PROSE AID ONLY: equations and tables can be damaged by extraction. Use the original PDF/source. PAGE 1 QUANTUM CM RESEARCH / VERIFIED BASELINE 1.0 Correctness Audit of Boolean CM Modal Models Derivations, exhaustive finite checks, model boundaries, and retained failures Prepared for Brian / Quantum CM project Audit date 18 September 2026 Core operations Boolean AND and XOR contraction; permutations, rotation, transpose, and explicitly typed tensor products Deliverables This PDF, editable LaTeX source, independent code, raw evidence, historical source snapshots, and reproducibility instructions Principal finding. The reversible Boolean/modal superposition, interference, Bell/GHZ, teleportation, and dense-coding constructions have valid algebraic realizations. However, they do not all belong to one unchanged operational model. Several earlier conclusions require narrower assumptions, and some do not survive the move from a shared coefficient module to independent phase registers with unrestricted Boolean-linear gates. Most consequential corrections. The coefficient-to-operator embedding is not a tensor equivalence; nonzero shared-module preparations can tensor to zero; the literal universal resource includes phase entanglement; the 64-outcome protocol uses an enlarged gate set; finite-copy no-activation fails in that enlarged model; and the Bell support table has no support-faithful no-signalling probability extension. This is a mathematical and computational audit, not a certification of a physical quantum device, a novel quantum algorithm, or every theorem that could be formulated about CMs. No signed lift, complex amplitude, normalization, or Born rule is used in the core constructions. PAGE 2 Boolean CM: correctness and model boundaries 18 September 2026 Contents 1 Executive findings and the corrected baseline 3 1.1 What survives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Corrections that must accompany those results . . . . . . . . . . . . . . . . . . . 3 2 Audit scope, evidence, and reproducibility 4 2.1 Inputs actually available . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 What “exhaustive” means here . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Native CM logic and a necessary type ledger 5 3.1 The Boolean contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 LM extraction does not supply the modal postulate . . . . . . . . . . . . . . . . 6 3.3 Four models, not one silently changing model . . . . . . . . . . . . . . . . . . . . 6 4 Rotation, the derived phase algebra, and interference 7 4.1 Rotation is native; its algebra is a different type from a CM . . . . . . . . . . . . 7 4.2 The corrected nilpotent and the limit of the imaginary-number analogy . . . . . 7 4.3 Reversible superposition and phase-sensitive XOR cancellation . . . . . . . . . . 8 5 Shared-module structure: what is proved and what is problematic 8 5.1 Units, gates, and Smith classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.2 Separability and the scope of gate classification . . . . . . . . . . . . . . . . . . . 9 5.3 A fundamental preparation-closure failure . . . . . . . . . . . . . . . . . . . . . . 9 6 Literal tensoring is not an entanglement-preserving change of notation 10 6.1 The Choi-style embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 6.2 The universal resource contains more correlations . . . . . . . . . . . . . . . . . . 10 6.3 Local equivalence changes under the larger gate set . . . . . . . . . . . . . . . . . 11 7 Measurement assumptions, Bell tables, and contextuality 11 7.1 The exact operational contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 7.2 Bell contradiction with only three settings per party . . . . . . . . . . . . . . . . 11 7.3 The complete restricted 192-basis result . . . . . . . . . . . . . . . . . . . . . . . 12 7.4 Why the shared and literal counts agree . . . . . . . . . . . . . . . . . . . . . . . 13 8 GHZ: exact successes and retained negatives 13 9 Universal exact teleportation: construction and proof 14 9.1 What “universal” requires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1 PAGE 3 Boolean CM: correctness and model boundaries 18 September 2026 9.2 The conditional branch formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 9.3 A small, explicit, search-free Bell analyzer . . . . . . . . . . . . . . . . . . . . . . 14 9.4 What was executed, not merely inferred . . . . . . . . . . . . . . . . . . . . . . . 15 9.5 The older four-outcome shared protocol . . . . . . . . . . . . . . . . . . . . . . . 16 10 Dense coding, readout, and resource accounting 16 11 Restricted states, quotients, ancillas, and copies 17 11.1 The shared hierarchy remains a restricted theorem . . . . . . . . . . . . . . . . . 17 11.2 The literal restricted hierarchy is different . . . . . . . . . . . . . . . . . . . . . . 17 11.3 Single-copy postselection and product ancillas . . . . . . . . . . . . . . . . . . . . 18 11.4 Finite-copy activation: the old no-go cannot be exported . . . . . . . . . . . . . . 18 12 Orthogonality and the proposed symplectic repair 19 13 No-cloning, exact readout, and what Boolean logic can still do 19 14 A probability obstruction, not just an unspecified Born rule 20 15 Consolidated class and evidence tables 21 15.1 All 15 structured classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 15.2 Exact contextuality coverage counts . . . . . . . . . . . . . . . . . . . . . . . . . 21 15.3 Do not merge the two resource hierarchies . . . . . . . . . . . . . . . . . . . . . . 22 15.4 Evidence ledger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 16 Literature, novelty, and useful research directions 24 17 What may now be used as a foundation, and what remains open 25 17.1 A safe construction contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 17.2 Open or deliberately unclaimed items . . . . . . . . . . . . . . . . . . . . . . . . 25 A Reproducing and inspecting the audit 26 A.1 Raw matrix convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 A.2 Why the evidence is reproducible but not a universal certification . . . . . . . . . 26 2 PAGE 4 Boolean CM: correctness and model boundaries 18 September 2026 1 Executive findings and the corrected baseline 1.1 What survives The most defensible operational baseline is theliteral independent-register modal model: each local system has an eight-dimensional vector space over Boolean XOR/AND arithmetic, nonzero vectors are states, composite systems use the ordinary Boolean tensor product, reversible evolution is an explicitly allowed invertible Boolean-linear map, and measurement is an added possible/impossible basis-readout rule. This is an instance of finite-field modal quantum theory, with a distinguished CM rotation and optional rotation-compatible restrictions [3, 5]. Within that baseline, the following constructions are valid: reversible modal superposition and XOR interference; tensor nonseparability; the tested Bell and GHZ contradictions; universal exact one-copy transfer with a full-rank correlated resource and a complete 64-outcome analyzer; and 64-message dense coding using one transmitted eight-dimensional subsystem and a shared resource. They arealgebraic/modal results. They are not consequences of truth-table notation alone. Claim Correct final statement Status CM rotation and phase algebra The order-four permutationP generates a 16-element commuting operator algebra by XOR and composition. Proved; exhaustive centralizer scan. Superposition and interference Reversible shears create multibranch vectors and later XOR cancellation. Proved; explicit tests. Entanglement analogue Non-simple tensors exist. Literal bipartite separability is matrix rank one, not the old ring criterion. Proved; both classifications checked. Bell/GHZ The stated modal tables have no deterministic global assignment. Explicit contradictions; exhaustive finite checks. Universal teleportation With full literal resource rank eight, a fixed 64-outcome Boolean analyzer and invertible corrections recover every nonzero local state. Proved; direct tensor execution. Dense coding 64 local encodings give 64 linearly independent joint states, all decoded exactly. Proved; 64/64 checks. Physical quantum theory Neither a physical implementation nor the probabilistic structure of quantum mechanics is established. Not claimed; a probability obstruction is proved. Table 1: Positive results do not erase the operation, tensor, and measurement assumptions. 