Major revision, 19 October 2026. A total ordering of tensor-product sites permits the standard Jordan-Wigner construction of anticommuting dressed operators. We retain the explicit exchange-sign calculation and the finite K_3 colour-label matrix identity, while separating both from particle statistics and colour confinement. An ordering-dependent parity string does not by itself select a physical fermionic algebra or local transport law. The earlier same-register product of orthogonal colour projectors is not a baryon creation operator, and the symmetric vector of a three-colour label space is not an SU(3) singlet. Fermionic matter remains an explicit premise in the later framework; the calculation is a conditional representation of exchange algebra, not a derivation of the spin-statistics connection. Changes from earlier versions: The revision withdraws derivation of baryon statistics, local fermion transport and colour confinement from XOR closure. The revision distinguishes Hermitian Majorana-like operators from nilpotent creation operators; a product of orthogonal same-register colour projectors is zero. A symmetric three-colour vector is not an SU(3) singlet; an anti-Hermitian square is not an energy argument. Retained result: Finite Jordan-Wigner anticommutation calculation and K3 matrix identities. Previously titled: Pauli Antisymmetry from F2 XOR-Closure: A Discrete-Geometric Derivation of Composite Baryon Statistics. The preceding version and earlier versions remain available. This is a revised preprint, not an independently peer-reviewed result. It uses existing evidence; no new scientific calculations were run and research branches remain frozen. AI assistance is acknowledged in the manuscript. Revision guide for all twelve papers. The attached PUBLICATION_UPDATES.csv provides old and new titles, DOIs and replacement descriptions for catalogue and website updates.
Major revision, 19 October 2026. A total ordering of tensor-product sites permits the standard Jordan-Wigner construction of anticommuting dressed operators. We retain the explicit exchange-sign calculation and the finite K_3 colour-label matrix identity, while separating both from particle statistics and colour confinement. An ordering-dependent parity string does not by itself select a physical fermionic algebra or local transport law. The earlier same-register product of orthogonal colour projectors is not a baryon creation operator, and the symmetric vector of a three-colour label space is not an SU(3) singlet. Fermionic matter remains an explicit premise in the later framework; the calculation is a conditional representation of exchange algebra, not a derivation of the spin-statistics connection. Changes from earlier versions: The revision withdraws derivation of baryon statistics, local fermion transport and colour confinement from XOR closure. The revision distinguishes Hermitian Majorana-like operators from nilpotent creation operators; a product of orthogonal same-register colour projectors is zero. A symmetric three-colour vector is not an SU(3) singlet; an anti-Hermitian square is not an energy argument. Retained result: Finite Jordan-Wigner anticommutation calculation and K3 matrix identities. Previously titled: Pauli Antisymmetry from F2 XOR-Closure: A Discrete-Geometric Derivation of Composite Baryon Statistics. The preceding version and earlier versions remain available. This is a revised preprint, not an independently peer-reviewed result. It uses existing evidence; no new scientific calculations were run and research branches remain frozen. AI assistance is acknowledged in the manuscript. Revision guide for all twelve papers. The attached PUBLICATION_UPDATES.csv provides old and new titles, DOIs and replacement descriptions for catalogue and website updates.
A strictly local one-dimensional lattice-register analysis shows how fermionic exchange statistics arise as a gauge constraint on discrete encodings of indistinguishable occupancy information, rather than as an independent postulate. The mechanism starts from a hard-core on-site Hilbert space and imposes operational locality axioms on the physically measurable even observable algebra—on-site number locality, strictly nearest-neighbor hopping locality, and compositional commutativity on disjoint regions—then demands that single-excitation (odd) representatives realize fermionic exchange through the canonical anticommutation structure. A sharp local-commutant lemma identifies on-site parity as the unique intermediate-site “sign resource” compatible with number-commuting locality while still flipping odd operators, forcing any consistent encoding to carry a parity-string tail. This yields a necessity-and-uniqueness-up-to-gauge result: the only way to implement fermions on a tensor-product 1D register without injecting nonlocal artifacts into physical observables is to pay the Jordan–Wigner positional-bookkeeping cost, with the apparent ordering dependence promoted to a gauge choice acting by conjugation on odd representatives while leaving the even observable algebra invariant. The paper further links exclusion to an information-theoretic capacity constraint induced by hard-core truncation and highlights an open-system consequence: under local noise and loss channels, occupancy-limited (hard-core/fermionic) dynamics restrict accessible local error weight and alter entropy-production pathways relative to unconstrained bosonic loss, making the encoding constraint operationally testable in discrete lattice settings.
