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.
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.
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.
Papers I and II derived gravitational geometry and second-order dynamics from finite accessibility without invoking measurement, observation protocols, or collapse as primitive notions. This paper extends the operational program to the measurement sector, deriving the structure of record formation and probabilistic weights from the same finite accessibility and compositional constraints. Finite accessibility restricts admissible operations on a subsystem to a proper operator subalgebra. Because bounded certification excludes operators requiring nonlocal coordination or arbitrarily fine phase resolution, the accessible algebra is strictly smaller than the full operator algebra. When a system becomes correlated with orthogonal environment sectors, off-diagonal operators between those sectors fall outside the accessible algebra. The reduced state is therefore indistinguishable, under all admissible operations, from its diagonal ensemble. Definiteness emerges as an algebraic consequence of sector restriction, not as a collapse postulate. The weights of these sectors are fixed uniquely by compositional consistency. Phase invariance, multiplicativity under independent refinement, and stability under record coarse-graining constrain any admissible assignment of sector weights to satisfy a functional equation whose only continuous, non-contextual solution is quadratic in amplitude. The Born rule therefore arises as the unique invariant measure compatible with finite accessibility. Irreversibility follows from action scaling: reversal requires operators outside the admissible tensor algebra and incurs action cost proportional to the number of correlated subsystems. Since admissible operations are bounded by the action gap, macroscopic record reversal is excluded. Measurement introduces no new physical primitives beyond finite accessibility, locality, and compositional stability. Record formation, classical definiteness, and Born probabilities are structural consequences of algebraic restriction under finite accessibility. **This paper is part of the Finite Accessibility series, which reconstructs gravitational geometry, quantum dynamics, gauge structure, cosmological perturbation theory, and the dark sector from a single operational primitive: a finite certification ceiling ω* governing the maximum rate at which physical relations can be maintained across extended domains.** **Introductory papers** - Geometric Limits to Coherent Timekeeping in Extended Reference Frames — https://doi.org/10.5281/zenodo.18824556 - Accessibility Selection Law (ASL) — https://doi.org/10.5281/zenodo.18824649 - Phase Boundary Gravity — https://doi.org/10.5281/zenodo.18824707 **Pillars (numbered I–IV)** - Paper I: Deriving Gravitational Geometry and Schrödinger Dynamics from Finite Accessibility — https://doi.org/10.5281/zenodo.18601600 - Paper II: Deriving Inertia, Motion, and Wave Propagation from Finite Accessibility — https://doi.org/10.5281/zenodo.18625174 - Paper III: Accessible State Spaces, Record Stability, and the Emergence of Quantum Statistics — https://doi.org/10.5281/zenodo.18625192 - Paper IV: Frame Dissolution, the Certification Ceiling, and Observable Deformations — https://doi.org/10.5281/zenodo.18654458 **Extension papers** - Dark Matter from Finite Accessibility — https://doi.org/10.5281/zenodo.18654433 - Dark Energy from Finite Accessibility — https://doi.org/10.5281/zenodo.18654523 - No-Go Theorems and the Admissibility Corridor — https://doi.org/10.5281/zenodo.18654493 - Gauge Structure from Finite Accessibility — https://doi.org/10.5281/zenodo.18928152 - Perturbation Theory from Finite Accessibility — https://doi.org/10.5281/zenodo.18928213 **Experimental paper** - Experimental Probes of Finite Accessibility: GW Dispersion and Vacuum Path Independence — https://doi.org/10.5281/zenodo.18931126 **Keywords:** Born rule derivation, measurement problem, quantum statistics, finite accessibility, operator subalgebra, record formation, definiteness emergence, wave function collapse, compositional consistency, accessible state spaces, action gap, irreversibility, sector restriction, quantum foundations, algebraic quantum mechanics, decoherence
The Enter Calculus derives the Born rule and the Lüders update from five operational requirements on evaluation strategies. This manuscript studies the quantitative regime in which those requirements hold only approximately. The analysis identifies the local variables that may legitimately control a per-measurement tolerance, proves a linear residual-state estimate, and shows that outcome maps inherit an almost-Lipschitz modulus without requiring an independent Hölder assumption.The record axiom is separated into three inequivalent scopes: pair, protocol, and controlled. This distinction determines the accumulation law for adaptive protocols. On the Lüders-exact residual class, the optimal adaptive deviation is exactly 1 minus (1 minus epsilon) raised to the mth power. A rotated-residual construction proves that this class is maximal.A noisy Born witness gives the crossover profile 1 minus (1 minus epsilon divided by 2) raised to the mth power. The reflected direct/suspended gap has the closed form: 1 minus epsilon, multiplied by the quantity 1 minus (1 minus epsilon divided by 2) raised to the mth power. This expression is independent of coherent copy multiplicity.A multi-axiom tolerance geometry yields a cube-root transfer into derived anchoring, while credit constraints produce a scaling collapse onto the function F of x, equal to 1 minus e raised to the negative x divided by 2.The accompanying born-rates package verifies the closed forms and falsifies invalid scope promotions, but it does not replace proof. The resulting geometry is governed by three features: the local budget, the scope of the record axiom, and the residual class.
This derivationally self-contained sector connection instantiates Return Theory (RT): Foundations 1.0.0 (doi:10.5281/zenodo.23286579) in finite-dimensional quantum-record systems. It studies a friend–Wigner laboratory, general two-branch systems, and an exactly solvable branching environment. Quantum mechanics remains the unchanged host law. The RT connection explicitly separates candidate generation, record admission, validation reference, representation reference, observer access, image fibres, inter-observer agreement, and operational objectivity. The release establishes four mathematical results: complementarity for one access, complementarity for two disjoint systems, agreement bounds for jointly realised admitted records, and the equal-tolerance threshold for disjoint content and phase observers. Nine host-law consequences include the friend’s exact content record, Wigner’s exact phase record and content non-identifiability, persistence of internal agreement with loss relative to an outside record, the exact cost of a partial inter-observer copy, agreement without a record, fragment admission and redundancy, phase exclusion from fragments, the classical mutual-information plateau, and an overlapping-observer code construction. The discriminating RT-connection prediction class is explicitly empty in version 1.0.0: every observable distribution in the declared systems is fixed by quantum theory together with the stated preparations, channels, and measurements. Four prospective protocols, V1–V4, are therefore classified as tests of host-law consequences and calibrations rather than as independent RT physical predictions. The reproducibility package contains the release PDF and LaTeX source, exact state-vector code, fixed-seed validation calculations, JSON results, generated tables and figures, machine-readable B1–B10 declarations, the three-way result ledger, checksums, and release-audit scripts. No external data are required. This release continues the earlier public Quantum Records record (doi:10.5281/zenodo.23282709), which is cited only for lineage and timestamping and is not used as a premise.