The Watabe-Claude Method reads a signal's structure by assigning 4D-embeddedwindows to the 24 vertices of the 24-cell and decomposing the occupancy with thegroup B4. This note takes the 24-cell the other way -- into a qubit. Its 24vertices, as unit quaternions, are exactly the binary tetrahedral group 2T, afinite nonabelian subgroup of SU(2); under the quaternion-to-SU(2) map they are24 distinct single-qubit gates. Three exact facts, all verified by classical simulation of one or two qubits(no external quantum or group-theory libraries): 1. Foundation. The 24 vertices form 2T (order 24): closed Cayley table, Latin square, 24-cell inner-product signature {-1,-1/2,0,1/2,1}. As SU(2) matrices they are 24 distinct, nonabelian gates containing the Pauli set. 2. An honest negative. Encoding the vertex assignment for a quantum optimiser is a dead end: the 5-qubit binary QUBO has 31 terms (vs 300 one-hot) but up to five-body couplings, and a shallow QAOA recovers the right vertex only 3-10% of the time -- the assignment is a classical argmax with no role for a quantum optimiser. 3. The finding. Applied to a single probe qubit, the 24 gates collapse to 12 distinct outputs, because 2T's center {+-1} is exactly the global phase a lone qubit cannot observe (U(-q)=-U(q) verified for all 24); the observable group is the projective image 2T/{+-1}, the order-12 tetrahedral rotation group. A controlled application makes the center's phase relative and separates all 24. Optimal single-shot (square-root) discrimination succeeds 16.7% over the 12 classes and 12.1% over all 24 -- 2.0x and 2.9x the random baselines, ratios fixed by the 24-cell geometry. The through-line with the classical WCM series: there, B4 reads which symmetrychannel carries a signal's structure; here, the same polytope's group structuregoverns which distinctions a quantum measurement can make. No quantum speedup oradvantage of any kind is claimed; the contribution is an exact correspondencebetween a polytope's group structure and the observability limits of a qubit.Self-audit: 12 findings, all NUMERICAL / NEGATIVE / STRUCTURAL, none overclaimed. v1.0.1 (2026-08-27): Adds an explicit, end-to-end-verified measurement circuit. The optimal square-root discrimination is certified minimum-error (Yuen-Kennedy-Lax conditions plus an independent iteration from random measurements), realised as an explicit Naimark unitary, embedded in a power-of-two qubit register, and compiled to an atomic gate list of RY, RZ, X, and CNOT only -- verified end to end (every gate one of those four types, the product reproducing the target measurement within floating-point accumulation; ~2e-11 over the ~84,000-gate 24-way circuit). The lone-qubit collapse is sharpened to an orbit hierarchy (24 -> 12 generic -> 6 symmetry-axis), naming the measure-zero exceptional probes (including the computational basis). Terminology and scope clarifications throughout (B4 vs 2T; central quotient; coherent-control assumption; honest symmetry-blind CNOT upper bounds 25,210 / 2,022). No quantum speedup is claimed. Status: preprint, not peer-reviewed. This is an independent research note; its revisions were shaped by critique from an AI collaborator, which is not a substitute for formal peer review. Text CC-BY-4.0; code AGPL-3.0-or-later.