1.2 Corrections that must accompany those results Earlier interpretation Audited correction The CM identity squares to zero. False. The 2× 2 truth patternδCM =I2 remains idempotent. The different 4× 4 operator I4⊕P 2 squares to zero. Removing a primitive cyclic product invalidates its entire algebra. Too strong. The cyclic rule is exactly composition of the derived rotation operators. It is not pointwise AND or native2× 2 composition. 3 PAGE 5 Boolean CM: correctness and model boundaries 18 September 2026 Earlier interpretation Audited correction The shared and literal constructions are merely the same theory in different coordinates. False. Dimensions, tensor products, preparations, entanglement, and operational resources differ. All 256 local vectors are states. The zero vector is only a null regression input. There are 255 nonzero local modal states. The same logical Bell pair teleports the entire literal local system. False. A phase-anchored logical Bell pair has rank two. Universal eight-dimensional transfer uses rank eight and includes phase correlations. Four Bell outcomes still suffice after independent tensoring. False for the old logical-only analyzer. Each of its four labels leaves 16 unresolved phase sectors with distinct Bob maps. Everything in the literal protocol is generated by the commutingP algebra alone. False. A shearT not commuting withP is used;P,T generate all of GL(4, 2). Joint analyzers also require allowed cross-register wiring. The 15 Smith classes classify the enlarged literal theory. Only under the restricted rotation-compatible action. Under full local GL(8, 2), matrix rank classifies bipartite pure-state orbits. A singular (0, 2) resource cannot be activated with finite copies. True only for the earlier rotation-linear, tensor-over-A architecture. In the literal model two copies can heraldedly yield a rank-eight resource. Different literal/shared contextuality tables reveal an unexplained invariant hierarchy. Their entire restricted empirical models are related by conjugating Bob’s settings and relabeling outcomes. The matching totals have an explicit explanation. No-cloning is a restriction on ordinary Boolean computation. False. It is a restriction on the chosenlinear modal tensor-state dynamics. Stored coefficient strings can be copied; nonlinear Boolean coefficient maps can construct their tensor square. Only a Born rule remains to be supplied. Too optimistic. No no-signalling probability distribution can reproduce the tested Bell table with positive probability on every modal-possible event. 2 Audit scope, evidence, and reproducibility 2.1 Inputs actually available Four mounted ZIP archives were inspected and preserved: • Pure_Boolean_Modal_Quantum.zip • Native_Boolean_CM_Rotation_Phase.zip • Native_Boolean_CM_Block_Quantum.zip • Independent_Phase_CM_Tensor_Audit.zip The user’s Windows directory was not accessible. Original archive hashes are recorded indata/ input_archives.json. The original CM manuscript and the revised operator-level manuscript were consulted through File Library excerpts, including the displayed CM/LM definitions and 4 PAGE 6 Boolean CM: correctness and model boundaries 18 September 2026 measurement discussion [1, 2]. Those library PDFs are not silently represented as files read from the Windows directory. The historical sources are retained undersource_snapshots/. Their original content is evidence to audit, not an authority that makes their interpretations true. All three supplied SHA-256 manifests checked without missing or changed entries: 43, 25, and 19 listed files respectively. The rotation-phase package had no corresponding manifest. 2.2 What “exhaustive” means here A proved theoremis justified for its stated mathematical domain. Anexhaustive finite search covers the stated finite universe, or uses an exact decision procedure over that universe. A regression/spot checkverifies selected instances. Aliterature comparisonis not a novelty proof. An open itemhas not been established. All four historical test suites passed again:22+10+15+12 = 59 tests. A new implementation, not importing historical computational modules, recomputed the main inventories, tables, protocols, and counterexamples. A separate 31-test suite checks the new kernel and evidence. The total is 90 passing named tests, in addition to the millions of explicit cases in the drivers. A test count is not a substitute for the proofs and domain descriptions below. The independent kernel uses packed Boolean rows. A second scalar AND/XOR contraction checks the packing; independent rank algorithms are compared on all 65,536 binary4×4 matrices. The contextuality calculation uses bitset implication closure rather than the historical solver, and is cross-checked by brute force in small scenarios. Teleportation is verified both from full tensor contractions and from the derived branch-map formula. The dense-coding calculation is newly recorded as executable evidence rather than merely asserted in prose. Limits of certification.This audit is not a proof-assistant formalization. It does not exhaust all 264 literal bipartite state vectors by enumeration, allGL(8, 2) measurement contexts, or arbitrary ancilla instruments. Universal claims in those larger domains are made only where a proof is given. The earlier signed/complex lift is outside the requested pure-Boolean branch and is not certified here. 3 Native CM logic and a necessary type ledger 3.1 The Boolean contraction Let B ={0, 1}. Every scalar entry in the core calculation is a bit. Define (A◦B)ij = ⨁ k (Aik∧Bkj), (A◦v)i = ⨁ j (Aij∧vj). (1) The symbol◦ always means this AND/XOR matrix contraction. Pointwise conjunctionA∧B is a different operation. The notationF2 is a mathematical identification of XOR with field addition and AND with field multiplication; it introduces no numerical amplitudes. All ranks below are over this XOR/AND field,not the Boolean OR–AND factorization rank sometimes also called “Boolean rank.” Integers used as array indices, packed bit masks, counts, dimensions, and external rank diagnostics are not evolving amplitudes. Likewise, the real-valued linear program in Section 14 is an external consistency test and is not part of state evolution. 5 PAGE 7 Boolean CM: correctness and model boundaries 18 September 2026 In the original true-first CM convention, |x⟩ = ( x ¬x ) , ⟨x|C|y⟩ =Cxy, x,y ∈{ 1, 0}. The contraction selects one cell. There are 16 possible 2× 2 truth tables. Pointwise AND/XOR/NOT make the same set a Boolean algebra; OR, NAND, NOR, implication, and XNOR are derived pointwise operations. Under XOR and◦ it is insteadM2(F2), with six invertible elements. These are two structures on the same set, not interchangeable compositions. 