We prove that the discrete Wigner function of the canonical $N$-qubit state $|\psi_N\rangle = \mathrm{CZ}{01}\cdot(T\otimes I^{\otimes N-1})\cdot|{+}^{\otimes N}\rangle$ satisfies an exact conservation law for all $N \geq 2$. Three quantities are independent of system size: the total negative Wigner mass $\sum{W<0} W_N(u) = -\sqrt{2}/8$, the total Wigner mass $\sum_u W_N(u) = \cos^2(\pi/8) = (2+\sqrt{2})/4$, and the total variation $\sum_u |W_N(u)| = (1+\sqrt{2})/2$. The conserved quantity $(1+\sqrt{2})/2 = \delta_S/2$, where $\delta_S = 1+\sqrt{2}$ is the silver ratio, reflects the $T$-gate angle $\pi/4$ via the half-angle identity $\cos^2(\pi/8) = (2+\sqrt{2})/4$. The $T$ gate imprints a fixed Wigner defect of magnitude $\sqrt{2}/8$ into phase space; adding $|{+}\rangle$ spectator qubits spreads this defect across $2^{N-2}$ cells but never changes its total. The conservation follows from the factorisation $W_N = W_2 \otimes W_{|+\rangle}^{\otimes N-2}$ and the fact that $\sum_u W_{|+\rangle}(u) = 1$ exactly for any stabiliser state. The conserved constants $\cos^2(\pi/8)$ and $(1+\sqrt{2})/2$ are the tier-boundary constants of the three-tier magic resource theory (Buckley 2026, Hot Logic). The conservation law explains why these constants are fundamental: they are $N$-independent invariants of the entire single-$T$ family, not properties of any specific state. The dark-magic tier is stable under adding spectator qubits. All five quantities are confirmed to machine precision for $N = 2, 3, 4, 5, 6$ by experiment x471b. Keywords: Wigner function · magic states · resource theory · T gate · stabiliser states · discrete Wigner function · silver ratio · conservation law · Clifford hierarchy · quantum computation · Origami ISA · dark magic
The Wigner Defect Conservation Law: N-Independence of Magic Imprinted by a Single T Gate
We prove that the discrete Wigner function of the canonical N-qubit state ∣ψN⟩=CZ01⋅(T⊗I⊗N−1)⋅∣+⊗N⟩ satisfies an exact conservation law for all N≥2. Three quantities are independent of system size: the total negative Wigner mass ∑W<0WN(u)=−2/8, the total Wigner mass ∑uWN(u)=cos2(π/8)=(2+2)/4, and the total variation ∑u∣WN(u)∣=(1+2)/2. The conserved quantity (1+2)/2=δS/2, where δS=1+2 is the silver ratio, reflects the T-gate angle π/4 via the half-angle identity cos2(π/8)=(2+2)/4. The T gate imprints a fixed Wigner defect of magnitude 2/8 into phase space; adding ∣+⟩ spectator qubits spreads this defect across 2N−2 cells but never changes its total. The conservation follows from the factorisation WN=W2⊗W∣+⟩⊗N−2 and the fact that ∑uW∣+⟩(u)=1 exactly for any stabiliser state. The conserved constants cos2(π/8) and (1+2)/2 are the tier-boundary constants of the three-tier magic resource theory (Buckley 2026, Hot Logic). The conservation law explains why these constants are fundamental: they are N-independent invariants of the entire single-T family, not properties of any specific state. The dark-magic tier is stable under adding spectator qubits. All five quantities are confirmed to machine precision for N=2,3,4,5,6 by experiment x471b. Keywords: Wigner function · magic states · resource theory · T gate · stabiliser states · discrete Wigner function · silver ratio · conservation law · Clifford hierarchy · quantum computation · Origami ISA · dark magic
The Wigner Defect Conservation Law: N-Independence of Magic Imprinted by a Single T Gate
We prove that the discrete Wigner function of the canonical N-qubit state ∣ψN⟩=CZ01⋅(T⊗I⊗N−1)⋅∣+⊗N⟩ satisfies an exact conservation law for all N≥2. Three quantities are independent of system size: the total negative Wigner mass ∑W<0WN(u)=−2/8, the total Wigner mass ∑uWN(u)=cos2(π/8)=(2+2)/4, and the total variation ∑u∣WN(u)∣=(1+2)/2. The conserved quantity (1+2)/2=δS/2, where δS=1+2 is the silver ratio, reflects the T-gate angle π/4 via the half-angle identity cos2(π/8)=(2+2)/4. The T gate imprints a fixed Wigner defect of magnitude 2/8 into phase space; adding ∣+⟩ spectator qubits spreads this defect across 2N−2 cells but never changes its total. The conservation follows from the factorisation WN=W2⊗W∣+⟩⊗N−2 and the fact that ∑uW∣+⟩(u)=1 exactly for any stabiliser state. The conserved constants cos2(π/8) and (1+2)/2 are the tier-boundary constants of the three-tier magic resource theory (Buckley 2026, Hot Logic). The conservation law explains why these constants are fundamental: they are N-independent invariants of the entire single-T family, not properties of any specific state. The dark-magic tier is stable under adding spectator qubits. All five quantities are confirmed to machine precision for N=2,3,4,5,6 by experiment x471b. Keywords: Wigner function · magic states · resource theory · T gate · stabiliser states · discrete Wigner function · silver ratio · conservation law · Clifford hierarchy · quantum computation · Origami ISA · dark magic
Magic -- the resource enabling quantum computational advantage beyond stabilizer circuits -- has a clean phase-space characterization in odd prime dimensions that qubits notoriously lack. We study C(rho), the l_1 distance from a state's discrete Wigner function to the stabilizer polytope, and determine its exact geometry. For a single qubit with Bloch vector r we prove the closed form C = max(0, ||r||_1 - 1)/2, so the maximum is (sqrt(3)-1)/2, attained at the eight face states. For products of single-qubit states, C(rho_1 x ... x rho_n) <= prod_i (1 + C(rho_i)) - 1, with equality whenever every factor has Bloch sign product r_x r_y r_z <= 0; this includes the exact tensor powers ((1+sqrt(3))/2)^n - 1 and the exact maximum of C over fully separable n-qubit states.
