A phase-space census of the states arising in Shor's algorithm for thestandard textbook instance N=15, a=7, together with a contrasting analysisof the non-abelian hidden shift problem over dihedral groups. For N=15, a=7 the period r=4 divides the register size 2^n=16 exactly. Weshow by direct computation of the discrete Wigner function that the fullpre-measurement QFT output is therefore a stabiliser state, with zeronegative Wigner cells. This is an arithmetic property of the choseninstance rather than of Shor's algorithm in general, and it illustrates aknown hazard: N=15 is degenerate in ways that make small-scale factoringdemonstrations prove less than they appear (Smolin, Smith and Vargo, Nature499, 163 (2013)). Resolving the four post-measurement coset branches showswhere the negativity is: three of the four carry exactly 8 negative Wignercells each, while the zero-measurement branch remains stabiliser. Thespeedup on that branch comes from high-weight stabiliser generatorscompressed by coherent cancellation, not from non-stabiliser resources. For the non-abelian hidden shift over the dihedral group D_N, the Wignerfunction is non-negative before the non-abelian Fourier transform andacquires negativity after it, for all N >= 3. The abelian cases D_1 = Z_2and D_2 = Z_2 x Z_2 are the exceptions. The mechanism is representation-theoretic: D_N for N >= 3 has 2-dimensional irreducible representationswhose Fourier blocks do not normalise the Pauli group, and so lie outsidethe Clifford group. We propose the eigenphase spectrum of the non-abelianQFT as a basis-independent, group-theoretic diagnostic that avoids thequbit-padding artefacts affecting Wigner-based measures for non-dyadicgroups. Scope. All computations are exact numerics on the specific instancesstated. No general claim is made about the magic requirements of Shor'salgorithm at cryptographically relevant sizes, and the classificationframing of earlier versions has been withdrawn. Version note (2026-08): retitled and rescoped. The earlier version framedthis as a general "lifting procedure" for classifying arbitrary algorithms;that framing overstated what two worked instances support. Two companionpapers previously cited here have been withdrawn by the author, and theirreferences are removed. Quantities are reported as negative Wigner cellcounts, computed directly, rather than in the derived units used in thewithdrawn work. 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
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
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
Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup H of a group G must be determined from a quantum state
ψ over G that is uniformly supported on a left coset of H. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of
ψ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups H of the q-hedral groups, i.e., semidirect products
Zq⋉Zp, where
q∣(p−1), and in particular the affine groups
Ap, can be information-theoretically reconstructed using the strong standard method. Moreover, if
∣H∣=p/polylog(p), these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently.
Nonstabilizerness (magic) is the quantum resource that, together with stabilizer operations, makes universal quantum computation possible. However, quantifying nonstabilizerness is difficult. The standard measure sums over exponentially many Pauli operators, and for quantum codes no quantitative theory has been available. A structural observation removes this obstacle: the Clifford sector carries no magic and can be removed by a Clifford transformation, leaving an order-three object whose magic is an exactly evaluable fourth moment. Removing this inert background yields closed forms for several code families: a formula for all Dicke states, reducing a 100-qubit case from
4100 terms to a short binomial sum; a bound for cubic-phase codes (twisted quantum doubles and non-Abelian topological order), saturated only by the
D4 code; and a cyclic/zero criterion for group multiplication states. For codeword-stabilized (CWS) codes, the companion paper \cite{Liu26arXiv} carries this reduction to its conclusion: the most magical codes are exactly the Sidon sets, and the nonstabilizerness bounds the code's transversal non-Clifford power. Together these results reduce the computation of nonstabilizerness for a broad class of codes to finite classical counting problems, and identify the maximally magical codes among them.
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
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 63-point 3-qubit symplectic polar space
W(5,2) contains the Fano plane
PG(2,2) as the 7-point GHZ stabiliser subspace. This paper proves that the complementary 56 points decompose into exactly seven orbits of eight points each under the action of the GHZ stabiliser group
Z23, with one orbit per Fano line. This decomposition has a three-level nested structure. Level 0: The Fano plane
PG(2,2) itself — the 7 GHZ stabilisers, the Wigner support of the GHZ state, and the difference set for all orbits. Level 1: The 7 orbit labels
L=0,…,6 form a second copy of
PG(2,2) under bitwise XOR: orbits
OL,
OM,
OK satisfy
OL⊕OM⊆OK iff
L⊕M=K in
Z23. This is the XOR-Fano composition rule governing magic state combination. Level 2: Each orbit
OL splits into two disjoint Lagrangian 4-cliques
OLA⊔OLB, each a maximal abelian subspace of
W(5,2) — the same type of structure as the Fano plane itself. The Wigner negativity of any 3-qubit state decomposes as
N(ρ)=∑LNL(ρ), where
NL is the negativity on orbit
OL. The orbit index
L simultaneously identifies which Fano line is broken (activated) in the resource-theory sense, which magic valence the state carries, and which line is asymmetric in the 6-7 directed-routing architecture. Magic is broken-line activation in phase space. A remark establishes the formal analogy with spontaneous symmetry breaking: the
PSL(2,7)→S4 symmetry reduction of the 6-7 architecture is the discrete finite-group analogue of
SU(2)×U(1)→U(1) in the electroweak sector. The 6 intact Fano lines play the role of the massless photon; the 1 broken line plays the role of the massive gauge bosons; the activated orbit
OL plays the role of the Goldstone mode;
η>0 is the order parameter. The 7 choices of broken line are the 7 discrete vacua. The FMO complex is the biological realisation: the symmetric (
η=0) configuration is evolutionarily unstable, and the broken-symmetry configuration is the stable ground state selected over a billion years — the closest natural analogue to a thermodynamically driven symmetry-breaking transition.