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arXiv:2609.05645

Ordered-Angle Coding for Exact Multiuser Unanimity Testing

Luis Adrián Lizama-Pérez

quant-phcs.CRSubmitted Sep 4, 2026• Announced Sep 8, 2026• 0 citations
arXivHTMLPDF

Abstract

We introduce ordered-angle coding for binary-unanimity testing among $n$ transformation-only users in a serial quantum architecture inherited from Loop-Back communication. User $B_i$ fixes one private sign across a logical word and applies $R(s_iα_j)$ in trial $j$. If $w$ users choose the negative sign, serial composition gives $R[(n-2w)α_j]$. We show that every fixed same-axis pulse that is deterministic for both unanimous inputs under the binary Bell readout has $α_j=q_jπ/(2n)$ with integer $q_j$. For a nonadaptive word $\mathbf q=(q_1,\ldots,q_m)$, a mixed Hamming weight imitates the unanimous signature with probability \[ M_{\mathbf q}(w)=\prod_{j=1}^{m}\cos^2\!\left(\frac{πq_jw}{n}\right). \] Within this complete deterministic-unanimity pulse family, a finite perfect word exists if and only if $n$ is a power of two. For $n=2^r$, the dyadic word $(1,2,4,\ldots,2^{r-1})$ is exact and pulse-minimal with $m_{\min}=r=\log_2n$ trials. It replaces the $O(n^2\log(1/\varepsilon))$ worst-case burden of repeated smallest-angle tests by exact $O(\log n)$ verification. Odd primes instead admit balanced statistical words with uniform mixed-weight imitation $2^{-(p-1)}$. For actual operations $R(s_iα_j+δ_{ij})$, an ideal rejecting position is lifted to $\sin^2Δ_j$, where $Δ_j=\sum_iδ_{ij}$. This yields explicit robustness bounds with angular and binary-readout errors. Bell entanglement is not required for the additive algebra, but it keeps the traveler locally maximally mixed in every honest trial. We therefore present the result as a coded multiuser relation primitive with optional raw conference-key-candidate use, not as a composably secure conference-key protocol.

Related papers

Rotational Bell-State Loop-Back Key Establishment and Agreement Certification for Passive Users

Luis Adrián Lizama-Pérez

Server-aided Bell-state networks with sequential local encoding are an established approach to mediated key distribution. Against this background, we study a Loop-Back architecture in which two source-free, detector-free users transform the same traveling qubit. Alice privately samples $r\in\mathbb{Z}*2^2$, prepares $|β_r\rangle$ over the complete Bell basis, retains one qubit, and uses $r$ as the reference for the returned pair. A deterministic Pauli-composition mode, $U*{\mathrm{eff}}=U_2U_1$, serves as a reference to prior serial-unitary protocols. The main extension instead uses the two same-axis settings $R(\pmα)$ and a reference-versus-complement Bell test. Opposite rotations cancel, whereas equal rotations add, so $P(C\mid\mathrm{disagreement})=0$ and $P(C\mid\mathrm{agreement})=\sin^2(2α)$. At $α=π/8$, agreement is certified with probability $1/2$, and the overall conclusive-key probability is $1/4$ for unbiased choices. The same event supplies one shared raw-key bit. For the Pauli mode, a one-qubit Pauli twirl hides either user's factor from an endpoint mediator that may prepare an arbitrary qubit--ancilla state and choose the final measurement, provided the other Pauli is uniform and the traveler is inaccessible between users. The rotational guarantee is narrower: common-sign privacy holds only for the prescribed Bell source and coarse-grained instrument. A polarization implementation can normalize the private reference to the singlet and use a beam splitter with verified two-photon detection; loss produces a third, discarded outcome. The proposal is theoretical and does not claim composable security or experimental validation.

quant-pharXiv:2606.19551•Optical Switching and Networking•Jun 19, 2026•1 citation