The Watabe-Claude Method reads a signal's structure by assigning 4D-embeddedwindows to the 24 vertices of the 24-cell and decomposing the occupancy with thegroup B4. This note takes the 24-cell the other way -- into a qubit. Its 24vertices, as unit quaternions, are exactly the binary tetrahedral group 2T, afinite nonabelian subgroup of SU(2); under the quaternion-to-SU(2) map they are24 distinct single-qubit gates. Three exact facts, all verified by classical simulation of one or two qubits(no external quantum or group-theory libraries): 1. Foundation. The 24 vertices form 2T (order 24): closed Cayley table, Latin square, 24-cell inner-product signature {-1,-1/2,0,1/2,1}. As SU(2) matrices they are 24 distinct, nonabelian gates containing the Pauli set. 2. An honest negative. Encoding the vertex assignment for a quantum optimiser is a dead end: the 5-qubit binary QUBO has 31 terms (vs 300 one-hot) but up to five-body couplings, and a shallow QAOA recovers the right vertex only 3-10% of the time -- the assignment is a classical argmax with no role for a quantum optimiser. 3. The finding. Applied to a single probe qubit, the 24 gates collapse to 12 distinct outputs, because 2T's center {+-1} is exactly the global phase a lone qubit cannot observe (U(-q)=-U(q) verified for all 24); the observable group is the projective image 2T/{+-1}, the order-12 tetrahedral rotation group. A controlled application makes the center's phase relative and separates all 24. Optimal single-shot (square-root) discrimination succeeds 16.7% over the 12 classes and 12.1% over all 24 -- 2.0x and 2.9x the random baselines, ratios fixed by the 24-cell geometry. The through-line with the classical WCM series: there, B4 reads which symmetrychannel carries a signal's structure; here, the same polytope's group structuregoverns which distinctions a quantum measurement can make. No quantum speedup oradvantage of any kind is claimed; the contribution is an exact correspondencebetween a polytope's group structure and the observability limits of a qubit.Self-audit: 12 findings, all NUMERICAL / NEGATIVE / STRUCTURAL, none overclaimed. v1.0.1 (2026-08-27): Adds an explicit, end-to-end-verified measurement circuit. The optimal square-root discrimination is certified minimum-error (Yuen-Kennedy-Lax conditions plus an independent iteration from random measurements), realised as an explicit Naimark unitary, embedded in a power-of-two qubit register, and compiled to an atomic gate list of RY, RZ, X, and CNOT only -- verified end to end (every gate one of those four types, the product reproducing the target measurement within floating-point accumulation; ~2e-11 over the ~84,000-gate 24-way circuit). The lone-qubit collapse is sharpened to an orbit hierarchy (24 -> 12 generic -> 6 symmetry-axis), naming the measure-zero exceptional probes (including the computational basis). Terminology and scope clarifications throughout (B4 vs 2T; central quotient; coherent-control assumption; honest symmetry-blind CNOT upper bounds 25,210 / 2,022). No quantum speedup is claimed. Status: preprint, not peer-reviewed. This is an independent research note; its revisions were shaped by critique from an AI collaborator, which is not a substitute for formal peer review. Text CC-BY-4.0; code AGPL-3.0-or-later.
Prior work by the authors [1] showed that the 24 vertices of the 24-cell, as unit quaternions, form the binarytetrahedral group 2T, that a controlled probe lifts the phase-blind projective group 2T/{±1} to all 24 gates, and thatthe optimal single-shot measurement discriminating them compiles, gate by gate, to an explicit RY/RZ/X/CNOTcircuit. That compilation was symmetry-blind: a textbook Givens synthesis of a generic 32×32 unitary, emitting25,210 CNOTs. Here we ask what the group structure of the input buys back. We report four findings. (A) The 24controlled-probe states span only a rank-3 subspace, so a reader need be correct only there; restricting thesynthesis to that support cuts the emitted CNOT count from 25,210 to 5,980 (4.2×) with the same single-shotarchitecture and the identical optimum. (B) The optimal POVM is exactly 2T-covariant: its 24 elements are onegroup orbit of a single seed, and the analysis map is a genuine tight-frame isometry that intertwines the supportrepresentation with the regular representation of 2T. (C) Consequently the reader factors as a data-independent2T group-Fourier transform followed by a placement whose data-dependent part is tiny (72 CNOT); the rank-3signal occupies just 5 of the 24 Fourier lines, which is the structural reason (A) works. (D, honest negative) Acheap synthesis of that group-Fourier transform is not achieved: the naive tensor product of the C3 and Q8transforms fails to block-diagonalize because 2T = Q8 : C3 is a crossed product, and the missing twiddle absorbsall the cost. We make no quantum-advantage claim; every number is a real emitted gate count or a verified matrixidentity from exact, numpy-only statevector simulation. ※ The license for codes is AGPL3.0-or-later
