Everettian branch multiplicity is often discussed as though it were a count of wavefunction terms or decoherent alternatives. This paper argues that the relevant object is an algebra-relative packing number on physically admissible history classes. Candidate histories, already assumed to belong to a decoherent or approximately decoherent family, are first quotiented by exact physical symmetries and then compared only through distinctions visible in a specified operator system of admissible effects. The resulting effective branch packing number \( N_{\mathrm{eff}}(t;{\mathcal A},\varepsilon) \) is the maximal number of mutually \( \varepsilon \)-distinguishable symmetry classes of histories relative to \( \mathcal A \). The paper proves a sequence of structural results: an orbit-packing reduction theorem; monotonicity under refinement of the admissible operator system; metric packing bounds; polynomial and, under the natural affine-rank condition, sharp asymptotic record-compression theorems for collective linear records of identical bosonic qudits; contraction of channel distinguishability under recovery; and a logical cluster theorem for recovered channels. In particular, \( r_0 \) independent collective record directions give \( \Theta(N^{r_0}) \) record growth rather than the labeled \( d^N \) growth; and separated clusters of recovered logical channels, rather than microscopic histories, determine logical branch multiplicity. Applications include the three-qubit bit-flip code, bosonic qudit occupation sectors, and a formal orbit-record model for dipositronium annihilation histories. The framework relates decoherent histories, Quantum Darwinism, exchange symmetry, and quantum error correction without asserting literal merger of globally orthogonal branches.
Everettian branch multiplicity is often discussed as though it were a count of wavefunction terms or decoherent alternatives. This paper argues that the relevant object is an algebra-relative packing number on physically admissible history classes. Candidate histories, already assumed to belong to a decoherent or approximately decoherent family, are first quotiented by exact physical symmetries and then compared only through distinctions visible in a specified operator system of admissible effects. The resulting effective branch packing number ( N_{\mathrm{eff}}(t;{\mathcal A},\varepsilon) ) is the maximal number of mutually ( \varepsilon )-distinguishable symmetry classes of histories relative to ( \mathcal A ). The paper proves a sequence of structural results: an orbit-packing reduction theorem; monotonicity under refinement of the admissible operator system; metric packing bounds; polynomial and, under the natural affine-rank condition, sharp asymptotic record-compression theorems for collective linear records of identical bosonic qudits; contraction of channel distinguishability under recovery; and a logical cluster theorem for recovered channels. In particular, ( r_0 ) independent collective record directions give ( \Theta(N^{r_0}) ) record growth rather than the labeled ( d^N ) growth; and separated clusters of recovered logical channels, rather than microscopic histories, determine logical branch multiplicity. Applications include the three-qubit bit-flip code, bosonic qudit occupation sectors, and a formal orbit-record model for dipositronium annihilation histories. The framework relates decoherent histories, Quantum Darwinism, exchange symmetry, and quantum error correction without asserting literal merger of globally orthogonal branches.
Abstract Decoherent-histories methods identify time-extended alternatives whose diagonal weights may support probabilistic interpretation, while environmental-record approaches ask which alternatives a specified observer can recover. We formulate their intersection as a finite branch-record packing problem generated from one cross-branch construction. A codebook is admissible only if its declared coarse-graining obeys a bounded probability-sum-rule error and its accessible states can be decoded under a specified observer policy with finite error. Uniform reweighting defines a geometric packing number; a communication interpretation additionally requires a preparation instrument. Retrospective resolution is stricter: the base history family is exhaustive, selected histories must cover sufficient physical diagonal weight and be individually recoverable, and the same declared quantum-to-classical measurement channel used for decoding must preserve approximate consistency after histories are refined by its registered outcomes. This measurement-refined condition quantitatively controls the difference between the actual outcome distribution and the normalized diagonal classical-quantum prediction. For uniform branch-record packing and its communication realizations, general converses follow from accessible dimension, fixed outcome count, Holevo-Fano information, and hypothesis testing. Exact models quantify record compression, access, memory, measurement policy, temporal dependence, and shared latent errors. In a jointly realized four-bit benchmark, scalar consistency and recovery admit six histories, whereas direct measurement-refined consistency under the declared local Helstrom instrument admits four at the same tolerances. Finite-sample procedures distinguish held-out recovery achievability, structure-conditional consistency certification, structural converses, independently constructed parameter intervals, and abstention.
