For compiled nested-measurement circuits, or reflection towers, the companion paper established that every discrepancy between the coherent Born law tr(M rho) and the record-dephased collapse self-model tr(M Delta_J rho) is governed by reduced difference operators derived from X_J = rho − Delta_J rho. This paper extends that exact but rigid subsystem theory into an operational theory of accessibility. First, we prove the channel-interface theorem: for any quantum channel Lambda modeling access, the two laws agree for every effect available after Lambda if and only if Lambda(X_J) = 0; the maximal bias is delta_J(Lambda) = (1/2)||Lambda(X_J)||1, attained by a Helstrom measurement. By data processing, this quantity is an information monotone. Second, we introduce operator-valued Mobius elements Z_J and prove a localization theorem showing that Z_J is supported exactly on branch pairs whose records differ on every element of J. This yields a corrected hierarchy of classicality orders: state-level Mobius support is always the simplicial complex generated by the branch-difference sets, and therefore has order one whenever two branches coexist, whereas the Frauchiger–Renner obstruction has order two at the question level, equivalently at the interface level of Wigner’s measurement channel, whose Mobius weights are exactly |eta_M(J)|. Third, we prove that pairwise visibility equals the Uhlmann fidelity of the inaccessible-side branch states, v_rr′(S) = F(sigma{\bar S,r}, sigma_{\bar S,r′}), thereby placing screening, independent-noise product laws—which become exact equalities—and correlated record environments, including all-or-nothing secret-sharing records, within a single formula. Fourth, we construct an explicit compiled tower exhibiting exact operator-level cancellation at a budget-complete interface with unit pairwise visibilities, proving that the family of agreeing interfaces is a down-set but is not closed under unions. Fifth, we prove robustness: diamond-norm gate errors epsilon_n change every discrepancy by at most the sum over n of epsilon_n, converting exact screening into quantitative approximate screening. Finally, we develop cost-constrained discrepancies, including an exact exponential implementability gap of 2^(−c) for product measurements on a depth-c read-chain; the accessibility hypergraph, in which budgets correspond to covering and screening to hitting; the parameter geometry of the Frauchiger–Renner family, where the order-two stratum has codimension two and full classicality is impossible on the open simplex; and two-sided asymptotic detection exponents controlled solely by the single-copy discrepancy.
For compiled nested-measurement circuits, or reflection towers, the companion paper established that every discrepancy between the coherent Born law tr(M rho) and the record-dephased collapse self-model tr(M Delta_J rho) is governed by reduced difference operators derived from X_J = rho − Delta_J rho. This paper extends that exact but rigid subsystem theory into an operational theory of accessibility. First, we prove the channel-interface theorem: for any quantum channel Lambda modeling access, the two laws agree for every effect available after Lambda if and only if Lambda(X_J) = 0; the maximal bias is delta_J(Lambda) = (1/2)||Lambda(X_J)||1, attained by a Helstrom measurement. By data processing, this quantity is an information monotone. Second, we introduce operator-valued Mobius elements Z_J and prove a localization theorem showing that Z_J is supported exactly on branch pairs whose records differ on every element of J. This yields a corrected hierarchy of classicality orders: state-level Mobius support is always the simplicial complex generated by the branch-difference sets, and therefore has order one whenever two branches coexist, whereas the Frauchiger–Renner obstruction has order two at the question level, equivalently at the interface level of Wigner’s measurement channel, whose Mobius weights are exactly |eta_M(J)|. Third, we prove that pairwise visibility equals the Uhlmann fidelity of the inaccessible-side branch states, v_rr′(S) = F(sigma{\bar S,r}, sigma_{\bar S,r′}), thereby placing screening, independent-noise product laws—which become exact equalities—and correlated record environments, including all-or-nothing secret-sharing records, within a single formula. Fourth, we construct an explicit compiled tower exhibiting exact operator-level cancellation at a budget-complete interface with unit pairwise visibilities, proving that the family of agreeing interfaces is a down-set but is not closed under unions. Fifth, we prove robustness: diamond-norm gate errors epsilon_n change every discrepancy by at most the sum over n of epsilon_n, converting exact screening into quantitative approximate screening. Finally, we develop cost-constrained discrepancies, including an exact exponential implementability gap of 2^(−c) for product measurements on a depth-c read-chain; the accessibility hypergraph, in which budgets correspond to covering and screening to hitting; the parameter geometry of the Frauchiger–Renner family, where the order-two stratum has codimension two and full classicality is impossible on the open simplex; and two-sided asymptotic detection exponents controlled solely by the single-copy discrepancy.