3.2 LM extraction does not supply the modal postulate A formula-valued LM is a symbolic operator built from the polarity pairsX,¬X and Y,¬Y. Its evaluated pairing has the identity ⟨A|[MXΘY ]|B⟩ = (A↔X) Θ (Y ↔B). (2) Valuation commutes with this expression because it preserves AND, XOR, and NOT. The revised manuscript explicitly calls this logical pairing, rather than a physical measurement [2]. The original discussion similarly describes logical truth extraction and does not supply probability densities [1]. The new modal rule “a nonzero output coefficient is a possible outcome” is therefore anadditional operational interpretation. It is not derived from Equation(2). The same warning applies to selective measurement, discarding, classical outcome messages, and the claim that a coefficient pattern cannot be read nondestructively as an ordinary stored bit string. Evidence: 16,384 same-frame truth-fusion cases, 512 signed-axis alignment cases, and 256 general LM pairing valuations independently checked.. 3.3 Four models, not one silently changing model Model Objects Composition and interpretation C Native CMs/LMs Truth tables, formula-valued operators, pointwise Boolean logic, and typed logical pairing. R Rotation operatorsA Selected XORs ofI4,P,P 2,P 3; ordinary Boolean operator composition. S Shared moduleA2n One four-coordinate coefficient register for all logical branches; tensor product overA;A-linear operations. L Literal registers (F8 2)⊗n A separate four-position phase factor per party; tensor product over F2; restricted or full Boolean-linear gates must be declared. Table 3: Forn = 1, 2, 3, modelShas Boolean dimensions8, 16, 32, whereas modelLhas dimensions 8, 64, 512. In model L, therestrictedP-compatible family consists of2× 2 block matrices with blocks inA. The enlarged family admits arbitrary local invertible8× 8 Boolean matrices and appropriate joint analyzers. Results proved under one family are not automatically results under the other. 6 PAGE 8 Boolean CM: correctness and model boundaries 18 September 2026 4 Rotation, the derived phase algebra, and interference 4.1 Rotation is native; its algebra is a different type from a CM Write the four atomic CM positions clockwise asE0,E 1,E 2,E 3, starting at the upper left. Thus E0 = [ AND], E1 = [ UP], E2 = [ NOR], and E3 = [ DOWN]. With v(C) = ( c0,c 1,c 2,c 3)T, clockwise rotation is P =   0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0   , P 4 =I4, P 2̸=I4, P T =P 3. (3) Here powers mean repeated◦. Transposition of the underlying CM induces the permutationK interchanging coordinates 1 and 3, andK◦P◦K =P 3. Define the operator selected by a CM pattern: Λ(c) = ⨁ j:cj=1 P j, A ={Λ(c) :c∈ B4}. (4) This uses only selection by bits and XOR of actual matrices. Composition is still Equation(1). Expanding a composition of two such operators gives the cyclic coefficient rule from the earlier work. Thus the previously introduced cyclic product isderivable; it is not an extra primitive in the audited implementation. Theorem 4.1(Exact centralizer). The matrices commuting withP in M4(F2) are exactly the 16 operators inA. Proof. The four vectorse0,Pe 0,P 2e0,P 3e0 form a basis. IfA commutes withP, its value one0 determines its values on this whole basis:AP je0 =P jAe0. There are 16 choices forAe0. The 16 matrices in Equation (4) realize all these choices and commute withP. Evidence: All 65,536 Boolean 4× 4 matrices scanned; exactly 16 commute. All 256 phase- operator pairs checked with two contraction implementations.. 4.2 The corrected nilpotent and the limit of the imaginary-number analogy Let N =I4⊕P. Then N 2 =I4⊕P 2, N 4 = 0, rank⊕,∧(N j) = 4−j (0≤j≤ 4). The operatorsI4,N,N 2,N 3 form a basis ofA. ConsequentlyA is isomorphic toF2[u]/(u4), with u naming N and its product interpreted as◦. The truth patternδCM =E0⊕E2 =I2 satisfies δCM∧δCM =δCM, δ CM◦δCM =δCM. Its selected rotation operator is insteadΛ(δCM) =N 2, whose square is zero. No contradiction remains once the types are distinguished. Importantly,Λ is not a homomorphism for pointwise conjunction or native2× 2 composition. The cycleI,P,P 2,P 3 is a phase-clock analogy, not complex scalar multiplication. In characteristic two, additivenegationisidenticaltotheoriginalscalar; P 2 isnotadistinctscalar −1. Complement of a coefficient vector is affine,v↦→v⊕ 1, and is not complex phase negation. 7 PAGE 9 Boolean CM: correctness and model boundaries 18 September 2026 There is also a correction to a previous minimal-dimension suggestion. No invertible2×2 Boolean matrix has order four, but J3 =   1 1 0 0 1 1 0 0 1   satisfies J4 3 =I3, J2 3̸=I3. Four coordinates are natural for the literal four-position CM permutation; they are not the minimum dimension for an arbitrary order-four Boolean-linear operator. 4.3 Reversible superposition and phase-sensitive XOR cancellation The shear C = ( 1 0 1 1 ) satisfies C2 =I2, Ce 0 =e0⊕e1. This is a multibranch vector capable of later cancellation, not a probabilistic mixture. On two four-coordinate branches, use Csh = ( I4 0 I4 I4 ) , S k = ( I4 0 0 P k ) . The circuitCsh◦Sk◦Csh maps (e0, 0) to (e0,e 0⊕ek). The second-branch packed patterns are 0, 3, 5, 9 for k = 0, 1, 2, 3. Identical paths cancel; distinct rotations leave distinguishable exact patterns. A coarse nonzero detector distinguishes only zero from the other three patterns. A symmetric splitter is Hrot = ( I4 N 2 N 2 I4 ) , H 2 rot =I8, H T rot◦Hrot =I8. It splits each elementary Boolean coordinate into both logical branches. It doesnot split every nonzero phase pattern:N 2v = 0 for three nonzero patterns. Twelve of the 15 nonzero phase patterns split. The phase-kickback identity also survives: swapping the branches of(e0,e 2) equals applying P 2 to both branches. 5 Shared-module structure: what is proved and what is prob- lematic 5.1 Units, gates, and Smith classes An element ofA is invertible exactly when its selector has odd parity. There are eight units. Reduction modulo the ideal(N) gives the fieldF2. A 2× 2 block gate is invertible exactly when its reduction is inGL(2, 2), so | GL2(A)| = 6· 84 = 24, 576. Among the 65,536 possible gates, 512 are Boolean-orthogonal after expansion to8× 8 matrices. There are 128 monomial invertible gates and 24,448 nonmonomial ones. Nonmonomial here does not mean every basis input splits; it means the gate is not just an invertible block scaling and permutation. Row-unit rescalings and outcome permutations reduce complete invertible two-row bases to 192 unordered modal bases, of which four arise from the orthogonal subgroup. 8 PAGE 10 Boolean CM: correctness and model boundaries 18 September 2026 Theorem 5.1(Two-party rotation-linear normal form). Every 2× 2 matrix overA is equivalent, under independent invertible row and column operations overA, to exactly one Da,b = diag(N a,N b), 0≤a≤b≤ 4, whereN 4 = 0. There are 15 classes. Proof. Every nonzero operator is a unit timesN a. Choose an entry of minimum valuation, move it to the first diagonal position, and remove its unit factor. Because it divides every other entry, Boolean row/column shears clear its row and column. Reduce the remaining diagonal entry similarly. The ideals generated by entries and the elementary divisor data, equivalently the ranks of successiveN-filters in this two-generator case, determine the exponents uniquely. The expanded Boolean rank profile is rj = max(4−a−j, 0) + max(4−b−j, 0), j = 0, 1, 2, 3. (5) All 65,536 matrices were classified independently using these profiles. Table 5 records every orbit size. 5.2 Separability and the scope of gate classification A shared two-party state is CM-separable when its2× 2 coefficient matrix is an outer product of twoA-vectors. This is equivalent tob = 4 in the normal form. Indeed, a vector over the chain ring can be transformed to(N s, 0) by an invertible change of coordinates; applying this on both outer-product factors leaves at most one nonzero diagonal entry. Conversely,(N a, 0) is explicitly an outer product. Thereare5,266suchvectorsincludingzero, hence60,270nonseparablevectors. Azerodeterminant alone isnot a separability test over this ring. Local invertibles preserve separability in both directions, since they transform the two factors separately and have local inverses. The stabilizers of all 15 canonical states were calculated by exact fiber counting, and each satisfies |Stab(Da,b)|| Orbit(Da,b)| = 24, 5762. The full values are indata/local_stabilizers_audited.csv. In addition, all 24 two-party basis permutations were tested on all 5,266 separable inputs: 16 create nonseparability for some input; eight preserve it for every input. This isnot an enumeration of all two-party invertible gates overA. 