Everettian quantum mechanics describes alternative histories within one evolving quantum state, while quantum Darwinism relates their accessibility to information retained in environmental records. I examine how finite record memory constrains the number of distinguishable histories and the organisation of their conditional ancestry. When d + 1 histories must each be decoded from a d-dimensional fragment with success d/(d + 1), the established discrimination optimum forces a regular-simplex geometry. Its conditional mixtures then determine exact ancestral edge costs and a joint criterion for the permissible topology, depth, total variation and largest step. In the three-history binary example, equally weighted labels decoded with success 2/3 require 61.6% greater minimum conditional variation in a qubit than in an optimally designed three-dimensional record. For six histories, minimising total variation and minimising the largest step select different genealogies. Quantitative stability estimates show that the additional constraint persists under sufficiently small departures from ideal decoding and weights. At this threshold, trace-distance contraction prevents a fixed channel acting only on an unrestricted optimum from compressing it into the smaller record at unchanged accuracy, although an explicit controlled preparation realises the alternative encoding. A von Neumann-inspired application distinguishes hereditary function, present configurations and retained mutation histories, and a comparison of finite and infinite counts separates these quantities from cosmological multiplicity. The resulting theory makes the limitations of environmental memory consequential for readable branching while keeping conditional variation distinct from physical rates of change. Its bounds apply to specified records and histories without assigning an absolute number of worlds since the universe’s beginning.
Overview This preprint develops a finite-history framework for quantifying how coherently stored environment records reveal logical quantum information. Instead of considering only which error occurred, the analysis treats the full complementary environment channel associated with a labeled family of finite quantum-error histories. For two selected history labels, the leakage is expressed exactly through the numerical-range diameter of the compressed relative operator. This yields global finite-history compression bounds, a zero-leakage characterization equivalent to Knill–Laflamme scalarity, and lower and upper bounds for recovery by a single common channel. Main results For a balanced independent two-step dephasing model on one logical qubit encoded into two physical qubits, the paper solves the worst-case optimization over the full rank-two code Grassmannian: infV∈Gr(2,4)supϕ,ψLhist(V)=21. The transverse leakage is determined exactly by Ltr(η,t)=max{21,2η,41−η21+t}. All global minimizers are classified. Besides the parity-isoclinic orthogonal-axis family, the minimizer set contains a genuine non-isoclinic region characterized by 0<η≤41,ez=oz=0,∣e⋅o∣≤1−4η24η2. A universal quartic inequality needed for the all-code theorem is reduced analytically to an exact polynomial-positivity problem and closed using rational Bernstein-basis verification. Distributional stability On a fixed labeled history-operator family, the complementary channel, worst-parameter leakage, and all-code minimax value are Lipschitz-stable under changes of the history probability distribution. With dA(p,q)=[∑h(ph−qh)2]1/2, the minimax value satisfies ∣Λ(p)−Λ(q)∣≤2dA(p,q). Consequently, key quantitative implications of the balanced complete solution—including the minimax value, principal near-minimizer coordinate constraints, and a universal operational recovery lower bound—remain stable under nearby biased or correlated perturbations.
A Kernel-Role Determinacy Theorem for Record Fixation, Fixed-Domain Quotients, and Boundary-Trace Normal Form This paper specializes the AASC kernel to the construction and use of quantum measurement records. Its central question is not initially how outcomes are produced, whether a wavefunction collapses, or which interpretation of quantum theory is correct. It asks what must be fixed for a measurement record to function as the same determinate physical and evidential object across retention, prospective use, deliberation, comparison, and report. For any non-degenerate determinate physical object, state, carrier, transformation, record, comparison, or endpoint, the paper takes the governing kernel to be: NDObj(x) ⇒ K(x) = (Adm, Stand, Ref, Irr). Admissibility, standing, reference, and irreversibility are therefore treated as necessary role conditions of determinate record use, not as an optional interpretive overlay added to an otherwise complete physical description. Measurement-record determinacy The manuscript constructs a fixed-domain measurement-record target in which a record must preserve: reference to the same construction token; standing-bearing evidential eligibility; admissible reuse; original act-time qualification; lawful redescription; retained and prospective record use; deliberative and comparative use; reportability and downstream inheritance. The resulting assignment space is quotiented by equivalence of standing-bearing measurement-record content. A rival quotient claiming exactly the same content collapses to the canonical measurement-record quotient. Representational differences that preserve the complete role content are treated as skin; differences affecting record identity, evidential ordering, admissibility, continuation, or downstream use remain physically relevant. Physically non-selective quantum regimes The paper then studies a physically non-selective quantum regime containing unitary branching, entanglement, decoherence, branch-local records, and stable local reports. The central result is conditional and target-specific: The bare unitary-entangling-decohering package can support multiple stable branch-local records without supplying a unique standing evidential record for the fixed construction-relative target. This is not a claim that branch-local records are unreal, unstable, or physically useless. It is a distinction between: physical occurrence of branch-local record structures; local