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.
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.
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.
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.
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.
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.
Which changes of frame preserve a source together with the records it can carry? For the native eight-state cell of Recognition Science, we classify the unitary transformations preserving its six-dimensional contact space and admitting covariance of the full populated-record source. The source forces commutation with the literal half-cycle. In the contact coordinates this is the conformal symplectic equation BᵀJB = wJ. The represented group is (Sp(2) × U(1))/ℤ₂, with an eleven-dimensional Lie algebra. We derive the source commutant, the faithful representation and its observable kernel, and show how one shared control transports both retained references while leaving the readers unchanged in operator closure. The mechanism is explicit: record mixing identifies three source components, and the half-cycle turns that identification into an alternating form on the contact carrier.
This technical note studies selected reset resource trade-offs for a fixed source-tagged quantum-record apparatus at independently known source length m = 65. It preserves the already published target preparation, complete-reader initialization, calibration and acquisition results as inherited interfaces. For exact unitary reset with a pure initialized environment and arbitrary inert-reference correlations, the environment dimension must be at least the 131-dimensional source dimension. Under the staged-isolation contract in which a dedicated source bank is the only system allowed to retain the source label during the source stage, five qutrits are necessary among qutrit tensor banks because 3^4 < 131 <= 3^5. An explicit five-qutrit encoded-bank construction attains that count. The paper compares two constructions satisfying the same full-joint reset interface. A nearest-neighbor-transport circuit uses 65 source qutrits and 2,159 finite slots; its source stage combines neighboring transfers with 65 globally vacuum-conditioned extraction pulses. An encoded-bank circuit uses five qutrits and 15 slots, but assumes one nonlocal, coherently phased 130-block all-site extraction Hamiltonian. Under the declared supply, control, generator-error and isolation conditions, the sufficient reset bounds are below 0.000083661 and 0.000081959, respectively, and both remain within the existing downstream initialization allocation. The contribution is an apparatus-specific comparison of selected feasible points in environment dimension, control connectivity, pulse count, strength and slew, elapsed model time, calibration burden and complete-input error. It does not prove a Pareto frontier, a general reset resource theory, a local synthesis of the encoded Hamiltonian, an approximate or catalytic optimum, an all-length theorem or practical implementation.