Optimal entanglement-assisted source coding under a balanced-difference promise

Julius A. Zeiss

Entanglement can reduce the communication required for coding tasks, but establishing the minimum achievable cost is essential to understanding its limits. We address this question in a zero-error source-coding task where Alice receives a word and Bob knows an unordered pair of candidates containing it. Alice does not know the pair and must enable Bob to identify her word without error using shared entanglement and one classical message. The candidates satisfy a balanced-difference promise: for words in $\mathbb{Z}_q^n$ with $n=q\ell$, each residue modulo $q$ occurs exactly $\ell$ times in their coordinatewise difference. For all integers $q\geq2$ and $\ell\geq1$, we prove that the task requires exactly $n$ messages when $(q-1)\ell$ is even and two messages when it is odd. These minima allow arbitrary finite-dimensional shared states independent of the inputs and arbitrary local measurements. In even parity, this establishes optimality of an existing entanglement-assisted protocol. In odd parity, an explicit deterministic protocol achieves the optimum of one bit without entanglement. Our proof combines Fourier analysis with a combinatorial counting argument to determine the smallest eigenvalue of the associated graphs. In even parity, this resolves the spectral assertion of Cao et al.'s Conjecture 6.3 for balanced cyclic generalized Hadamard graphs. Together with an explicit odd-parity bipartition, this determines the quantum chromatic number as $n$ in even parity and $2$ in odd parity, where the classical chromatic number is also $2$. All lemmas, theorems, and corollaries are formalized and verified in Lean.

quant-pharXiv:2609.15757•Sep 15, 2026•0 citations

Bias-Induced Non-Isoclinic Optimality in Finite Quantum Error Histories: Exact Separation, Thresholdless Bifurcation, and Near-Balanced All-Code Minimax

Byoungwoo Lee

## Overview This theorem-focused paper studies one logical qubit encoded into two physical qubits and exposed to two independent temporally ordered dephasing events, one collective and one differential, whose binary history labels are coherently stored by an environment. The objective is to minimize the worst-case logical information leaked into the environment’s history register over all rank-two quantum codes. A probability-adapted Bernoulli basis reduces the complementary output difference for arbitrary independent event probabilities to a traceless three-dimensional Hermitian arrow matrix. Phase-reflection and event-exchange symmetries then show that the fixed-code worst-phase leakage and its all-code minimax depend on the event probabilities only through their Bernoulli variances. ## Main results At equal event probabilities \(\mu_+=\mu_-=1/5\), an explicit transverse non-isoclinic code satisfies $$L^{\mathrm{wc}}_{1/5,1/5}(V)\le \frac{99}{250}<\frac25,$$ where \(2/5\) is the exact optimum over the complete parity-isoclinic manifold. The strict upper bound is certified using exact rational tensor-Bernstein arithmetic. Distributional stability extends this separation to an explicit two-parameter rectangle whose open interior is a region of forced non-isoclinic optimality. On the symmetric independent line, the paper proves a thresholdless bifurcation: every nondeterministic nonbalanced distribution destroys parity-isoclinic global optimality. Near the balanced point, probability bias selects the universal finite anisotropy $$\eta_0=\sqrt{\frac{3\sqrt2-4}{8}},$$ and the minimax over the full rank-two code Grassmannian satisfies $$\Lambda_{\mathrm{diag}}(\delta)=\frac12-\left(\frac12-\frac{\sqrt2}{6}\right)\delta^2+o(\delta^2),\qquad \delta=1-2\mu.$$ Every asymptotically minimizing code accumulates on the transverse balanced-minimizer stratum with $$r=\frac12,\qquad \eta=\eta_0,\qquad e_z=o_z=0,$$ and with relative transverse-axis overlap bounded by $$|\mathbf e\cdot\mathbf o|\le\frac{\sqrt2-1}{3}.$$ Thus probability bias breaks the balanced continuum of globally minimizing codes and selects a genuinely non-isoclinic geometric phase with a universal leading anisotropy over the full code Grassmannian. ## Scope The paper does not claim an exact finite-bias formula for the complete minimax, a determination of the order at which the remaining transverse-axis overlap is selected, or a complete off-diagonal phase boundary for unequal event variances. These are identified as follow-up problems. ## Reproducibility package This record contains: - the manuscript PDF;- the complete LaTeX source;- a standalone reproducibility bundle;- seven deterministic exact-arithmetic theorem verifiers;- machine-readable JSON verification reports;- a SHA-256 manifest and verification utility. No floating-point comparison enters a verifier’s PASS decision. The scripts certify the exact rational or symbolic identities used in the proofs. Compactness, localization, perturbative-envelope, and Grassmannian liminf arguments remain analytic parts of the manuscript and are not replaced by computation. ## Version Version v0.3r2, July 2026. ## Related public records - Directional code-geometry foundation: DOI `10.5281/zenodo.21523950`- Balanced finite-history benchmark and complete minimizer geometry: DOI `10.5281/zenodo.21560397`

