocessed subsets, and GO(X) transport preserves tractability and common presentations while a normalization ledger tracks pullbacks over the frozen initial field.Continuing normalizations are recorded as coordinate changes, and the inverse accumulated change returns the state to the caller’s coordinates.Source: §2.15.2 · pp. 47–49
• At a finite-projective-order residual node, native edges and chaining realize a group of transfer classes because each transfer inverse is a positive power.The identity transfer is realized by the native edge, and chaining realizes products.Source: §2.15.2 · p. 49
• Global normalization leaves G(A, Λ) unchanged, while retained augmentation can only add realizable transfers.Source: §2.15.2 · p. 49
• If Tract(KΛ) holds, downward inheritance and Lemma 2.25 imply Tract(KG0).Source: §2.15.2 · p. 49

3.1 Binary Interpolation

• A rank-one binary B factors into nonzero algebraic unaries, while a nonsingular transfer matrix A of infinite projective order yields a rank-one binary H by interpolation.For m designated H occurrences, interpolation uses m + 1 oracle queries and O(m^3) added vertices in total.Source: §3.1 · p. 50

3.2 Single-occurrence carriers and unary terminals

• The universal one-pair reduction evaluates an instance with one ∆-vertex using at most three oracle queries over Γ.The reduction uses ordered ports; its fixed witness network has constant size.Source: §3.2 · pp. 50–51
• Single-occurrence replication uses a carrier-preserving context to turn access with one g-occurrence into unrestricted access to h, with an O(k)-size nested context for k copies.For h-only instances, the construction adds a fixed nonzero-valued instance before applying the one-g reduction.Source: §3.2 · pp. 51–52
• If adding a nonzero unary is reducible to the original problem, the original problem is #P-hard unless its signature set is tractable; a unary companion with nonzero contraction is also available for any nonzero unary g.The common unary exit applies to both ordinary Holant and K-Holant.Source: §3.2 · p. 52
• Access to EQ_d for d ≥ 4 exposes both unary point masses in ordinary Holant or K-Holant, enabling the common unary exit.The corollary states that access to one nonzero unary is enough to invoke Lemma 3.4.Source: §3.2 · p. 53

3.3 Equality and generalized-equality anchors

• An equality anchor or a directly realized pure G_q = c0δ0^q + c1δ1^q with q ≥ 4 yields #P-hardness or Tract(KG).For the pure generalized-equality case, c0 and c1 are nonzero algebraic scalars.Source: §3.3 · p. 54

4 Endpoint-Nondegenerate Eight-Vertex Signatures

• Endpoint-nondegenerate eight-vertex signatures vanish on odd-weight inputs and are nonzero at both endpoints; their quaternary interface supports the stable matching-deck proof and independent quaternary dichotomy.Source: §4 · p. 55

4.1 Product and local-affine exceptions

• P-transformable signatures are pure generalized equalities, products of two nonsingular weighted equalities, or full-even-support dense signatures whose realizing sets also realize equality anchors.Every such signature has support size 2, 4, or 8. The local-affine case collapses to affine.Source: §4.1 · pp. 55, 57

4.2 The finite affine quaternary certificate

• The affine base case admits an algebraic diagonal normalization under which support is a generalized-equality pair, a four-word block pattern, or all even words.The finite certificate reduces the analytic classification to eighteen cores. The support-four case yields a genuine factorization into nonsingular weighted equalities, Turing-exposed only after retained factor saturation.Source: §4.2 · pp. 59–60
• Legal diagonal normalizations are tracked in one ledger, while direct constructions reach already-resolved routes, including a hard support-six signature outside the A-, P-, and L-transformable classes.The support-six signature is hard by Theorem 2.20; the retained proper-factor route composes back to the original signature set.Source: §4.2 · pp. 61–62
• The finite affine certificate leaves only support four with AD = BC as a continuing route, where the original signature factors across two two-port blocks into nonsingular weighted equalities.A finite retained, factor-saturated augmentation keeps the complete factor batch and a weighted-equality matrix as a Turing-exposed proper factor, not a directly realizable signature.Source: §4.2 · pp. 61–62

4.3 Eight-Vertex Synthesis: The Signature-Set-Wide Interface

• The signature-set-wide interface gives three outcomes: #P-hardness, a tractable presentation, or a retained augmentation carrying a weighted-equality factor from the original signature.The equality is the sole unresolved output at this stage; the interface avoids invoking the later equality-accessible closure, and hardness reductions compose toward the input set.Source: §4.3 · pp. 62–63
• For a tensor-prime endpoint-nondegenerate quaternary, failure of the tractability criterion implies #P-hardness, and the criterion is decidable.The argument uses the interface to rule out its tractable and continuing factorization outcomes under tensor-primality, then transfers hardness through preprocessing.Source: §4.3 · p. 63

5 Equality-Accessible Matching Decks

• Literal binary equality makes the native X-wire interface equivalent to ordinary equality wires with X available as a binary signature.The conversion preprocesses X-components into equality or X kernels, using an EX edge for each X kernel and accounting for cycle scalars.Source: §5 · p. 64

5.1 Binary transfers, complete groups, and local quaternary dispatch

• If every directly realizable nonzero nonsingular binary has finite projective order, their projective transfers form a finite subgroup of PGL2(L).At a binary-residual state, this yields a finite complete safe transfer group.Source: §5.1 · pp. 65–66
• The binary completeness test dispatches singular binaries to a factor/unary exit, infinite-order binaries to interpolation, and otherwise detects strict safe-group enlargement via an actual binary.If the groups are equal, every nonzero actual binary is projectively one of the fixed representatives.Source: §5.1 · p. 66
• The finite quaternary dispatcher either reaches an established terminal or successor, witnesses strict group growth, or continues with a minimum-arity nonbinary prime whose cards are zero or represented in the safe group.On this continuing branch, every nonzero card has a trivial factor forest; the boundary does not invoke the global matching-synthesis measure.Source: §5.1 · p. 67

5.2 Stable matching decks and restart control

• A matching deck contains zero and all direct one-copy proper cards formed by dressing selected ports, closing one ordered port pair, and retaining at least two ports.Cards use fixed actual binary representatives and are direct gadget outputs, not analytic linear combinations.Source: §5.2 · pp. 68–69
• Stabilization either resolves a terminal or tractable outcome, or returns a successor whose nonbinary primes remain even and whose deck contains only zero or group-matching signatures.The construction starts from a finite factor-saturated algebraic signature set containing literal equality and a minimum-arity even-arity tensor-prime nonbinary signature.Source: §5.2 · p. 70
• Under frozen-token coverage, same-stratum transitions must process a frozen queue token and decrease its counter or factor potential; otherwise, the group enlarges or arity decreases.This prevents stable-state restarts from cycling.Source: §5.2 · pp. 71–72

Refine it to the full within-node queue measure

• A lexicographic measure decreases through group or arity changes, frozen-token completion, or factor-potential decrease; equal-arity prime replacement is charged to a separate outer induction.“Protocol-discharged” applies only after concrete physical closure and does not itself establish realizability, factor access, or stratum closure.Source: Refine it to the full within-node… · pp. 73–74

5.3 The main stable-state theorem and four-way routing

• If Tract(K(G)) holds, K-Holant(G) is in FP; otherwise it is #P-hard, with TractX(K(G)) giving the equivalent tractability condition.The deck-stable state contains the distinguished literal equality I.Source: §5.3 · p. 74
• Every nonhard rigidity survivor has a common product, affine, or flat-Lagrangian K-presentation, which also places the transformed edge in that class.Source: §5.3 · p. 74
• The fixed-equality marks refine the finite projective-group classification into four families, dispatching stable tuples to the arguments in Sections 6–9.The routing uses the marks’ membership and conjugacy-class data without implicit conjugacy normalization.Source: §5.3 · p. 75
• With literal equality fixed, Theorem 5.14 routes complete groups into four marked cases: dihedral, even cyclic, V4, or tetrahedral, octahedral, and icosahedral forms.The dihedral forms are Mm, Bm, Am, and A0m; the V4 forms are standard Pauli and exotic; the Platonic forms include T, Text, four S4 forms, and two fixed-I A5 displays.Source: §5.3 · p. 75
• The classification retains the ordered physical marks x=[X] and j=[Z], rather than re-anchoring the supplied literal equality or identifying forms merely by abstract group type.The marked routing refines unmarked group types; its displays are physical refinements, and the paper independently proves each marked stratum.Source: §5.3 · pp. 75, 78, 79
• Each group-specific theorem either determines complexity at its stable tuple or yields a transition to a strictly smaller active tuple; any enlargement of G requires stabilization and rerouting first.The outer worklist induction separately handles movement between distinct same-arity primes.Source: §5.3 · p. 76

