We present a self-contained, auditable framework for reconstructing discrete gauge con nections from loop-based records (e.g. interferometric phase/visibility readouts) on a finite 2- complex. The core contribution is a Minimal Verification Suite (MVS) that separates three logically distinct questions: (i) curvature consistency via plaquettes, (ii) non-commutativity via commutator loops, and (iii) constraint compatibility via a discrete Gauss closure defect. We prove that a commutator suite passing on a generating set implies Abelian reducibility of the reachable subgroup, and we specialize this statement to SU(2), showing commuting controls lie in a common maximal torus (a single rotation axis). A central technical result is an exact closed-form identity for the SU(2) commutator Wilson scalar: Wcomm = 1 2Tr(UV U −1V −1 ) = 1 − 2 sin2 (α/2) sin2 (β/2) ∥n × m∥ 2 , where U = exp(iα n · σ/2) and V = exp(iβ m · σ/2). This yields a hard, record-only axis misalignment bound under a measurable non-degeneracy gate. We then harden MVS sound ness into explicit, certificate-friendly inequalities: loop-fit errors propagate to chord-edge errors without path amplification in spanning-tree gauge, Projects can be compared by a constructive gauge-alignment algorithm, and Gauss-closure defects admit a calibrated envelope law decom posing statistical, calibration, leakage, and non-Abelian contributions. Finally, we introduce tiered certificates (MVS-A Abelian, MVS-N Non-Abelian, MVS-B Boundary-limited) and pro vide a fully worked 2 × 2 plaquette patch example with replayable YAML templates.
We present a self-contained, auditable framework for reconstructing discrete gauge con nections from loop-based records (e.g. interferometric phase/visibility readouts) on a finite 2- complex. The core contribution is a Minimal Verification Suite (MVS) that separates three logically distinct questions: (i) curvature consistency via plaquettes, (ii) non-commutativity via commutator loops, and (iii) constraint compatibility via a discrete Gauss closure defect. We prove that a commutator suite passing on a generating set implies Abelian reducibility of the reachable subgroup, and we specialize this statement to SU(2), showing commuting controls lie in a common maximal torus (a single rotation axis). A central technical result is an exact closed-form identity for the SU(2) commutator Wilson scalar: Wcomm = 1 2Tr(UV U −1V −1 ) = 1 − 2 sin2 (α/2) sin2 (β/2) ∥n × m∥ 2 , where U = exp(iα n · σ/2) and V = exp(iβ m · σ/2). This yields a hard, record-only axis misalignment bound under a measurable non-degeneracy gate. We then harden MVS sound ness into explicit, certificate-friendly inequalities: loop-fit errors propagate to chord-edge errors without path amplification in spanning-tree gauge, Projects can be compared by a constructive gauge-alignment algorithm, and Gauss-closure defects admit a calibrated envelope law decom posing statistical, calibration, leakage, and non-Abelian contributions. Finally, we introduce tiered certificates (MVS-A Abelian, MVS-N Non-Abelian, MVS-B Boundary-limited) and pro vide a fully worked 2 × 2 plaquette patch example with replayable YAML templates.
This work introduces Distributed Verification of Lie Closure (DVLC), a gauge-invariant certification protocol for testing whether a set of experimentally accessible control primitives generates a desired target Lie algebra. DVLC is designed for settings—such as holonomic and geometric quantum control—where only conjugacy-invariant data (e.g. traces, characters, Wilson-loop observables) are naturally available, and where full Hamiltonian reconstruction or explicit commutator closure tests are impractical. Rather than reconstructing generators or computing nested commutators, DVLC compares experimentally estimated class functions of short, near-identity word holonomies against universal Baker–Campbell–Hausdorff (BCH) predictions for a chosen compact target group (e.g. SU(d)). The protocol operates directly on the adjoint quotient (conjugacy classes), avoiding gauge fixing by construction, and parallelises naturally across words and invariants. Under explicit and physically transparent assumptions—local separation of conjugacy classes by finitely many class functions, uniform BCH truncation on a finite word suite, and a finite Lie presentation of the target algebra by low-degree relators—a soundness theorem is proven: agreement of invariant data implies that the system generators satisfy the same defining relators as the target, inducing a Lie-algebra homomorphism into the system (and an embedding for simple targets). Quantitative robustness results, including explicit stability for SU(d) trace invariants, and finite-sample, distribution-agnostic guarantees suitable for distributed experiments are provided. Minimal SU(2) and SU(3) examples demonstrate how DVLC detects noncommutativity and higher-bracket structure using only a small number of scalar, gauge-invariant observables—without tomography, commutator computation, or basis-dependent rank conditions. This preprint is intended as a self-contained theoretical contribution to quantum control, holonomic quantum computation, and Lie-algebraic certification, and the author welcomes feedback and discussion.