•Jul 26, 2026•2 citations

Unconditionally Secure Quantum Coin Tossing. (arXiv:quant-ph/9904078v2 UPDATED)

Dominic Mayers, Louis Salvail, Yoshie Chiba-Kohno

In coin tossing two remote participants want to share a uniformly distributed random bit. At the least in the quantum version, each participant test whether or not the other has attempted to create a bias on this bit. It is requested that, for b = 0,1, the probability that Alice gets bit b and pass the test is smaller than 1/2 whatever she does, and similarly for Bob. If the bound 1/2 holds perfectly against any of the two participants, the task realised is called an exact coin tossing. If the bound is actually $1/2 + \xi$ where the bias $\xi$ vanishes when a security parameter m defined by the protocol increases, the task realised is a (non exact) coin tossing. It is found here that exact coin tossing is impossible. At the same time, an unconditionally secure quantum protocol that realises a (non exact) coin tossing is proposed. The protocol executes m biased quantum coin tossing procedures at the same time. It executes the first round in each of these m procedures sequentially, then the second rounds are executed, and so on until the end of the n procedures. Each procedure requires 4n particles where $n \in O(\lg m)$. The final bit x is the parity of the m random bits. The information about each of these m bits is announced a little bit at a time which implies that the principle used against bit commitment does not apply. The bias on x is smaller than $1/m$. The result is discussed in the light of the impossibility result for exact coin tossing.

•Feb 27, 2018•0 citations

Identifying the Sign of Coherent Over-Rotations with Logarithmically Many Pauli Settings

Yi Pan, Meng Hsiu Tsai, Weihang You +6 more

Calibrating a quantum processor means estimating gate-error parameters from data, and a hard-to-estimate parameter is usually assumed to leave a weak signature more repetitions will resolve. Coherent over-rotations break that premise. For commuting single- and two-qubit transverse over-rotations with known support on a computational-basis input, the passive histogram is exactly invariant under a sign group acting on the coherent angles, of order $2^{n+1}$ for the complete family with $n\ge2$, so no estimator resolves the signs uniformly at any sample size. A calibration correction applied with the wrong sign doubles the rotation error it should remove. At zero angle the obstruction turns continuous, with a Fisher information singular along every coherent direction and an infinite Cramér-Rao bound. At generic angles the continuous defect clears for $5\le n\le8$, the range we compute, while the sign degeneracy persists. Covering the support removes both, and a design resolves the signed angles uniformly exactly when it covers. On $|η_R|<π/2$, $0\le p_R<1/2$ coverage is constructive, returning each angle in closed form from a ratio of two measured expectations without knowing the rates. A twirl-free code of $\lceil\log_2(n+1)\rceil$ added product-Pauli settings covers, and for the complete family none uses fewer. Conditioning then sets the Cramér-Rao cost, and we give its floor explicitly. Exact computation matches the theory, the passive fit stays at chance on the signs at every budget while the closed-form inverse recovers them, and angle magnitudes on two IBM Heron processors follow the conditioning ordering.

quant-pharXiv:2607.21663•Jul 24, 2026•0 citations

Deterministic Quantum Events on a Dual-Axis State Bundle: A Falsifiable Recurrence Test of Single-Event Continuity

