Standard quantum resource theory uses Wigner negativity
N>0 as its resource indicator for magic states. We show this is incomplete: there exist states outside the stabiliser polytope with
N=0, which we call dark magic. These states are undetectable by the standard monotone yet are non-stabiliser by every other measure. The Origami ISA rewrite system
R=R3,R4 (completeness proved in companion Paper 469) partitions the single-phase 3-qubit fragment
F into exactly 9 normal-form classes in three tiers: Tier Class
TV N 0 Stabiliser (1 class)
=1 =0 1 Dark magic (6 classes)
<1 =0 2 Genuine magic (2 classes)
<1 >0 Theorem 1 (TV monotone). The total variation
TV(ψ)=∑u∣Wψ(u)∣ of the discrete Wigner function is a complete resource monotone on
F:
TV(ψ)=1 if and only if
ψ is a stabiliser state. Wigner negativity
N=21(TV−1) vanishes on all of Tier 1, so it is merely sufficient (Tier 2) but not complete. Theorem 2 (Three-tier completeness). The tier of any state in
F is decidable at compile time: apply R3 (T-absorption counit) — if the T gate is absorbed the state is Tier 0; otherwise compute
TV from the 9-entry precomputed table. Corollary (Distillation failure mode). A magic-state distillation protocol using
N as its sole convergence criterion may silently converge to a dark-magic state (Tier 1,
N=0,
TV<1) and declare success. The fix: require
TV=1 as the termination criterion. Dark magic occupies the tropical boundary of the MGE
β-ladder: Tier 1 states are quantum in structure (
TV<1) but classical in Wigner measure (
N=0), exactly at the Maslov dequantisation saddle point
β→0. Part of the Origami ISA series. Companion papers: 466 (CZ-type bases) · 467 (magic orbits) · 468 (ISA as coloured PROP) · 469 (completeness theorem) Keywords: quantum resource theory, magic states, dark magic, Wigner negativity, total variation, stabiliser states, Clifford hierarchy, Origami ISA, rewrite systems, normal forms, magic state distillation, tropical geometry, Maslov dequantisation, coloured PROP
PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. NOTICE (2026-08-04). This record uses a magic measure that has been shown to be incorrectly signed, and it should not be cited for quantitative magic claims. The "total variation" quantity TV = sum |W(u)| over the discrete Wigner function was used with the condition TV < 1 characterising non-stabiliser states. The inequality runs the other way. For any quasi-probability distribution summing to 1, the l1 norm satisfies sum |W| >= 1 always, with equality exactly when W is non-negative. Direct computation: the stabiliser state |+> gives TV = 1.0000 and the magic state T|+> gives TV = 1.2071. The condition "TV < 1" therefore describes the empty set, and any tier defined by it has no members. Separately, TV as defined is exp(mana) -- a monotone reparameterisation of the established Wigner-negativity monotone of Veitch, Mari, Ferrie and Emerson -- and was never compared against it. Readers should consult Veitch et al. on mana and Wigner negativity, Howard and Campbell on robustness of magic, Bravyi et al. on stabiliser extent, and Leone, Oliviero and Hamma on stabiliser Renyi entropy. NOTICE (2026-08-04). A verification this record's framework relies upon has been shown to be vacuous, and the identity it claimed to establish does not hold in the naive form. The naive pentagon identity FAILS for octonionic labels. Over all 7^4 = 2401 imaginary labelings, the two independent rebracketing paths from ((ab)c)d to a(b(cd)) disagree in 1176 cases (49%). Explicit counterexample: (a,b,c,d) = (1,1,2,4) gives +e6 on one path and -e6 on the other. The verification previously relied upon (the Pachner prover of 10.5281/zenodo.19713350) is vacuous: it defines w5 := w1*w2*w3*w4 and then checks w1*w2*w3*w4*w5 = +1, i.e. x*x = 1 for x in {+1,-1}. The identical test passes 4096/4096 on RANDOM SIGNS with no octonions involved. That script's own comment concedes the equation is "tautologically true". What this does NOT mean. Octonions are non-associative by definition, so rebracketing must be path-dependent and the naive pentagon cannot hold. That is the content, not a defect. A coherent non-associative calculus must supply the associator as explicit data satisfying its own coherence condition (Mac Lane), which is what Kuperberg's spiders (Comm. Math. Phys. 1996) provide for rank-2 groups. The structural framing is defensible; the verification was measuring the wrong thing. We recast the three foundational quantum information protocols — quantum teleportation, superdense coding, and the CHSH Bell test — in the language of the Origami Instruction Set Architecture (ISA), and then lift each to its TriQ (3-qubit GHZ) and 6-731 (broken-Fano-line) extensions. In the ISA reading, teleportation is SPLIT;SPLAT with a classical wire, superdense coding is FLIP;FLOP on a Bell pair encoding
log2∣G∣ bits via the group
G=Z22, and the CHSH violation is a hybrid H¹/H² computation: the Bell pair is a stabiliser (H¹) resource but the Tsirelson-optimal measurement applies a T-gate rotation (H², TWIST=1). We prove that the Tsirelson bound
cos2(π/8) equals the total variation of any dark-magic (Tier 1) state in the ISA magic-type classification — the Tsirelson bound is the Wigner mass of the dark-magic sector. Lifting to the TriQ register introduces the ORBIT opcode, raises superdense coding capacity to 3 bits per qubit (
