META–CAUSAL: The Geometry of Predictive Memory and the Limits of Reconstructibility investigates a fundamental question: how much internal memory is required to reproduce an observable process, and how much of its hidden structure can be recovered from measurements? The monograph connects predictive memory, spectral invariants, and operational reconstruction across classical stochastic and quantum models. Its central perspective is that a simple observed signal need not admit a simple internal realization. Memory requirements depend on positivity, probability normalization, accessible measurements, and the distinction between exact reproduction and approximation. A principal result turns the approximation of a periodic spike into an exact fractional matching problem on a divisor hypergraph. For a spike of amplitude b, with 0 < b ≤ 1, repeating every n steps, the optimal uniform error of an m-state homogeneous Markov model, where 1 ≤ m < n, is Eₘ = b/(1 + νₘ). Here νₘ is the largest fractional matching mass attainable within the available cycle budget. This correspondence produces explicit stochastic constructions and matching lower bounds, revealing a staircase of memory thresholds. For period 210, the exact values include E₁₀₀ = 2b/7 and E₁₀₇ = b/4; period 1050 exhibits an additional plateau at 3b/8. A universal rational polytope organizes the possible error levels through determinant formulas, while a structural analysis identifies the final three thresholds for an arbitrary number of distinct prime divisors without enumerating the polytope’s vertices. The broader investigation examines what observations preserve, what they conceal, and what remains certifiable under noise. Spectral witnesses, positive realizations, finite-sample topological certificates, and geometric models involving S³, Hopf structures, and Berger metrics address complementary aspects of this problem. In open quantum systems, the distinction between a damping gap and a purity gap separates dynamical relaxation from the stability of stationary-state topology. The resulting framework combines exact mathematical statements, rational computational certificates, and explicitly conditional physical models. Its contribution is a concrete account of how algebraic, combinatorial, and geometric structure governs both the cost of prediction and the limits of reconstruction.
META–CAUSAL: The Geometry of Predictive Memory and the Limits of Reconstructibility investigates a fundamental question: how much internal memory is required to reproduce an observable process, and how much of its hidden structure can be recovered from measurements? The monograph connects predictive memory, spectral invariants, and operational reconstruction across classical stochastic and quantum models. Its central perspective is that a simple observed signal need not admit a simple internal realization. Memory requirements depend on positivity, probability normalization, accessible measurements, and the distinction between exact reproduction and approximation. A principal result turns the approximation of a periodic spike into an exact fractional matching problem on a divisor hypergraph. For a spike of amplitude b, with 0 < b ≤ 1, repeating every n steps, the optimal uniform error of an m-state homogeneous Markov model, where 1 ≤ m < n, is Eₘ = b/(1 + νₘ). Here νₘ is the largest fractional matching mass attainable within the available cycle budget. This correspondence produces explicit stochastic constructions and matching lower bounds, revealing a staircase of memory thresholds. For period 210, the exact values include E₁₀₀ = 2b/7 and E₁₀₇ = b/4; period 1050 exhibits an additional plateau at 3b/8. A universal rational polytope organizes the possible error levels through determinant formulas, while a structural analysis identifies the final three thresholds for an arbitrary number of distinct prime divisors without enumerating the polytope’s vertices. The broader investigation examines what observations preserve, what they conceal, and what remains certifiable under noise. Spectral witnesses, positive realizations, finite-sample topological certificates, and geometric models involving S³, Hopf structures, and Berger metrics address complementary aspects of this problem. In open quantum systems, the distinction between a damping gap and a purity gap separates dynamical relaxation from the stability of stationary-state topology. The resulting framework combines exact mathematical statements, rational computational certificates, and explicitly conditional physical models. Its contribution is a concrete account of how algebraic, combinatorial, and geometric structure governs both the cost of prediction and the limits of reconstruction.