branch-relative reportability; evidential role fixation for a record reused as the same standing object across the measurement practice. The paper shows that non-selective structures may agree on branch weights, density content, decoherence, correlations, instrument sums, channel closure, effect weights, projector orthogonality, and repeatability without selecting one fine record as the standing evidential record. Fixed-domain quotient and fixation commutation The theorem ladder establishes: fixed-domain assignment-space forcing; exact quotient construction for standing-bearing record content; same-scope equivalence collapse; fixation commutation under admissible transformation; inheritance of the original qualification requirement; persistence-bearing reuse; asymmetry exhaustion; anti-weakening; selector exhaustion. If evidential ordering varies across admissible assignments while the target requires one fixed-domain ordering, the ordering is not defined on that scope merely because numerical values or branch-local reports happen to coincide. The paper therefore treats measurement-record fixation as prior to the theory-level evidential reuse that depends on it. Boundary-trace normal form After internal and provenance/indexical routes have been exhausted, the residual same-domain route is a singleton stable boundary trace. Such a trace is not collapse dynamics, hidden-variable selection, or a new physical mechanism. It is an admissibility-preserving role anchor that can fix a record-bearing assignment only when its channel, incidence, transport law, original authority, and singleton-positive condition are independently warranted. The result distinguishes: sound partial selection from complete role fixation; unique trace value from unique original occupant; class-valued record fixation from raw-token fixation; later discovery of an original qualification from retroactive repair; physical record occurrence from standing evidential identity. Finite quantum specialization The paper occupies the abstract structure with an explicit finite two-branch quantum measurement domain containing: two outcomes; two fine records and one merged record; denominator-two branch weights; a balanced integer-amplitude state; density-table realization; selective and non-selective instrument maps; environment-record maps; projective branch maps; complementary effects; zero-product projectors; repeatability operations. On the declared balanced witness, amplitude weights, density weights, branch-map traces, effect weights, pointer records, instrument records, environment records, and channel records agree on the same branch architecture. Selective surfaces preserve the fine-record distinction. Non-selective surfaces agree only on the merged record. The fine records therefore remain separated, while the declared non-selective surfaces supply no singleton fine-record selector. The finite selector-boundary theorem eliminates the disjunction of candidate hidden selectors across the state, instrument, environment, channel, effect, and repeatability surfaces. The resulting diagnosis is precise: the missing structure is admissible fine-record role fixation, not further branch production, weighting, decoherence, stabilization, or repeatability. Scope and non-claims The manuscript establishes a finite, conditional, manuscript-facing theorem. It does not claim: to derive Hilbert-space quantum mechanics; to derive general POVM theory; to derive full projection-valued-measure theory; to reconstruct the full operator algebra; to provide a general Kraus representation theorem; to reconstruct complete positivity; to derive the continuum Born rule; to derive universal physical decoherence theory; to introduce a new collapse dynamics; to prove that no physical collapse process occurs in nature; to prove physical branch selection; to establish interpretive truth; to classify every possible physical measurement mechanism. The absence of a hidden selector on the declared finite non-selective surfaces is therefore not a theorem that no physical collapse or selection process can occur elsewhere. It is the finite theorem that those declared surfaces do not themselves supply the required fine-record selector. Relation to quantum interpretations The result does not deny the practical success of quantum theory, decoherence, branch-relative reporting, instrumental updates, or decision-theoretic methods. A practice aimed only at prediction, branch-local reporting, or instrumental updating may be a different target from the construction-relative record use studied here. If a theory claims to underwrite retained, prospective, deliberative, and evidential reuse of the same record-bearing construction token, it must preserve the relevant kernel roles and admit a standing-bearing fixation regime. Bare Everettian dynamics, without additional dynamics, ontological privileging, or admissible restriction to one constructional class, supplies no fine-record selector on the declared target. A strategy that preserves non-selective physics while restricting admissible record-fixation assignments belongs to a restriction-to-one-constructional-class route. Instrumentalist and QBist strategies instead reduce or redirect the target explanatory burden. Role in the AASC programme This paper provides the record-fixation layer of the AASC Quantum Determinacy Pair. Its companion Bell-scope work addresses the compatibility and no-signaling side of quantum witness determinacy. Together, the pair treats quantum records and witnesses as kernel-governed standing structures rather than as automatically determinate physical occurrences. The paper’s principal contribution is to identify the first failure point in a physically non-selective measurement regime: The initial problem is not necessarily the production of branch-local outcomes. It is the admissible fixation of the record role through which outcomes are retained, reported, remembered, compared, and used as evidence. The public Lean 4 formalization is synchronized with the finite measurement-boundary release v0.9.1 and provides the finite artifact for the declared effect, projector, compatibility, closed-stack, boundary-certificate, and audit layers. Keywords AASC; measurement problem; quantum measurement; measurement-record fixation; physically non-selective regimes; Everettian non-selection; record-role determinacy; fixed-domain quotient; standing equivalence; boundary-trace fixation; selector exhaustion; decoherence; branch-local records; admissibility and irreversibility; Lean 4 formalization