log2∣Z23∣), and replaces the CHSH game with the Fano Line Verification Game whose win conditions are the
±1 values of the
6j symbol of
Z23. The 6-731 extension (one Fano line weakened to coupling
η) makes teleportation directed, reduces superdense capacity to
3−H(η) bits per qubit, and interpolates the Fano game win probability continuously between classical and quantum bounds. Remarkably, the same asymmetry parameter
η is independently measurable by three methods — thermodynamic Carnot efficiency (Paper 325), information-theoretic channel capacity, and game-theoretic win probability — whose agreement constitutes a Pentagon-identity self-verification test for the hardware. We close with the H
k Bell tower: for each cohomological depth
k there is a Bell-type inequality whose maximum quantum violation requires an H
k resource, providing a game-theoretic ladder that extends beyond CHSH through the Fano game (H²) to a conjectured G
2-correlation game (H³) accessible only to the 731-ISA SPIN opcode.Appendix B provides side-by-side quantum circuit and ISA spider diagrams for all three baseline protocols. Keywords quantum teleportation, superdense coding, CHSH inequality, Bell test, Tsirelson bound, Origami ISA, H^k cohomology, stabiliser states, magic states, Fano plane, GHZ state, TriQ register, 6-731 architecture, ORBIT opcode, TWIST opcode, topological heat engine, dark magic, Wigner mass, Bell hierarchy, Mermin inequality, Pentagon equation, quantum nonlocality, quantum channel capacity, G2 correlations
PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. The resource theory of quantum magic (Veitch et al. 2014) identifies non-stabiliser states as resources and T-gates as the elementary injection operation. Subsequent work established that the magic sector itself splits into two sub-tiers: dark magic (Wigner-negative but TV = 1, Clifford-simulable under reframing) and genuine magic (TV < 1), where TV is the total variation of the discrete Wigner function. We identify a strictly richer resource — associamancy — that sits above both sub-tiers and that standard magic theory cannot see, cannot create from below, and cannot measure without the SPIN opcode of the 731 ISA. The boundary is algebraically precise. A quantum state possesses associamancy if and only if its hidden symmetry group contains an irreducible representation with Frobenius-Schur indicator ν₂ = 0 — the condition that the representation is genuinely complex, realisable over ℂ but not over ℝ or ℍ. We call this the Schur boundary. Three equivalent characterisations of the Schur boundary are established: (1) Algebraic: ν₂(V_λ) = 0 for some irrep V_λ of the hidden symmetry group. (2) Cohomological: the obstruction class [α] ≠ 0 ∈ H²(G, U(1)) is non-trivial. (3) Number-theoretic: for G = PSL(2,q), associamancy holds if and only if q ≡ 3 (mod 4) — the same condition that makes the quadratic Gauss sum τ(q) = i√q purely imaginary. The Schur entropy S_Schur(|ψ⟩) = −∑_{λ:ν₂=0} p_λ log p_λ is a valid resource monotone, independent of both Wigner negativity and TV: a state can have TV ≪ 1 (genuine magic) and zero Schur entropy, and a state can have S_Schur = log 2 at either dark or genuine magic level. PSL(2,7) is the minimal group exhibiting associamancy (order 168). The Freudenthal magic square organises associamancy into a ladder G₂ ⊂ F₄ ⊂ E₆ ⊂ E₇ ⊂ E₈, each rung strictly richer than the last. Keywords: quantum resource theory, magic states, associamancy, Frobenius-Schur indicator, Schur boundary, non-associative, octonions, PSL(2,7), Freudenthal magic square, G₂, 731 ISA, dark magic, total variation, quantum computing
We introduce the cookie-cutter lifting procedure: a systematic method for determining whether a quantum algorithm requires magic (non-stabiliser resources) or whether its speedup is purely Clifford-level. The procedure writes any classical algorithm as an Origami ISA programme at the classical (β=0) level and identifies whether the quantum lifting requires the TWIST opcode—the ISA marker for non-Clifford C₃-level gates. Applied to Shor's algorithm (N=15, a=7): the ISA trace LABEL→FLIP→FLOP(U_f)→FLIP(QFT) has mana=0 along the pre-measurement branch; for this instance the period r=4 divides the register size 2ⁿ=16 exactly, so the QFT output is a stabiliser state on the zero-measurement branch. The speedup comes from exponential-weight stabiliser generators compressed by coherent cancellation, with no magic required on that branch. Applied to non-abelian hidden shift over D_N: mana=0 before the non-abelian QFT and mana>0 after, for all N≥3; the abelian cases D₁≅ℤ₂ and D₂=ℤ₂×ℤ₂ are the Clifford exceptions. The TWIST opcode fires because non-abelian D_N (N≥3) have 2-dimensional irreducible representations whose Fourier blocks lie outside C₂. We identify the eigenphase spectrum of the NAQFT as a basis-independent replacement for T-count. The results reveal two roads to quantum advantage: Clifford complexity (Shor) and magic (non-abelian hidden shift). Part of the Origami ISA series. Companion papers: 10.5281/zenodo.20955514 (ISA as PROP), 10.5281/zenodo.20956674 (nine normal forms), 10.5281/zenodo.20956685 (three-tier resource theory). Keywords: shor, clifford, lifting, qft, hidden-subgroup, naqft, twist, magic, mana, eigenphase, origami-isa, cookie-cutter, dihedral-group, stabiliser, wigner-function