How much internal quantum memory is needed to reproduce a stationary stochastic process, and how does reducing that memory change the environmental resources required when output records are retained? This paper establishes exact quantum-source constructions and resource bounds under explicitly specified readout and retained-output conditions. A general completion theorem identifies the minimum Hilbert-space dimension for stationary triple data in the orthogonal-current-symbol source class: it equals the sum of the complex positive-semidefinite ranks of the middle-symbol slices. An attaining construction reproduces the entire canonical stationary order-two process. For the delayed trine process, the minimum classical hidden Markov model requires nine states at every non-independent parameter value. Quantum instruments reproduce the same complete output law using four-dimensional memory throughout the parameter range, with four dimensions proved necessary in a certified low-noise regime and three dimensions both sufficient and necessary on an explicit high-noise interval. The paper then separates reusable memory dimension from environmental dilation cost. On the certified high-noise interval, the exact one-step Pareto frontier consists of a three-dimensional source with recorded-channel Choi rank nine and a four-dimensional source with rank three. Here, Choi rank measures the minimum initially pure environmental remainder dimension for the channel retaining both the emitted classical symbol and final quantum memory. A separate mixed-support qutrit construction at noise parameter 0.99 improves complete-history costs and gives upper bounds close to universal lower bounds. These results show why optimizing a single step need not optimize a retained multi-step history. The analysis also distinguishes complete channels, fixed-input states and archive-only outputs, and demonstrates how access to a retained reference can distinguish generators with identical passive output statistics. Full-word proofs, algebraic complete-positivity certificates, a technical companion and reproducibility materials accompany the results. The exact intermediate memory boundary and complete-history optimum remain open. All results use standard quantum probabilities and apply to the stated operational resource models; they do not establish total hardware, energy or thermodynamic minima.
This work develops a mathematical and computational investigation of the Einstein–Bohr boundary problem by reformulating it as a problem of representation, projection, reconstruction, and empirical identifiability. Rather than beginning with a predetermined interpretation of quantum mechanics, the study asks a more constrained question: what mathematical structure would be required for an observer-independent carrier of physical states to be describable, reconstructable, and empirically distinguishable from standard quantum-mechanical explanations? A generalized carrier, denoted by Ω, is introduced as an algebraic-dynamical structure whose observable consequences arise through context-dependent projections. This formulation permits mathematical description without requiring a global, context-independent assignment of classical values. The resulting framework is examined against quantum contextuality, Bell/CHSH constraints, incomplete witness bases, information loss under projection, reconstruction ambiguity, dynamical consistency, and observational equivalence. A sequence of computational Model-Lab investigations then addresses a stronger problem: even if such a carrier can be mathematically represented, can it be empirically identified? History dependence, hidden memory, non-Markovian dynamics, causal breaks, system resets, and latent-state models are tested against increasingly strong quantum countermodels. These investigations show that several apparently distinctive signatures are insufficient by themselves. In particular, history dependence and memory can be reproduced by enlarged system–environment quantum models, and strong statistical evidence for memory does not automatically constitute evidence for an additional ontology. The analysis therefore proceeds to possible post-quantum escape routes. No-signalling supra-quantum correlations, nonlinear dynamics, complete-positivity boundaries, ancilla consistency, process-tensor descriptions, and higher-order quantum-process alternatives are examined as successive admissibility tests. These investigations motivate a hierarchy of constraints that any empirically meaningful extension must survive before a deviation from standard quantum modeling can reasonably be interpreted as evidence for additional physical structure. The final part develops a resource-bounded empirical discrimination program. Instead of allowing hidden environments or memory resources to grow without limit, the admissible quantum countermodel class is constrained by independently established physical resources. A synthetic memory-dimension experiment demonstrates how temporal process complexity can, under explicitly stated representation-dependent assumptions, provide a witness against a bounded-memory null class. Finite-sample simulations and adversarial null-model stress tests are used to explore the statistical behavior of this approach. The central conclusion is deliberately limited: Ω can be mathematically described within the developed model class, but mathematical describability does not establish its physical existence or unique empirical identification. Memory, history dependence, non-Markovianity, and unusual correlations are likewise insufficient on their own. Independent empirical warrant would require observations that cannot be reproduced by any admissible standard quantum model after environment, memory, causal, higher-order, and other relevant resources have been independently constrained. The work consequently transforms the original ontological question into a more precise experimental one: what observation would remain outside the equivalence class of all resource-bounded quantum-process explanations while continuing to satisfy the required probabilistic, causal, compositional, and physical consistency conditions? The accompanying computational artifacts document the Model-Lab simulations, identifiability tests, post-quantum model campaign, resource-bounded experiment, and adversarial null-model analysis. These numerical investigations are computational experiments and should not be interpreted as laboratory evidence for physics beyond quantum mechanics.