Major revision, 19 October 2026. This revision separates three objects previously conflated: a classical [8,4,4] code, its one-code CSS stabilizer state, and a nonlinear set of 48 allowed eight-bit register words. The register admits a 32+16 partition by one generation bit. A declared evolution preserves that partition if its operators commute with the corresponding parity. This elementary conservation statement is retained; it does not establish a concatenated quantum code, a noise threshold or topological protection. The channel-friction ansatz, golden-ratio fixed point and XOR bookkeeping are given their proper conditional scope. Claims of CKM predictions and a derived particle-mass mechanism are withdrawn, as is the claim that the legacy walk is native to the corrected geometry. Changes from earlier versions: The revision removes concatenated-code/topological-protection claims and CKM/mass predictions unsupported by the declared register and retired transport law. The revision separates 16 codewords, a one-dimensional CSS stabilizer space and 48 allowed register words; the latter has Hamming distance one. The revision corrects the geodesic/line-graph count. The revision states a real recovery criterion and the hypotheses of the commutation argument. Retained result: Classical code facts, the nonlinear register count, a conditional sector-conservation proposition and explicitly conditional scalar/graph identities. Previously titled: A Concatenated Quantum Code with Topologically-Protected Outer Structure on the Q3 Hypercube: Channel-Friction Semantics and Applications to Particle Physics. The preceding version and earlier versions remain available. This is a revised preprint, not an independently peer-reviewed result. It uses existing evidence; no new scientific calculations were run and research branches remain frozen. AI assistance is acknowledged in the manuscript. Revision guide for all twelve papers. The attached PUBLICATION_UPDATES.csv provides old and new titles, DOIs and replacement descriptions for catalogue and website updates.
Non-Associative Quantum Error Correction (NA-QEC): The Exceptional Jordan-KL Condition and Rank-1 Idempotents in G2 Topologies
CORRECTION NOTICE (2026-08). No error-correction simulation was performed for this work; the Knill–Laflamme conditions are asserted rather than verified. Treat the construction as a proposal. A revised version is in preparation. Standard quantum error correction (QEC) is founded on the Knill-Laflamme (KL) conditions, which implicitly assume that physical error operators compose associatively. Over the non-associative Octonions O and the exceptional Lie group G2, this assumption fails: the elementary identity (PEa†)(EbP)=P(Ea†Eb)P no longer holds, generating a chiral shear that breaks the Hermitian reality of observable eigenvalues. We resolve this obstruction by replacing ordinary matrix multiplication with the Jordan product A∘B=21(AB+BA), which restores Hermiticity by symmetrisation while retaining non-associativity. Working over the 27-dimensional Exceptional Jordan Algebra J3(O) (the Albert algebra), we prove the Exceptional Jordan-KL Condition: for the Furey projector P=diag(1,0,0)∈J3(O), the Jordan U-operator satisfies {P,Ea†∘Eb,P}=cabP for every symmetrised error element Ea†∘Eb∈J3(O), where cab∈R is a scalar determined by the normalised trace. The proof rests entirely on the standard Peirce decomposition of J3(O) and McCrimmon's classification of primitive idempotents; no novel algebraic machinery is required. We apply this result to the 731-QPU: the Steane [[7,1,3]] code operates as the hardware-native QEC layer, and the Fano-plane incidence structure enforces distance d=3 geometrically via the Associator Penalty. NA-QEC monitors Jordan-algebraic coherence continuously rather than measuring discrete Pauli parities, trading measurement-induced decoherence for a geometric error-detection channel. We carefully delineate what is proved from what is conjectured: the claim that NA-QEC evades the Eastin-Knill theorem is stated as an open conjecture, motivated by the observation that the Eastin-Knill proof requires the logical subspace to carry a continuous Lie-group representation, a condition that is absent when the logical space is the 1-dimensional Peirce-1 subspace of J3(O).
