This record freezes the analytical and computer-assisted successor protocol to the HLV Golden Gauge Forge finite-family study. Previous work established an exact golden 6→3 cut-and-project carrier fingerprint, a nested bounded-locality exhaustion, maximum vertex degree 12, maximum edge–face incidence 10, volume-independent positive HLV coefficient bounds, and a strictly positive physical-sector gap for every fixed finite connected complex satisfying the declared hypotheses. A preregistered exact Z3 finite-family study successfully constructed the controlled carrier sequence but did not support the frozen positive volume-uniform-gap candidate. This negative result motivates a locality-based analytical programme rather than further unrestricted enlargement of finite matrices. The primary system is a fixed-lattice Z3 gauge Hamiltonian on a declared class of finite oriented two-complexes with uniformly bounded vertex degree, face incidence, interaction range, and positive electric and magnetic coefficients. The protocol freezes a sequential theorem programme consisting of: an explicit volume-uniform gap for the electric product-state backbone; derivation of system-size-independent locality and interaction constants; a strong-coupling perturbative stability theorem with a volume-independent spectral gap; exponential clustering of spatially separated gauge-invariant observables; a van-Hove/Følner theorem for the canonical HLV origin-connected exhaustion; and, only if these gates are closed, construction of a fixed-cutoff thermodynamic-limit state. The computer-assisted component freezes an eight-radius geometry audit, graph-boundary layers of thickness one to three, radius-two rooted patch-frequency diagnostics, exact local norm and interaction-overlap calculations, a global exact-diagonalization cap at cycle rank 10, independent numerical verification, and complete deterministic replay. HLV carrier specificity is tested against matched controls consisting of uniform weights on the same complex, 50 exact histogram-preserving type-stratified weight permutations, 12 deterministic internal-window phason shifts, and three rational Fibonacci-slope periodic approximants. The protocol explicitly forbids replacing a local perturbation estimate by the global magnetic operator norm, which grows with system size and cannot establish a bulk spectral gap. It also keeps the thermodynamic limit, representation-cutoff removal, and continuum lattice-spacing limit strictly separate. A positive result under this protocol would establish, at most, a fixed-Z3, fixed-lattice strong-coupling theorem for a class of aperiodic bounded-locality gauge complexes and a separate finite test of whether the HLV geometric assignment produces effects not reproduced by matched controls. It would not establish full SU(3) dynamics, representation-cutoff removal, a continuum Yang–Mills theory, or the Yang–Mills existence and mass-gap problem.
This record freezes the analytical and computer-assisted successor protocol to the HLV Golden Gauge Forge finite-family study. Previous work established an exact golden 6→3 cut-and-project carrier fingerprint, a nested bounded-locality exhaustion, maximum vertex degree 12, maximum edge–face incidence 10, volume-independent positive HLV coefficient bounds, and a strictly positive physical-sector gap for every fixed finite connected complex satisfying the declared hypotheses. A preregistered exact Z3 finite-family study successfully constructed the controlled carrier sequence but did not support the frozen positive volume-uniform-gap candidate. This negative result motivates a locality-based analytical programme rather than further unrestricted enlargement of finite matrices. The primary system is a fixed-lattice Z3 gauge Hamiltonian on a declared class of finite oriented two-complexes with uniformly bounded vertex degree, face incidence, interaction range, and positive electric and magnetic coefficients. The protocol freezes a sequential theorem programme consisting of: an explicit volume-uniform gap for the electric product-state backbone; derivation of system-size-independent locality and interaction constants; a strong-coupling perturbative stability theorem with a volume-independent spectral gap; exponential clustering of spatially separated gauge-invariant observables; a van-Hove/Følner theorem for the canonical HLV origin-connected exhaustion; and, only if these gates are closed, construction of a fixed-cutoff thermodynamic-limit state. The computer-assisted component freezes an eight-radius geometry audit, graph-boundary layers of thickness one to three, radius-two rooted patch-frequency diagnostics, exact local norm and interaction-overlap calculations, a global exact-diagonalization cap at cycle rank 10, independent numerical verification, and complete deterministic replay. HLV carrier specificity is tested against matched controls consisting of uniform weights on the same complex, 50 exact histogram-preserving type-stratified weight permutations, 12 deterministic internal-window phason shifts, and three rational Fibonacci-slope periodic approximants. The protocol explicitly forbids replacing a local perturbation estimate by the global magnetic operator norm, which grows with system size and cannot establish a bulk spectral gap. It also keeps the thermodynamic limit, representation-cutoff removal, and continuum lattice-spacing limit strictly separate. A positive result under this protocol would establish, at most, a fixed-Z3, fixed-lattice strong-coupling theorem for a class of aperiodic bounded-locality gauge complexes and a separate finite test of whether the HLV geometric assignment produces effects not reproduced by matched controls. It would not establish full SU(3) dynamics, representation-cutoff removal, a continuum Yang–Mills theory, or the Yang–Mills existence and mass-gap problem.