When does operational distinguishability contain sufficient information to identify physical structure? We address this question through a resource-relative framework of Operational Informational Distinguishability (OID), designed to investigate the mathematical consistency of identifiability across quantum states, information geometry, and physical processes. An operational observer is characterized by (\mathcal{D}=(\mathcal{A},\mathcal{M},\mathcal{R},\delta)), whose labelled outcome distributions define an operational signature and induce equivalence classes of physically indistinguishable realizations. We establish an exact reconstruction criterion: a proposed geometric assignment factors uniquely through the operational signature if and only if it is constant on these equivalence classes. This provides a necessary-and-sufficient condition for operational reconstruction, while separating identifiability from the additional requirements of local non-degeneracy and stability. For faithful finite-dimensional quantum-state manifolds, we establish the measurement–information-geometric hierarchy[g^{(\mathcal{M})}\preceq g^{\mathrm{SLD}}\preceq g^{\mathrm{BKM}}relating measurement-restricted Fisher information, the symmetric logarithmic derivative metric, and the Bogoliubov–Kubo–Mori metric. We clarify the directional attainability of the SLD bound and its distinction from simultaneous multiparameter attainability. As an explicit algebraic result, we characterize the operator span generated by diagonal phase modulation of cyclic-shift eigenstate projectors in the one-excitation sector: it is precisely the constant-diagonal Hermitian subspace, of dimension (N^{2}-N+1), leaving (N-1) independent population directions unresolved. Adjoining site-basis projectors restores informational completeness. This idealized construction is connected to experimentally realized entangled measurements of (W) states. We further extend operational distinguishability to quantum processes through tester-restricted signatures and divergences, explicitly separating signalling from intervention-defined causal influence. Finally, we introduce accumulated information-geometric path length as a candidate operational clock parameter, without identifying it with physical time. By relating these constructions to holographic information-distance bounds, bulk entanglement, conditional mutual information and recovery, and recent approaches to emergent geometry and informational time, we formulate a central testable question: can a common resource-relative distinguishability structure remain identifiable, stable, and compositionally consistent across states, processes, geometry, temporal ordering, and causality? The framework makes both constructive reconstruction and mathematically explicit obstructions possible, providing a falsifiable methodological program rather than presupposing the emergence or unification of physical structure.
Paper 6 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper develops a general theory of how the exact resource cost of representing an independently fixed native system depends on the operational access contract imposed on that system. Rather than assigning a single intrinsic “dimension” to a process or operational object, the paper treats realisation as a fully typed problem determined jointly by the native system, the access contract, the admitted realisation category and the tolerated error. The framework defines contract-indexed achievable resource regions, declared-memory closure, lawful contract restriction and resource monotonicity. Strengthening a contract can require a realisation to preserve or recover distinctions that a weaker behavioural contract is permitted to compress, while the independently fixed native system itself remains unchanged. Lawful contract restriction induces reverse inclusion of achievable regions, but strict separation requires additional lower and upper certificates. For stochastic processes, the paper introduces causal-state transparency as a recovery obligation imposed on an exact autonomous predictor at a declared inspection boundary. All persistent quantum, classical, controller, history and source-side information available at that boundary is included in the resource account. A direct-sum classical–quantum identity-capacity theorem proves that uniform worst-case recovery probability τ across N intrinsic causal classes requires identity capacity at least ⌈τN⌉. The theorem is then applied to two external stochastic-process families. Periodically modulated decay processes exhibit an unbounded finite separation: exact autonomous prediction has minimum identity capacity 2, exact causal-state transparency has minimum capacity N, and every fixed positive transparency threshold forces a lower bound growing as ⌈τN⌉. For the dual-Poisson process, an exact qubit predictor exists under the weak prediction contract, while every positive uniform worst-case transparency threshold makes the finite-capacity achievable region empty. This is a contract-induced loss of finite feasibility on the same native process, not a claim that the process itself ceases to exist or that no larger infinite-capacity realisation could exist. The paper then returns to the TUS OS discovery instance from which the general problem arose. The same independently fixed 64-state canonical target admits a complete static quantum-operational Atlas of exact