The parent paper [1] reduced the circuit synthesis of the optimal 24-outcome reader for the binarytetrahedral group 2T to a single fixed operator, the 2T group Fourier transform, and identified its dominantcost as FQ8, the 8×8 group Fourier transform of the quaternion group Q8. The companion addendum [2]reported, as an honest negative, that on the parent paper's generic Givens-based emitter astructurally-built FQ8 still costs 218-234 CNOT -- far above the O(101) structural estimate -- and framed thegap as a binary-embedding and generic-decomposition tax: an emitter artifact rather than arepresentation-theoretic obstruction. This note discharges that claim with verified circuits. Using only thestructure of FQ8 -- the plus/minus pairing induced by the center Z2 of Q8, which lets a single Hadamardseparate the character sector from the two-dimensional-representation sector -- we build an exactfactorization in which the sole entangling object is a single controlled two-qubit gate Cq2[Wr]. We thenshow that Wr has Cartan (KAK) coordinates (π/4, 0, 0), the CNOT class, so its two-qubit nonlocal core isone CNOT. Three constructions of FQ8 are emitted and verified equal to FQ8 up to global phase atmachine precision, giving a monotone progression of 234 → 33 → 23 emitted CNOT. Every number belowis a real emitted, verified circuit, not a cost-model estimate. We are explicit about what is not achieved: thesingle-digit floor implied by "one CNOT core" is not reached, because lifting Wr to its controlled form costsone Toffoli plus controlled-local flanks (14 CNOT as emitted); reducing this further is a compiler task, not anew representation-theoretic result. The result establishes that the 200-CNOT wall was largely an emitterartifact. ※ The license for codes is AGPL3.0-or-later
The parent paper [1] reduced the circuit synthesis of the optimal 24-outcome reader for the binary tetrahedralgroup 2T to the synthesis of a single fixed operator: the 2T group Fourier transform F, whichblock-diagonalizes the regular representation into irreducible blocks [1,1,1, 2,2,2, 2,2,2, 3,3,3]. That paperreported, as an honest negative (its Result D), that the naive tensor product FC3 ⊗ FQ8 fails to realize F,because 2T = Q8 : C3 is a crossed (semidirect) product, and it named the missing piece — the twiddle fromthe Clifford theory of Q8 : C3 — as a remaining construction. Here we construct it. We show (i) the twiddle hasan exact closed form: in coset-times-fiber coordinates, each two-dimensional-sector column of F equals (ωmM0)t applied to the Q8-Fourier column, where M0 is the intertwiner of the two-dimensional irrep ρ with itsC3-conjugate (rebuild residual 1.7 × 10-16); (ii) a semidirect-product Fourier obstruction: nodirect-product-form two-factor split of F exists, because the C3-action moves the Fourier dual of Q8 nontrivially(M0 ≠ I) — stated and verified as a direct-versus-semidirect contrast, not as an observed residual; and (iii) adirect measurement of the binary-embedding tax: the abstract group-FFT gate count is O(101), but ageneric-Givens emitter on the 24-into-32 five-qubit embedding cannot realize it, so the O(102) target isreachable in principle yet unreached with a structure-blind emitter. The line's firmly established compressionremains the parent paper's Lever-1 (25,210 → 5,980 CNOT, 4.2×). No quantum-advantage claim is madeanywhere. All results are exact statevector-level numpy computations at machine precision. ※The license for codes is AGPL3.0-or-later
