Can fixed recognition interactions supply the control needed to write an exact record? We prove full unitary controllability on the 1,536-dimensional neutral sector of two source–reference pairs. The allowed controls are simultaneous sums of explicit local braid and paired-contact Hamiltonians, with adjustable positive strengths and durations; the original holding and positive scalar energy charges remain present. The main algebraic step converts effective control on a 1,024-dimensional matched subspace into control supported there, then uses a contact with an explicit right inverse to reach the remaining coordinates. Compact-group control gives actual finite words. An exact eight-label coherent readout follows, together with arbitrary encoded interventions and preparation of each label from one encoded blank. The four active cells sit in a fourteen-cell register: the other ten evolve by a retained unitary, with the same identity valid for every finite external reference. The result establishes exact existence within the prepared encoding, without a uniform synthesis-time or pulse-count bound. The paper and source archive, including verify_certificates.py, are available at https://doi.org/10.5281/zenodo.23875346.
Can fixed recognition interactions supply the control needed to write an exact record? We prove full unitary controllability on the 1,536-dimensional neutral sector of two source–reference pairs. The allowed controls are simultaneous sums of explicit local braid and paired-contact Hamiltonians, with adjustable positive strengths and durations; the original holding and positive scalar energy charges remain present. The main algebraic step converts effective control on a 1,024-dimensional matched subspace into control supported there, then uses a contact with an explicit right inverse to reach the remaining coordinates. Compact-group control gives actual finite words. An exact eight-label coherent readout follows, together with arbitrary encoded interventions and preparation of each label from one encoded blank. The four active cells sit in a fourteen-cell register: the other ten evolve by a retained unitary, with the same identity valid for every finite external reference. The result establishes exact existence within the prepared encoding, without a uniform synthesis-time or pulse-count bound. The paper and source archive, including verify_certificates.py, are available at https://doi.org/10.5281/zenodo.23875346.
Can fixed recognition interactions supply the control needed to write an exact record? We prove full unitary controllability on the 1,536-dimensional neutral sector of two source–reference pairs. The allowed controls are simultaneous sums of explicit local braid and paired-contact Hamiltonians, with adjustable positive strengths and durations; the original holding and positive scalar energy charges remain present. The main algebraic step converts effective control on a 1,024-dimensional matched subspace into control supported there, then uses a contact with an explicit right inverse to reach the remaining coordinates. Compact-group control gives actual finite words. An exact eight-label coherent readout follows, together with arbitrary encoded interventions and preparation of each label from one encoded blank. The four active cells sit in a fourteen-cell register: the other ten evolve by a retained unitary, with the same identity valid for every finite external reference. The result establishes exact existence within the prepared encoding, without a uniform synthesis-time or pulse-count bound.
A symmetry specifies which interactions are compatible with a system; it does not specify which interactions its controls can produce. We resolve this distinction for a finite recognition apparatus with a main signal, a retained record and an independent clock. An explicit unitary change of coordinates turns the two signal ports into four equivalent copies of a four-dimensional symplectic representation. The invariant source Hamiltonians are therefore exactly the Hermitian 4×4 coefficient matrices acting between those copies. The original controls preserve one decoded record component, excluding even a simple exchange allowed by the symmetry. Adding forward waits under one fixed positive-strength recorded-exchange Hamiltonian removes this obstruction: every invariant source unitary, with identity on the clock, lies in the operator-norm closure of the resulting finite programs. The proof derives the required operations from the stated inventory: a clock echo removes unwanted clock motion, and a record-restoring subroutine supplies control of the main register. We also carry the original energy readout through the same programs, retaining its reference and positivity conditions.
A symmetry specifies which interactions are compatible with a system; it does not specify which interactions its controls can produce. We resolve this distinction for a finite recognition apparatus with a main signal, a retained record and an independent clock. An explicit unitary change of coordinates turns the two signal ports into four equivalent copies of a four-dimensional symplectic representation. The invariant source Hamiltonians are therefore exactly the Hermitian 4×4 coefficient matrices acting between those copies. The original controls preserve one decoded record component, excluding even a simple exchange allowed by the symmetry. Adding forward waits under one fixed positive-strength recorded-exchange Hamiltonian removes this obstruction: every invariant source unitary, with identity on the clock, lies in the operator-norm closure of the resulting finite programs. The proof derives the required operations from the stated inventory: a clock echo removes unwanted clock motion, and a record-restoring subroutine supplies control of the main register. We also carry the original energy readout through the same programs, retaining its reference and positivity conditions. The deposit includes the paper and its LaTeX source with six vector schematics. AI-use disclosure: OpenAI Codex assisted with mathematical formalization, manuscript drafting and editing, schematic code, and source and layout checks. The results are supported by explicit mathematical arguments and machine-checked proofs in the author's research project.