5.3 A fundamental preparation-closure failure Proposition 5.2(Nonzero factors can have a zero shared tensor). If every nonzero vector in the shared module is declared an allowed state, independent preparation is not closed under the tensor product overA. Proof. Takex =N 3e0̸= 0 andy =Ne 0̸= 0 as coefficient vectors on a fixed logical branch. Their shared tensor coefficient is selected byN 3◦N =N 4 = 0. Thus two nonzero preparations yield the zero joint vector. In the literal tensor product their product is nonzero: a pair of nonzero Boolean coordinates contributes a one. 9 PAGE 11 Boolean CM: correctness and model boundaries 18 September 2026 In packed phase coordinates this witness isx = 15, y = 3. The literal product is nonzero (packed value 13,107), whereas the shared product is zero. This doesnot invalidate the ring identities or the explicit teleportation contractions. It does invalidate treating the unrestricted collection of all nonzero shared-module vectors as an automat- ically coherent independent-preparation operational theory. Restricting to primitive/unimodular preparations might repair part of this problem, but would remove many of the previously counted states and requires a separate closure analysis. That repair is not silently assumed here. 6 Literal tensoring is not an entanglement-preserving change of notation 6.1 The Choi-style embedding For a shared coefficient matrixm = (mij), form the8× 8 Boolean block matrix M = ( Λ(m00) Λ( m01) Λ(m10) Λ( m11) ) . Its literal bipartite vector is |M⟩⟩= ⨁ j,k:Mjk=1 |j⟩A⊗|k⟩B. (6) This is an injective linear encoding of 65,536 shared coefficient vectors into a space with264 Boolean vectors. It is not a monoidal equivalence between the two tensor theories. A literal nonzero bipartite state is a simple tensor exactly when its coefficient matrix has rank one. For the canonical resources, rank⊕,∧M = 8−a−b. Thus the structured Choi family has just nine nonzero literal product states, rather than 5,265; 65,526 of its vectors are literally nonseparable. For example, the shared product state with coefficient matrixdiag(1, 0) maps to the literal matrixdiag(I4, 0), of rank four. The encoding has introduced phase correlations. It cannot be used to argue that the two notions of entanglement are identical. 6.2 The universal resource contains more correlations The phase-anchored logical Bell state |0,e 0; 0,e 0⟩⊕| 1,e 0; 1,e 0⟩ has literal rank two. The full resource used for universal transfer is |Φ8⟩ = 7⨁ j=0 |j,j⟩, M =I8, (7) which has rank eight. It includes a correlated four-position phase factor as well as the logical Bell correlation. Increasing the analyzer from four to 64 outcomes is therefore not the only change: the operational resource has changed too. Equation (7) is preparable using reversible Boolean gates in the enlarged model. Regard a local label as three binary labels, create a modal superposition on each of Alice’s three factors 10 PAGE 12 Boolean CM: correctness and model boundaries 18 September 2026 using C, and copy thebasis labelsinto Bob’s initially fixed factors with three controlled-XOR permutations. This is the usual tensor-basis preparation, not cloning an arbitrary unknown modal vector. 6.3 Local equivalence changes under the larger gate set Under full localGL(8, 2), ordinary row/column reduction gives U◦M◦V = diag(Ir, 0), r = rank⊕,∧M. Hence rank alone classifies literal bipartite pure-state orbits. In particular(0, 2) and (1, 1) both have rank six and are locally equivalent. Explicit invertible witnesses are saved indata/equal_ rank_class_equivalence.json. The old distinction between their strong/logical contextuality is therefore a distinction relative to the restricted 192-basis family, not an invariant under arbitrary full Boolean-linear local gates. Enlarging the measurement family can expose additional contextuality. No exhaustive full-GL(8, 2) contextuality classification is asserted in this report. 7 Measurement assumptions, Bell tables, and contextuality 7.1 The exact operational contract In literal modal quantum theory, a pure state is anonzero vector. A complete measurement is a basis of the dual space, or equivalently an invertible analyzer followed by coordinate readout. An outcome is possible precisely when its evaluated coordinate is one. If only logical outcomes are recorded, the four phase coordinates are coarse-grained: an outcome is possible when its entire output block is nonzero. Coarse-graining is anOR of possibilities, not XOR of outcome amplitudes. These operations must not be confused. A selective outcome leaves the corresponding nonzero conditional vector; disregarding an outcome is naturally represented by the span of conditional vectors. No probabilities are attached to these alternatives. This measurement contract is standard in modal quantum theory [3]; its use here is an explicit extension of CM logical extraction. Proposition7.1 (Possibilisticno-signalling). For literal tensor states and complete local invertible analyzers, the set of possible outcomes at one party is independent of the other party’s choice of local analyzer. Proof. Fix an Alice effect and contract it into the joint state. The result is a Bob vector (or block of vectors). Bob’s invertible basis change cannot transform a nonzero vector to zero or zero to nonzero. Therefore whether at least one Bob outcome is possible is unchanged. The same reasoning works with the parties interchanged and with block coarse-graining. This is a support statement. It is weaker than the existence of a no-signallingprobability distribution; see Section 14. 7.2 Bell contradiction with only three settings per party Use the following ordered dual bases on the logical two-coordinate factor: Z = ((1, 0), (0, 1)), X = ((1, 1), (1, 0)), Y = ((0, 1), (1, 1)). 11 PAGE 13 Boolean CM: correctness and model boundaries 18 September 2026 The letters are setting names, not complex Pauli operators. For the support-Bell state, the possible/impossible table is: Alice\ Bob Z X Y Z 1001 1110 0111 X 1110 0111 1001 Y 0111 1001 1110 Table 4: Each string lists outcomes00, 01, 10, 11; one means possible. Both the shared support- Bell table and the literal correlated construction give this table. Theorem 7.2(Explicit modal Bell contradiction). No deterministic assignment to the six local settings satisfies Table 4. Proof. The contextsZZ,XY,YX require z =AZ =BZ, x =AX =BY, y =AY =BX. The remaining contexts implyz∨x, z∨y,¬(x∧z),¬(y∧z), x∨y, and¬(x∧y). Ifz = 0, then x =y = 1, contradicting the last condition. Ifz = 1, thenx =y = 0, contradictingx∨y. Evidence: All 26 = 64 deterministic assignments independently rejected. No expectations or CHSH probabilities are used.. The contradiction rules out a local relational hidden-variable explanation with simultaneous setting assignments and the stated support semantics. It does not rule out a conventional computer globally calculating these tables, nor derive spatially nonlocal physics from a Boolean truth table. 