Paper 1 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper establishes an exact tetrahedral qubit-state geometry within the deterministic six-bit construction underlying the TUS OS Triad. For every lawful six-bit anchor A, the full-state-space maximin construction selects companions B,C such that the resulting triple has exact pairwise Hamming separation (4,4,4). Every such triple has a unique equidistant completion in the same six-bit state space, given by D = A ⊕ B ⊕ C. Under the normalised sign embedding, the four completed states have unit norm, pairwise inner product −1/3, zero centroid and rank three. They therefore form the vertices of a regular tetrahedron. The paper constructs an exact isometric representation of this geometry as four qubit Bloch vectors forming a tetrahedral symmetric informationally complete structure, while affine centring and common renormalisation of the original three-state face yield exact qubit-trine geometry. The analytical construction is verified exhaustively over the complete finite state space. The census covers all 64 six-bit anchors, 249,984 ordered anchor-conditioned candidate pairs, 41,664 unordered triples, 960 mutual-distance-four triples, and 240 completed four-state sets. The same geometric recovery is demonstrated in two frozen public TUS receipt packages. The result separates the mathematical completion from the protocol object. Only the three states (A,B,C) are emitted as active Triad starts; the fourth state (D) is a uniquely determined mathematical completion and does not become an additional receipt-bearing protocol member. The tetrahedral and trine constructions therefore provide exact qubit-geometric representations of an already-fixed finite classical structure rather than additional states or operations in the native TUS OS protocol. The claim boundary is explicit. The paper does not claim that TUS OS physically prepares qubits, performs SIC or trine measurements, exhibits contextuality or other nonclassical behaviour, or obtains quantum computational advantage. Its result is an exact finite geometric correspondence and protocol linkage between the completed six-bit Triad structure and canonical qubit tetrahedral and centred-face trine geometry. Supporting General Theory Programme This paper is Paper 1 and the opening theorem paper in the TUS OS quantum-operational and Answerability research programme. It proves that the canonical six-bit TUS OS Triad has exact (4,4,4) Hamming geometry, a unique XOR completion, regular-tetrahedral geometry, an exact qubit-SIC representation, and centred-face trine geometry. Paper 2 develops the exact operational calculus and full-carrier qubit prepare–measure representation. Paper 3 extends the result to the complete relational Atlas and exact global static quantum dimension. Paper 4 develops the complete-future, orthogonality, process, LANDING/GUIDANCE, and native Triadic synthesis theory. Paper 5 introduces the Anchored Analog Quantum System framework, while Papers 6–7 generalise the contract-relative resource and operational-observability structure. Papers 8–9 then develop generic Answerability theory, Paper 10 proves the complete protected and executable realisation of TUS OS in the Answerability Calculus, and Paper 11 closes the programme with A General Theory of Answerability, establishing Structural Identity, Recognition, the canonical mandatory protected core, and lawful Realisation Freedom. Paper 1 therefore provides the mathematical starting point for the complete programme. Paper 1 — Canonical Tetrahedral Qubit-State Geometry in the TUS OS Triad XOR Completion, Receipt Verification and the Qubit-Geometric Semantic WitnessEstablishes the canonical tetrahedral geometry, unique XOR completion and exact qubit-SIC representation. The canonical maximin construction yields a unique same-space completion whose normalised sign embedding forms a regular tetrahedron, while the centred and commonly renormalised emitted face has exact qubit-trine geometry. The emitted three-state face itself is not asserted to be a trine before that affine centring and normalisation. Paper 2 — The TUS OS Operational CalculusClassical Construction of Exact Qubit Prepare–Measure Behaviour from a Receipt-Custodied Engine ArchitectureExtends the Paper 1 geometry to exact full-carrier qubit prepare–measure behaviour from the deterministic classical engine under the declared Coordinate-Incidence Operational Harness. For each lawful foundation, the 64 six-bit carriers map to exactly 27 anchor-relative qubit preparations, while the separately declared face Harness gives the exact trine table. Paper 3 — From Finite Relational Structure to Qubit Operational FormAmbient Foundations, Canonical Subatlas, Exact Global Quantum Dimension, GUIDANCE Witness, Sequential Closure and Global Representation in TUS OSExtends the local construction to the completed-foundation family and canonical TUS OS subatlas, establishes exact carrier-reconstruction structure, proves that the repeated-common-effect static Atlas has exact minimum complex Hilbert dimension 4, and establishes deterministic predictive-state minimum 64 for the frozen complete principal future language. Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusNative Contract, Statistical Discrimination and the Invariant Probability ModuleFixes the formal native domain, complete principal terminal future-experiment language and one-common-word experiment class; establishes complete statistical separation of distinct carrier pairs, exact parity-resolved discrimination results and complement reachability; and proves that the rational invariant probability module has dimension 33. Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic SynthesisEstablishes universal complement-support orthogonality and its propagation under exact common-CPTP dynamics, yielding the certified same-parity support graph and the bound 32 ≤ d_CPTP^min ≤ 64, with cross-parity forcing and the exact state-only common-CPTP minimum remaining unresolved. It further establishes recovery-complete LANDING and GUIDANCE results, exact 64-dimensional selected-line instrument/readout minima at their declared contracts, an exact classical–quantum hybrid construction, and the native structural/predictive quotient correspondence. Paper 5 — TUS OS as an Anchored Analog Quantum SystemNative Witnesses, Predictive Synthesis, and Access-Contract Resource SeparationDefines Analog Quantum Views and Anchored Analog Quantum Systems, proves that TUS OS is a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus, elevates the inherited structural/predictive correspondence into the Predictive Synthesis Identity Theorem, organises the inherited exact quantum-operational results through a typed access-contract resource law, and composes the recognition, predictive, resource and downstream-conservation results into the TUS OS Anchored Analog Quantum System Theorem. Paper 6 — Contract-Relative Quantum Realisations of Classical ProcessesPredictive Compression, Vanishing Worst-Case Causal-State Transparency, and Finite-Dimensional BreakdownGeneralises the access-dependent resource phenomenon beyond the TUS-specific discovery corpus by formalising the contract-relative achievable-resource object (S,C,𝔐,ε) ⟼ Ach(S,C;𝔐,ε), and establishes three exact contract-relative regimes: strict finite separation, unbounded finite separation and finite-capacity breakdown. These are instantiated respectively by the TUS OS 4→64 separation, the periodically modulated decay 2→N exact prediction-to-transparency separation, and dual-Poisson finite-capacity breakdown for every positive uniform worst-case transparency threshold. Paper 7 — Operational Observability and Contract-Induced Resource Transitions in Quantum RealisabilityStructural Classification, Family Scaling, and Finite-Resource BreakdownDevelops a structural classification theory for contract-induced representation-resource transitions, introducing the Contract Distinction System (CDS), CRC cofinal/refinement equivalence, and Certified Observability Transport Theory (COTT). It separates internal divergence from family divergence, classifies baseline-feasible strengthenings as resource-neutral, strict finite, finite-resource breakdown, or UNRESOLVED, and proves exact and asymptotic witnesses including an interface-induced 2→3 separation, the TUS OS 4→64 transition, incrementally unbounded finite separation for periodically modulated decay processes, and finite-resource breakdown for positive-threshold dual-Poisson transparency. Paper 8 — The Resource Geometry of Answerable InferenceIrreducible Answerability Structure, Minimal Sufficient Cores, and Certified Inference-Resource BoundariesDevelops the resource geometry of answerable inference. It formalises lawful-discharge semantics, source-level inferential obligations, answerability-sufficient reductions and irreducible sufficient cores, while keeping semantic structure, certified lower bounds and operational feasibility independently constituted. Gate E establishes a local answerability-to-resource threshold result; globally, exact closure identifies the Pareto-minimal inference-resource boundary. On the generated exact branch, L = ↑F, and under well-partial-order conditions the resulting lower-resource geometry has a unique finite canonical antichain normal form. Paper 9 — The Answerability CalculusLawful Reduction, Contract–Resource Representation, Composition and ActivationDevelops a general mathematical calculus of Answerability as a constituted, typed relation. Its Base Theorem states that no terminal proposition counts as an answer unless it lawfully discharges the independently constituted contract. The paper develops lawful reduction, contract–resource geometry, extremal representation, repai