This is a separate preprint record that develops the robustness layer of the outcome-resolved mid-circuit-measurement gauge analysis. It uses the exact complete-positivity benchmark of the preceding preprint as an input, but its main result concerns the composition of imperfect-control uncertainty, residual-gauge conditioning, and instrument-model error. The general ingredients—gate-set-tomography gauge freedom, commutant geometry, perturbation by singular-value gaps, quantum-instrument physicality, and mid-circuit-measurement learnability—are prior art. The paper does not claim a new general GST gauge principle, a dimension-free tomography theorem, or experimental validation. Related identifier ● Identifier: 10.5281/zenodo.23043202 ● Relation: Is derived from (isDerivedFrom) ● Resource type: Publication -> Preprint ● Title: Outcome-Resolved Mid-Circuit Measurements as Exact Gauge Constraints: Complete-Positivity Windows, Correlation Certificates, and Robustness Limits Description / Abstract Self-consistent quantum characterization is invariant under similarity transformations, so an experimentally fitted gate set is generally identified only up to gauge. Exact trusted controls can reduce this ambiguity, but realistic controls are imperfect; simultaneously, a mid-circuit measurement (MCM) instrument is itself only approximately described by a model. These two uncertainties act at different logical stages and should not be merged before the gauge geometry is reconstructed. This preprint gives an explicit finite-dimensional stability analysis. For a finite reference control family ({R_i}), define the commutator map (\mathcal C(B)=([B,R_i])_i) and let (\gamma) be its smallest positive singular value. If two exactly gauge-equivalent implemented gate families ({G_i}) and ({G’_i}) both lie within Frobenius errors (\delta_i) of the same references, then every trace-preserving gauge matrix (B) lies within [\operatorname{dist}{F}(B,\ker\mathcal C)\le \frac{2\Delta}{\gamma}|B|{\mathrm{op}},\qquad\Delta=\left(\sum_i\delta_i^2\right)^{1/2}.] For the two-qubit control set (H\otimes I), (S\otimes I), (I\otimes H), (I\otimes S), and CNOT, the exact commutant is (\operatorname{span}{\mathcal D,\mathrm{id}-\mathcal D}), so the residual scalar gauge is (B_c=\mathcal D+c(\mathrm{id}-\mathcal D)). In the declared unweighted Frobenius convention, direct computation gives (\gamma=0.830129650657\ldots). The main result closes two previously separate robustness budgets. Suppose an independent physical or calibration argument places the scalar projection (c) in a positive prior interval ([\ell,u]). If (\alpha=2\Delta/\gamma<1), a self-normalized estimate removes the otherwise unknown factor (|B|): [|B-B_c|_F\le \bar e=\frac{\alpha M_0}{1-\alpha},\qquad M_0=\max(1,u).] If (\bar e<m_0=\min(1,\ell)), the condition number of (B) is explicitly bounded. Composing this control-side bound with a Choi-Frobenius instrument-model error (\varepsilon) yields a closed outer complete-positivity condition for the symmetric noisy Lüders MCM benchmark, [(1-\eta)|1-2p|,|c-c^{-1}|\le \eta+2d\rho,] where (\rho) is an explicit function of (\Delta,\gamma,\ell,u), and (\varepsilon). Hence the imperfect-control problem again reduces to an explicit scalar gauge interval. The result separates transverse control uncertainty from instrument-model uncertainty and identifies the loss of robustness as the prior gauge envelope approaches singularity.