7.3 The complete restricted 192-basis result A localA-basis consists of two unimodular row effects, reduced by independent multiplication by units and by row permutation. Exactly 192 unordered bases remain. This is a complete family for thisdeclared block-basis quotient; it is not every fine-grained literal8-outcome measurement. For each pair of settings, forbidden outcomes produce clauses (Ai̸=a)∨ (Bj̸=b). A global assignment exists exactly when the resulting 2-SAT instance is satisfiable. A possible section extends globally exactly when adding its two outcome literals leaves the instance satisfiable. We use the distinctions of the possibilistic contextuality framework [6]: • Strong: no global assignment exists. • Logical, not strong: a global assignment exists, but some possible section has no global extension. • Relationally local: every possible section extends; one global assignment alone would not suffice. The zero state has empty supports and is excluded as a preparation rather than called strongly contextual. All 1922 = 36, 864 setting pairs were computed for every canonical class in both shared and literal models. Exact implication closure decides the constraints over all2384 deterministic 12 PAGE 14 Boolean CM: correctness and model boundaries 18 September 2026 assignments without pretending to enumerate them one by one. The result is four strong classes, six logical-not-strong classes, four nonzero local classes, and zero. Their structured state counts are 26,214, 34,056, 5,265, and one respectively. The counts and uncovered sections are in Table 6. 7.4 Why the shared and literal counts agree Define conjugation of a phase operator by ordinary transpose,¯A =AT. SinceP T =P 3, it is an automorphism ofA. In a shared contraction the output coefficient is represented by ⨁ ij Ei◦Mij◦Fj. In a literal Choi contraction the corresponding phase block is ⨁ ij Ei◦Mij◦F T j . Thus the literal table at(E,F ) is the shared table at(E, ¯F ), with the row-unit representatives and outcome order adjusted. Conjugation maps the complete 192-basis family to itself. Proposition 7.3(Empirical-model isomorphism). The two restricted support models are exactly isomorphic by a permutation of Bob’s settings and possible outcome swaps. Equal aggregate contextuality counts are a consequence of this relabeling. Evidence: Every context in all 15 classes checked under the explicit relabeling; 8,640 direct literal effect-pair matrix contractions also verify the formula. Brute-force tests on the four-basis family check the 2-SAT coverage algorithm.. 8 GHZ: exact successes and retained negatives The literal correlated three-party state is |GHZ8⟩ = 7⨁ j=0 |j,j,j⟩. Each bipartition flattening has rank eight, so it is nonseparable across all three cuts. This is not a claim about a density-matrix pairwise reduction; the theory here has no such density matrix. The 27 logicalZ/X/Y contexts contain 112 possible sections. All 512 deterministic assignments to the nine local settings fail. A six-context contradiction is ZZZ, ZXX, XZY, XXX, YYZ, YYY. No subset of one through five contexts from this particular 27-context family is contradictory. The exact subset counts tested were 27, 351, 2,925, 17,550, and 80,730: 101,583 subsets in total. Six is therefore minimalwithin this family, not among all possible CM measurements or all possible GHZ constructions. The four tested orthogonal block bases, by contrast, yield 1,024 global assignments and no uncovered possible sections for this state. Nonseparability is not sufficient to force contextuality in every restricted measurement family. For a weighted literal variant, applyN 2 to Charlie’s phase leg in the logical111 term, leaving the 000 term unweighted. Its three flattening ranks are8, 8, 6. TheZ/X/Y test is local (16 global assignments); the four orthogonal bases give logical contextuality (291 global assignments and 28 uncovered sections). This is one explicitly specified three-party lift, not a unique Choi lift of the old shared weighted GHZ. The numerical result is verified; selecting a canonical higher-party lifting principle remains a modeling question. 13 PAGE 15 Boolean CM: correctness and model boundaries 18 September 2026 9 Universal exact teleportation: construction and proof 9.1 What “universal” requires Fix a known shared resource, an input-independent analyzer, and an outcome-dependent cor- rection. Universal exact transfer means that everynonzero unknown input vector is recovered exactly on every possible complete outcome. The null input may be tested as a linear identity but is not an additional physical/modal state. No quotient by a global phase or unit is needed for the successful protocols. We analyze the standard one-copy architecture: Alice holds the unknownd-dimensional system and her half of a resource, performs a complete rank-oned2-outcome modal analysis, reports its label, and Bob applies a linear correction. Ancillas and filters will be treated separately rather than hidden in this definition. 9.2 The conditional branch formula Let the resource have coefficient matrixM, and reshape analyzer rowm into ad×d matrix Rm. With inputψ, the coefficient of Bob coordinatek is ⨁ i,j ( ψi∧ (Rm)ij∧Mjk ) . Consequently the conditional vector is Tm◦ψ, T m =M T◦RT m. (8) This equation is a Boolean contraction identity, not an assumed measurement probability formula. A complete analyzer is invertible when the flattenedRm form a basis of the entire matrix space. Theorem 9.1(Literal one-copy resource criterion). In the above literal Boolean-linear archi- tecture, universal exact transfer exists for ad×d resource whenever an invertible matrix-effect basis exists andrank⊕,∧M =d. Conversely, a resource with rank less thand has no nonzero universally exact success branch. Ford = 8, an explicit invertible effect basis is given below, so rank eight is necessary and sufficient. Proof. IfCm◦Tm =Id, thend = rank⊕,∧(Cm◦Tm)≤ rank⊕,∧Tm≤ rank⊕,∧M. For the converse, choose a complete basis with everyRm invertible. IfM is invertible, everyTm is invertible and Cm =T−1 m recovers every input. Postselecting a nonzero singular branch cannot evade the obstruction. Choosek̸= 0 in its kernel and x outside its kernel. Both x and x⊕k give the same nonzero branch output; no fixed correction can recover both distinct inputs. A zero branch is never a success event. 9.3 A small, explicit, search-free Bell analyzer Use four invertible2× 2 effect matrices L0 = ( 0 1 1 1 ) , L 1 = ( 1 0 1 1 ) , L 2 = ( 0 1 1 0 ) , L 3 =I2. (9) 14 PAGE 16 Boolean CM: correctness and model boundaries 18 September 2026 Their flattened rows form the invertible analyzer A4 =   0 1 1 1 1 0 1 1 0 1 1 0 1 0 0 1   . (10) The original modal literature already contains equivalent four-state Boolean Bell constructions and the corresponding teleportation/dense-coding protocols [3]; the present audit does not claim their discovery. Ford = 8 = 23, set Rijk =Li⊗Lj⊗Lk, i,j,k ∈{ 0, 1, 2, 3}. (11) Each matrix is invertible. Since the fourLi form a basis ofM2(F2), their triple tensors form a basis ofM8(F2). The 64-row analyzer with rowvec(Rijk)T is therefore invertible. This is a tensor construction of three mobit protocols, up to wiring permutations; an unexplained numerical search is unnecessary. The phase effectsLj⊗Lk can be synthesized usingP, its inverse, and T =   1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1   . Exact group generation gives⟨P,T⟩ = GL(4, 2), with 20,160 elements. Explicit words for the 16 canonical phase effects and independent checks of the old 16 synthesis words are included. Crucially,T◦P̸=P◦T. This is a genuine enlargement beyond the commuting phase algebra. The analyzer is an allowed joint operation on Alice’s systems, not a product of disconnected local actions on them. 9.4 What was executed, not merely inferred The fresh code constructs the