5.4 Strict growth of the complete binary group

• When recomputation gives G < G+, the marked type of G+ must be reachable along Figure 1’s directed path, but the arrows do not guarantee that every source state realizes each transition.Even if an exposed binary is already represented in G, augmentation can reveal additional multi-copy binaries, so G+ is the complete semantic closure rather than merely the subgroup generated by the first new class.Source: §5.4 · p. 79, Figure 1 · p. 80
• Successor routes depend on whether j enters the recomputed closure: cyclic CN routes to Am or A0m, while exotic A2 enters Am only when m ≡ 2 (mod 4).For the Pauli V4 form, a dihedral overgroup routes to Mm or Bm with even m; when 4 divides m, adjoining j to A2 first gives the fixed-I A0 form.Source: §5.4 · pp. 79, 81
• Every continuing transition strictly increases the complete binary group, so bounded group orders make the restart graph acyclic; the finite-group classification also rules out further strict overgroups of S4 and A5.Text cannot grow to A5 because that would force an S4 subgroup despite 24 not dividing 60.Source: §5.4 · p. 81

5.5 Shared methods for the four branch proofs

• The effective-kernel algebra is either span{I, X} or Mat2(C): cyclic routed groups have the proper algebra, while dihedral, Klein-four, and Platonic groups have the full algebra.In the full-algebra case, evaluations against four actual chain matrices reconstruct every two-port fibre algebraically; in the proper case, the Frobenius-orthogonal complement is invisible.Source: §5.5 · pp. 82–83
• An actual gadget using at most two copies of q yields an endpoint-nondegenerate eight-vertex signature that is not G-matching under the theorem’s stated group cases.The resulting signature can be passed to the established quaternary interface.Source: §5.5 · p. 83

6.1 Trace covers and alternating-cycle localization

• Compatible trace-cover closures preserve nonvanishing and protected discrepancies while shortening alternating cycles or reducing residual typed tags.Two typed tags can be cross-closed while preserving a distinguished discrepancy, leaving at most one residual tag.Source: §6.1 · pp. 86–87
• Minimal disagreements localize to one of three bounded residues: a matching-disagreement six-port card, a two-tag six-port card, or a one-tag quaternary with a protected minor.The first two configurations cannot arise from cards of one fixed three-factor product.Source: §6.1 · p. 88
• Each residue has a consumer that reaches a quaternary boundary or factorization, and complete protocol closure discharges every typed matching, parity, or coefficient-line disagreement.A protected-cycle or two-tag six-port card contracts to an actual one-tag quaternary; iteration yields a resolved leaf or strict successor.Source: §6.1 · pp. 89–90
• On the nondischarged branch, each sector pencil’s nonzero sampled cards share one fixed perfect matching and have at most one varying Segre factor line; all other binary factors are fixed projectively.Source: §6.1 · p. 90
• Live-sample disagreements are protocol-discharged; otherwise each sector has one residual matching and at most one varying factor, whose samples lie on a single ruling.The Segre-line criterion gives a common ruling when there are at least three live samples; with at most two, their span is already the allowed one-factor ruling.Source: §6.1 · p. 90

6.2 Endpoint directions and the parallelogram test

• For m ≥ 3, a Möbius map permuting the mth roots of unity is ϕ(z) = αz or ϕ(z) = α/z; the sampled monomial pencil is proportional or has a complementary-axis normal form.The normal forms are K(t) = λ(E0 + αtE1) or K(t) = λ(tE0 + αE1); the binary factor is an actual factor of a higher-arity card, not an independently realized gadget.Source: §6.2 · pp. 90–91
• In the C/C branch with one common varying edge, actual live-card residual matchings agree exactly when the endpoint parallelogram condition holds.Failure is localized using actual cards, not endpoint slices.Source: §6.2 · p. 93
• Endpoint pairs containing Z or P are discharged through impossibility, exact factorization, a quaternary factor, or a typed disagreement.The quaternary factor is Turing-exposed as a strict lower-ν successor, not treated as a directly realized gadget.Source: §6.2 · p. 93
• The four-flat branch either has support equal to a three-edge matching coset or yields an actual contraction with a nonzero quaternary of support size six or eight.The latter output is not a product of current-group monomial binaries and cannot re-enter a six-port state.Source: §6.2 · p. 95
• The sole nondischarged endpoint survivor is a C/C pair with the endpoint parallelogram condition and one compatible residual matching.The subsection states this survivor for every deleted pair.Source: §6.2 · p. 95

6.3 Global support integration and the Reed–Muller alternative

• The global support-integration subsection begins from the survivor’s endpoint parallelogram and compatible residual matchings for every deleted pair.The supplied passage introduces the integration of these local conditions but does not state its outcome.Source: §6.3 · p. 95
• The nondischarged support is a global matching coset: support integration yields a self-dual code with matching and Reed–Muller alternatives, then excludes the latter.The Reed–Muller case requires n = 8 and is incompatible with the normalized μ_m-dihedral deck for m ≥ 3.Source: §6.3 · p. 97
• For m ≥ 3, an actual phase t ∈ μ_m \ {±1} produces a nonzero minor contradicting the fixed Segre ruling, excluding the Reed–Muller support residue.This obstruction establishes a support identity only; coefficient separation is handled in the following stages.Source: §6.3 · p. 98

6.4 The six-port coefficient calculation

• The six-port dispatcher resolves zero, partial-live, and support-disagreeing q4 cases rather than treating them as stopping outcomes.A zero q4 yields an exact factorization of its parent q6; unresolved cases continue through the factor-first consumer or retain the q6 only with all eight logical coefficients nonzero.Source: §6.4 · pp. 99–100
• On continuing branches, one-coordinate q6 minor localization forces rank-at-most-one flattening equations, or a current-group matching product.The bounds hold for all three choices of singled-out coordinate and are recorded before child tokens are completed.Source: §6.4 · pp. 100–101
• On the continuing full-support branch, actual parity contractions equate complementary coefficient products, while physical factor cards expose binary ratios α±1, β±1, and γ±1.The cards are formed by cycling through the three matching edges; noncurrent or unsafe factors exit the continuing branch.Source: §6.4 · p. 101
• For a nonproduct affine residue, the cross-ratio condition gives r + r^-1 = -6; at r = -1, two-copy wiring yields 4EQ4, and every other nonproduct case not already terminated enters Corollary 4.7.When r = 1, the parent six-port tensor is the product of its three matching binaries.Source: §6.4 · pp. 102–103

6.5 Polarized coefficient integration

• Phase-grid interpolation and rank-one flattenings establish an exact tensor product of binary signatures on the global matching.Mixed two-copy cards and cube-edge integration make each edge ratio independent of the other coordinates; for m = 4, phases must range over all of µ4 to avoid squaring away a nonproduct sign.Source: §6.5 · p. 104

6.6 Reduction semantics and completion

• For Stable(Λ, Mm, f), Even(Λ), arity(f) = ν(Λ) = 2k + 2, and m ≥ 3, Theorem 6.24 concludes Close{P}(Λ, Mm, f).The nondischarged branch has global matching support and an exact binary product; relative re-entry and well-founded occurrence induction extend the active-prime analysis across the unresolved-prime worklist.Source: §6.6 · p. 106