DOI 10.5281/zenodo.20410167 carries no abstract — its description contains only notes-to-self ("translate 'computational entropy' into the standard unit, add the baseline anchor"). This note attempts the requested baseline anchor and reports why it cannot be supplied as asked, together with a structural finding that supersedes the question. The central object encodes nothing The deposited decoder assigns self.n, self.k, self.d = 96, 6, 8 as literal constants. Computed from its own parity-check matrices over GF(2): rank(Hz)=23, rank(Hx)=73, 23+73=96=n, hence k=0. Equivalently rowspace(Hx) = ker(Hz) exactly, both of dimension 73 = the cycle-space dimension E−V+1: every cycle is a triangle sum and no logical operator exists. This is the expected topological answer — the 24-cell's boundary is a triangulated 3-sphere, simply connected, so it encodes nothing. Confirmed three independent ways. The CSS condition itself is satisfied exactly: the geometry is right, the label is wrong. The claimed d=8 is undefined rather than merely unverified. Consequence for the requested benchmark "Iso-accuracy against an off-the-shelf decoder" is undefinable when there are no logical qubits. Measured on the metric that survives (2000 trials per rate, versus PyMatching 2.4.0 exact MWPM on the same Hz): both always return to the codespace, but the deposit's decode_mwpm — named MWPM, in fact greedy nearest-pair — returns corrections 5.9% to 12.8% heavier across p=0.01–0.10, the gap widening with noise, never better. What could not be measured, and is therefore not reported Energy. RAPL exists on this machine but energy_uj is root-only and perf_event_paranoid blocks the power PMU. No FLOPs/joule figure is given; wall-clock, peak RSS (77,592 KB), matrix bytes and operation counts are reported instead. Separately, "computational entropy" has no operational definition anywhere in the corpus — every occurrence is unsourced marketing copy with mutually inconsistent percentages. The quantity the note-to-self asks to convert cannot be converted. Propagation This defect propagates to DOI 10.5281/zenodo.20407370, whose published claim is "composing six [[96,6,8]] 24-cell honeycomb codes": compose_24d_prototype.py is byte-identical in both deposits and imports the same k=0 class.
Short note with the circuit, an independent verifier and the search records. For ancilla-free Clifford preparation of the logical zero |0_G> = 2^{-11/2} Σ_{c ∈ C^⊥} |c> of the [[23,1,7]] quantum Golay code (Open Problem Zoo op_c847b635f53b0812, after Paetznick and Reichardt 2012), the best documented circuits were 45 two-qubit gates at two-qubit depth 7 and 44 gates at depth 8 (Doherty et al., arXiv:2603.17743). We exhibit an explicit circuit with 44 two-qubit gates (all controlled-Z, i.e. CNOT up to free single-qubit Cliffords) in six disjoint two-qubit layers of sizes 9, 10, 7, 6, 7, 5, so g_G(D) ≤ 44 for every D ≥ 6; it dominates both previous Pareto points. The circuit is given in the cyclic coordinates of the Golay code (generator polynomial 1+x^2+x^4+x^5+x^6+x^10+x^11) and verified two independent ways: a stabilizer-tableau proof (the output stabilizer group is CSS with all + signs, its Z-code is a binary [23,12,7] code, hence the Golay code by uniqueness, and an explicit wire permutation is exhibited) and a full 2^23-amplitude statevector overlap of 1.0 with the target. The circuit was found by an autonomous AI research agent (the depositor's EVO discovery harness, Claude Sonnet 5.5 proposer) by a local search started from the published 44-gate depth-8 circuit: a bit-packed C tableau simulator, a cost equal to the number of qubits left entangled when the circuit is run backwards from the target, and a random walk among 44-gate circuits that penalises gates in layers beyond the target depth. The same searches did not find 43 gates at any depth or 44 gates at depth 5 (about 10^10 and 10^9 evaluations respectively; these are negative search outcomes, not bounds). Claim boundary: an upper bound g_G(6) ≤ 44 witnessed by an explicit, machine-checkable circuit; no optimality or priority claim. Files: the note (PDF + LaTeX), the circuit in three equivalent JSON forms (agent wire labels; cyclic coordinates; cyclic coordinates with CNOT as the only two-qubit gate), the verifier and relabeling finder, the agent's notes and search code, the task text, SHA-256 sums.