We develop a complete first-order theory of observer-relative interference for nested, Wigner’s-friend–type measurement scenarios, modeled as compiled reflection towers: quantum circuits in which every internal measurement is implemented by a transversal record copy and every internal classical control is implemented by a record-controlled unitary. For any set J of records, the discrepancy between the coherent Born law, tr(Mρ), and the record-dephased self-model, tr(MΔ_Jρ), is determined by the single Hermitian traceless operator X_J = ρ − Δ_Jρ. Our central result, the exact interface theorem, states that an interface—either a subsystem S or a unital *-subalgebra A—is indifferent between the two laws if and only if the reduced difference operator tr_{\bar S}(X_J), respectively E_A(X_J), vanishes; moreover, the maximal bias over all effects equals the trace distance and is attained by a Helstrom measurement. Expanding the reduced difference in terms of branch cross-operators Γ^S_{rr′} identifies four distinct mechanisms: combinatorial screening by inaccessible record copies, residual which-path suppression, question-specific cancellation, and multipartite synergy. The combinatorial “copy budget” of earlier treatments is recovered as the zeroth-order approximation. We also prove a multiplicative suppression law for noisy records, replacing the hard combinatorial accessibility threshold with an exponential classicalization depth; a Möbius localization theorem, which assigns every certainty paradox a well-defined classicality order and shows that the Frauchiger–Renner obstruction is irreducibly second-order, with Möbius weight (1/2)√(p00 p11) = 1/6; a non-robustness theorem showing that single-record empirical adequacy in the Frauchiger–Renner state is a codimension-one coincidence; and an algebraic agreement theorem stating that, on the record-invariant algebra, coherent execution and fully collapsed sequential execution agree exactly at every depth.
A reflection tower iterates the compilation of an agent’s measurement protocol into coherent laboratory dynamics, producing a family of global states |Ψ_d⟩ indexed by nesting depth d. It has been conjectured that continuity of the outcome law in the depth index forces a finite critical depth D* beyond which coherence is destroyed—an operationally derived ultraviolet cutoff. We prove that no such cutoff exists within the calculus and identify precisely what does. Starting from six imported statements and finite-dimensional linear algebra, we establish: (i) a tower-agreement theorem—under an explicit sector-commutation hypothesis, the collapsed state at depth d is exactly the record-dephasing of the coherent state, so every record-block-diagonal observable has the collapsed Born law, and the joint law of the level-≤ k records is independent of d ≥ k; (ii) the conjectured cutoff fails for two independent reasons: the admissible strategy space is a singleton, so depth dependence can occur only in the protocol family, over which the continuity axiom does not quantify, while the padded depth-d and depth-(d + 1) states in the chain archetype have ray distance √3⁄2; (iii) an unrestricted interface detects a total-variation deviation of exactly 1⁄2 at every depth, with no decay, whereas an incompatible tower already deviates by 1⁄2 in the record law itself; (iv) a pairwise screening lemma whose correct index is the minimum, over branch pairs, of the number of distinguishing copy registers, yielding the sharp resource-indexed accessibility horizons D*(k; B) = k + B − 1 for the chain and D*(k; B) = k − 1 + ⌈log₂(B + 1)⌉ for broadcast, both diverging as B → ∞, with the horizon infinite whenever records are only partially copied; (v) a perturbative budget that degrades with depth, contrary in sign to the conjecture; and (vi) a four-part infinite-tower analysis with a formal mutual-exclusivity dichotomy. A self-contained artefact reproduces every numerical result, including a repaired horizon table.
Redundant records make macroscopic coherence progressively harder to witness without altering any executable probability. We make this quantitative. Working from five imported statements of an operational calculus—Born uniqueness with projective update, the classification of compiled primitives acting after a record, single-level agreement, internal no-signaling, and the path chain rule—together with finite-dimensional linear algebra, we prove: (i) a screening lemma showing that a witness of level-k coherence at depth d must have support on every one of the c(k,d) registers carrying a copy of the record, with exact first-crossing horizons D* chain = k + B − 1 and D* bc = k − 1 + ceil(log2(B + 1)) for the two archetypal copy schedules; (ii) an approximate-compatibility theorem, dTV(Pcoh, Pcoll) ≤ sum over j ≥ 2 of ηj, with constant one, where ηj is a state-dependent commutator norm, together with a two-parameter family showing that the optimal universal constant lies in the interval [1/√2, 1] and that no quadratic improvement is possible; (iii) an imperfect-record bound of √(p0 p1) times the product over i not in A of γi in the effect normalization, sharp for two-branch families, with effective horizons shifted by an explicit redundancy term L(γ, δ); (iv) light-cone and communication lower bounds, the latter requiring exactly c(k,d) − 1 classical bits; and (v) an infinite-tower analysis separating existence, convergence, block-diagonal superselection, and tail coherence, with divergence criterion given by the sum over j of (1 − γj) diverging. Every statement carries an explicit status label, and a self-contained computational artefact reproduces every quoted number.