Volume-Uniform Finite-Volume Bounds with the Representation Cutoff Removed: a Reconstruction and Computer-Assisted Improvement of Creation-Operator Constants We study the Kogut–Susskind Hamiltonian of pure SU(2) and SU(3) lattice gauge theory on three-dimensional cubic boxes, H = aΣ_e L_e + bΣ_p (1 - F_p), with L_e the electric Casimir on link e and F_p = N⁻¹Re tr U_p, at strong coupling (small b/a). An unrefereed preprint by Gauvin adapts the creation-operator method of Bravyi, DiVincenzo and Loss to this setting and states, for SU(3), a physical gap uniform in the volume, with the representation cutoff removed, for b/a ≤ 1/648. We give a self-contained reconstruction of the finite-volume argument, identify its external analytic inputs, and reproduce its constants. We then prove a sequence of refinements of the commutator estimate: a selection rule of Wilson-loop multiplication that restricts admissible creation placements, fractional witness counting, orthogonality of output sectors, Haar-measure bounds for vacuum projections before and after the plaquette multiplication, a two-link sandwich identity, a joint vacuum-row bound, and a size-weighted norm. The finite tuple constants and final sufficient inequalities are checked in exact rational arithmetic by two author implementations and a separate verifier written from the stated definitions. The operator inequalities that justify these arithmetic definitions are proved analytically. On open boxes the physical gap is at least 3a/2 for SU(2) whenever b/a ≤ 2/95, and at least 8a/3 for SU(3) whenever b/a ≤ 2/35 (1/16 with a sharper representation-theoretic bound), uniformly in the box. These sufficient couplings are 3072/95 and 1296/35 times those obtained from the preprint's constants (for SU(2), our transfer of them). The corresponding open-box endpoints in Hamiltonian units are β_H=√(4/95) ≈ 0.205 for SU(2) and β_H=√(6/35) ≈ 0.414 for SU(3), or √(3/16) ≈ 0.433 with the strengthened SU(3) bound. The basic open-box certificates also bound the static fundamental potential below linearly: for static charges at graph distance L ≥ 6, E_{x,y} - E_vac ≥ σ L with σ = 5/16 a for SU(2) at b/a ≤ 2/95 and σ = 5/9 a for SU(3) at b/a ≤ 2/35, uniformly in the box. A variable exclusion fraction and an independent local-decoupling upper bound bracket the first physical excitation energy above the vacuum. On open boxes containing an elementary plaquette, at b/a = 1/5000 this energy lies in [2.990, 3.050] a (SU(2)) and [5.326, 5.390] a (SU(3)). These are strong-coupling statements at fixed lattice spacing, not continuum statements. Files ym_strong_coupling_gaps.pdf: the paper. zenodo_ymgap_v1.0.0_2026-10-23.zip: the certificate bundle: two independent implementations of the certified tuple constants, the verifier, the tamper-test driver, the Section 8 upper-bound program and its exact checker, the number generator, manifests and verifier outputs for all 34 certificate configurations, the complete tuple files, and the paper source with every number generated from these files. Its SHA-256 is in the adjacent .sha256 file; SHA256SUMS inside covers every file. README.md, LICENSE.md: copies of the bundle's front matter. Verification ./replay.sh (Python 3, standard library only; a few minutes) regenerates every tuple file, compares it entry by entry with the second implementation, verifies every manifest in exact rational arithmetic, runs the tamper tests and the exact check of the two-sided bracket upper bounds, and regenerates every number of the paper; it exits non-zero on any failure. Scope: strong-coupling statements for the Kogut–Susskind lattice Hamiltonian at fixed lattice spacing, uniform in the volume, with the representation cutoff removed. They are not continuum statements and do not address the continuum Yang–Mills mass gap problem. Licence: CC BY-NC 4.0 (LICENSE.md), licensed by SmartLedger Solutions. Commercial-use enquiries: labs@smartledger.solutions. No patent licence is granted by this release.