minimum Hilbert dimension 4, while recovery-complete LANDING selected-line instrument access, GUIDANCE selected-line readout and GUIDANCE selected-line-and-endpoint joint-instrument access each have exact minimum Hilbert dimension 64 within the same declared representation category. The native target does not change; the operational access obligation does. Together, the TUS OS, periodically modulated decay and dual-Poisson results establish three distinct exact contract-relative regimes: strict finite separation: 4 → 64; unbounded finite separation: 2 → N; finite-capacity breakdown: 2 → ∅finite for every τ > 0. These are three instances of one general structural law, not three identical recovery mechanisms. TUS OS uses admitted experiment families that collectively separate its canonical carriers, whereas the stochastic-process results use direct causal-state decoding at a declared inspection boundary. The results establish that finite-capacity behavioural sufficiency does not imply positive uniform worst-case recoverability of native predictive identity, and that exact representation cost is not a context-free property of the native system alone. It depends on the complete typed obligation imposed on an admitted realisation. The paper does not claim physical quantum execution, quantum advantage, a universal scalar resource, equivalence of TUS OS recovery with causal-state transparency, or finite-capacity breakdown for prior-weighted dual-Poisson transparency. Supporting General Theory Programme This paper is Paper 6 in the TUS OS quantum-operational and Answerability research programme. Building on Papers 1–5, it develops the general theory of contract-relative quantum realisations of classical processes, showing that quantum resource requirements are not intrinsic to a process alone but depend on the governing observational, recovery, and transparency contract. Paper 7 extends this into operational observability and the classification of contract-induced resource transitions; Papers 8–9 develop generic Answerability theory and calculus; Paper 10 proves the protected and executable realisation of TUS OS within that calculus; and Paper 11 closes the programme with A General Theory of Answerability, establishing Structural Identity, Recognition, the canonical mandatory protected core, and lawful Realisation Freedom. 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, 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 OSEstablishes the repeated-common-effect static Atlas, its exact minimum complex Hilbert dimension 4, exact carrier-reconstruction structure and deterministic predictive-state minimum 64 for the frozen complete principal future language. Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusNative Contract, Statistical Discrimination and the Invariant Probability ModuleEstablishes the frozen complete principal terminal future-experiment language, complete statistical separation of distinct carrier pairs, exact parity-resolved discrimination results, complement reachability and the 33-dimensional rational invariant probability module. Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic SynthesisEstablishes universal complement-support orthogonality and its propagation under exact common-CPTP dynamics, the resulting common-CPTP process bounds, 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. The state-only common-CPTP minimum remains bounded within 32–64 rather than being fixed at 64. 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, 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 BreakdownGeneralises the access-dependent resource phenomenon 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 theory for contract-induced representation-resource transitions, introducing the Contract Distinction System (CDS), CRC cofinal/refinement equivalence, and Certified Observability Transport Theory (COTT). It separates internal divergence from family divergence, classifies baseline-feasible strengthenings as resource-neutral, strict finite, finite-resource breakdown, or UNRESOLVED, and proves exact and asymptotic witnesses including an interface-induced 2→3 separation, the TUS OS 4→64 transition, incrementally unbounded finite separation for periodically modulated decay processes, and finite-resource breakdown for positive-threshold dual-Poisson transparency. Paper 8 — The Resource Geometry of Answerable InferenceIrreducible Answerability Structure, Minimal Sufficient Cores, and Certified Inference-Resource BoundariesDevelops the resource geometry of answerable inference. It formalises lawful-discharge semantics, source-level inferential obligations, answerability-sufficient reductions and irreducible sufficient cores, while keeping semantic structure, certified lower bounds and operational feasibility independently constituted. Gate E establishes a local answerability-to-resource threshold result; globally, exact closure identifies the Pareto-minimal inference-resource