A single-qubit Ramsey or T2 measurement reports how much interference survives on a qubit, but not why the rest is missing. Phase information lost to an unobserved environment and phase information coherently redistributed into neighbouring qubits produce the same local signature, although only the second can be recovered. We study the interference that a target qubit regains when a controller measures a chosen set of helper systems and applies an outcome-dependent phase correction. For a fixed interference basis and any helper subset T, we prove that the maximum average l1 coherence recoverable this way equals 2‖C_T‖₁, twice the trace norm of the off-diagonal block of the reduced state, and that a projective measurement on T attains it. Imposing a budget on the number of helpers gives a monotone hierarchy I(0) ≤ I(1) ≤ … ≤ I(m) from local coherence to full assistance, from which we derive an access order d_η (the fewest helpers that recover a fraction η of the accessible interference) and an unlocking spectrum π_r. Exact examples show that states with identical local and fully assisted coherence can have opposite access structure, and that coherent leakage and dephasing with the same Ramsey visibility separate at the first helper layer. Exact state-vector calculations on eight qubits show the access order rising from 0.90 for depth-2 local circuits to 3.25 at depth 8 and 4.10 for Haar-random states. Applied retrospectively to published Bell-state tomography from the ibm_brisbane processor, the first layer shows that 91–96% of the accessible interference is invisible in the target marginal. We outline uses in hardware diagnostics: separating redistribution from decoherence, pairwise leakage maps from two-qubit tomography, and sizing mid-circuit readout for feed-forward. Files: the paper (PDF) and CRIA_code.zip, a Python package that regenerates every figure, table and number in the paper and numerically verifies its analytic results (run: python interference_accessibility_reproducibility.py --verify).
This paper proves a finite-dimensional collapse-and-obstruction theorem for the relation between Heisenberg observable evolution and Schrödinger state evolution. The starting datum is a completion-locked boundary-channel datum: a boundary Hilbert space, a locked boundary observable algebra, a Heisenberg comparison on boundary observables, and an admissible state-side channel. The basic duality law is the trace pairing Tr[X Φ(ρ)] = Tr[Φ*(X) ρ], restricted, when appropriate, to a locked boundary algebra A∂. The theorem proves that this pairing undergoes Heisenberg–Schrödinger collapse exactly in the full-algebra reversible case: A∂ = B(H∂) and the Heisenberg dual is a ∗-automorphism, equivalently the state-side channel is unitary conjugation. Outside this reversible full-algebra case, the same expectation pairing is a boundary-channel law rather than a unitary full-algebra equivalence. Three rigorous obstructions certify noncollapse relative to the locked boundary datum: a proper boundary algebra is state- and channel-compressive; a nonmultiplicative Heisenberg dual cannot be implemented by unitary conjugation on the full algebra; and a many-to-one bulk/history realization can induce identical boundary laws from inequivalent channels or histories. Thus finite-dimensional closed-system unitary quantum mechanics is identified as the special case in which the boundary algebra is the whole kinematic algebra and the observable-state interface has no quotient, no proper boundary restriction, and no channel-valued loss of reversibility. The theorem has immediate consequences for measurement theory. Nonselective projective measurement is a boundary-channel operation: it fixes the commutative measured algebra while removing off-boundary coherences. Selective measurement is conditioning on a boundary record, not state-uniform retuning of the unconditioned boundary law. Bipartite no-signaling is likewise a fixed-boundary theorem: a remote completely positive trace-preserving channel fixes the local boundary algebra in the Heisenberg picture. Random-unitary symmetry-selection channels and finite rigidity-gap quotient selections supply two realization mechanisms. Their conserved observables are determined by the active commutant, while their low-temperature or equilibrium limits are controlled by explicit channel-norm bounds. License note: Distributed under CC BY-NC-ND 4.0.
Abstract Decoherence suppresses interference, but an observer’s ability to reconstruct a classical pointer value depends on additional resources: distinguishable conditional environmental states, an allowed measurement, sufficient accessible fragments, and sufficiently independent records. We provide an explicit minimal quantum-to-operational bridge that places these requirements in one chain. A system qubit is correlated with conditional environmental qubit states, permitting exact expressions for residual coherence, quantum mutual information, pointer Holevo information, accessible information, and Helstrom discrimination error. Full density-matrix calculations validate the formulas across 280 endpoint and interior cases. An accessible-sector/hidden-bath construction shows that coherence can fall below 2.3 × 10⁻⁵ while the observer’s record information and error remain fixed at 0.0803 bits and 0.4009. Measurement policies shift recovery thresholds: for record overlap 0.8 and error tolerance 0.05, collective or optimized adaptive-local discrimination requires four fragments, a fixed canonical-basis measurement requires six, and repeated one-fragment Helstrom measurements followed by classical decoding require seven. More importantly, records matched at the one-fragment level can have radically different collective behavior. For a binary pointer, the common-mode classical channel has an exact accessible-information ceiling of 1 − h₂(0.1) = 0.5310 bits, while the noncommuting hidden-memory ensemble approaches the same Holevo-information ceiling; matched independent records approach one bit and zero error. Disjoint observer counts and unanimous agreement likewise depend on shared error structure. Finally, declared local measurements induce classical record channels that are subjected to held-out model selection, predictive checks, abstention, and simultaneous finite-sample threshold certification. The resulting framework supplies the validated quantum-to-operational record model used in the companion analysis of operational capacity bounds for distinguishable decoherent histories.