We give a self-contained explicit counterexample for the existential unregularized bounded-region fermion question on printed page 371 of Oberwolfach Report 7/2025, with pointwise operator-norm topology distinguished from strong-operator topology on page 365. In dimension three, one fixed smooth nonnegative radial compactly supported pair potential and one fixed interaction ball give a fixed smooth normalized annihilation mode whose Heisenberg norm differences tend to the maximal value 2. Explicit packed finite-particle Slaters and times pi/(2N+1-kappa) witness the same vector-norm limit, a stronger vector asymptotic, and changing normal-state values tending to 4. Both diagonal-density and normal-ordered interactions are covered with ordered-pair coefficient 1 and no ultraviolet smearing. A fixed smooth-cutoff commutator, Slater second moment and two-sector Duhamel comparison give error O(N^(-1/6)+N^(-1/3)). The kinetic energy acts on all of R³. Continuity on every fixed vector and in every fixed normal state remains valid. Siebert's arXiv:2609.34402v1, submitted 28 September 2026 before the original 7 October candidate, supplies the earlier high-density cluster and localized phase mechanism. Its stated theorem concerns full-space difference potentials, but its proof directly adapts to this bounded symmetric cutoff kernel and yields the qualitative existential conclusion. Its Proposition 2.6 explicitly gives maximal norm 2 for the full-space constant-number reference. The present note gives an explicit free-Slater transfer to the fixed spatial cutoff. A separately checked audit-time energy bootstrap also recovers the cutoff maximal limit through the earlier localized phase mechanism. These cutoff implications are deductions made in the audit, not literal cutoff theorems attributed to the prior paper. No first solution, first quantitative refinement, historically certified novelty, or continued open status through October 2026 is claimed. The bounded primary-source priority audit records access and indexing limitations. This five-page unrefereed research note is accompanied by its identical authored PDF/LaTeX source, unchanged 26-file historical candidate and independently authored analytic, operator-algebra, computational, source/model, priority and priority-mapping audits. The fresh computational family has 2,668 exact finite predicates and 558 floating diagnostics; finite tests supplement the universal written proof, and numerical diagnostics are not rigorous interval enclosures or formal proof certificates. Historical optimized-assertion and inventory weaknesses are repaired by externally pinned closed regular-file validation and isolated nonoptimized checker execution, while all original bytes and the original 1/5 candidate count are preserved. Historical result outputs are not claimed byte-identical to current replays, and runtime source-byte checks are explicitly omitted. AI tools were used extensively in development, verification, adversarial review and writing. Independent AI-assisted review is not human or journal peer review; neither is claimed. No copied third-party source bodies are distributed. No all-potential, all-region, dimension-one/two, fixed-density/energy, thermodynamic, fixed-state-discontinuity or CAR-invariance result is asserted.
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.
We construct complete Gauss cutoffs and certify spectral gaps for finite SU(3) lattice gauge Hamiltonians with positive electric coefficients and Wilson potentials. Each cutoff retains all allowed edge representations and invariant tensors. A separate finite extension retains every one-step Wilson output and determines the exact boundary Gram matrix. Decomposing these outputs by excess-edge sets gives local overlap and electric-energy- relative bounds. Under explicit uniform local hypotheses, they imply a volume-independent O(N−2) norm-resolvent error for the Hamiltonian before vacuum subtraction. A quadratic- form Schur argument combines the finite data with an analytic bound on the entire omitted space to certify the original untruncated operator. For two adjacent plaquettes, the complete cutoff has dimension nine and its full-output extension has dimension thirty-five. Exact rational arithmetic gives eigenvalue intervals, a gap lower bound 154/39 at coupling 1/2, and the bound 3/5 throughout [1/2, 5/6]. We also give an exact weighted-cycle formula for the electric gap and an explicit positive interacting gap on each fixed graph. As a representation result, a common Cauchy-relative Perron–Frobenius embedding transports the heat semigroup, Gauss constraints, and ordered observables. Its integer parameter K does not determine a physical gap law in log K. Conditional limit statements identify the additional vacuum, reconstruction, and uniform infrared estimates required for a four-dimensional Yang–Mills theory.