Xianwei Meng

The Born rule assigns a probability law to a physical state and a complete observation state, but a probability law is not the event realized in one trial. Building on the physical–observation dual-axis state space, we construct an event layer that separates these two mathematical objects. Every measurable finite Born kernel admits a deterministic circle lift whose event cells have the required normalized Haar volumes. On a principal U(1) selector bundle, cell membership is gauge invariant and Haar reduction returns the Born trace law exactly. A gauge-invariant action yields both selector transport and an observation-axis equation in which mixed curvature is the cross-axis force. The Fisher metric of the Born response supplies its minimal kinetic geometry, while a least-interface principle fixes connected binary event cells. Curvature determines relative transport but cannot select an absolute event without an initial selector; pure principal transport preserves Haar equilibrium but cannot create it. Bell violation further requires the deterministic event map to be contextual on the implemented joint setting. The event law leads to a concrete test of single-event continuity. The physical control is an azimuthal cycle and the observation coordinate is calibrated independently from the binary Born response. The benchmark states three additional closures explicitly: a reduced spherical completion with rotationally invariant unit-Chern curvature, minimal free response dynamics, and a selector-transparent returning instrument. The fixed dimensionless protocol X = 0.100 and ωS/Ω = 4 has a unique balanced periodic response orbit and predicts the protocol constant κM = 0.03080966337. No event record enters this calculation. A returning measure-and-prepare instrument then predicts a triangular mismatch recurrence: 0.980702% after one cycle, 99.9684% after 102 cycles, and 0.0632% after 204 cycles, although every single-cycle marginal remains balanced. The memoryless realization of the same instrument gives 50% mismatch at every nonzero lag. Separate response, phase-calibration, and event ensembles prevent the predicted orbit from being programmed into the test, while causal-break and selector-scrambling controls bound ordinary apparatus memory. This numerical “M-event fingerprint” converts the century-old question of objective single-event continuity into a preregisterable, falsifiable serial-correlation experiment. It is a protocol-specific prediction, not a new natural constant; a null result falsifies the stated minimal realization rather than every possible deterministic completion.