full512× 8 input-to-joint-state preparation and the full512× 512 analyzer acting on Alice’s pair and Bob’s identity factor. It extracts Bob’s 64 conditional8× 8 maps and checks Equation (8). For both the historical literal analyzer and the new tensor-product analyzer, all256×64 = 16, 384 input/outcome identities recover exactly. Of these,16,320 per analyzerconcern nonzero inputs and possible outcomes; 64 are null-vector regressions. The old raw received/corrected outputs also match the new independent execution exactly. For all 65,536 structured resources, all 64 canonical branch ranks were computed explicitly: 4,194,304 rank comparisons, with no mismatches. The equality is also proved from invertibility of Rm: rank⊕,∧Tm = rank⊕,∧M. Direct full-tensor contraction additionally checks all 15 canonical resources and 16 fixed-seed nonsymmetric resources, giving 1,984 branch-formula comparisons. Exactly 24,576 structured resources have rank eight. The rank criterion itself applies to arbitrary8× 8 Boolean resources, not only this structured subset; that broader statement is proved, not exhaustively enumerated. Finally,Cm◦Tm =I8 implies (Cm◦Tm)⊗Ir =I8r 15 PAGE 17 Boolean CM: correctness and model boundaries 18 September 2026 for any reference dimensionr. Thus the identity preserves arbitrary correlations with a reference system. All 64 maps were also explicitly checked with a two-dimensional reference. A full-rank resource is not merely correct on basis states. 9.5 The older four-outcome shared protocol The same logical analyzer acts in the shared coefficient module with four outcomes and recovers all coefficient pairs(A,B )∈A 2, as an exact module identity. All 1,024 input/outcome identities were checked. This includes four null-input regressions. Its operational interpretation retains the preparation-closure limitation identified above. In the literal model, applying only this logical analyzer leaves 16 separate Alice phase-pair sectors under each reported logical label. All 16 sector maps are nonzero, distinct, and of rank two. A four-valued logical label therefore does not identify a single universally invertible Bob map. The literal construction requires a full 64-outcome phase-and-logical analysis in this architecture. 10 Dense coding, readout, and resource accounting Use|Φ8⟩ and the 64 invertible encodingsRm of Equation(11). Alice acts only on her subsystem, yielding (Rm⊗I8)◦| Φ8⟩ =|Rm⟩⟩. These 64 joint vectors are linearly independent. LetQ be the matrix with these vectors as columns. After Alice transmits her subsystem, Bob appliesQ−1 and obtains the unique coordinate vectorem. All 64 states were constructed explicitly, their Boolean rank is 64, and all 64 messages decode exactly. The dense decoder isQ−1; it should not be confused with the teleportation analyzer, whoserows are the flattened effects. Their relationship depends on transpose and dual-basis conventions. Under a single complete modal basis readout, an unassistedd-dimensional carrier can perfectly distinguish at mostdpreparations. If effects distinguish preparationsψj, one can select functionals fi with fi(ψj) = 0 for i̸=j and fi(ψi) = 1; this forces linear independence. The constructed assisted protocol distinguishesd2 preparations. Thus the8 versus 64 zero-error basis-readout comparison is meaningfulinside the specified modal resource model. It is not a Shannon-capacity theorem, a physical bandwidth result, or compression of an eight-bit classical memory into three physical bits. An eight-dimensional Boolean modal vector is stored by eight ordinary coefficients in a conventional simulation. Reading all those coefficients as classical data is a different access model. Labels for 64 outcomes require six classical binary symbols; this is label accounting, not a probability law. With the restrictedP-compatible encodings alone, the invertible encodings span only a 16- dimensional subspace ofM8(F2), not a 64-dimensional one. Therefore 64 independent codewords cannot be obtained by that restricted encoding family acting onΦ8. The full dense-coding result depends on the declared gate enlargement. 16 PAGE 18 Boolean CM: correctness and model boundaries 18 September 2026 11 Restricted states, quotients, ancillas, and copies 11.1 The shared hierarchy remains a restricted theorem ForDa,b overA, the canonical coefficient map retains the quotient Qa,b =A/(N 4−a) ⊕ A/(N 4−b), |Qa,b| = 28−a−b. (12) HereA/(N 4) =A andA/(N 0) is the zero module. Equivalently, this is the quotient by the kernel of the canonical resource action. Cardinality is an external algebraic diagnostic, not physically communicable bits. The largest common exact family in the one-branchA-linear architecture has 256 vectors for (0, 0), 16 for(0,b ) with b >0, and only the zero vector fora >0. The same maxima can be attained by complete four-outcome analyzers for the free-line cases, with reversible corrections. For the upper bounds, ifa> 0, every entry of the corrected branch lies in(N); such a matrix is nilpotent, so it cannot fix a nonzero vector. Ifa = 0 0 ⇐⇒ (a,b|X,Y ) is modal-possible. The answer is no. This obstruction is known in modal quantum theory [4]; it was independently recomputed for the exact convention used here. Theorem 14.1 (No support-faithful no-signalling probability extension). Every normalized no-signalling nonnegative probability assignment respecting the forbidden events of Table 4 assigns probability zero to six events that the modal table calls possible. Proof. Solve the linear equalities for normalization, no-signalling, and the forbidden events. Write x =p(11|YZ ), y =p(01|YY ), andz =p(10|YY ). The equalities imply p(00|ZX ) =−x, p (11|YZ ) =x, p(11|XX ) =y +z− 1, p (00|YY ) = 1−y−z, p(11|ZY ) =x +y−z, p (00|XZ ) =−x−y +z. Nonnegativity of each displayed pair forcesx = 0, y +z = 1, andy =z. Hencey =z = 1/2 and all six displayed events have probability zero. Each is marked possible in the Boolean table. These equations use ordinary real arithmeticonly as an external diagnostic. No negative or fractional value has been inserted into CM state evolution. The resulting weak resolution assigns 1/2 to the other allowed events, but deletes six modal possibilities, so it is not a faithful probability realization of the original supports. Evidence: Linear programming separately maximized every allowed event and the common minimum allowed probability; an exact symbolic affine solution supplies the proof above. Raw results and exact constraints are included.. 20 PAGE 22 Boolean CM: correctness and model boundaries 18 September 2026 Thus a physical interpretation cannot simply append an ordinary no-signalling probability rule while retaining all these predictions unchanged. It would need to restrict states or measurements, alter the modal interpretation, or otherwise change the operational model. This is stronger than saying that a Born rule has not yet been discovered. 