Paper 3 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper develops an exact finite local-to-global analysis of the six-dimensional Hamming-cube foundation Atlas underlying the TUS OS construction. Every admissible completed foundation carries the same exact local qubit prepare–measure representation, while the complete repeated-common-effect Atlas has exact minimum complex Hilbert dimension four. The paper classifies the full ambient family of completed four-state foundations in the six-dimensional Hamming cube. Every foundation is uniquely an affine coset associated with a perfect matching of the six coordinates. There are exactly 240 unordered foundations and 5,760 ordered codebooks. Every foundation induces the same exact 27-class tetrahedral-SIC qubit operational quotient, with carrier-fibre profile 1⁸, 2¹², 4⁶, 8¹ and the corresponding Bloch-radius structure of eight pure states, twelve states of radius √(2/3), six states of radius 1/√3, and one maximally mixed state. Each ambient chart reduces losslessly to a signed perfect matching. The chart-coloured union of two matchings obeys an exact component-parity collision law, yielding the complete seven-value image spectrum over all 28,680 ambient foundation pairs. Exactly 10,800 pairs jointly reconstruct the full 64-state carrier, giving an ambient mathematical chart-cover number of two. The frozen TUS selector defines a canonical subatlas of 36 distinct foundations and 64 anchor-indexed chart instances across nine matching classes. Its selected-foundation reconstruction graph is bipartite and eight-regular with 144 edges, while selector multiplicity yields exactly 400 reconstructing pairs among the 2,016 canonical chart-instance pairs. Only this canonical subatlas inherits the native LANDING and GUIDANCE structure. At the sequential level, all 384 active map/chart descent tests fail: no active LANDING map descends to a state-only transformation on a fixed 27-class local chart quotient. Under the complete principal future language, every canonical chart refines to all 64 deterministic predictive states, with maximum separating depth two. The subordinate GUIDANCE witness is also classified: its endpoint map has image size 34, eight two-cycle attractors and maximum transient depth five, while the additional GUIDANCE-selected chart raises principal path reconstruction from 138/192 contexts to 169/192. The paper then establishes the exact global static common-effect Hilbert dimension. Although every individual foundation admits an exact qubit representation, no unrestricted qutrit state/common-effect family can reproduce all 4,096 Atlas probabilities while simultaneously satisfying all 240 overlapping foundation POVM identities. Together with an explicit chart-independent dimension-four construction, this proves d_q^atlas = 4 for the complete static Atlas. The attaining construction uses two declared qubit sectors; unrestricted uniqueness or rigidity of minimum representations is not claimed. The result therefore exhibits a precise local-to-global separation: exact qubit prepare–measure form exists on every individual foundation, but an exact chart-independent repeated-common-effect representation of the complete Atlas requires complex Hilbert dimension four, while the complete deterministic future language preserves all 64 predictive identities. Local operational state, global relational state and deterministic predictive state are consequently distinct mathematical structures. The paper explicitly keeps these distinct from the still-unresolved whole-position process state. The scope boundary is explicit. The paper does not establish a physical quantum carrier, operational access to all 240 ambient foundations, LANDING as a quantum channel, GUIDANCE as a quantum observable, endpoint channel or instrument, contextuality, nonlocality, quantum advantage, unrestricted rigidity of minimum representations, or whole-position process synthesis. Its conclusions are exact at the static Atlas, canonical-chart and deterministic predictive scopes defined in the manuscript. Supporting General Theory Programme This paper is Paper 3 in the TUS OS quantum-operational and Answerability research programme. It extends the results of Papers 1–2 from the canonical local geometry and exact qubit prepare–measure behaviour to the complete finite relational Atlas. It classifies the full family of admissible foundations, proves that every local chart carries the same exact 27-class qubit operational quotient, identifies the canonical TUS subatlas, and shows that the complete native future language refines those local quotients back to all 64 deterministic predictive states. Its principal global result is that the complete common-effect Atlas has exact minimum quantum dimension d=4, while remaining distinct from the larger 64-state predictive structure and from later process-level dynamics. The paper also classifies the subordinate GUIDANCE witness and its finite reconstruction contribution without promoting it into a channel or instrument. The wider programme then develops complete-future and process closure in Paper 4, the Anchored Analog Quantum System framework in Paper 5, contract-relative resource and operational-observability theory in Papers 6–7, generic Answerability theory in Papers 8–9, complete TUS OS formal and executable realisation in Paper 10, and culminates in Paper 11 with A General Theory of Answerability. Paper 1 — Canonical Tetrahedral Qubit-State Geometry in the TUS OS TriadXOR Completion, Receipt Verification and the Qubit-Geometric Semantic WitnessEstablishes the canonical tetrahedral geometry, unique XOR completion, exact qubit-SIC representation and centred-face trine geometry. Paper 2 — The TUS OS Operational CalculusClassical Construction of Exact Qubit Prepare–Measure