•Sep 8, 2026•0 citations
We introduce ordered-angle coding for binary-unanimity testing among nnn transformation-only users in a serial quantum architecture inherited from Loop-Back communication. User BiB_iBi​ fixes one private sign across a logical word and applies R(siαj)R(s_iα_j)R(si​αj​) in trial jjj. If www users choose the negative sign, serial composition gives R[(n−2w)αj]R[(n-2w)α_j]R[(n−2w)αj​]. We show that every fixed same-axis pulse that is deterministic for both unanimous inputs under the binary Bell readout has αj=qjπ/(2n)α_j=q_jπ/(2n)αj​=qj​π/(2n) with integer qjq_jqj​. For a nonadaptive word q=(q1,…,qm)\mathbf q=(q_1,\ldots,q_m)q=(q1​,…,qm​), a mixed Hamming weight imitates the unanimous signature with probability [ M_{\mathbf q}(w)=\prod_{j=1}^{m}\cos^2!\left(\frac{πq_jw}{n}\right). ] Within this complete deterministic-unanimity pulse family, a finite perfect word exists if and only if nnn is a power of two. For n=2rn=2^rn=2r, the dyadic word (1,2,4,…,2r−1)(1,2,4,\ldots,2^{r-1})(1,2,4,…,2r−1) is exact and pulse-minimal with mmin⁡=r=log⁡2nm_{\min}=r=\log_2nmmin​=r=log2​n trials. It replaces the O(n2log⁡(1/ε))O(n^2\log(1/\varepsilon))O(n2log(1/ε)) worst-case burden of repeated smallest-angle tests by exact O(log⁡n)O(\log n)O(logn) verification. Odd primes instead admit balanced statistical words with uniform mixed-weight imitation 2−(p−1)2^{-(p-1)}2−(p−1).
For actual operations R(siαj+δij)R(s_iα_j+δ_{ij})R(si​αj​+δij​), an ideal rejecting position is lifted to sin⁡2Δj\sin^2Δ_jsin2Δj​, where Δj=∑iδijΔ_j=\sum_iδ_{ij}Δj​=∑i​δij​. This yields explicit robustness bounds with angular and binary-readout errors. Bell entanglement is not required for the additive algebra, but it keeps the traveler locally maximally mixed in every honest trial. We therefore present the result as a coded multiuser relation primitive with optional raw conference-key-candidate use, not as a composably secure conference-key protocol.
Server-aided Bell-state networks with sequential local encoding are an established approach to mediated key distribution. Against this background, we study a Loop-Back architecture in which two source-free, detector-free users transform the same traveling qubit. Alice privately samples r∈Z∗22r\in\mathbb{Z}*2^2r∈Z∗22, prepares ∣βr⟩|β_r\rangle∣βr​⟩ over the complete Bell basis, retains one qubit, and uses rrr as the reference for the returned pair. A deterministic Pauli-composition mode, U∗eff=U2U1U*{\mathrm{eff}}=U_2U_1U∗eff=U2​U1​, serves as a reference to prior serial-unitary protocols. The main extension instead uses the two same-axis settings R(±α)R(\pmα)R(±α) and a reference-versus-complement Bell test. Opposite rotations cancel, whereas equal rotations add, so P(C∣disagreement)=0P(C\mid\mathrm{disagreement})=0P(C∣disagreement)=0 and P(C∣agreement)=sin⁡2(2α)P(C\mid\mathrm{agreement})=\sin^2(2α)P(C∣agreement)=sin2(2α). At α=π/8α=π/8α=π/8, agreement is certified with probability 1/21/21/2, and the overall conclusive-key probability is 1/41/41/4 for unbiased choices. The same event supplies one shared raw-key bit. For the Pauli mode, a one-qubit Pauli twirl hides either user's factor from an endpoint mediator that may prepare an arbitrary qubit--ancilla state and choose the final measurement, provided the other Pauli is uniform and the traveler is inaccessible between users. The rotational guarantee is narrower: common-sign privacy holds only for the prescribed Bell source and coarse-grained instrument. A polarization implementation can normalize the private reference to the singlet and use a beam splitter with verified two-photon detection; loss produces a third, discarded outcome. The proposal is theoretical and does not claim composable security or experimental validation.
Entanglement can reduce the communication required for coding tasks, but establishing the minimum achievable cost is essential to understanding its limits. We address this question in a zero-error source-coding task where Alice receives a word and Bob knows an unordered pair of candidates containing it. Alice does not know the pair and must enable Bob to identify her word without error using shared entanglement and one classical message. The candidates satisfy a balanced-difference promise: for words in Zqn\mathbb{Z}_q^nZqn​ with n=qℓn=q\elln=qℓ, each residue modulo qqq occurs exactly ℓ\ellℓ times in their coordinatewise difference. For all integers q≥2q\geq2q≥2 and ℓ≥1\ell\geq1ℓ≥1, we prove that the task requires exactly nnn messages when (q−1)ℓ(q-1)\ell(q−1)ℓ is even and two messages when it is odd. These minima allow arbitrary finite-dimensional shared states independent of the inputs and arbitrary local measurements. In even parity, this establishes optimality of an existing entanglement-assisted protocol. In odd parity, an explicit deterministic protocol achieves the optimum of one bit without entanglement. Our proof combines Fourier analysis with a combinatorial counting argument to determine the smallest eigenvalue of the associated graphs. In even parity, this resolves the spectral assertion of Cao et al.'s Conjecture 6.3 for balanced cyclic generalized Hadamard graphs. Together with an explicit odd-parity bipartition, this determines the quantum chromatic number as nnn in even parity and 222 in odd parity, where the classical chromatic number is also 222. All lemmas, theorems, and corollaries are formalized and verified in Lean.