6.7 The two nonstandard marked dihedral sectors

• Marked-sector coordinates transport the root pencils and fusion law to Bm and Am while keeping analytic rescalings distinct from physical gadgets.The transported monomial fusions preserve unique internal lifts and therefore introduce no cancellation.Source: §6.7 · pp. 107–108
• The six deleted pairs have exact support patterns: each nonempty solution is one total parity together with a nonempty subset of its four complementary word pairs.Actual sector-X dressings send a live complementary pair to 0000 and 1111, leaving other live pairs among the three complementary weight-two blocks.Source: §6.7 · pp. 108–109
• For nonsingular symmetric binaries E and F in A, P, or L, the pullback EFE remains in the same class.The proof reduces the affine and product cases directly; in the local-affine case, every nonsingular symmetric member of L lies in A.Source: §6.7 · p. 109
• The Bm-sector reduction yields a common witness: A and L cases resolve as tractable leaves, while P is the continuing rigidity leaf.The reduction is signature-set-wide; its whole-problem change is not a local gadget or a GO(X)-ledger update.Source: §6.7 · pp. 109–110
• The A0_m reduction preserves literal equality and transports the deck between native-X and Z-edge presentations, yielding a common witness for the original problem.The open-network equivalences preserve deck properties, and the A and L cases resolve as tractable leaves while P remains the rigidity leaf.Source: §6.7 · pp. 110–112
• Under the stated safe-deck condition, the Am quaternary analysis yields a terminal or strict group successor, a retained binary factorization, or an endpoint-nondegenerate eight-vertex signature.The last alternative is an actual signature in current physical coordinates; its original sector-coordinate signature also satisfies the stated condition.Source: §6.7 · p. 112
• In the Am quaternary analysis, two active blocks force a safe binary outside the group, one active block gives a 2 + 2 factorization, and no active block yields a generalized equality.The no-active-block branch also constructs an actual current-coordinate endpoint-nondegenerate eight-vertex signature using available phase rotations.Source: §6.7 · p. 113
• The marked q4 consumer retains and factor-saturates each actual q4, tests all three physical 2 + 2 flattenings, and dispatches exposed factors before invoking Lemma 6.30.On the remaining branch, q4 is tensor-prime with a complete physical deck that is zero or safe-group-valued; analytic endpoint support alone is not a physical endpoint-nondegenerate q4.Source: §6.7 · pp. 113–114
• A gauge-coherent comparison batch uses a shared ordered contraction multigraph, edge orientations, and external-port order, varying only designated phase or channel choices.The same parent-port set S is used throughout the batch; unrelated contraction skeletons need not induce the same set.Source: §6.7 · p. 114
• Gauge transport preserves support, zero or partial-live status, matching directions, flattening ranks, and exact factorization; coherent comparison batches share one locally gauged parent.Nonzero two-channel traces have at most one forbidden phase, reversal preserves the finite phase set, and safe ordered ratios lie in δ−1H.Source: §6.7 · p. 115
• Primitive transport preserves the root-pencil and support arguments, while fixed δ-weights rescale Reed–Muller obstructions and reparameterize polarization tests by nonzero factors.Safe logical ratios are normalized into H and the final physical quaternary is routed through the marked q4 consumer.Source: §6.7 · p. 116

Consequently every branch either

• Occurrence induction sends each deck branch to a resolved leaf or strict successor, or returns an exact factor batch to lower-arity processing; bounded outputs are physical q4 or q6.Each continuing branch decreases the occurrence completion measure, and constructed child-card factors are processed within the selected occurrence’s local queue.Source: Consequently every branch either · pp. 116, 118, 119
• Under Stable(Λ, Am, f), Even(Λ), and even m ≥ 4, Am-sector deck rigidity establishes Close{P}(Λ, Am, f) with one common product witness for the retained set.The excluded case m = 2 is a Klein stratum.Source: Consequently every branch either · pp. 118–119

7.1 Cyclic phase pencils and physical dressings

• For N ≥ 3, RN(a, b, c, d) holds exactly when every s ∈ µN gives a matrix that is zero or projectively Dη for some η ∈ µN.The equivalence includes zero members and the all-zero quadruple.Source: §7.1 · p. 120
• Under the phase-contraction conditions, an even Walsh table has either two nonzero endpoints with ratio in µN or zero endpoints and cyclic central blocks; exactly one nonzero endpoint is impossible.In the zero-endpoint case, each central block is zero or ρI(1, αI), with ρI ∈ Q× and αI ∈ µN; conversely, tables in these alternatives satisfy the phase-contraction property.Source: §7.1 · p. 121
• The cyclic boundary uses actual oriented binary gadgets attached through native X-edges, while the Walsh transform is analytic rather than a physical transformation.The fixed representatives carry recorded nonzero scalars, and the four-port dressing normalizes its direct-gadget boundary tensor by those scalars.Source: §7.1 · pp. 119–121

Define the map FN : µ4

• For odd N, F_N is bijective.The supplied construction chooses r1 with r1^2 = uvw/R, then sets r2 = u/r1, r3 = v/r1, and r4 = w/r1.Source: Define the map FN : µ4 · p. 122

7.2 Quaternary entry and theorem statement

• At the quaternary boundary, the proper-cyclic branch yields an unsafe binary, a strict group enlargement, or a tensor-prime endpoint-nondegenerate eight-vertex signature.The boundary returns a resolved leaf, an outer deck successor, or Tract(KΛ).Source: §7.2 · p. 122
• When zero endpoints occur, some minor ∆i is nonzero, yielding a tensor-prime endpoint-nondegenerate eight-vertex output.If all three minors vanished, the core would factor across a binary-by-binary split, contradicting the retained tensor-prime-core hypothesis.Source: §7.2 · p. 124
• With nonzero endpoints, successful dressing produces an endpoint-nondegenerate eight-vertex signature, while the square-obstruction case yields either such a signature or a strict group successor.For three active blocks, dressing works when the signed product ϵ1ϵ2ϵ3α1α2α3 is a square; if 4 | N, this is possible precisely when α1α2α3 is a square.Source: §7.2 · pp. 124–126
• Theorem 7.5 identifies a sole rigidity branch after the quaternary boundary resolves arity four or passes it to a strict outer successor.The stated theorem assumes a stable state with arity(f) = ν(G) and KG = span{I, X}, and concludes Close{FLag}(G, G, f).Source: §7.2 · p. 126

7.3 Reconstruction of the minimum Reed–Muller core

• For minimum-core arity n > 4, the zero-card graph has no edges; every connected component is otherwise complete bipartite.On the continuing branch, odd arity is handled by hardness or Tract(KG), so the remaining arity is even.Source: §7.3 · pp. 126–127
• For n ≥ 6 with no zero-card edges, the weight-two graph Wf is a matching, nonzero cards have equal support matchings, and every Wf edge is a global Walsh equality.The comparison also shows every component of Wf is a path or cycle.Source: §7.3 · pp. 127–128
• If W_f is nonempty, either n = 6 yields a constant-size gadget for the even-parity signature of arity four or f has a nontrivial tensor factorization; in particular, n ≥ 8 forces W_f = ∅.The nonempty branch therefore produces an equality anchor or a proper factorization before the rigidity argument continues.Source: §7.3 · pp. 129, 131
• When E(Z_f) = ∅ and W_f = ∅, common card matchings reconstruct the support, and the surviving core has arity eight with support RM(1, 3).The equal-pair slices have weight divisible by four, and the reconstructed support is the Reed–Muller code RM(1, 3).Source: §7.3 · pp. 131–132
• On the eight-port RM(1, 3) support, the coefficient atlas reduces coefficients to a linear sign character, and available X-dressings yield the standard core RM8.The passage identifies RM8 in lexicographic order as the Shao–Cai signature f8.Source: §7.3 · pp. 132–133
• The within-C2 continuation proves signature-set-wide rigidity: the retained signatures and their edge share one flat-Lagrangian presentation.Its exact Reed–Muller network calculus, Radon access, localization, affine patching, and recognition also supply structural tools used in later constructions and proofs.Source: §7.3 · pp. 133–134
• On the continuing matching-deck branch, the coefficient atlas rules out every safe transfer class beyond [I] and [X].This is the safe-group collapse after the Reed–Muller core has been fixed.Source: §7.3 · p. 134
• The quotient r_zp=b_T(z)/b(z) is a character of D taking values ±1, so evaluating at 18 identifies the physical transfer classes as [I] and [X].Source: §7.3 · p. 134