Behaviour from a Receipt-Custodied Engine ArchitectureEstablishes exact full-carrier qubit prepare–measure behaviour from the deterministic classical engine under the declared Coordinate-Incidence Operational Harness. For each lawful foundation, the 64 six-bit carriers map to exactly 27 anchor-relative qubit preparations under one foundation-defined tetrahedral-SIC Harness, while the separately declared face Harness gives the exact trine table. Paper 3 — From Finite Relational Structure to Qubit Operational FormAmbient Foundations, Canonical Subatlas, Exact Global Quantum Dimension, GUIDANCE Witness, Sequential Closure and Global Representation in TUS OSExtends the local prepare–measure construction to the complete ambient family of 240 completed foundations and the canonical TUS OS subatlas. Every completed foundation has the same 27-class qubit operational quotient; pair relations between charts provide exact carrier-reconstruction structure; the complete repeated-common-effect static Atlas has exact minimum complex Hilbert dimension 4; and the frozen complete principal future language has deterministic predictive-state minimum 64. The dimension-4 result is a static representation theorem and does not by itself supply common quantum process dynamics. Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusNative Contract, Statistical Discrimination and the Invariant Probability ModuleFixes the formal native domain, complete principal terminal future-experiment language and one-common-word experiment class; establishes complete statistical separation of distinct carrier pairs, exact parity-resolved discrimination results and complement reachability; and proves that the rational invariant probability module has dimension 33. Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic SynthesisEstablishes universal complement-support orthogonality and its propagation under exact common-CPTP dynamics, yielding the certified same-parity support graph and the bound 32 ≤ d_CPTP^min ≤ 64, with cross-parity forcing and the exact state-only common-CPTP minimum remaining unresolved. It further establishes recovery-complete LANDING and GUIDANCE results, exact 64-dimensional selected-line instrument/readout minima at their declared contracts, an exact classical–quantum hybrid construction, and the native structural/predictive quotient correspondence. Paper 5 — TUS OS as an Anchored Analog Quantum SystemNative Witnesses, Predictive Synthesis, and Access-Contract Resource SeparationDefines Analog Quantum Views and Anchored Analog Quantum Systems, proves that TUS OS is a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus, elevates the inherited structural/predictive correspondence into the Predictive Synthesis Identity Theorem, organises the inherited exact quantum-operational results through a typed access-contract resource law, and composes the recognition, predictive, resource and downstream-conservation results into the TUS OS Anchored Analog Quantum System Theorem. Paper 6 — Contract-Relative Quantum Realisations of Classical ProcessesPredictive Compression, Vanishing Worst-Case Causal-State Transparency, and Finite-Dimensional BreakdownExtends the programme beyond the TUS-specific discovery corpus by formalising the contract-relative achievable-resource object (S,C,𝔐,ε) ⟼ Ach(S,C;𝔐,ε), and establishes three exact contract-relative regimes: strict finite separation, unbounded finite separation and finite-capacity breakdown. These are instantiated respectively by the TUS OS 4→64 separation, the periodically modulated decay 2→N exact prediction-to-transparency separation, and dual-Poisson finite-capacity breakdown for every positive uniform worst-case transparency threshold. Paper 7 — Operational Observability and Contract-Induced Resource Transitions in Quantum RealisabilityStructural Classification, Family Scaling, and Finite-Resource BreakdownDevelops a structural classification
We construct an operational quantum model using the free overlap Hamiltonian on a complete nonsingular aperiodic D₆/H₃ carrier and a specified Fibonacci clock. From the quantitative resolvent estimate we derive O(h) continuum error for actual H⁴ cell-average preparations without a fixed spectral cutoff; variable-step midpoint has O(h+δmax²) error. Compatible mixed states and finite measurement protocols converge in trace norm and total variation. Postselected states converge when the limiting success probability is positive, with an explicit denominator bound. The carrier supports an exact two-aperture interference identity and a controlled continuum, paraxial and far-field construction of two-slit fringes. Uncertainty is stated through quadratic forms for finite-variance states, and marker complementarity includes mixed, unequal-prior and nonideal readouts on consistently weighted ensembles. An explicit number-conserving quartic interaction, calibrated with pulse strength O(h⁻¹), prepares a mode singlet during one actual Fibonacci lapse with O(h) state error for fixed smooth source modes while free dynamics remains active. Transport from a common source and quantitative truncation produce complete binary readouts in disjoint laboratories at finite spacing. A preparation family with momentum k=h⁻¹ᐟ⁵ satisfies |Sh+2√2|≤Ch¹ᐟ⁵, retaining all events. The specified local algebras obey the Tsirelson bound and operational no-signalling exactly. Operator-norm calibration bounds also control imperfect readouts and combine with state and finite-sample errors. These are realization and approximation theorems under stated Hilbert-space, probability, composition, preparation and instrument assumptions. Established quantum phenomena are implemented with explicit regulator errors; quantum probability and apparatus dynamics are not inferred from static geometry alone.