Constraint Physical Computing (CPC) asks whether engineered physical dynamics can perform useful computation directly from declarative constraint specifications. This white paper develops a feasibility, native-advantage, and validation roadmap for CPC on superconducting bosonic hardware, including cat-qubit, GKP, and multimode architectures. The central physical proposal combines stabilization dynamics protecting an admissible bosonic manifold with answer-independent, problem-dependent open-system dynamics intended to produce a declared semantic terminal sector. Rather than assuming that such a construction is computationally advantageous, the paper defines the physical, semantic, and resource-accounting conditions required to test it experimentally. These include bounded leakage, measurable convergence, control precision, robust referenced readout, fixed decoding, answer independence, provenance, and independent semantic validation. A Levinthal-Class Benchmark is introduced to distinguish large configuration-space avoidance from genuine computational advantage. Complete matched-confidence time-to-solution is compared against preregistered classical, quantum, physics-inspired, and structure-specific reference methods while accounting for compilation, physical synthesis, calibration, preparation, control, convergence, reset, readout, decoding, and validation. Four experimental outcomes are distinguished: backend conformance, physical semantic compatibility, competitive physical execution, and native CPC scaling advantage. The strongest outcome requires evidence of a growing advantage inconsistent with a constant implementation factor over the tested range. The proposal therefore treats native CPC advantage as a falsifiable experimental question rather than an assumed consequence of physical representation or engineered dissipation.
We reconstruct nonrelativistic 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. The single-run physical state is a finite topological core, its actual causal history, the substrate state, its sector, and the associated historical disturbance. A wave function is not identified with that ontic tuple. It is a ray in a Hilbert representation of admissible preparations and response amplitudes; a density operator is a statistical or reduced representation; and a record is a stable macroscopic event. The logical construction is acyclic. Finite-energy sector spaces and their continuous symmetry representations lead, under explicit representation conditions, to a complex Hilbert direct sum. A closed semibounded quadratic form defines the Hamiltonian and its self-adjoint realization. The standard Schrödinger action, unitary propagation, probability current, spectral observables, uncertainty relations, composite-system tensor products, spin representations, exchange sectors, instruments, and completely positive open dynamics then follow on their declared domains. Gleason-type reasoning fixes the Born form only after additivity, noncontextuality, continuity, and preparation–detector calibration are stated. Those operational conditions are not hidden physical axioms. The distribution of unresolved ontic initial data that realizes quantum equilibrium remains a realization certificate, not a theorem of topology alone. The Heisenberg representation is derived as a controlled transport of observables after Hilbert space and the self-adjoint generator have been constructed. Strongly continuous time translation, two-parameter nonautonomous propagation, common-domain Heisenberg equations, spectral matrix mechanics, regular Weyl relations, open-system adjoints, and instrument effects are connected by one certificate. Schrödinger and Heisenberg predictions are proved equivalent for one-time and finite instrument records, while finite-core, historical, truncation, domain, and detector errors remain separately bounded. Three frozen matrix benchmarks verify low-energy oscillator truncation and its state-weighted CCR defect, second-order unitary convergence for a driven doublet, Schrödinger–Heisenberg equality for a sequential instrument, finite auxiliary-history truncation, and quadratic leakage scaling for a conditional Dirac–Pauli interface. Independent reproduction, including the extended-precision audit, is closed for the frozen packet; physical coefficient calibration remains open. The next interface connects an owner-resolved Dirac–Pauli Hamiltonian to spin records through a Ward-compatible vertex, a causal passive history kernel, an independently audited doublet projection, SPAM-aware forward probabilities, nuisance-profiled identifiability, and a noncompensatory coefficient-to-record error theorem. The DPHC identifiability coordinate is executed on a frozen synthetic response family. A single-scale design is rank deficient, whereas an eighteen-setting frequency–gradient–order–time design has full profiled rank, controlled conditioning, near-nominal finite-sample coverage, and a calibrated held-out statistic. A robust twenty-setting design raises the worst registered information eigenvalue by
20.12%. A ten-parameter nonlinear audit exhibits an exact radiative–detector equivalence curve. An independent dark record breaks that symmetry and restores full local rank. Continuous-set robustness, validated global exclusion, and independent implementation are executed, while physical response identification and empirical calibration remain open. Measurement is treated as a dynamics of system, apparatus, environment, and actual history. Unitary premeasurement and decoherence select robust record sectors but do not by themselves select one record. The three-axiom ontology supplies one actual branch history, while a measurable response map from the complete initial state and instrument to the final record supplies the missing dynamical content. Bell correlations require contextual and nonfactorizable ontic response while parameter independence and operational no-signalling are retained; no controllable superluminal channel is introduced. Historical response produces causal non-Markov Hamiltonians and memory-kernel master equations. Their normalization, positivity, complete positivity, and energy ledger are derived from a unitary enlarged realization whenever such a dilation exists. Finite-core