7.4 Flat-Lagrangian networks and Radon access

• RM8/I/X networks with I/X internal connections have zero or affine-coset flat-Lagrangian boundaries, and polynomial-size networks realize every such relation.Every nonzero boundary produced this way has even arity.Source: §7.4 · pp. 134, 136
• Orthogonal maps preserve parity and fix 1n, so neither 0 nor 1n lies in the orbit of a weight-two vector; this establishes the necessity of excluding those directions.Source: §7.4 · p. 137

7.5 Relative flat-Lagrangian closure and completion of Theorem 7.5

• Every flat-Lagrangian signature of arity at most six is a product of projective I/X binaries on a perfect matching; a nonmatching flat-Lagrangian proper card can first occur at arity ten.The appearance of such a card is not by itself a protocol exit.Source: §7.5 · p. 138
• For even arity n≥10, if both accessible Radon cards are zero or flat-Lagrangian, nonaffine orbit-sum gluing yields a nonzero nonaffine signature of arity at most eight via an O(n^2)-size RM8/I/X context.The two charts glue affinely exactly when their support directions agree and their phase ratio lies in µ4.Source: §7.5 · pp. 138–140
• For even n ≥ 10, if every accessible orbit sum Fv is affine, affine hyperplane patching implies that g is affine.The proof uses Walsh-transform invariance of the affine class to pass from affine restrictions to a global conclusion.Source: §7.5 · pp. 140–141
• An affine signature of even arity at least eight is flat-Lagrangian if every accessible-direction Radon card is zero or flat-Lagrangian.The recognition lemma uses the dimensions and phases of the actual cards to force self-dual support and constant coefficients.Source: §7.5 · pp. 141, 143
• Every retained tensor-prime occurrence is certified flat-Lagrangian, so the retained signature set and its edge share one flat-Lagrangian presentation.The closure argument rules out surviving nonflat occurrences; tensor closure and retained factor saturation give the common presentation.Source: §7.5 · p. 145

7.6 Character-flat Pauli tools for the later standard- V4 argument

• Networks of RM8 and ordered Bell binaries with native X-edges have zero or character-flat boundaries, and every nonzero arity-2k character-flat signature has an O(k^2)-size network realization up to scalar.The boundary support is self-dual; cancellation either removes each feasible boundary fiber or leaves a common scalar and a boundary character.Source: §7.6 · p. 146
• Every nonzero character-flat signature of arity at most six is, up to port permutation and nonzero scalar, a product of ordered Bell binaries on a perfect matching.Character-flat signatures are also closed under nonzero scalars, tensor products, port permutations, one-port ordered-Bell attachments through native X-edges, and nonzero tensor factors; the network calculus separates H6 from such networks.Source: §7.6 · p. 146
• Ordered Bell attachments implement shifts and linear characters; splitting self-dual directions into repetition lines realizes their factors as ordered Bell signatures.Nonzero factors of a character-flat tensor are character-flat as well.Source: §7.6 · pp. 146–147
• No network using only RM8, actual ordered Bell binaries, and native X-edges can have boundary proportional to H6.Such nonzero boundaries are character-flat, with support dimension three, whereas H6 has the six-bit even-parity support of dimension five.Source: §7.6 · p. 147
• With RM8, actual ordered Bell binaries, and native X-edges, one copy of even-arity g directly realizes signed cards whose orbit numerators are g(x)+g(x+v) and g(x)−g(x+v).The realization is up to a known nonzero scalar and a recorded exterior Bell sign.Source: §7.6 · p. 147
• If the orbit-sum signature Fv is not in A, one copy of g yields a nonzero signature outside A of arity at most eight.This holds for even-arity g of arity at least 10 under the two-chart character-flatness hypothesis.Source: §7.6 · pp. 148, 150
• If nonzero g in A has even arity at least 10 and every signed card is zero or character-flat, then g belongs to CFlat.The proof reduces g to a linear sign character on a self-dual support.Source: §7.6 · pp. 150, 152

8 Klein-Four Matching Decks

• The standard-Pauli and exotic Klein subcase theorems directly synthesize the main theorem, with no separate generic Tract survivor in either stratum.The section treats the standard and exotic finite-certificate statements separately.Source: §8 · p. 153

8.1 The standard V4 form: Pauli cards and exact fusion calculus

• Exactly four nonzero coefficients in the M-Bell basis characterize a single alternating four-cycle between M and N; every such recoupling has four nonzero coefficients.After affine reparametrization, the two varying M-edge labels share an invertible linear part, while other labels are constant.Source: §8.1 · p. 155

8.2 The Pauli cores

• Under the no-exit condition, a normalized full-V4 even core is quaternary or locally Pauli-equivalent to H6 or RM8; other outcomes are matching products or have a genuine tensor factor.For fixed-ruling tensors, no irreducible case exists at arity at least ten.Source: §8.2 · pp. 156–158

8.3 The direct H6-to-RM8 circuit

• The four-copy H6 circuit realizes RM8 using legal I-paths, with each connection forcing equal Bell labels without cancellation.The circuit retains the complete safe group V4 and does not enter the proper-C2 branch.Source: §8.3 · p. 158, Figure 2 · p. 158
• For a terminal-clean RM-only full-V4 state, Theorem 8.8 provides a retained RM8 augmentation whose stabilized protocol yields a common character-flat presentation or another listed terminal outcome.The common affine presentation is signature-set-wide.Source: §8.3 · p. 159
• On the nondischarged branch, the induction places each reachable prime of even arity at least ten in CFlat, yielding a common character-flat presentation for the transported signatures and edge tensor.The argument uses affine hyperplane patching and tensor closure; a single global normalization ledger keeps all occurrences in the same coordinates.Source: §8.3 · p. 161

8.4 The H-anchored actual family of realized signatures

• With literal H6 and RM8 anchors retained, the subsection defines actual H6-anchored residual and signed-Radon cards, then completes a finite realized family with recorded networks, wirings, port orders, and realization scalars.The residual cards use selected disjoint port pairs and actual Pauli dressings; the family includes ordered projective flat-Lagrangian signatures of bounded arity.Source: §8.4 · pp. 162–163

8.5 The coherent Bell-syndrome projector

• The coherent Bell-syndrome projector yields an affine-coordinate lift of W ♯, and its complete family of actual contexts supports the reduction K-Holant(Λ, h, Π(Λ ∪{h})) ≤T K-Holant(Λ, h) ≤T K-Holant(Λ).The displayed projector tensor has support p0 + p1 + q0 + q1 = δb with phase (−1)^ℓbq0; affine changes, a free coordinate, and slicing preserve A.Source: §8.5 · pp. 163–165

8.6 Five-spread gluing

• The five-spread gluing lemma is set up for a nonzero even-arity-at-least-ten signature in a retained, factor-saturated, normalized full-V4 set containing literal H6 and RM8 anchors with predecessor provenance.The subsection defines five two-dimensional direction subspaces forming a spread and uses live masks and comparison tensors for cells lying in a spread coset.Source: §8.6 · p. 166
• Under the prime-factor condition, all live cells share one stabilizer encoding, and their orbit labels are affine up to at most two directions spanning the quotient.The condition applies to every nonzero proper Pauli card, H6-package, and RM8 fixed-syndrome cross-card.Source: §8.6 · p. 166
• A correctable single-site erasure has one of two normal forms: a factored-out stabilizer state or an auxiliary qubit absorbed into a stabilizer isometry.The normal forms hold after a fixed one-qubit Clifford on the erased port.Source: §8.6 · p. 168
• If no single-site erasure is correctable, the encoder has a full-rank column normal form with every column nonzero and a nowhere-zero stabilizer quadratic phase.The displayed form is V|z⟩ = ζ(z)|c + M^Tz⟩, with rank M = k and m_r ≠ 0 for every output port r.Source: §8.6 · p. 168