15 Consolidated class and evidence tables 15.1 All 15 structured classes Table 5: Complete structured family. The shared separability column refers to modelS; literal nonzero product states occur only at rank one. The zero vector is bookkeeping, not an allowed modal preparation. a b Orbit size Literal rank Shared sep. 192-basis status Universal 0 0 24,576 8 No Strong Yes 0 1 18,432 7 No Logical No 0 2 9,216 6 No Logical No 0 3 4,608 5 No Logical No 0 4 4,608 4 Yes Local No 1 1 1,536 6 No Strong No 1 2 1,152 5 No Logical No 1 3 576 4 No Logical No 1 4 576 3 Yes Local No 2 2 96 4 No Strong No 2 3 72 3 No Logical No 2 4 72 2 Yes Local No 3 3 6 2 No Strong No 3 4 9 1 Yes Local No 4 4 1 0 Yes Excluded No 15.2 Exact contextuality coverage counts Table 6: Per-class counts in the complete 192-basis family, identical under the shared/literal setting relabeling. “Extend- able” counts allowed joint sections having some compatible global assignment. a b Possible Extendable Uncovered Status 0 0 141,312 0 141,312 Strong 0 1 139,264 65,536 73,728 Logical 0 2 137,216 100,352 36,864 Logical 0 3 135,168 119,808 15,360 Logical 21 PAGE 23 Boolean CM: correctness and model boundaries 18 September 2026 a b Possible Extendable Uncovered Status 0 4 131,072 131,072 0 Local 1 1 135,168 0 135,168 Strong 1 2 131,072 65,536 65,536 Logical 1 3 126,976 98,304 28,672 Logical 1 4 118,784 118,784 0 Local 2 2 122,880 0 122,880 Strong 2 3 114,688 65,536 49,152 Logical 2 4 98,304 98,304 0 Local 3 3 98,304 0 98,304 Strong 3 4 65,536 65,536 0 Local 4 4 0 0 0 Excluded 15.3 Do not merge the two resource hierarchies Table 7: Largest exact families count vectors including zero. Subtract one for nonzero modal preparations. The last column is the minimum literal-copy count forheralded extraction of I8 under unrestricted local Boolean-linear processing. a b Rank Shared exact Literal exact Literal copies 0 0 8 256 256 1 0 1 7 16 128 2 0 2 6 16 64 2 0 3 5 16 32 2 0 4 4 16 16 2 1 1 6 1 64 2 1 2 5 1 32 2 1 3 4 1 16 2 1 4 3 1 8 2 2 2 4 1 16 2 2 3 3 1 8 2 2 4 2 1 4 3 3 3 2 1 4 3 3 4 1 1 2 never 4 4 0 1 1 never 15.4 Evidence ledger 22 PAGE 24 Boolean CM: correctness and model boundaries 18 September 2026 Table 8: Executed universes and proof scopes. All counts are diagnostics, not amplitude values. Item Universe or evidence Result Boundary CM/LM identities 16,384 truth-fusion, 512 alignment, 256 pairing cases Confirmed Native Boolean logic, not modal measurement. Packed Boolean kernel All 256 pairs of 2x2 matrices; all 65,536 4x4 ranks and inverses; rectangular checks Confirmed Independent scalar and packed implementations. Rotation centralizer All 65,536 binary 4x4 matrices 16 operators Exactly the XOR span of I, P, P squared, P cubed. Rotation powers Exact identities and ranks 4,3,2,1,0 Confirmed Different types for the CM identity and its selected operator. Order-four minimum All six GL2 maps; explicit 3x3 witness Correction Four is not the minimum arbitrary linear dimension. Gate inventory All 65,536 phase-block gates 24,576 / 512 Invertible / Boolean-orthogonal; 128 monomial. Two-party separability All 65,536 coefficient states and all 65,536 factor pairs Confirmed 5,266 shared products including zero, versus 10 literal structured vectors of rank at most one. Local stabilizers All 15 classes; exact relation-fiber counts Confirmed Orbit-stabilizer equality in GL2(A) squared. Basis-permutation gates 24 permutations on all 5,266 shared product vectors 16 entanglers Not all possible two-party gates. Restricted contextuality 15 classes x 36,864 contexts x two models Confirmed Exact implication-closure coverage; not brute enumeration of 2384 assignments. Shared/literal relation Every restricted context; 8,640 direct effect-pair checks Relabeling Bob coefficient conjugation explains all table changes. Bell Nine contexts, all 64 assignments Contradiction Modal local-assignment assumptions explicitly stated. GHZ 27 contexts, all 512 assignments; 101,583 subsets below six Minimum six Only this stated context family. Weighted GHZ One specified literal lift; Z/X/Y and four orthogonal bases Mixed Local in the first family; logically contextual in the second. P and shear generation Exact closure of P, inverse P, and T 20,160 All GL4; T leaves the commuting phase subalgebra. Shared teleportation 256 inputs x four outcomes 1,024 identities Includes four null regressions; module preparation caveat remains. Literal teleportation Two analyzers, each 256 inputs x 64 outcomes 32,768 identities 16,320 meaningful nonzero cases per analyzer. Direct branch formula 15 canonical plus 16 fixed-seed resources, all 64 branches 1,984 checks Uses explicit full tensor matrices, including nonsymmetric resources. All resource ranks 65,536 resources x 64 branches 4,194,304 checks No branch-rank mismatch. Reference preservation 64 corrected identity maps tensor I2 Confirmed Proof extends to every reference dimension. 23 PAGE 25 Boolean CM: correctness and model boundaries 18 September 2026 Item Universe or evidence Result Boundary Dense coding 64 encoded joint states and full inverse decoder 64/64 Basis rank 64; not a physical bandwidth claim. Old logical-only analyzer Four labels, each with 16 phase sectors Fails Sixteen distinct rank-two Bob maps per logical label. Shared restricted transfer 983,040 correction fixed-space cases Confirmed 256, 16, or one vector depending on class. Shared fixed quotient 983,040 analyzer-row solvability tests 4 / 2 / 3 labels Scope permits arbitrary A-linear corrections; zero quotient is trivial. Literal restricted transfer Rank 1 through 8, complete analyzers 32,640 checks Attains all 2r vectors in an r-dimensional subspace. Literal finite-copy activation All 15 canonical classes; explicit filters Positive Two copies of rank six can heraldedly yield rank eight. Shared tensor closure Explicit two nonzero factors Fails N cubed times N is zero under the shared tensor. Shared orthogonal rows All 65,536 four-block rows Zero correctable Among rows meeting the necessary orthogonal condition. Orthogonal Bell basis Even-dimensional hyperplane proof No-go Complete rank-one analyzer with all corrections orthogonal. Common bilinear form All 65,536 four-dimensional forms Only zero No nondegenerate form preserves both P and T. No-cloning Proof and all 256 basis-cloner inputs Linear no-go 247 multi-support failures; coefficient copying remains possible. Probability extension External real LP and exact affine solution Fails faithfully Six modal-possible events forced to zero by no-signalling. Historical regression 22 + 10 + 15 + 12 named tests 59 pass Passing old tests does not certify their interpretation. New test suite 31 named independent tests 31 pass In addition to all driver enumerations. 16 Literature, novelty, and useful research directions Modal quantum theory already supplies finite-field state spaces, general invertible evolution, support-based measurements, entangled states, no-cloning, Bell phenomena, teleportation, and dense coding [3, 5]. The elementary four-state protocol underlying the presentd = 8 construction is not a new quantum-information primitive. The independent tensor construction makes that relationship particularly explicit. The global-section distinction between logical and strong contextuality is established prior work [6]. Generalized/categorical treatments also separate the algebra of protocols from their complex- Hilbert-space realization [8]. Werner’s classification concerns tight complex quantum teleportation and dense coding with the appropriate unitary resources; the Boolean full-rank criterion must not be imported as a deterministic physical-quantum criterion for arbitrary partially entangled states [7]. “Quantum mechanics over sets” uses a related but different formalism and should not be used to supply a probability rule to this model without additional changes [10]. Matrix-logic square-root- 24 PAGE 26 Boolean CM: correctness and model boundaries 18 September 2026 of-NOT constructions that introduce complex matrices likewise do not make the native Boolean P a complex scalar [11]. Finite rings and finite-chain-ring coding constructions already have a literature [12]; no novelty claim is based merely on using a ring with nilpotents. The best-supported distinct research object is narrower: a typed CM rotation-operator subalgebra, its exact restricted gate and measurement families, the finite classification relative to those families, and the comparison with the enlarged literal modal theory. The fresh corrections themselves are useful: they identify which properties depend on rotation-linearity, a tensor choice, an access assumption, or an extra correlation resource. Practical uses presently supported by evidence are exact finite-model experimentation, verification examples, symbolic Boolean operator analysis, and educational/foundational separation of assumptions. Coding theory, error correction, algorithmic advantage, physical implementations, and resource-efficient simulation remain prospective applications, not demonstrated outcomes of these tests. A bounded literature search is not a proof of novelty. 