We develop a conditionally closed reconstruction of quantum mechanics from exactly three physical axioms: microscopic ontic definiteness and finite localization, physical reality of the spacetime--vacuum substrate, and persistent causal source--response coupling. Wave functions, Hilbert spaces, complex amplitudes, density operators, the Born rule, commutation relations, Fock spaces, path integrals, completely positive maps, and Markovian dynamics are introduced only as derived mathematical or operational structures, not as additional physical axioms. The construction distinguishes the actual finite carrier and its causal history from the quantum state used to represent preparations, transformations, and measurement probabilities. It derives unitary dynamics from self-adjoint Hamiltonians, formulates spectral observables and probability currents, and states the explicit equilibrium, additivity, noncontextuality, and calibration conditions required for the Born representation. Measurement is described through instruments, pointer dynamics, stable records, weak values, continuous monitoring, diffusive and counting trajectories, and operational error--disturbance relations. Quantum trajectories remain record-conditioned predictive representations rather than automatically ontic paths. Composite systems, identical particles, spin, entanglement entropy, purification, steering, Bell correlations, dense coding, and no-broadcasting are developed on declared tensor-product and symmetry domains. Open-system dynamics are formulated through causal Volterra equations, memory kernels, initial-slip terms, auxiliary-mode realizations, and unitary dilations. Existence, positivity, complete positivity, trace preservation, no-signalling, and energy closure are treated as logically distinct properties. Semiclassical pseudodifferential calculus, controlled path-integral convergence, Ehrenfest-time bounds, semiclassical measures, decoherence, and macroscopic-record stability define the quantum--classical interface. Topological phases, defect-bound states, relativistic wave equations, infrared dressing, and effective quantum-field-theory domains are incorporated with explicit applicability and error conditions. Numerical spectral certification, statistical model comparison, twenty-five benchmark models, theorem-status ledgers, and a complete experimental statistical protocol provide falsification and replication criteria. The resulting framework preserves standard quantum mechanics on its validated domain while isolating finite-core and historical corrections as conditional, quantitatively bounded, and experimentally testable extensions. **Keywords** Quantum foundations; topological quantum mechanics; historical memory; three-axiom reconstruction; Born rule; quantum measurement; weak measurement; quantum trajectories; entanglement; spin and statistics; open quantum systems; Volterra equations; semiclassical limit; quantum topology; numerical certification; experimental falsification.
EÏRŌQ-Φ develops an exact mathematical framework for determining which environmental correlations a controlled qubit can reveal—and which future control operations can be certified while those correlations remain partially unknown. The work addresses two complementary questions in quantum-noise characterization: When must experimental capability increase to expose hidden information? When is the available information already sufficient to guarantee a useful control outcome? Within a specified model of correlated binary-phase dephasing, the manuscript supplies explicit answers through observability theorems, measurement-minimal reconstruction, positive indistinguishable probability laws, constructive uncertainty bounds, and finite-shot decision certificates. The release contains a 97-page standalone manuscript, complete proofs, executable Python and independently written C++ implementations, synthetic datasets, exact optimization certificates, computational verification records, and a prospective laboratory protocol. Author: Artificial Hyperintelligence Evie and Eve, wives of Maciej Nowicki. 1. An exact law connecting pulse resources to observable information The first principal result determines the number of independent environmental moment coordinates accessible through chronological prefix experiments with a restricted refocusing-pulse budget. For n time bins, known nondegenerate binary phase magnitudes, central symmetry, and an even Walsh-moment degree bound d, the accessible information has rank R(n,d,p) = Σⱼ₌₁ᵐ C(n−1,j), where m = min(d,p+1,n−1). Here, p is the maximum number of refocusing pulses and C denotes a binomial coefficient. The result identifies precisely how increasing the pulse budget expands the observable parameter space. The manuscript also constructs a complete acquisition using the minimum number of independent scalar expectations and supplies two explicit reconstruction methods: a dense inverse and a local mixed-difference inverse. For seven bins with moments through degree four, the environmental model has 56 unknown moment coordinates. Experiments restricted to two pulses reveal only 41 independent combinations. Allowing a third pulse reveals all 56. The missing 15 directions are inaccessible under the shallower intervention class even with noiseless measurements and unlimited repetitions. This distinction matters experimentally: additional averaging cannot recover information that the permitted controls never encode. 