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 $4\times4$ 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.
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.
A durable quantum record, the location of its coherence, and a uniquely actual outcome are different physical questions. We study their separation in a supplied finite-dimensional quantum theory. An explicit, time-independent apparatus couples a system–detector encoding to a controller, a distinct archive, and a finite returning source. A conditional energy penalty suppresses release of the original record without deleting its return coupling. An invariant reduction and a one-sided spectral-gap estimate yield a certified interval with archive occupation above 0.98438, controller-latched occupation above 0.99161, and filled-but-unlatched weight below 2.402 × 10^-7 in one stated design. The same record statistics admit different source-label overlaps and hence different coherence on a specified retained boundary. A conserved product of complementary exchanges accesses that coherence when the finite source is included. Given a calibrated multiregister reader, its finite-duration response separates the coherent apparatus from an independently added archive-gated reduction law, with the added process left active during readout. A conservative example retains a response gap above 0.01717 after separately stipulated one-percent error allowances. The contribution is the finite construction and its quantitative retention, access, and readout interfaces. Quantum carriers, Hamiltonian couplings, trace readout, source preparation, and the reader are supplied; local reader synthesis, empirical calibration, absolute rate selection, and unique-event actualization are not established. Paper code: CFQR-P1 (Complementarity-First Quantum Records — Paper 1). The associated bundle contains the technical supplement, editable LaTeX, numerical checking programs and rational certificates, and selected Lean sources with returned build and axiom-audit evidence. The 39 supporting declarations are partially formalized; eleven recorded type excerpts contain printing elisions. This is not end-to-end formalization, external human peer review, an experimental detection of fundamental collapse, or a derivation of unique-event actualization. Author-owned manuscripts, documentation and mathematical data are CC BY 4.0; author-owned software is MIT, subject to the bundle’s licence map and third-party notices.
A two-mode quantum harmonic oscillator is reconstructed from the axiom that a single distinction is the primitive degree of freedom. The construction forces an internal reference witness, Euclidean geometry and a positive irrational remainder δ=2−1. The binary count remains I=1, while its complete account retains the identifying relations and succession that the count alone omits. Symmetry conservation and recoverable closure establish persistence, with a next lawful step always available. Circular closure supplies the exact area increment g=(π−2)/4. Its accumulation forces escapement: an additional complete unit is admitted with its witness and origin retained. Applying the primitive construction to this unit supplies the joint Archimedes trammel and its three-dimensional family. The completion event fixes the relative phase and recoverable shadow amplitudes. Subsequent comparisons complete the cell account, whose positive irrational remainder Ω=1−π/(32) permits further admission. Representing complete accumulated area as squared ladder weight uniquely fixes wn2=(n+1)g and the dimensionless action coefficient ℏ∗=g. Source and child quadratures satisfy the canonical relations on a common invariant domain. At escapement, the completed-unit contribution restores the commutator term missing from the source remainder. The Hamiltonian Htick=(πℏ∗/2)(nα+nβ+Id) has a complete discrete spectrum; ni counts oscillator occupations. Its unitary evolution reproduces the trammel's quarter-turn, and an explicit Hermite transform supplies the equivalent Schrödinger representation. The geometric birth comparison becomes an entangled source–child state. Its complete amplitudes and retained context recover the producing account. Each finite generated retained history admits a faithful quantum memory representation satisfying structural record coherence with sequential weak value one.
This preprint studies operational equivalence and recovery in a twenty-dimensional residual system with a fixed four-sector representation of dimensions 8, 8, 2 and 2. It distinguishes what a declared observation algebra can detect, what a channel preserves, and what a retained classical record makes recoverable. Record-resolved criteria separate useful reversible-error records from unrelated labels and measured-sector records, even when the discarded channel is the same. For a specified coarse-record model, the work identifies exactly recoverable observable algebras and proves an optimal entanglement-fidelity expression on a fixed four-state orbit code. Preparation-aware recovery is then classified over all one-use isometric subspaces in the stated family. An explicit ten-dimensional marker code attains exact recovery under a specified stochastic record where the whole-sector support restriction permits only eight dimensions. Its encoding, nonbaseline marker measurement, and conditional inverse operations remain supplied resources. Scoped Lean evidence verifies the explicit arbitrary-complex-matrix recovery construction and related finite identities; it does not formalize the general complete-positivity/reference framework or optimal-dimension theorem. The recorded-error model is not derived from the companion exchange-timing model, and no protection of its long pulse sequence is claimed.
A measurement apparatus must do more than produce the correct final outcome frequencies: its records must faithfully preserve its own earlier declarations. This paper constructs autonomous quantum measurement chains that address both requirements within a specified nonrelativistic Schrödinger–Bohmian model. Assuming Bohmian guidance, complete initial quantum equilibrium, and a finite apparatus stock prepared independently of the unknown input, the manuscript gives a Hamiltonian bounded below that implements each fixed finite measurement programme to any prescribed positive tolerance. The complete model retains the source, pointers, internal clock, coherent resource products, copied archives, reset receivers, and an inaccessible quantum reference. Forced harmonic traps supply exact pointer motion. Retained orthogonal keys permit stationary record storage, while explicit copying and transfer preserve earlier records before working registers are reset. A finite-mass quantum clock replaces external time-dependent control. Weighted derivative estimates quantify its approximation error and bound the absolute probability current through archive decision surfaces. The resulting theorem separates final quantum-output accuracy from historical-record fidelity, including preservation throughout designated holding intervals. It retains coherent null outcomes and provides reproducible finite examples. Every positive tolerance uses finite resource choices for the fixed programme and observation horizon, although Gaussian packets retain unbounded tails. This is a conditional implementation and error-control result. It does not derive the initial Born law, uniquely determine Bohmian guidance, establish closeness of complete microscopic trajectories, or demonstrate laboratory realization of the admitted interactions.