Abstract We develop Ω TRINITAS ENGINE v6, a mathematical and computational framework for constraint-preserving projected unitary dynamics, with particular emphasis on invariant physical subspaces, gauge-constraint leakage, local unitary evolution, and reproducible Hamiltonian lattice gauge theory. The central problem is to distinguish approximation error in discrete or product-formula time evolution from genuine violation of a prescribed physical constraint. For an orthogonal projector P defining the physical subspace and a unitary evolution U, the framework quantifies constraint leakage through ||(I-P)UP|| and establishes the exact identity ||PUx||^2 = ||x||^2 - ||(I-P)Ux||^2 for physical initial states. In finite-dimensional Hilbert spaces, exact preservation of the physical subspace is characterized by invariance under U, equivalently by vanishing leakage. For products of local unitaries, leakage admits an additive bound in terms of the individual factors, while Hamiltonian-generated evolution is controlled through commutators between the projector and the local generators. In particular, when every local Hamiltonian term preserves the Gauss-law physical subspace, finite-depth product-formula dynamics can preserve the gauge constraint exactly even when the approximation to the target continuous-time evolution remains imperfect. We investigate these results computationally using an explicitly constructed gauge-enriched successor model based on the 1+1-dimensional U(1) lattice Schwinger model with staggered fermions, periodic boundary conditions, finite electric-flux truncation, and Gauss-law reduction. Independent constructions of the physical basis and Hamiltonian agree exactly in the audited finite systems, with zero matrix and spectral discrepancies and no detected Gauss-violating hopping transitions. A symmetric local product formula exhibits approximately third-order single-step error and second-order fixed-time convergence while maintaining numerical unitarity near machine precision. Sparse low-energy calculations at fixed physical volume reproduce the massless Schwinger rest-gap benchmark: an even-power continuum extrapolation gives 0.5642377786 in units of the gauge coupling, compared with the exact continuum value 1/sqrt(pi) = 0.5641895835. In the representative electric-flux cutoff test, variations are below the reported numerical resolution. Momentum-resolved T2 sectors further reproduce the expected relativistic dispersion with strong numerical agreement at the lowest momenta and a larger, explicitly retained deviation at higher momentum. The numerical analysis constitutes finite-regulator calibration and continuum extrapolation rather than certification of a controlled continuum limit. Systematic uncertainties associated with fit family, fit range, finite physical volume, regulator ordering, and higher-momentum behavior remain distinct from the proved constraint-preservation results. This separation prevents numerical agreement with a known continuum theory from being promoted to a stronger mathematical or physical claim. The work explicitly distinguishes mathematical proof, finite computational verification, numerical observation, calibration, and physical interpretation. Agreement with the Schwinger model is treated as a nontrivial calibration of the gauge-enriched successor construction, not as a derivation of gauge theory from canonical Ω TRINITAS dynamics. The manuscript does not establish endogenous U(1) gauge emergence from canonical Ω TRINITAS v9 and does not claim correspondence between Ω TRINITAS and the microscopic dynamics of the actual universe. Negative results, failed constructions, limitations, and unresolved questions are retained as part of the epistemic record. Ω TRINITAS ENGINE v6 therefore provides a reproducible reference framework for projected quantum dynamics, invariant subspaces, Gauss-law preservation, constraint leakage, Hamiltonian lattice gauge theory, local unitary circuits, Trotter and product-formula evolution, sparse exact diagonalization, electric-flux truncation, continuum extrapolation, and momentum-sector spectroscopy. The canonical reference implementation and machine-readable numerical results are designed to preserve a strict distinction between proved statements, computationally verified finite cases, numerical evidence, open problems, and unsupported physical correspondence, providing an auditable foundation for subsequent mathematical and computational investigations.