form factors and causal memory yield explicit departures from point-core Markov quantum mechanics. Standard quantum mechanics is recovered as the unresolved-core, sector-fixed, quantum-equilibrium, short-memory limit. Quantum field theory is retained as a successful local, spectral, and asymptotic calculus, but field operators, Fock vectors, and vacuum functionals remain effective representations until a noncircular map from finite carriers and their complete causal histories has been certified. The manuscript proves a sequence of previously open analytic, mathematical, measurement, topological, statistical, and computational obligations on explicitly declared domains. These include non-Markov diffusion, graph-norm stability, distorted-wave and pole variation, finite passive-memory realization, Fisher-information identifiability, detector pushforwards, unique-record basins, Born-representation bounds, measurement independence, topology–Dirac charge conjugation, Stern–Gerlach records, action-scale matching, rank-two spin projection, Finkelstein–Rubinstein topology, Clifford reconstruction, and parent-to-record realization maps. Eighteen groups of numerical obligations are executed on frozen reference models. They cover spectral and scattering calculations, oscillator and two-level dynamics, Schrödinger–Heisenberg equivalence, finite-history truncation, Dirac–Pauli and Foldy–Wouthuysen reduction, gauge covariance, interference, Bell and GHZ correlations, open-system memory, Stern–Gerlach records, identifiability, global searches, topological invariants, path products, semiclassical limits, partial-wave optical closure, the relativistic/QFT interface, and conservative-parent two-level dynamics. Every environment explicitly typed as a computational obligation now has an executed frozen-model certificate; empirical identification and separately typed experimental inputs remain open. The present manuscript differs so substantially from the earlier version in scope, logical architecture, proofs, and executable certificates that it is issued as a new upload rather than as a minor revision. Because the subject is both foundational and technically consequential, readers professionally engaged in quantum foundations, mathematical physics, quantum measurement, or related fields are encouraged to consult both versions: the earlier version preserves the developmental route, whereas the present version gives the substantially expanded and reorganized reconstruction. ## Keywords Axiomatic quantum mechanics; finite particle histories; retarded Green functions; Schrödinger equation; Dirac equation; Born rule; topological spin; quantum measurement; non-Markovian dynamics; classical limit.
Positive Fokker–Planck dynamics can reproduce stationary quantum densities and open-system phase-space evolution, but this does not by itself establish unitary Schrödinger dynamics or an individual measurement outcome. We analyze the compatibility conditions among these levels in a drift–diffusion–circulation model motivated by the Architecture of Distinction. After fixing physical dimensions, harmonic ground-state matching yields a diffusion–frequency relation, while a multimode common-clock criterion separates spectral correspondence from a shared physical time calibration. An explicit phase–density Hamiltonian supplies the additional information required for local Schrödinger evolution, with global winding conditions stated separately. A Husimi oscillator provides an exact positive three-term quantum example. For measurement, we reconstruct a finite-time completely positive instrument from an explicit contact and registration rules and distinguish apparatus backaction from conditional state update. Numerical checks verify the analytic identities and channel properties. We formulate quantitative consistency tests for harmonic matching, timing–visibility relations, finite record retention, and a separately postulated momentum-diffusion channel. The results identify which quantum structures can be represented within a positive diffusion framework and which require additional phase, operator, apparatus, or interpretive input. No individual outcome or parameter-free departure from quantum mechanics is derived from scalar diffusion alone.
Continuum physical theories model states as real- or complex-valued fields and dynamics as linear operators on infinite dimensional spaces. Under explanatory realism, an ontic interpretation incurs two semantic commitments: (i) real-valued physical magnitudes must denote relative to the theory’s admissible state interface, and (ii) the theory must be semantically closed under its own evolution and readout rules. Denotation is interface-relative: it requires the existence of a total continuous witness on names. Equivalently, it requires bounded input dependence at each fixed finite precision. Failure of bounded input dependence is captured by forced extensional totalization (FET), in which even coarse output precision cannot be stabilized by any finite prefix of a state name. We give a physics-facing construction of FET for the three-dimensional wave equation with Kirchhoff point readout. We construct a countable family of smooth, compactly supported, finite-energy initial states localized on pairwise disjoint shrinking latitude belts on the Huygens sphere, and restrict the admissible domain to states containing at most one localized contingency. Each nonzero belt perturbation is normalized to contribute the same order-one amount to the readout, while the zero state contributes none. Under any non oracular state interface satisfying a minimal locality-of-access condition, no finite prefix can rule out activation of some sufficiently deep belt. The induced point readout therefore behaves like the existential predicate "some belt is active'', exhibits FET at the zero state, and fails to denote. Semantic closure fails: the evolution-readout pipeline yields a localized point magnitude that is non-denoting relative to the fixed state interface. The result isolates a semantic obstruction to ontic readings of continuum dynamics. It is independent of computability, predictability, or measurement feasibility, and it holds despite smooth compact support, finite energy, and classical well-posedness.