This paper is the seventh in a sequence developing an interpretive framework that redescribes physical reality as a causally consistent history of information updates rather than as a collection of objects independently existing within spacetime. Two prior results set up its question. The third paper (Uchida, 2026c) characterized public finality—recoverability of a committed fact by many mutually independent accessors, non-disturbingly and repeatably—as a broadcast redundancy: an effectively classical pointer value proliferated across many disjoint environmental fragments, approximately in the form of spectrum broadcast structure. The fourth paper (Uchida, 2026d) read the evaporating black hole as the regime in which a committed interior record crosses, at the Page time, from hidden to recoverable encoding, reconstructible from the radiation through entanglement-wedge reconstruction; and the parent framework (Uchida, 2026a, §10) had flagged holographic quantum error correction as a place its language of protected, redundantly encoded, locally recoverable records might find content. The present paper makes that content precise as a structure theory of recoverable records, kept separate from any claim of new physics or any suggestion that the framework derives the underlying results. The central claim is conditional and structural. The framework does not claim that every recoverable piece of information is a code, that the universe is literally a quantum error-correcting code, or that AdS/CFT describes our universe. It claims that when information qualifies as an erasure-recoverable record because an encoding preserves a logical degree of freedom—or a logical operator algebra—against a specified family of physical erasures, then that recoverability is formalized as an exact or approximate quantum error-correcting structure, in which each correctable erasure leaves an authorized complementary region supporting a recovery map. That formalization is itself an established mathematical classification; the paper's interpretive contribution is to read the series' recoverable record through it, and to set it against a second, distinct architecture. Broadcast redundancy makes the same effectively classical value legible from many disjoint fragments; erasure-correcting redundancy protects noncommuting logical quantum information by keeping each correctable erased region ignorant of—approximately private with respect to—the protected logical content, while suitable remaining regions recover it. The formal engine of this distinction is the duality between correctability and privacy (the information–disturbance tradeoff), not the no-cloning theorem; no-cloning, with the access structure of quantum secret sharing, survives as the reason independently usable recovery regions for arbitrary unknown logical quantum information must overlap. Three commitments keep the reading honest. First, it separates three senses of "record"—protected logical quantum information, a committed classical record, and a publicly final record—and insists that recovery or decoding restores access to logical quantum information but does not, by itself, commit a classical value; commitment additionally requires measurement, pointer imprinting, decoherence, and stabilization (Uchida, 2026f), and public finality a still later broadcast. Second, holographic quantum error correction is offered as the sharpest worked model of the distinction, with exactly solvable tensor-network toy codes (exact within the model) kept distinct from realistic semiclassical holographic reconstruction beyond those limits (generally treated as approximate), and both from the further question of state-dependence. Third, code-recoverability does not by itself imply public finality: protected quantum information can be recoverable from authorized regions while remaining non-public. The framework relabels code structure as the architecture of recoverability and derives none of it, making no independent empirical prediction. Its disciplined thesis is that codes protect recovery; broadcasts create publicity—different architectures that a single physical history may nonetheless realize in sequence.
We develop a field-theoretic framework for quantum error correction centred on fusion-space codes in unitary fusion categories. Admissible clusters determine total-charge sectors and orthogonal footprint projectors recording locally visible data left by error histories. The central distinction is between diagnostic footprint algebras and syndrome-admissible commuting algebras: the latter can be measured without revealing logical information and resolve chosen error representatives into measured sectors. For such algebras, exact correctability is equivalent to fibrewise Knill--Laflamme conditions, yielding a measure-then-recover factorization. Under a contractible-vacuum locality hypothesis, closed neutral composites give a categorical sufficient criterion for scalar action on the code. In the Ising theory, four σ punctures show that pair-charge footprints can be complementary logical diagnostics and realize an exact one-qubit Clifford shadow. A proper six-σ code instead admits a syndrome-admissible pair-charge measurement and exact recovery from an explicit Majorana bilinear error. A second bilinear has the same measured footprint but differs by a logical bit flip, producing a concrete nontrivial footprint fibre and genuine decoding ambiguity. We also formulate conformal-block likelihood data and compute geometry-dependent Ising four-point weights. For growing code families, we prove a conditional Peierls-type threshold theorem: bounded connected-region growth, local stochastic noise, local neutralizability of small residual components, and componentwise decoder balance imply PrL(fail)≤C∣ΩL∣e−cL below a nonzero constant error rate. We conclude with representation-theoretic and algebro-geometric directions involving tube and Hopf algebras, Yangian-type structures, Higgs bundles, spectral curves, Jacobians, and abelian varieties.