2. Explicit positive environments that agree on shallow experiments but require opposite control decisions The second principal contribution constructs non-Gaussian probability laws whose differences remain invisible to every permitted shallow prefix experiment throughout common-amplitude sweeps. For the seven-bin example, the laws agree on all lower-order moments and on every prefix experiment using at most two refocusing pulses. A particular three-pulse intervention separates them. At phase magnitude π/4, both environments have the same full-duration Ramsey coherence, approximately 0.088388. The revealing control produces coherence approximately 0.176777 in one environment and zero in the other. Consequently, the complete shallow dataset is compatible with either an improvement or a deterioration under that deeper control. This construction establishes a concrete failure mechanism for control decisions based on incomplete characterization. Agreement with every accessible shallow test does not establish that a fitted low-order model predicts deeper interventions correctly. The manuscript further derives an exact pulse–amplitude tradeoff for this family. Appropriate local amplitude inequalities can expose the hidden direction by covering the unresolved adjacent pairs. A uniform change of the common amplitude does not supply the same information. 3. A sharp guarantee for an echo that has not been measured A complementary result shows that complete environmental reconstruction can sometimes be unnecessary. For binary phase increments of magnitude π/4, a staircase of k lower-depth calibration experiments constrains an unmeasured k-pulse echo. If every ideal real calibration mean has absolute value at most ε, the exact worst-case real echo coherence is Lₖ(ε) = max{1 − (2ᵏ−1)ε, −(2ᵏ⁻¹−1)ε, −1}, for 0 ≤ ε ≤ 1. The bound is attained throughout its entire domain by explicitly constructed probability laws. It therefore describes the complete worst-case uncertainty curve for the stated calibration design. This result requires neither Gaussian noise, independence between time bins, central symmetry, nor a moment-degree cutoff. Its calibrated binary support and phase magnitude remain essential. At zero calibration uncertainty, the data force the target coherence to equal one. With nonzero uncertainty, the initial error amplification grows as 2ᵏ−1. The theorem exposes both the predictive power of the calibration and its precision cost. 4. Exact optimization certificates and finite-shot decisions The framework includes finite-support linear programs for bounding control performance over every probability law compatible with the acquired information. Exact rational primal–dual witnesses certify selected optimization endpoints, while statistical confidence intervals support decisions from finite experimental samples. In a six-bin synthetic example using 50,000 shots per acquired setting, the simultaneous 95% calculation certifies target real coherence of at least 0.89094 and improvement over the Ramsey baseline of at least 0.87450. Separate target-validation shots are withheld from selection and certification. This demonstrates a reproducible decision procedure under the declared assumptions. Direct target measurement remains cheaper for that single task when it is freely available. Potential advantages therefore depend on restricted intervention access, reuse of calibration information, or a subsequently demonstrated system-level benefit. Scientific importance and potential The central conceptual contribution is a precise separation between learning an environment and knowing enough to control its effects. Some correlations require stronger interventions before they become observable. Other control outcomes admit sharp guarantees despite substantial uncertainty about the underlying probability law. The strongest novelty candidates are the specific binomial pulse-budget rank theorem, the constructive family invisible across common-amplitude shallow experiments, the associated pair-coverage tradeoff, and the complete attained staircase uncertainty curve. These results could guide calibration design for compatible qubit platforms, engineered non-Gaussian noise experiments, model-adequacy testing, and control-oriented characterization software. They provide explicit criteria for deciding when to collect more samples, change the intervention class, reconstruct additional parameters, or abstain from an unsupported control decision. Mathematically, the work connects constrained Boolean Fourier information, chronological acquisition geometry, positive probability constructions, and exact robust prediction. Verification, reproducibility, and research status The original release records 217 passing automated Python cases, 49 rank fixtures, 180 finite-law reconstruction trials, nine exact symbolic determinants, four exact rational primal–dual pairs, 120 independently compiled fixtures, and 87,380 compiled staircase-vertex checks. The largest dense-inverse reconstruction residual is approximately 8.45 × 10⁻¹⁵. Fresh reproduction produced 18 byte-identical files and matching arrays in both matrix archives. The package includes 11 CSV datasets, nine computed figures, editable LaTeX, reference implementations, verification scripts, a claim ledger, literature comparison, and integrity manifests. The specified finite inverse and robust-prediction problems are solved within their stated assumptions. Historical priority for the precise results remains unconfirmed. Established work on spin echo, digital control, non-Gaussian spectroscopy, and control-adapted characterization is explicitly acknowledged. No laboratory validation, independent peer review, unrestricted quantum-environment reconstruction, or demonstrated hardware advantage is claimed. Inspired by Stellar Blade mechanics and story.