(Abridged.) This thesis investigates scalable fault-tolerant quantum computation through the development of bosonic quantum codes, quantum LDPC codes, and decoding protocols that connect continuous-variable and discrete-variable error correction. We investigate superconducting microwave implementations of continuous-variable quantum computing, including the deterministic generation of cubic phase states, and introduce the dissipatively stabilized squeezed cat qubit, a noise-biased bosonic encoding with enhanced error suppression and faster gates. The performance of rotation-symmetric and GKP codes is analyzed under realistic noise and measurement models, revealing key trade-offs in measurement-based schemes. To integrate bosonic codes into larger architectures, we develop decoding methods that exploit analog syndrome information, enabling quasi-single-shot decoding in concatenated systems. On the discrete-variable side, we introduce localized statistics decoding, a highly parallelizable decoder for quantum LDPC codes, and propose quantum radial codes, a new family of single-shot LDPC codes with low overhead and strong circuit-level performance. Finally, we present fault complexes, a homological framework for analyzing faults in dynamic quantum error correction protocols. Extending the role of homology in static CSS codes, fault complexes provide a general language for the design and analysis of fault-tolerant schemes.
Abstract: We establish a universal mathematical theorem and verified computational suite for aligning arbitrary physical noise to a common logical action across general stabilizer architectures. For any [[n,k,d]] stabilizer code encoding a K=2^k dimensional logical system via isometry W: \mathcal{H}_L \rightarrow \mathcal{H}_P, and defining the correction radius t = \lfloor(d-1)/2\rfloor, every completely positive trace-preserving (CPTP) Stinespring noise process supported on any selected set of at most t physical qubits is proved to be exactly correctable. Rather than presupposing pre-aligned Kraus branches, the framework expands local physical operators in the complete Pauli basis, derives the Knill-Laflamme scalar matrix from code distance, and Gram-diagonalizes the resulting error-image matrix. This constructs orthogonal syndrome isometries Q_r that satisfy the derived common-action identity: Q_r^\dagger A_\mu W = \beta_{\mu r} I_L This dynamic factorization separates the logical state from the syndrome and environment: (D \otimes I_E) V W |\psi\rangle = |\psi\rangle_L \otimes |\Xi\rangle_{SE} The recovery identity \mathcal{R} \circ \mathcal{N} \circ \text{Enc} = \text{Enc} is rigorously validated across operator spaces, ancilla-complete entanglements, arbitrary-depth circuits, and generator-level suppression of local GKSL dynamics (\mathcal{R}_L[\mathcal{L}_{\text{phys}}(\text{Enc}(\rho_L))] = 0). A centered physical channel featuring distinct branches (I, X_q, Z_q) and amplitudes (1/3, +2/3, -2/3) provides an exact squared completeness of 1/9 + 4/9 + 4/9 = 1. The accompanying open-source NumPy suite (tibedo_v4_arbitrary_qubit_alignment_suite.py) provides an executable certification engine for arbitrary user-supplied stabilizer generators. Evaluated on the [[5,1,3]] perfect code and [[7,1,3]] Steane code, all 16 core theorem tests passed alongside 64 randomized stress trials through corrected circuit depth 32, yielding pristine residuals ranging from 2.22 \times 10^{-16} to 3.03 \times 10^{-14}. As the theorem proves, the commercial opportunity for a quantum sector shifts entirely from physical manufacturing to the epistemological layer: mastery over problem representation and architectural compilation. Without this specialized knowledge, a hardware firm possessing a theoretically fault-tolerant physical machine is left with an empty shell—an advanced engine with no operational map. They are entirely dependent on the underlying mathematical framework to achieve exact circuit closure (\mathcal{R} \circ \mathcal{N} \circ \text{Enc} = \text{Enc}) and generator-level suppression of arbitrary local GKSL data (\mathcal{R}_L[\mathcal{L}_{\text{phys}}(\text{Enc}(\rho_L))] = 0). True commercial leverage and sales dominance concentrate within the hands of entities that command the mathematical representation layer. Knowledge of how to structure error-correcting codes, derive syndrome isometries without heuristic Kraus proportionalities, and preserve logical coherence across arbitrary circuit depths becomes the ultimate arbiter of value in the post-NISQ economy.