8.7 Physical support and phase localization

• Under Lemma 8.10’s hypotheses, if g is outside A, then an actual proper Pauli card is nonaffine, a genuine tensor factor occurs, or arity(g) is at most 8.The correctable-erasure case yields either a factor or a proper nonaffine actual card, including when the nonaffinity is phase-only.Source: §8.7 · pp. 170–171
• In every no-witness branch with nonaffine support, the number N of residual code ports is at most 5.The proof rules out N ≥ 6.Source: §8.7 · p. 172
• In the affine-support branch, comparison tensors determine all relative magnitudes and phases after absorbing the standard Bell- and code-encoder phase.Source: §8.7 · p. 172
• A nonaffine tensor has an actual proper Pauli card outside A, a genuine tensor factor, or arity at most 8; in a no-witness phase branch, N is at most 4.The exposed-output cross-card preserves a nonstandard cubic phase, yielding the localization bound.Source: §8.7 · p. 174

8.8 The relative implication

• Under the stated filtering conditions, every continuing proper tensor-prime factor of a full-V4 eight-port base is in A; the realized-signature theorem likewise places qualifying even tensor-prime companions of arity at least 10 in A.The eight-port theorem assumes every nonzero proper actual card is filtered and continuing proper tensor-prime factors are standard affine; the arity-at-least-10 proposition assumes filtered proper cards, H6-packages, and RM8 fixed-syndrome cross-cards have only prime factors in A.Source: §8.8 · pp. 174–175
• After complete protocol closure, the relative-rigidity proof reduces the remaining cores to H6 and RM8, then concludes that every companion and retained signature lies in one normalized affine class.The forward actual circuit supplies literal RM8 for the H-anchored argument; the proof uses bases at arities six, eight, and above.Source: §8.8 · p. 176

8.9 The exotic Klein deck

• The exotic Klein calculus defines a Frobenius-dual basis with pairing ⟨Ea, Eb⟩F = da1[b = ϑ(a)] and (d0, d1, d2, d3) = (2, 2, 4, 4).Its matching-basis recoupling coefficients lie in Q(i)×; support four corresponds to one alternating four-cycle and two varying matching edges.Source: §8.9 · p. 177, Thus define · p. 177

Thus define

• If every exotic pair card of a nonzero six-port tensor is zero or a product of two exotic binaries, the tensor is a product of three exotic binaries, so there is no exotic analogue of H6.The six-port interface covers live, zero, scale, and cancellation cases.Source: Thus define · p. 178
• For even-arity tensors of arity at least eight whose proper exotic cards are zero or matching products, all live cards for each deleted pair use one residual perfect matching.The proof contracts actual exotic cards while preserving two nonzero matching products until a six-port disagreement would result.Source: Thus define · p. 178
• All live p-cards use one residual perfect matching, and the fixed-ruling fiber dichotomy forces permutation label maps when the tensor has no nontrivial positive-arity factor.Nonempty fibers have size one or four; a four-point fiber exposes a genuine edge factor, so under the no-factor hypothesis all fibers are singletons.Source: Thus define · p. 179
• An arity-eight signature whose exotic pair cards are zero or products of three exotic binaries must have a genuine tensor factor.The conditional interface theorem gives each survivor a proper rank-one flattening, hence a genuine proper tensor factor.Source: Thus define · p. 179
• Every signature of arity at least ten whose proper exotic cards are zero or exotic matching products has a genuine proper tensor factor.A dual-cross contraction yields four terms varying in at least three coordinates, contradicting the two varying coordinates permitted for a four-term matching product.Source: Thus define · p. 180
• Every nondischarged rigidity survivor has one common affine presentation; all other branches resolve to a Tract-leaf or an outer deck successor.The arity-four branch is resolved or advances to a strict outer successor, while arity six contradicts primality and arities eight and above yield a factor.Source: Thus define · p. 180

9 Platonic Matching Decks

• For an A4 safe group containing literal I, the marked classification includes the standard T form, where x=[Q1] lies in Text but j=[Q3] and xj=[Q2] do not.The opposite icosahedral display has a coefficient dictionary that preserves zero cards, tensor factors, flattening ranks, matching labels, endpoint support, and transfer exclusion from the complete group, without changing the retained state or literal I.Source: §9 · p. 182
• Deck stability closes internal A4 only for T, while the other listed Platonic groups close with the empty marking.The stable groups are T, Text, O, O′, Odt, Otd, I, and I−; closure is Close_A for T and Close_∅ for the rest.Source: §9 · p. 183

9.1 Common card-map interface and quaternary boundary

• The card-map interface dispatches binary outputs to resolved leaves, strict complete-group successors, or completion of the selected one-copy token.Zero outputs are discarded; nonsingular safe outputs are tested through transfer BX, while rank-one factors are exposed when a concrete binary is adjoined.Source: §9.1 · p. 183
• A directly realized endpoint-nondegenerate eight-vertex signature with matching-flattening ranks (2,2,2) or (3,3,3) implies #P-hardness or Tract(KΛ).In the continuing case, factorization into two nonsingular binaries would force ranks (1,4,4), contradicting either assumed triple.Source: §9.1 · p. 185

If every fixed actual representative Kg satisfies

• If every fixed representative yields a zero or safe transfer for all six canonical pairs, internal A4 and either A5 display have no tensor-prime survivor; arity four factors or reaches a terminal.The possible terminals are unsafe-binary, pure generalized equality, or endpoint-nondegenerate eight-vertex signature; the arity-four process never recurses.Source: If every fixed actual representative… · pp. 185–186
• The classification has no rank-three case: rank zero is zero, ranks one/four factor, and rank two occurs only for S4, where actual representatives expose the stated reductions.For the two rank-two forms, actual inverse dressings expose EQ4 or directly reconstruct the tensors of Equation (142).Source: If every fixed actual representative… · p. 186
• Quadratic interpolation on the tetrahedral quaternion points forces a fixed-matching card-map image into one Segre ruling, so its reconstructed tensor has a proper factor.The interpolation also allows zero test values; it does not require a nonzero test point.Source: If every fixed actual representative… · p. 187
• A deficient six-port card map factors or yields an actual binary terminal or strict complete-group successor, leaving no deficient incidence branch.The finite exits are bounded local consumers: branches resolve, factor to lower arity, or pass to a strict successor, without invoking queue-termination or restart lemmas.Source: If every fixed actual representative… · p. 188

9.2 Rainbow localization and matching-contraction closure

• A rainbow line of three matching products localizes on a unique four-port set; physical kernel closure yields a six-port signature whose other actual cards remain zero or safe matching products.The three tensors share projective binary factors outside the four ports.Source: §9.2 · p. 188
• Across the seven marked rows, rich-line incidence excludes type (2, 1) and mixed S4 four-lines or A5 five-lines; rainbow lines form one actual orbit in the non-external forms.External A4 has only monochromatic three-lines and no rainbow deck line.Source: §9.2 · p. 189
• For a high-arity tensor-prime core, either all nonzero cards share one residual matching and the tensor factors, or actual binary kernels realize a six-port signature retaining the distinguished pair.In the second route, one vertex and exactly (n −6)/2 kernels realize the signature, whose actual cards are zero or safe matching products.Source: §9.2 · p. 190

9.3 The four fixed-equality marked S4 forms

• Each octahedral full-rank map has a resolved local exit: an eight-vertex signature, pure generalized equality, or binary whose transfer lies outside the current group.The exits are realized using each marked form’s actual kernel and at most four actual port dressings.Source: §9.3 · p. 192
• In octahedral deck lines, varying card matchings yield resolved exits, whereas a fixed matching yields a proper factor after form-specific coefficient pullback.The finite factor forest closes through lower-arity nonbinary factors, terminal singular or unsafe binaries, or safe binaries that strictly enlarge the recomputed group.Source: §9.3 · pp. 192–193