Paper XXIII of the Six-Throat Directed Geometry program extends the one-particle quantum structure developed in the preceding paper to composite and multipartite systems. Starting from the conserved positive Hilbert product and unitary one-particle dynamics of the directed spacetime, the construction derives the composite Hilbert tensor product, Schmidt decomposition, reduced density operators, local expectation values, purity, entanglement entropy, correlation tensors, normalized joint probabilities, and the CHSH structure. For identical subsystems, the swap operator produces orthogonal symmetric and antisymmetric permutation sectors together with their dimensions, invariant dynamics, exclusion rule, and occupation counting. The central directed result is that the exchange condition retains two directed branches rather than collapsing them into a single undirected sign. Directed branch and permutation parity therefore remain distinct structural labels throughout the multipartite construction. The resulting two-particle and N-particle laws establish a consistent directed permutation representation. Reversal of the directed branch changes every odd directed exchange while leaving even permutations, branch-blind reduced states, permutation sectors, exclusion, and occupation counting unchanged. The complete numerical closure audit contains 74 checks across nine derivational blocks, with all 74 conditions satisfied. Representative numerical residuals range from exact zero to approximately machine precision. The resulting fixed-particle-number theory contains tensor composition, entanglement, correlations, identical-particle statistics, dual directed exchange branches, and a consistent multipartite permutation structure. Field-level Fock construction, creation and annihilation operators, variable particle number, vacuum structure, and interacting quantum fields are intentionally left to the subsequent paper.
A supplementary calculational note to the interpretational program "Spacetime as Register, Not Stage". The note reconstructs the full structure of spin-1/2 singlet correlations — the joint distribution P = 1/4(1−αβ·a·b), the correlation E(a,b) = −a·b, and CHSH S = 2√2 — from the program's picture of the "entangled" pair as one indivisible complement unit read out locally at two sites by independently oriented detector settings. The note proves sufficiency, not derivation, and keeps an explicit ledger of its epistemic status: two rules are added and underived (a half-angle spinor readout structure, and the square law for amplitudes), two are derived conditionally (antisymmetry, relative to an explicit phase bridge; normalization, fixed by completeness with no free parameter), and every remaining gap is recorded in a debt register. The model exits the factorizable (Bell) class exactly where quantum mechanics does: conditioned on the full preparation state, local marginals remain 1/2 (parameter independence holds per preparation state), while outcome independence fails — with no superluminal transport claim derived from statistics, no superdeterminism (the preparation state carries no free parameter that could co-vary with later settings), and operational no-signaling verified numerically. A deterministic hidden-orientation contrast model is included, reproducing the linear correlation −1+2θ/π and S ≤ 2, to show that the cosine requires a genuine exit from the factorizable class rather than a change of vocabulary. The English text is a full translation of the author's Polish source version (both files included; frozen 28.07.2026, changes tracked in the program's correction register, Series 33).
Within the sealed-records program developed in Part I, we derive the structures of particle statistics and gauge theory as outputs rather than inputs. The results hold at stated scope and are accompanied by machine receipts. First, univalence superselection is derived within the stated record class: the silence of the relative sign is established by exhaustive verification within that class, so the term "derived" carries exactly that scope. The silent-seam axiom then excludes superpositions across the double cover, making the (-1)^F grading central. Second, spin quantization, 2m in Z, follows from the (1,3) signature together with the belt-trick identification, with record anyons confined to genuinely two-dimensional screens. Third, under the named framed-transport premise, namely that exchange is transport of the fiber with its frame, sealed positivity on the exchange collar forces the exchange phase (-1)^{2m}: wrong-statistics sectors carry strictly negative weights, and Pauli exclusion appears as an exact zero. Fourth, parastatistics is excluded at pair-exchange level: the transport commutant on V tensor V is exactly span{1, P}, and the +/-1 options are excluded by the ledger. The full n-strand Doplicher-Haag-Roberts parastatistics question is explicitly scoped. Fifth, the internal symmetry group of the fiber is reconstructed as the commutant of record exchange. Once the ledger forces the statistics, classical Schur-Weyl duality, imported with no novelty claim, delivers the full unitary group of the fiber: U(3) for three-strand fibers, with an implementation check 165 = 165 and the determinant slot included, and U(2) for the weak fiber, with checks 10 = 10 and 20 = 20. This is in the direction of Doplicher-Roberts reconstruction: group as output of statistics. What is delivered here is the global symmetry group; promotion to a local gauge symmetry with connection and dynamics is an additional step, graded as input where Part III uses it. Color receipts include exact Pauli exclusion for color, ||A_4|| = 0, and uniqueness of the baryon singlet. We state the paper's two named premises, framed transport and collar presentability, and the scope of each result. The matter-content selection built on these structures, together with its falsifiable consequences, appears in Part III.