We derive an explicit information-retaining evolution from full-current recorded recognition dynamics on the unrestricted square-summable field. Its adjoint recovers every input, and conjugation retains every bounded input action. An eight-posting Z-pattern reads less information: an infinite-dimensional family of normalized circulating states shares the entire window reading, including distinct orthogonal packets. Two commuting output families cannot both reproduce all native input actions. The contribution is a construction and separation result for these dynamics: the native coefficients satisfy the hypotheses of isometric recovery, while explicit witnesses distinguish full retention from a coarser reading and from coherent preparation. From a nonzero definite-counter input, recording alone preserves only one part in 512 of the squared norm in the uniform-reference sector and cannot prepare that sector. A finite native control word prepares it, up to one unit phase, for every source input. A finite rational-angle word attains any positive relative tolerance uniformly over all inputs and finite subsequent recording depths. This bound concerns preparation error followed by exact recording. The result establishes native reachability without a gate-count or synthesis-complexity bound. Physical implementation must supply the controls and resources used by the construction.
AbstractRetained measurement records constrain recoverability, but their formation also depends on the available physical controls. We give exact, constructively attained answers to both questions in two specified models. For a balanced nondemolition qubit process with actual binary observer memories unavailable to the decoder, we determine the maximum recovery compatible with average readout accuracy and pair agreement. The finite-record solution includes its feasible region and a common memory-writing and recovery process, without assuming exchangeability of observed records. Under a complete conditional-pairwise-independence promise at fixed informative accuracy, the optimal coherence has a sharp inverse-square-root asymptotic; fully conditionally independent records instead give exponential decay. Thus identical single-record and pair-record calibrations need not justify the product-law recovery bound. We then fix a logical code in four spin-3/2 systems and sequential diagonal contacts with exact controlled-swap endpoints. Hard budgets for net contact action and added spectral control charge yield an exact peak-excursion frontier within a declared quadratic-plus-cubic control class. A single positive-amplitude preparation jointly attains the record, recovery and occupation requirements, connecting the two results through an exact finite-dimensional optimization. A closed three-record example distinguishes fixed complete records from fixed averaged data. Separately calibrated imperfect contacts admit one-sided finite-error extensions. Together, the results characterize both informational compatibility and control-dependent formation through common attaining apparatuses.Preprint status and related materialsPREPRINT - AUTHOR-SUBMITTED MANUSCRIPT; NOT PEER REVIEWED OR ACCEPTED FOR PUBLICATION. This theoretical and computational work addresses the compatibility of retained binary quantum records with optimized accessible-system recovery, and the control-dependent resources required to form those records in a specified spin encoding. The article provides exact finite-record bounds, attainable recovery and resource frontiers, and calibrated finite-error qualifications. The manuscript has been submitted to Foundations of Physics for journal consideration; this Figshare record is not a journal publication. The technical supplement is included as a second PDF in this same record. Related software, verification code, exact certificates, and selected computational evidence are available separately at https://doi.org/10.6084/m9.figshare.34886079 under the MIT licence. When a journal version of record becomes available, a link to its DOI will be added here.AuthorKent Nimmo; Independent Researcher, Vancouver, British Columbia, Canada; ORCID 0009-0003-8503-8891.
A source-inclusive coherence observable is useful only when its measurement resources are admitted. This preprint constructs a bounded local reader for the conserved exchange observable of a specified finite quantum-record apparatus. A degree-three graph uses one independently prepared probe, twenty relays with arbitrary joint states, and finite-duration two-register extraction and full-SWAP interactions. The original apparatus Hamiltonian and an independently added archive-gated process remain active throughout their respective implementations. A prefix-dependent control-frame comparison, symmetry-resolved effect estimate, and explicit gain, phase and finite-slew waveform bounds yield a sufficient response separation greater than 0.005139889 for the stated fixed, fully accessible sixteen-position source and remaining response-error budgets. This is a conditional mathematical result, not measured hardware calibration, a partial-source-access theorem, or a unique diagnosis or derivation of fundamental collapse. The package contains the main manuscript, technical supplement, editable LaTeX, direct numerical reproduction sources, and accepted returned Mac evidence for 56 paper-specific supporting Lean declarations. That subset verifies selected algebra, scalar inequalities and the actual-constant numerical expression; it does not formalize the full physical/continuous-time theorem end-to-end. Separate-session AI review, numerical reproduction and returned kernel evidence are distinguished. Author-owned manuscripts, documentation and mathematical data are CC BY 4.0; newly authored numerical and Lean software is MIT. Predecessor and third-party notices remain unchanged; the reproduction package contains the licence map. No external funding; no competing interests declared. Public preprint v1.0; edition date 18 October 2026; DOI 10.5281/zenodo.23442579.