9.4 The internal A4 form

• The tetrahedral bridge has exactly twelve ordered maps: six rank-two permutations and six rank-four permutations, with the latter split into two orientation classes.For one class, every deleted-pair contraction by each tetrahedral kernel is a matching product; in the other, actual contraction and dressings yield endpoint-nondegenerate eight-vertex signatures of ranks (2, 2, 2).Source: §9.4 · pp. 193–194
• Every orientation-preserving tetrahedral bridge tensor is standard affine, and directly realizes literal H6, with four copies directly realizing RM8.These are reversible actual port-action identities; the stated reductions proceed from the bridge tensor to H6 and RM8.Source: §9.4 · p. 194
• An eight-port tetrahedral signature whose physical pair cards all lie in C6 either factors across a 2|6 cut or is affine; every tensor-prime survivor is affine.The exhaustive card audit implies any nonaffine parent has an actual card outside C6.Source: §9.4 · p. 195
• If all twelve tetrahedral test values are zero or standard affine, a Clifford change places the map’s image in a stabilizer code with at most four logical qubits.The result applies to a linear map from Mat2(C) to m output qubits.Source: §9.4 · p. 195
• Prepared internal quaternary occurrences resolve into factor, generalized-equality, endpoint-nondegenerate eight-vertex, or outside-binary outcomes, with no tensor-prime survivor in internal tetrahedral or icosahedral form.Under stable tetrahedral hypotheses, relative rigidity yields common affine closure; continuing six-port occurrences lie in M6 ∪ H6, while eight-port occurrences factor or are affine.Source: §9.4 · pp. 199–200
• Tetrahedral deck rigidity closes the stated Six4 state under the affine form, using a stable residue and transporting the common affine presentation back to Λ.The bridge rank is recomputed on the materialized six-port signature; the proof does not require a global restart lemma.Source: §9.4 · p. 200

9.5 The external A4 form

• External two-sided preservers have rank distribution (144, 0, 0, 576) across ranks (1, 2, 3, 4), and rank-one or rank-four maps reconstruct binary-pair factorizations.The reconstructed crossing-binary identities are analytic tensor identities, not actual gadgets.Source: §9.5 · pp. 201–202
• External tetrahedral deck rigidity leaves no tensor-prime survivor or common-affine leaf: the stratum reaches a terminal, group-enlargement restart, or genuine proper factor.At arity four, rank zero gives f = 0, ranks one and four factor, and ranks two and three do not occur.Source: §9.5 · p. 202

9.6 The A5 form √

• The normalized icosahedral bridge has six deficient and thirty full-rank solutions; each full-rank solution yields a quaternary with matching-flattening ranks (3, 3, 3) and an actual outsider.For the standard I display, the outsider lies outside I; the opposite display uses the fixed-equality card and network dictionary.Source: §9.6 · pp. 202–203
• Icosahedral deck rigidity closes the stated state: varying matchings yield a uniform outsider, while fixed matchings expose a proper factor whose forest leads to a resolved leaf or strict outer successor.The argument applies for G = I and, after the dictionary substitution, G = I−; its separator uses rank recomputed on the actual six-port coefficients.Source: §9.6 · p. 203
• At arity four, the cited lemmas and corollary, together with the local-consumer conclusion, close every output.The argument then considers the case n := arity(f) = ν(Λ) ≥6.Source: §9.6 · p. 204
• Proper factor branches close by lowering ν, reaching a binary terminal or strict group successor, or yielding only safe current-group binaries that contradict tensor primality.For the mixed route, one f-vertex and exactly (n −6)/2 actual kernels materialize a nonzero six-port h.Source: §9.6 · p. 204
• On the unchanged prime six-port branch, regenerating the complete oriented deck makes every card zero or a safe current-group matching product, yielding a stable tuple.The proof retains card factors and routes the stable residue by rank and recomputed marked group.Source: §9.6 · p. 204
• Octahedral and icosahedral forms leave no rigidity survivor, while the internal tetrahedral form leaves only the stated common affine class.The internal relative theorem classifies the retained set; the other marked forms have no common-presentation leaf.Source: §9.6 · pp. 204–205

10.1 Synthesis of the Equality-Accessible Matching-Deck Classification

• Frozen-token coverage ensures same-stratum continuations strictly progress: completing a token decreases the queue, while an intermediate factor split decreases its stored factor potential.Across restarts, strict group enlargement, lower minimum arity, queue closure, and genuine factor splits decrease the protocol’s measures.Source: §10.1 · pp. 206–207
• Materialized endpoint-nondegenerate tensor-prime quaternaries close by Theorem 1.3; Corollary 4.7 is reserved for a factorization or weighted-equality successor still requiring processing.After the finite factor/card queue is exhausted, the outcome is a terminal, a successor with smaller measure, or TractX.Source: §10 · p. 205, §10.1 · p. 207
• The stabilized synthesis exhausts the kernel algebras and marked routing, ending each branch in a terminal, direct TractX leaf, or common product or affine presentation.The flat-Lagrangian survivor is a structured affine subcase, while common local-affine conclusions are direct TractX leaves.Source: §10.1 · p. 208
• For retained, factor-saturated signature sets containing equality, Corollary 10.2 gives FP when TractX holds and #P-hardness otherwise.Factor access is by Turing exposure, with each deck regenerated over the augmented signature set and its recomputed complete group.Source: §10.1 · p. 208

10.2 Proofs of the Two Main Dichotomies

• The TractX tractable side comprises arity-two tensor closure and common affine, product, or local-affine alternatives; the flat-Lagrangian survivor is structured affine, not a third case.The equality-accessible closure leaves exactly the two cases of TractX.Source: §10.2 · p. 208
• After preprocessing and equality-wire equivalence, Theorem 1.1’s decidable criterion gives the ordinary binary-disequality problem an FP-versus-#P-hard dichotomy.When the criterion fails, the proof transfers hardness through an equivalent retained signature set containing equality.Source: §10 · p. 205, §10.2 · p. 209

11 Conclusion

• Theorems 1.1 and 1.2 characterize ordinary Holant(F, X) and K-Holant(F) as in FP exactly under their respective TractX criteria, and #P-hard otherwise.The formulations are equivalent: the ordinary-Holant criterion applies to K(F ∪ {I}), while the K-Holant criterion assumes F contains I.Source: §11 · p. 209
• With the binary anchor, Tract reduces to four TractX alternatives: arity-two tensor closure, or common affine, product, or local-affine transformability.The proof uses terminal identification, stable-pair recording, and group-specific analysis of the finite safe binary transfer group, while preserving gadget provenance and reduction direction.Source: §11 · p. 209
• Without assuming I, Theorem 1.3 gives a Tract(K(F)) FP-versus-#P-hard dichotomy when F contains an endpoint-nondegenerate tensor-prime quaternary signature, generalizing Cai and Fu’s single-signature eight-vertex dichotomy [CF23, Theorem 3.1].In the main dichotomy it supplies the quaternary terminal; Reed–Muller, Pauli, and Platonic structures provide rigidity mechanisms without adding tractable cases.Source: §11 · p. 209

Exact finite certificate claims

• The five certificate checks cover endpoint-eight-vertex, equality-accessible, normalized-dihedral, Klein-four, and Platonic claims corresponding to Sections 4–6, 8, and 9.The proper-cyclic branch in Section 7 has no separate finite certificate.Source: A Exact Finite Certificates · p. 210

Verification and scope

• Exact verifiers and deterministic audit payloads in the repository’s certificates/ directory reproduce the specified finite calculations but do not replace analytic normalizations, physical gadget and provenance arguments, or reductions.A coverage map records which entry points check the finite claims.Source: A Exact Finite Certificates · p. 210

Conventions

• The appendix fixes conventions: histogram entry sm means exactly m objects of size s, projective representatives use the first nonzero coordinate, and rank, span, and equality tests use the stated exact characteristic-zero field.The terminology follows specified sections, while the appendix specializes it to the displayed finite state spaces.Source: A Exact Finite Certificates · p. 210

• The subsection provides two exact finite statements used in Section 4.Source: §A.1 · p. 210