An Executable Record Grammar for Quantum Correlations: A finite compression certificate for stabilizer records, Wilson holonomy, magic, detector readout, and bounded residuals. A finite, executable demonstration of a 'record grammar': a disciplined separation of the full Hilbert-space state from the smaller algebra of detector-readable, stable, reusable records, in which every correlation left outside the grammar must be exactly zero by a selection rule, bounded by a certified trace-norm residual, or promoted to a named collective record. The central result is a four-qubit compression certificate: Bell/stabilizer endpoint records, a closed-loop T-holonomy (magic) record, and a finite detector branch are preserved exactly under the conditional expectation E(rho) = sum_i P_i rho P_i, while incompatible off-record witnesses are bounded by the trace norm and a sign witness saturates that bound. The paper is explicit that this is not a proof of polynomial scalability: a maximal abelian record basis still has 2^n sectors, and the scalable target is a generator-form ledger built from stabilizer/QEC generators, sparse magic resources, and bounded-correlation tensor-network data. Companion scripts demonstrate Wilson dressing of gauge readouts, magic retained as a named holonomy resource, noisy-detector channel laws, plaquette-chain correlation-length triage, and byte-level results on the eight-bit [8,4,4] record cell, including single-error syndrome detection and U(1) and SU(2) closed-loop flux readouts. All scripts are self-checking and reproduce to machine precision. This record is part of the finite-QEC substrate / It-from-Bit publication refresh.
We propose a simple ansatz that reduces the sourceless SU(2) Yang--Mills equations in (3+1) dimensions to nine algebraic constraints. Solving these constraints yields three closed-form families of exact wave solutions. \textbf{Family I} embeds linear electromagnetic waves into the non-Abelian theory, with vanishing commutators and dispersion ω=kc. \textbf{Family II} describes genuinely nonlinear self-interacting waves that also propagate at the speed of light but exhibit a constant, gauge-invariant offset in the color-electric field, nonvanishing commutators, and a discrete topological parameter ξη=±1 that controls the position of energy-density nodes (θ=0 or θ=π). This provides an observable signature with no analogue in Abelian electromagnetism. \textbf{Family III} is a pure gauge solution with vanishing field strengths, valid for arbitrary k and ω without any dispersion relation. These exact solutions offer new insights into how non-Abelian self-interactions fundamentally alter wave propagation and serve as benchmarks for numerical simulations, perturbative studies, and experiments on synthetic non-Abelian gauge fields.
This repository contains the computational reproducibility and verification package accompanying the frozen Version v1.0 manuscript: “Phase Curvature and Assumption-Budgeted Certification in Gravity-Mediated Entanglement.” The associated manuscript develops an operational certification framework for gravity-mediated entanglement experiments. Its central aim is to quantify when an observed entanglement signal excludes a specified neighborhood of local classical-mediator models under explicit assumptions and separately controlled nuisance channels. The package supports the manuscript’s analytical and numerical results concerning: gauge-invariant nonseparable phase curvature for diagonal two-path interactions; exact concurrence and separability-distance formulas for pure two-qubit outputs; assumption-budgeted exclusion bounds around the classical-mediator free class; primal and dual PPT/semidefinite-program certification; finite-shot confidence bounds for experimentally accessible Pauli correlators; certified residual bounds for physically compensated coherent nuisance interactions; stochastic geometry and Casimir–Polder uncertainty budgets; matter-bypass and common-auxiliary-channel bookkeeping; circular-shield mode analysis, including the azimuthal matched-null obstruction; thermal shield-channel factorization; finite-footprint high-mode suppression; an analytic positive high-mode tail bound for the shield benchmark; and a conservative outward-rounded audit of the finite numerical mode block. The frozen manuscript does not claim that a particular experimental apparatus has already closed every systematic-error channel. Quantities such as preparation error, matter-channel leakage, shield-model discrepancy, control error, and readout error remain apparatus-dependent inputs to the certification framework. Package contents The archive includes: the frozen Version v1.0 manuscript source; deterministic verification scripts for the theorem-core identities; finite-shot and Casimir–Polder statistical/geometry checks; shield-mode and analytic-tail verification; conservative floating-point interval/audit checks; submission-interface regression checks; machine-readable verification reports; and a SHA256 manifest for file-integrity verification. All verification scripts distributed with the Version v1.0 package return PASS under the documented software environment. The numerical audit is intended as a transparent reproducibility layer for the stated model calculations. It should not be interpreted as a formal interval-special-function proof or as certification of a realized experimental apparatus.