17 What may now be used as a foundation, and what remains open 17.1 A safe construction contract For future literal-model work, specify a nonzero Boolean state vector, its actual subsystem tensor factors, the allowed fixed Boolean-linear gates, a basis or coarse-grained modal readout, the selective outcome rule, and the classical message/resource accounting. Record explicitly when leaving the rotation-compatible subgroup. Under that contract, the full-rank teleportation theorem, literal nonseparability criterion, and dense-coding construction are established building blocks. For rotation-linear work, retainA as a derived operator algebra, not the native truth-pattern composition. Treat its Smith and quotient results as module algebra, and do not declare all nonzero coefficient states independently preparable until the zero-tensor problem has been addressed. Exact coefficient recovery and recovery modulo units or annihilator ideals are different tasks. 17.2 Open or deliberately unclaimed items The present audit does not establish a canonical repair of shared-module preparation closure; a complete contextuality classification under all literalGL(8, 2) fine-grained measurements; a canon- ical higher-party Choi lift; a deterministic/probabilistic optimality theory for activation; a replace- ment symplectic protocol satisfying an alternative operational structure; monogamy inequalities; a general Clifford/stabilizer subtheory with physical Pauli interpretation; a developed CM error-correcting code; a new efficient quantum-computing algorithm; or hardware/experimental predictions. Multiplication of a complete shared state by a unit preserves zero/nonzero support under the matchingA-linear readout, but it does not preserve an exact CM pattern. No global-unit quotient is imposed on the successful exact protocols. Under enlarged literal readout, such a transformation is generally a detectable operator, not automatically a harmless global phase. Final assessment.The positive Boolean protocol identities are real. The earlier tendency to combine them, their counts, and their limitations into one unrestricted “CM quantum theory” was not justified. The corrected baseline is a family of explicitly related algebras and modal 25 PAGE 27 Boolean CM: correctness and model boundaries 18 September 2026 models. That is a stronger basis for future work than either discarding the verified mathematics or treating every earlier interpretation as confirmed. A Reproducing and inspecting the audit The archive contains this LaTeX source and compiled PDF underpaper/, the new kernel/drivers under code/, raw CSV/JSON underdata/, a 31-test independent suite undertests/, historical source snapshots, and execution logs. It includes a machine-readable claim ledger and a SHA-256 manifest. No font files or credentials are distributed. From the extracted root, with the listed Python packages installed, run: python run_all.py --historical The historical flag reruns the four preserved old suites as well as the fresh calculations. Without it, the command reruns the independent calculations, evidence validation, and new tests. The report can be compiled without Python: cd paper latexmk -pdf -interaction=nonstopmode CM_Correctness_Audit.tex Alternatively, runpdflatex at least three times from a clean directory, until the table of contents and references stabilize. All generated table source is included, so compilation does not require importing a data file through LaTeX. The empirical timing logs are execution receipts, not performance benchmarks. A.1 Raw matrix convention Matrices in JSON are lists of packed rows, with bit zero representing column zero. A vector bit zero represents coordinate zero. Flattening is row-major: matrix entry(i,j ) occupies bitin +j for ann-column matrix. This convention differs from visually printing an ordinary binary integer most-significant-bit first; use the declared bit positions rather than guessing column order from a printed integer. A.2 Why the evidence is reproducible but not a universal certification The code contains executable assertions, independent kernel cross-checks, exhaustive inventories, exact implication solving, explicit tensor contractions, and mathematical proofs in this document. The old implementations were not imported to make the new numerical answers. Historical data were used for regression comparisons and to verify supplied synthesis words. Nevertheless, these are implementations and human-readable derivations, not machine-checked proofs of all possible claims. The complete ledger distinguishes the verified universes from the open domains. References [1] B. Droncheff. Correspondence Matrices; Algorithms for Propositional Logic. Research manuscript, May 2018, version 1.0.1. DOI: 10.13140/RG.2.2.28036.37764. Original CM, rotation, XOR, and LM definitions; Sections 3–4 consulted through the supplied File Library material. 26 PAGE 28 Boolean CM: correctness and model boundaries 18 September 2026 [2] B. Theory.Operator-Level Boolean Computation with Correspondence Matrices. Unpublished manuscript supplied in File Library, uploaded 17 September 2026. Sections 2 and 8 and Appendices A–B: Boolean contraction, formula-valued LMs, logical pairing, and evidence boundaries. [3] B. Schumacher and M. D. Westmoreland.Modal quantum theory. Foundations of Physics 42, 918–925 (2012); arXiv version 2, 2011. arXiv:1010.2929. Finite-field modal states, measurements, Bell/no-cloning, and dense-coding/teleportation constructions. [4] B. Schumacher and M. D. Westmoreland.Non-contextuality and free will in modal quantum theory. 2010. arXiv:1010.5452. Modal constraints and the obstruction to a support-faithful no-signalling probability assignment. [5] R. P. James, G. Ortiz, and A. Sabry. Quantum Computing over Finite Fields. 2011. arXiv:1101.3764. Finite-field interference and information-processing analogues. [6] S. Abramsky and A. Brandenburger.The Sheaf-Theoretic Structure of Non-Locality and Contextuality. New Journal of Physics13, 113036 (2011). arXiv:1102.0264. Global sections and possibilistic contextuality. [7] R. F. Werner.All Teleportation and Dense Coding Schemes. Journal of Physics A34, 7081–7094 (2001). arXiv:quant-ph/0003070. Tight complex-quantum protocol classification. [8] S. Abramsky and B. Coecke.A categorical semantics of quantum protocols. 2004. arXiv:quant- ph/0402130. [9] D. Gottesman. Stabilizer Codes and Quantum Error Correction . PhD thesis, 1997. arXiv:quant-ph/9705052. Stabilizer and binary symplectic machinery, not a Boolean- amplitude replacement for complex quantum mechanics. [10] D. Ellerman.Quantum mechanics over sets. 2013. arXiv:1310.8221. A related but different set-based formalism. [11] E. Mizraji.Generalized “Square roots of Not” matrices, their application to the unveiling of hidden logical operators and to the definition of fully matrix circular Euler functions 2021; revised 2024. arXiv:2107.06067. Matrix-logic comparison; no claim of a native Boolean complex unit. [12] X. Liu and H. Liu.Quantum Codes from Linear Codes over Finite Chain Rings. 2017. arXiv:1704.06375. Prior finite-chain-ring coding constructions, not a derivation of the present CM resource interpretation. 27