A.1.1 The fourth-root odd quaternary residue

• The twelve-loop test declares a rank-one exit for a pair with exactly one zero, an infinite-order exit when nonzero entries have different Gaussian norms, and otherwise a stalled state, retaining zero–zero pairs.The test's exits are exact and disjoint.Source: §A.1.1 · pp. 210–211
• Exactly 64 stalled states satisfy the stated condition; joining ports 0 and 1 of two copies by native X-edges yields an even-parity, support-six eight-vertex signature.The construction contracts two odd-parity signatures along two X-edges, with nonzero endpoint values.Source: §A.1.1 · p. 211
• For the other stalled states, exactly two phase shifts yield standard affine phases: δ∈{0,2} for 176 states and δ∈{1,3} for 128 states.The criterion is that the quadratic coefficients are even and the cubic coefficient vanishes.Source: §A.1.1 · pp. 211–212
• The complete affine odd family has a=(1,A,B,C) and b=(xyzABC,zBC,yAC,xAB), with A,B,C∈μ4 and x,y,z∈{1,−1}, giving 512 distinct normalized states.Source: §A.1.1 · p. 212
• One listed component of the no-rank-one-native-X-loop locus has x,y,z not all equal and (A^2,B^2,C^2)=(xy,xz,yz), comprising 48 states.Source: §A.1.1 · p. 212
• For the remaining 176 states, at least one G0j product is dense product-transformable, and the stated square-equality formulas ensure a test succeeds.Dense product-transformability means g(x)=ρr^wt(x)(−1)^{s·x1}[wt(x) is even] for some ρ, r≠0, and s∈F4.Source: §A.1.1 · p. 212

Proposition A.2 (Exact affine endpoint-nondegenerate eight-vertex certificate)

• The certificate enumerates 36,720 normalized affine signatures across five support counts (16, 120, 140, 30, 1) and retains 4,200 of 36,720 × 18 core-signature pairs.Six cores retain 612 inputs each and the other twelve retain 44 each.Source: §A.1.2 · pp. 212–213
• The three reference-state support strata form 1, 24, and 256 diagonal orbits, and J(g) is a complete invariant of those orbits.The diagonal equivalence is g′ = γ diag(1,t)^⊗4 g for γ,t ∈ Q×.Source: §A.1.2 · p. 213

Proposition A.2(iii) and Theorem A.5(ii)

• For support-four references, AD/BC is 1 or −1: the first case factors into nonsingular weighted equalities, while the second yields a pure generalized equality by a two-copy X-edge gadget.The proposition specifies the literal wiring for the two-copy construction.Source: §A.1.2 · pp. 213, 216

Proposition A.2(iv) and Theorem A.5(iii)

• The 512 full-even references reduce to dense product-transformable forms, rank-one native X-self-loops, or one of three crossed two-copy contractions realizing cE_r ⊗ E_r.The first support-four products among the remaining states occur for 96, 48, and 16 new states, respectively.Source: §A.1.2 · pp. 214, 216

Remark A.3 (Scope and trust boundary)

• The finite coefficient certificate does not establish transform-witness existence, upstream physical realization, global transformation legality, or signature-set-wide reduction; those interfaces remain in the main structural proof.Its support-four identity is checked algebraically rather than by a graph parser, and its implementation and exact integer/rational semantics form the stated trust boundary.Source: §A.1.2 · p. 215

Theorem A.4 (Affine endpoint normal-form interface)

• Under Theorem A.4’s endpoint condition, q = K^⊗4f admits a normalized affine representative up to scalar and diagonal transformation, with support among three specified forms.The alternatives are the complementary endpoint pair, a four-point plane {0^4,1^4,1^I,1^Iᶜ} with |I| = 2, or full even-parity support.Source: §A.1.2 · p. 215

Proposition A.2(v) and Theorem A.6

• Every full-support fourth-root odd table follows a loop-based route; stalled cases yield a support-six endpoint-nondegenerate signature or a certified phase shift with a rank-one loop or dense product-transformable output.Among stalled states, 64 meet the exceptional test; for the other 304, exactly two phase shifts make the table affine, and the resulting family has 336 rank-one-loop and 176 dense-output states.Source: §A.1.2 · pp. 214, 216

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

• For the stated equality-accessible groups, an actual gadget using at most two copies of a tensor-prime quaternary core realizes an endpoint-nondegenerate eight-vertex signature that is not G-matching.The certificate covers zero cards, both port orientations, all six contracted pairs, and fixed nonzero representative scalars.Source: §A.2 · pp. 219–220

A.3 Certificates for Section 6: Marked Dihedral Matching Decks

• Exactly 96 labelled normalized-dihedral supports are matching cosets; every other case admits an injective contraction to a six- or eight-word quaternary support that is not a product of two nonsingular monomial binaries.The exhaustive atlas has 1152 support-six cases and 1056 support-eight cases; the selected actual parity kernel gives a cancellation-free contraction.Source: §A.3 · pp. 220–221

A.4 Certificates for Section 8: Klein-Four Matching Decks

• The appendix presents the standard-Pauli and exotic finite certificate statements separately, in the order used in Section 8.The supplied passage provides no further result for this subsection.Source: §A.4 · p. 221

A.4.1 The standard Pauli V4 form

• For a nonzero six-port tensor with Pauli pair-cards zero or matching products, a live card exists, and dressings and port permutations can normalize its pair while preserving tensor reducibility.At a fixed pair, the four Pauli kernels form a basis, so they cannot all yield zero cards unless the tensor itself is zero.Source: §A.4.1 · p. 222
• Among nonempty allowable five-spread masks, 2,044 have residual dimension zero, 576 dimension one, and 31 dimension two; the last group comprises thirty affine size-eight masks and the full mask.The associated quotient is defined from functions vanishing on spread cosets modulo affine functions.Source: §A.4.1 · p. 224
• For S00=H6, pairwise-disjoint Frobenius supports of S01, S10, S11 and S00, with tensor-prime g, leave only H6: exactly sixteen ordered projective triples with scales (1, ϵ1, ϵ2, ϵ3), ϵj ∈ {1, −1}, yield 128 projective parent lines, all standard affine.Source: §A.4.1 · p. 227
• For five-spread-safe masks, residual dimensions are at most two; nonaffine fibre families have size at most four, while affine nonaffine quadratics have polar rank two or four and respectively three or zero splitters.The affine-support phase atlas further restricts phase-splitting set sizes to 0, 1, 3, or 7; size seven consists of all nonzero forms in a three-dimensional subspace.Source: §A.4.1 · pp. 226–228
• For the standard Pauli V4 form, |S(Q)| is 0, 1, 3, or 7; in the size-seven case, Q is projectively 2L1L2L3 for any basis of the three-space H.The terminal-clean eight-port interface gives a dichotomy between 00 ⊗ H6 and one of 128 certified projective parent lines; if g is tensor-prime, it lies in A.Source: §A.4.1 · p. 228

A.4.2 The exotic Klein form Vex

• In the exotic Klein form, qualifying six-port signatures factor into three exotic binaries; qualifying all-four-live eight-port signatures have a proper tensor factor.The six-port result holds for arbitrary algebraic scales, including cancellation loci, and leaves no tensor-prime survivor. The eight-port claim assumes the all-four-live fixed-ruling normal form and makes no partial-live claim.Source: §A.4.2 · pp. 229, 231

A.5 Certificates for Section 9: Platonic Matching Decks

• The Platonic matching-deck setup fixes a quaternion-basis normalization for localized mixed three-point incidences and defines rich-line matching-label types by residual matching counts.It identifies rainbow three-lines as type (1, 1, 1) and monochromatic lines as types (3), (4), and (5).Source: §A.5 · p. 231

A.5.1 The four fixed-equality octahedral forms O, O′, Odt, Otd

• For fixed-equality octahedral forms, actual kernels and local group dressings certify rank-(2, 2, 2) endpoint-nondegenerate signatures, with additional exits for mixed forms.For each mixed form, the certified alternatives are an endpoint-nondegenerate signature of ranks (2, 2, 2), a pure generalized equality, or a nonsingular binary B with [BX] outside the current group. Coefficient similarities identify tables only; they are not used as gadgets.Source: §A.5.1 · pp. 233–234
• Actual mixed octahedral exits have nonzero determinants and lie outside their groups, with endpoint rows of rank pattern (2, 2, 2); the quadratic test leaves no new octahedral survivor.The mixed exits use actual current-form kernels and dressings; in all four octahedral forms the quadratic evaluation matrix has full column rank, forcing a genuine common factor.Source: §A.5.1 · pp. 235–236