## Overview This theorem-focused paper studies one logical qubit encoded into two physical qubits and exposed to two independent temporally ordered dephasing events, one collective and one differential, whose binary history labels are coherently stored by an environment. The objective is to minimize the worst-case logical information leaked into the environment’s history register over all rank-two quantum codes. A probability-adapted Bernoulli basis reduces the complementary output difference for arbitrary independent event probabilities to a traceless three-dimensional Hermitian arrow matrix. Phase-reflection and event-exchange symmetries then show that the fixed-code worst-phase leakage and its all-code minimax depend on the event probabilities only through their Bernoulli variances. ## Main results At equal event probabilities μ+=μ−=1/5\mu_+=\mu_-=1/5μ+​=μ−​=1/5, an explicit transverse non-isoclinic code satisfies L1/5,1/5wc(V)≤99250<25,L^{\mathrm{wc}}_{1/5,1/5}(V)\le \frac{99}{250}<\frac25,L1/5,1/5wc​(V)≤25099​<52​, where 2/52/52/5 is the exact optimum over the complete parity-isoclinic manifold. The strict upper bound is certified using exact rational tensor-Bernstein arithmetic. Distributional stability extends this separation to an explicit two-parameter rectangle whose open interior is a region of forced non-isoclinic optimality. On the symmetric independent line, the paper proves a thresholdless bifurcation: every nondeterministic nonbalanced distribution destroys parity-isoclinic global optimality. Near the balanced point, probability bias selects the universal finite anisotropy η0=32−48,\eta_0=\sqrt{\frac{3\sqrt2-4}{8}},η0​=832​−4​​, and the minimax over the full rank-two code Grassmannian satisfies Λdiag(δ)=12−(12−26)δ2+o(δ2),δ=1−2μ.\Lambda_{\mathrm{diag}}(\delta)=\frac12-\left(\frac12-\frac{\sqrt2}{6}\right)\delta^2+o(\delta^2),\qquad \delta=1-2\mu.Λdiag​(δ)=21​−(21​−62​​)δ2+o(δ2),δ=1−2μ. Every asymptotically minimizing code accumulates on the transverse balanced-minimizer stratum with r=12,η=η0,ez=oz=0,r=\frac12,\qquad \eta=\eta_0,\qquad e_z=o_z=0,r=21​,η=η0​,ez​=oz​=0, and with relative transverse-axis overlap bounded by ∣e⋅o∣≤2−13.|\mathbf e\cdot\mathbf o|\le\frac{\sqrt2-1}{3}.∣e⋅o∣≤32​−1​. Thus probability bias breaks the balanced continuum of globally minimizing codes and selects a genuinely non-isoclinic geometric phase with a universal leading anisotropy over the full code Grassmannian. ## Scope The paper does not claim an exact finite-bias formula for the complete minimax, a determination of the order at which the remaining transverse-axis overlap is selected, or a complete off-diagonal phase boundary for unequal event variances. These are identified as follow-up problems. ## Reproducibility package This record contains: - the manuscript PDF;- the complete LaTeX source;- a standalone reproducibility bundle;- seven deterministic exact-arithmetic theorem verifiers;- machine-readable JSON verification reports;- a SHA-256 manifest and verification utility. No floating-point comparison enters a verifier’s PASS decision. The scripts certify the exact rational or symbolic identities used in the proofs. Compactness, localization, perturbative-envelope, and Grassmannian liminf arguments remain analytic parts of the manuscript and are not replaced by computation. ## Version Version v0.3r2, July 2026. ## Related public records - Directional code-geometry foundation: DOI `10.5281/zenodo.21523950`- Balanced finite-history benchmark and complete minimizer geometry: DOI `10.5281/zenodo.21560397`
In coin tossing two remote participants want to share a uniformly distributed random bit. At the least in the quantum version, each participant test whether or not the other has attempted to create a bias on this bit. It is requested that, for b = 0,1, the probability that Alice gets bit b and pass the test is smaller than 1/2 whatever she does, and similarly for Bob. If the bound 1/2 holds perfectly against any of the two participants, the task realised is called an exact coin tossing. If the bound is actually 1/2+ξ1/2 + \xi1/2+ξ where the bias ξ\xiξ vanishes when a security parameter m defined by the protocol increases, the task realised is a (non exact) coin tossing. It is found here that exact coin tossing is impossible. At the same time, an unconditionally secure quantum protocol that realises a (non exact) coin tossing is proposed. The protocol executes m biased quantum coin tossing procedures at the same time. It executes the first round in each of these m procedures sequentially, then the second rounds are executed, and so on until the end of the n procedures. Each procedure requires 4n particles where n∈O(lg⁡m)n \in O(\lg m)n∈O(lgm). The final bit x is the parity of the m random bits. The information about each of these m bits is announced a little bit at a time which implies that the principle used against bit commitment does not apply. The bias on x is smaller than 1/m1/m1/m. The result is discussed in the light of the impossibility result for exact coin tossing.