Because no discrete Wigner function for qubits is Clifford covariant, any such measure is frame dependent, and we identify which of its features are not. For a single qubit there are exactly two valid frames up to phase-space translation and C is identical on both. Pauli-covariant two-qubit frames number 2048 up to translation, and across them the ratio C(rho_Rx)/C(rho_Ry) takes the values 1/2, 1 and 2. In every one of these frames some stabilizer state has negative Wigner entries, a consequence of the Mermin-Peres square, so the two-qubit bound Gamma >= 1 + C/2 against the robustness of magic Gamma is frame independent.
Within the Wootters frame we prove integer values 1, 2, 1 of the tightness ratio kappa := (Gamma-1)/C for three families in the two-qubit repetition codespace, with explicit certificates. For n-qubit GHZ phase states we prove Gamma = |sin phi| + |cos phi| for all n and C >= 2^(floor(n/2)-1) (|sin phi| + |cos phi| - 1), tight for n = 2 and, by exact linear programming, for n = 3, 4, so kappa decays exponentially. Finally, C is not a magic monotone, so it cannot replace Gamma in distillation bounds.
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 TVN 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. 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
Coherent errors leave stabiliser-code-encoded magic states in superpositions of syndrome sectors. We ask how much magic these coherent errors add, and whether syndrome extraction depletes logical magic. Using Robustness of Magic, we show that, for a unitary error U on a correctable region which supports no stabiliser, the encoded state has the same robustness as the logical state plus one copy of the Choi state of U , in every stabiliser code encoding. Syndrome extraction removes this copy, returning the logical robustness. The removed magic is positive for every non-Clifford error: adding a non-stabiliser state to any state multiplies one plus the robustness by a (tightly- bounded) factor above one. The magic capacity of two-qubit gates Clifford-equivalent to diagonal or single-qubit gates is equal to their Choi-state robustness. We give a linear-algebraic test for when a Clifford error makes syndrome extraction measure a logical operator, and give magic lifetimes of stored single-qubit magic states under Pauli noise. Choosing the plane of the stored state extends these lifetimes about 1.5x on rotated surface codes under low code-capacity depolarising noise. We identify states and channels in odd-prime dimension for which Wigner negativity and robustness show equal lifetimes.
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 standard resource theory of magic treats magic as a scalar quantity — the total Wigner negativity of a quantum state. This paper shows that for 3-qubit systems, magic has additional structure: a valence, encoded in a 7-component orbit label indexed by the seven lines of the Fano plane PG(2,2). The 63-point 3-qubit symplectic phase space W(5,2) decomposes into the Fano plane (7 classical points) plus 7 orbits of 8 points each under the GHZ stabiliser group; each orbit corresponds to a distinct Fano line and carries a distinct magic valence. Just as chemical valence determines what bonds an atom can form, the magic valence of a quantum state determines what gates it can natively implement and what other magic states it can combine with. The central result is the CS Magic Composition Theorem: for any subset of controlled-S (CS) gates applied to the all-plus state, the negative-chi support is determined exactly by a symmetric-difference rule on the Fano geometry. Single CS gates produce primary-orbit magic; pairs produce secondary-orbit magic via the XOR-Fano rule; all three CS gates together produce magic exclusively in the three secondary orbits. The valence composition rule is bitwise XOR in Z₂³ — the Clebsch-Gordan rule of the representation ring R(Z₂³). The orbit label is stated in terms of the Pauli characteristic function χ rather than the Gross-Wigner function — a necessity, not a limitation, as a theorem of Raussendorf, Okay, Zurel and Feldmann establishes cohomological obstructions to Clifford-covariant Wigner functions for qubits. The characteristic function χ is the canonical magic indicator forced by these obstructions. Operational consequences: the orbit valence label distinguishes coherent gate-type errors (wrong CS gate applied to the wrong qubit pair) that are invisible to total negativity. The 7 stabiliser measurements of the Fano self-test simultaneously certify the GHZ state and read the magic valence label at zero additional circuit cost. Note: Earlier records (v1.0) used the term "magic type." Version 1.1 adopts the preferred term "magic valence" throughout, and adds a footnote connecting the concept to Queen's A Kind of Magic (1986). The mathematics is identical.