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.
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.
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.
Keeping a record changes which alternatives can interfere. On a pair of eight-state clock channels, we derive three recording operations whose identical final measurement has probabilities 1/4, 9/32, and 3/8. The full-current instrument retains fifteen distinct histories and transfers incoming occupation phase into their relative phases. Requiring its physical realization to conserve both native clock energy and occupation has a stronger consequence: every collective one-channel observable is conserved, and all admissible operations lie in the permutation commutant. This yields an exact reference-count balance, a sharp state-dependent disturbance bound, and a finite-reference obstruction to exact neutral recording. We characterize the complete finite-reference attainable region by an explicit finite algebraic synthesis problem, including its no-reference endpoint. A conserving collision construction then supplies a concrete attainable family, with an exact diamond-error formula and a uniform reference-return bound. These results connect the choice of record to observable dynamics and to the reference that pays for their implementation. A finite discrimination protocol states the required preparation, error tolerance, sample count, and independently calibrated time and energy scales. First public preprint release, version 1.0. The files include the PDF, the main TeX file, and a complete source archive with editable schematics and relevant verification supplements. AI-use disclosure: AI assistance was used for source analysis, mathematical exploration, manuscript drafting and revision, diagram preparation, and computational verification support. GPT-6 Astra reviews assessed arguments, scope, readability, and schematics. These were internal model reviews, not external peer review. Jonathan Washburn is the sole listed author and is responsible for the released manuscript.
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 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.
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 small quantum arrangement — a reader qubit Q₁ coupled to a remainder qubit Q₂ through one fixed gate, with declared operations on Q₁ only — has access to some distinctions of the remainder and not to others. This manuscript follows that arrangement through fourteen research units — thirteen with a model, exact proofs and an executable checker (with certified interval enclosures wherever a propagation is read), and one, the maintenance criterion, a definition with checked decision logic — and organizes what they establish around four distinctions. First, protecting an access restriction is different from preserving the distinctions a reader can use: a universal leakage bound and an exact one-gate value are proved; a reset-based maintenance pulse prolongs limited leakage at the price of previously accessible information, while a reversible pulse prolongs it and preserves the full declared input class. Second, meeting protection criteria does not identify a mechanism: a prepared finite clock-controller meets four declared criteria at four sizes, and so does a static field with the clock decoupled, and so does the clock's free mean field at the largest size. Third, restoring a supporting condition does not necessarily recover the capability: coupling alignment is shown to be a supporting condition of the capability, and a restoring process that improves alignment in six matched realizations meets the full restoration standard in one, misses it narrowly in one and fails in four, while useful access is preserved in none. Fourth, retaining a relation, delivering it to a reader and using it internally are different achievements, and a supplied design's advantage can reverse when its context changes without the comparison identifying the mechanism: a sector record restores access at three of four endpoints under an added permission; an internal coupling improves the primary endpoint without reaching the reference; a two-by-two of axis term and gating cell reverses both contrasts. Each result is stated with what it does not establish. No conclusion about self-maintenance, ownership or experience is drawn. The deposit is a technical manuscript, not peer-reviewed, that consolidates fourteen closed research notes and adds no theorem; the supporting proofs are in the notes. The package contains the manuscript (PDF and Markdown), the fourteen notes with two errata, every checker and its run record, the failure-path harness with its recorded outputs and a release adapter, the two earlier baseline notes, a claim-to-proof map, a correction ledger, a source lock and a manifest with the SHA-256 of every file. Every number printed in the manuscript was read by script from a packaged run record. Running checks/run_all.sh reproduces every checker's recorded summary line with the declared dependencies only (Python 3.11 or later; sympy 1.14, mpmath 1.3, numpy); the harness coverage, including the checkers it does not cover, is stated in §10 and Appendix A. Drafting and computation were AI-assisted under the author's direction. Archive SHA-256: 119d835b2d2e19463f3eb6478ea34ddbdf4deea220a5d6d6fe528dcf7061bedb.