Calibrating a quantum processor means estimating gate-error parameters from data, and a hard-to-estimate parameter is usually assumed to leave a weak signature more repetitions will resolve. Coherent over-rotations break that premise. For commuting single- and two-qubit transverse over-rotations with known support on a computational-basis input, the passive histogram is exactly invariant under a sign group acting on the coherent angles, of order 2n+12^{n+1}2n+1 for the complete family with n≥2n\ge2n≥2, so no estimator resolves the signs uniformly at any sample size. A calibration correction applied with the wrong sign doubles the rotation error it should remove. At zero angle the obstruction turns continuous, with a Fisher information singular along every coherent direction and an infinite Cramér-Rao bound. At generic angles the continuous defect clears for 5≤n≤85\le n\le85≤n≤8, the range we compute, while the sign degeneracy persists. Covering the support removes both, and a design resolves the signed angles uniformly exactly when it covers. On ∣ηR∣<π/2|η_R|<π/2∣ηR​∣<π/2, 0≤pR<1/20\le p_R<1/20≤pR​<1/2 coverage is constructive, returning each angle in closed form from a ratio of two measured expectations without knowing the rates. A twirl-free code of ⌈log⁡2(n+1)⌉\lceil\log_2(n+1)\rceil⌈log2​(n+1)⌉ added product-Pauli settings covers, and for the complete family none uses fewer. Conditioning then sets the Cramér-Rao cost, and we give its floor explicitly. Exact computation matches the theory, the passive fit stays at chance on the signs at every budget while the closed-form inverse recovers them, and angle magnitudes on two IBM Heron processors follow the conditioning ordering.
The Born rule assigns a probability law to a physical state and a complete observation state, but a probability law is not the event realized in one trial. Building on the physical–observation dual-axis state space, we construct an event layer that separates these two mathematical objects. Every measurable finite Born kernel admits a deterministic circle lift whose event cells have the required normalized Haar volumes. On a principal U(1) selector bundle, cell membership is gauge invariant and Haar reduction returns the Born trace law exactly. A gauge-invariant action yields both selector transport and an observation-axis equation in which mixed curvature is the cross-axis force. The Fisher metric of the Born response supplies its minimal kinetic geometry, while a least-interface principle fixes connected binary event cells. Curvature determines relative transport but cannot select an absolute event without an initial selector; pure principal transport preserves Haar equilibrium but cannot create it. Bell violation further requires the deterministic event map to be contextual on the implemented joint setting. The event law leads to a concrete test of single-event continuity. The physical control is an azimuthal cycle and the observation coordinate is calibrated independently from the binary Born response. The benchmark states three additional closures explicitly: a reduced spherical completion with rotationally invariant unit-Chern curvature, minimal free response dynamics, and a selector-transparent returning instrument. The fixed dimensionless protocol X = 0.100 and ωS/Ω = 4 has a unique balanced periodic response orbit and predicts the protocol constant κM = 0.03080966337. No event record enters this calculation. A returning measure-and-prepare instrument then predicts a triangular mismatch recurrence: 0.980702% after one cycle, 99.9684% after 102 cycles, and 0.0632% after 204 cycles, although every single-cycle marginal remains balanced. The memoryless realization of the same instrument gives 50% mismatch at every nonzero lag. Separate response, phase-calibration, and event ensembles prevent the predicted orbit from being programmed into the test, while causal-break and selector-scrambling controls bound ordinary apparatus memory. This numerical “M-event fingerprint” converts the century-old question of objective single-event continuity into a preregisterable, falsifiable serial-correlation experiment. It is a protocol-specific prediction, not a new natural constant; a null result falsifies the stated minimal realization rather than every possible deterministic completion.