A.5.2 The internal tetrahedral triality certificate

• The tetrahedral deck contains 432 projective points and 1728 matching-class three-point lines, while the internal triality sum equation has exactly twelve maps: six of rank 2 and six of rank 4.The 16 three-point lines on the standard twelve points form one actual left/right A4-orbit; the 1728 deck lines also form one orbit under actual dressings and port permutations.Source: §A.5.2 · pp. 236–237
• The three rejected tetrahedral orientations contract to outputs that are not hidden two-binary matching products, while each admits actual dressings to an endpoint-nondegenerate eight-vertex signature of ranks (2, 2, 2).The three surviving orientations have standard affine ancestors, and the twelve actual binary representatives are affine in the same normalized basis.Source: §A.5.2 · pp. 237–238

A.5.3 The icosahedral forms I and I−

• The icosahedral set is the projectivized binary icosahedral group, and its 10,800-point deck has 200 three-point and 72 five-point coset lines, with no mixed five-line.Each group point lies on ten three-lines and six five-lines; the opposite-chirality form adds no distinguished-involution case.Source: §A.5.3 · pp. 238–239
• Exhausting finite set B3 against sum equation (201) gives exactly 36 ordered solutions: six permutations of (0, A/2, B/2), plus six orderings of each of five unordered frames.The set B is {0, A/2, B/2} ∪ {Ru : [u] ∈ H}.Source: §A.5.3 · p. 239
• The bounded-interface icosahedral classification is a trichotomy: rank lowering, an actual nonmatching quaternary, or a genuine common factor.The common-factor alternative follows symbolically from Lemma 9.6, while the bounded calculation supplies the rank-two solutions and explicit nonmatching cards.Source: §A.5.3 · p. 240

A.5.4 The complete Platonic incidence atlas

• Deck sizes are T 432 points/1728 lines, O 1728/13824, Text 432/1152, and I 10800/216000; mixed octahedral forms share the O counts.Matching-label line partitions are T 1728(1,1,1), 1152(3); O 1728(1,1,1), 4608(3), 2592(4); Text 1152(3); and I 216000(1,1,1), 72000(3), 25920(5).Source: §A.5.4 · p. 242
• The line-count tuples are T (1728, 1152, 0, 0), Text (0, 1152, 0, 0), and I, I− (216000, 72000, 0, 25920), with no type (2,1) lines.The tuple entries count rainbow three-lines and monochromatic lines of sizes three, four, and five, respectively; no octahedral deck has a mixed four-line, neither icosahedral deck has a mixed five-line, and the external tetrahedral deck has no rainbow line.Source: §A.5.4 · p. 244

A.5.3 The icosahedral forms I and I−

• Every rainbow line in the listed Platonic forms lies in one orbit under actual local dressings and port permutations, with witnesses recording the port order and nonzero scalar.The mixed icosahedral forms each have 1728 deck points and 13824 rainbow three-lines in one actual local-dressing orbit.Source: §A.5.3 · p. 241, §A.5.4 · p. 244

A.5.4 The complete Platonic incidence atlas

• After deleting the domain points in any complex subspace of dimension at most two, the rich-line domain graph remains nonempty and connected.The atlas also reports connected intersection graphs of maximal rich domain lines for T, O, and I.Source: §A.5.4 · pp. 242, 244

A.5.5 The hereditary tetrahedral eight-port parent audit

• Across ten representative rich-line orbits, each bridge set has six elements and yields twelve ordered triples, reconstructing 240 projectively distinct eight-port parents.The 84 rich lines through each base point split into three and seven actual-stabilizer orbits, respectively; the representative calculations cover every rich line.Source: §A.5.5 · pp. 245–246
• Four parents also have an exact nontrivial rank-one flattening; for each of the other 180, the audit records a specific physical pair and actual tetrahedral kernel producing the outside card.The outside-card assertion for those 180 parents is existential, not a histogram of all 240 · 336 cards.Source: §A.5.5 · p. 246
• Among 240 audited tetrahedral parents, none has a rank-three distinguished card map; 180 have an actual card outside C6, while 60 have all 336 cards safe and belong to the standard affine class.The audited orbits have sizes |M6| = 25920, |H6| = 31104, and |S6| = 57024.Source: §A.5.5 · pp. 246–247

A.5.6 Deficient mixed-card localization

• A deficient mixed-card map yields a kernel-avoiding rainbow rich line and actual contractions to six support-ten quaternaries in the standard tetrahedral, first-octahedral, and icosahedral forms.For the second octahedral form, corresponding transported quaternaries need not have support ten, but invertible local transport preserves the three flattening ranks.Source: §A.5.6 · pp. 247–248

A.5.7 Two-sided root geometry at the minimum quaternary core

• The two-sided preserver rank counts (ranks 0–4) are A4: 1, 144, 0, 0, 576; S4: 1, 576, 2592, 0, 1152; and A5: 1, 3600, 0, 0, 7200.Thus rank two occurs only for S4, and rank three is absent; the rank-two maps form one actual local-octahedral orbit.Source: §A.5.7 · pp. 248–249
• Every full-rank preserver yields a product of two nonsingular binaries on a crossing matching, while the A4 rank-two zero depends on imposing both preserver conditions.The rank counts apply to all four fixed-equality octahedral forms, whose actual rank-two terminals are specified separately.Source: §A.5.7 · pp. 248, 251

A.5.8 Terminal certificates for the Platonic quaternaries

• Each listed Platonic quaternary has a directly realized terminal: an endpoint-nondegenerate eight-vertex signature, a pure generalized equality, or a nonsingular binary whose transfer class lies outside its group.The certificates cover deficient-rank, orientation-reversing tetrahedral, octahedral separator, and icosahedral frame outputs; each uses one quaternary vertex and at most four actual group binaries.Source: §A.5.8 · pp. 251–252
• Direct current-group kernels certify the deficient Odt/Otd rows; tetrahedral and octahedral calculations exclude their transfer directions, while second-form dressings preserve endpoint nondegeneracy and ranks (2, 2, 2).The second octahedral form is checked in its own coordinates, and its analytic similarity is not used as a physical gadget.Source: §A.5.8 · pp. 252–254
• Exact enumeration of 10,800 actual icosahedral cards found zero zero outputs, 1,260 inside-group outputs, 9,360 outside-group outputs, and zero nonzero singular outputs.The 30 resulting binaries are nonsingular, have distinct projective transfers, and none belongs to the complete 60-point set.Source: §A.5.8 · p. 254
• Each marked octahedral form has exactly six deficient and 24 full-rank maps; actual dressings give the unmixed forms rank-(2, 2, 2) signatures and the mixed forms one of three certified exits.The mixed exits are an eight-vertex signature, a pure generalized equality, or a nonsingular outside-group binary.Source: §A.5.8 · p. 255
• The internal tetrahedral bridge has exactly six deficient and six full-rank solutions; all 180 actual pair cards are safe for Σ+, while Σ− yields a rank-(2, 2, 2) endpoint-nondegenerate signature.The Σ− construction contracts ports 1 and 3 with the actual kernel Q0 = I and applies four actual dressings.Source: §A.5.8 · p. 255
• The 60 displayed icosahedral classes form a projective group isomorphic to A5, whose bridge has six deficient and 30 full-rank solutions; fixed actual kernels give every full-rank solution a rank-(3, 3, 3) quaternary and an outside-group binary.The result holds for arbitrary complex scales and cancellations, with corresponding actual kernels for the opposite display.Source: §A.5.8 · p. 255
• Under the stated card conditions, quaternary ranks cannot be three: ranks zero, one, or four yield the specified zero or binary-pair alternatives, while rank two is confined to octahedral forms and has the stated equality or signature outcome.If two nonzero cards in a pair family use different residual matchings, actual localization instead produces a rank-(3, 3, 3) quaternary and a nonsingular outside-group binary.Source: §A.5.8 · p. 256
• Every enumerated deficient quaternary realizes an endpoint-nondegenerate eight-vertex signature of ranks (2, 2, 2), a pure generalized equality, or a nonsingular binary whose transfer lies outside its group, using at most four actual group binaries.This representative-wise terminal statement covers the tetrahedral-orientation, octahedral-frame, and icosahedral-frame outputs.Source: §A.5.8 · p. 256