PR2E v1.2 states the finite-access ontology as an active boundary transducer: finite access → filter → mixer → metric + kernel return → observable readout. The filter selects retained channels and weights. The mixer orients the reciprocal phase/counterphase pair and preserves the induced metric. The kernel is invisible to K_Π = Π†Π and returns dynamically by exact Schur–Feshbach elimination. Scale, direction and Hilbert geometry are three distinct data of one asymmetric map; an observable is not fixed by a scale alone. Within a stated scalar one-pole class the readout has the recursive form R = A/(1−B), with A the retained contribution and B the unresolved return. Completeness of that class is left open. Palindromic reciprocity constrains retained spectral structure; Schur elimination owns the resolvent denominator. The two are not claimed to follow from one generating polynomial. The coherent-return criterion τ_rec ≪ τ_drift is an operational [C] condition only; no numerical response time is published here.
PR2E v1.2 states the finite-access ontology as an active boundary transducer: finite access → filter → mixer → metric + kernel return → observable readout. The filter selects retained channels and weights. The mixer orients the reciprocal phase/counterphase pair and preserves the induced metric. The kernel is invisible to K_Π = Π†Π and returns dynamically by exact Schur–Feshbach elimination. Scale, direction and Hilbert geometry are three distinct data of one asymmetric map; an observable is not fixed by a scale alone. Within a stated scalar one-pole class the readout has the recursive form R = A/(1−B), with A the retained contribution and B the unresolved return. Completeness of that class is left open. Palindromic reciprocity constrains retained spectral structure; Schur elimination owns the resolvent denominator. The two are not claimed to follow from one generating polynomial. The coherent-return criterion τ_rec ≪ τ_drift is an operational [C] condition only; no numerical response time is published here.
Companion technical paper of the Quantum Gauge Theory of Time (QGT). Starting from a single global physical generator and a finite non-invertible access to a conserved whole, this work derives the exact retained-sector reduction of the dynamics. The unread sector produces a retarded Volterra memory kernel and, equivalently, a Schur–Feshbach self-energy. For the scalar boundary return, positivity and spectral determinacy select the minimal cyclic SU(1,1) realization associated with the sech return amplitude. The paper further develops the corresponding generation readout while keeping the physical identification of downstream particle and field sectors explicitly conditional. The project hosts the canonical manuscript, reproducibility material, verification scripts, provenance records, and versioned QGT release alignments.
A conserved whole is taken as primitive and is represented by a conservative dynamics subject to finite non-injective access. The resulting observable/unread decomposition yields an exact memory kernel and self-energy, an exchange law between readable and unread sectors, and a dynamically defined distinction between a returning sector and a silent complement. Finite observations can identify invariants of the returning dynamics without uniquely reconstructing the physical internal partition. Inside a declared elementary category, the analysis then selects a reciprocal elementary kinematic type. Two explicit physical identifications connect this kinematics to a constrained hyperbolic pair action and to a cyclic unitary scalar readout. Bochner, Lanczos and Carleman reconstruction then determine a unique minimal essentially self-adjoint Jacobi generator on the cyclic returning sector. The paper distinguishes foundational premises, declared mathematical typing, theorem-level consequences, physical identifications and open physical gates. It does not determine the absolute physical scale, prove physical occupancy of the elementary category, fix the silent-sector completion, or derive Standard-Model and gravitational phenomenology.
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.
Physics may have mistaken a quotient for an origin. Quantum theory represents a preparation by a trace-one positive operator ρ. This is exact for relative outcome weights, but it is produced by the many-to-one map N(σ)=σ/Tr(σ), for which N(λσ)=N(σ) for every λ>0. If total response scale is inert under every admissible continuation, the identification loses nothing relevant. If one declared continuation can distinguish σ from λσ, however, normalisation has erased a future-relevant distinction. The trace-one state remains an exact terminal quantum reading, but it cannot by itself be a faithful future-sufficient carrier. The reason this order is easy to miss is structural: once the quotient is taken as the complete input, every downstream construction that factors through it is necessarily blind to the positive-ray scale already identified away. The mathematical ingredients used below are largely established and are not presented as isolated novelties. Jordan–von Neumann, Frobenius, and finite-dimensional Gleason–Busch-type results serve instead as consecutive links in a conditional dependency-order theorem. Equality of complete future-response profiles first fixes the minimal predictive quotient. Under explicit locks, positive refinement fixes square-root access; quadratic recomposition fixes a real inner-product carrier; and the stated return-algebra, positive-involution, and central exhaustive-phase locks then select its complex Hilbert representation. Positive additive effect weights acquire the unique pre-normalised trace form Wω(E)=Tr(σωE); and only after the non-zero total sω=Tr(σω) is closed does one obtain ρω=σω/sω and the Born reading. The familiar finite-dimensional Hilbert–trace–Born structure therefore reappears, but in a different dependency position. Two no-go results make this dependency claim exact within their stated domains: canonical ray normalisation cannot by itself faithfully carry active scale, and, in dimension at least two, a normalised invertible congruence is affine on the full state space only when its transport is scale-uniform. A common-profile theorem states when independently reconstructed readings are uniquely profile-equivalent over one fixed common future-profile image, rather than merely compatible descriptions. Under locks L0–L9 on one typed finite-dimensional assembly datum, these results form a single conditional genealogy; the paper neither derives quantum theory from NIR alone nor asserts that nature realises the locks. Its central result is not a new quantum rule. If two realised or admissibly generated representatives σ and λσ on the same positive ray, with σ≠0 and λ≠1, have different complete declared future-response profiles, no faithful representation of those profiles can factor solely through canonical ray normalisation N. The trace-one state is not wrong: it is the exact answer to a later, conditional question.
Problem What must a physical record preserve to reveal temporal direction, measure duration, and support a calibrated experiment? This two-volume mathematical working paper distinguishes causal order, clock readings, and forward–reverse evidence, and studies their joint realization under one history law. Principal results Relative-entropy disintegration identifies evidence lost through observation. Graph potentials characterize sufficient current sensors by the absence of affinity on unobserved cycles; paired histories extend the criterion to silent events and indistinguishable reaction channels. Anonymous records, finite memory, and stopped decisions are treated explicitly. Compensated currents control clock calibration under shared environmental dynamics. A preparation-and-work identity bounds directional records and terminal organization together. Exact elimination retains hidden memory, while inverse and energy margins control response. A moving-sector comparison bounds the complete quantum apparatus record, including interactions, leakage, and transport; a spectral gap gives an explicit operating-time estimate. Analytical constructions and counterexamples cover graph–matter selection, ring and torus persistence, localized field clocks and receivers, and source-correlated records. A kinetic parent combines a selected ring, an inertial field, and a finite reversible receiver. Preparation frontiers separate information cost from attainable initial laws of a fixed dynamics. Methods, scope, and open questions The proofs use conditioning, path potentials, martingale compensation, generating functions, and stable operator comparison. The project includes editable sources, bibliography, 47 reproducible finite verification programs, provenance, and a separate corrections report. The results concern specified model classes and conditional external inputs. Selected spatial geometry is not a derived Lorentzian spacetime; the joint ring receiver uses supplied capture materials and diffusive transport. A common microscopic derivation of Lorentzian geometry, apparatus formation, two distinct proper-time clocks, constrained gravitational response, and typical directional preparation remains open.