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.
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 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.
The Enter calculus [EC] derives the Born rule and the projective update from five operational requirements: composability, no-signaling, nullity, continuity, and records. This paper equips the calculus with a strictly linear resource geometry and develops its finite-resource metatheory, stating each result at precisely the strength established. An update grammar of four primitive moves makes every cut-indexed causal potential a conservation law, as shown in Theorem E through a case analysis of the grammar, and derives scalar monotonicity rather than assuming it. A latent-active split with fully specified typing rules turns monogamy of causal change into a typing invariant, as established in Theorem F. Resource-relative admissibility, with its quantification made explicit, is downward closed along runs, ensuring that the forcing threshold is well defined. The minimal register cost of the decohered fine-graining construction is computed in closed form in Theorem H, and the resulting realizable rational net improves the capacity-four rigidity mesh from the vacuous value delta equals one to delta star equals one-sixth in Theorem C star. Two rigidity layers are proved unconditionally: boundary rigidity, consisting of monotonicity, antipodal symmetry, the Sandwich and Seal lemmas, and a descent proposition, which holds for every admissible strategy without an update hypothesis; and diagonal-faithful rigidity, established in Theorem I through a fully detailed single-qubit case with an explicit Pexider argument and a multi-qubit induction, forcing the Born rule on every sub-capacity configuration at every capacity N greater than or equal to three. The residual question—whether diagonal-unfaithful update slack can survive at full depth—remains unresolved. Instead, it is reduced in both directions to sharply delimited objects: an effective hierarchy of relaxations whose finite levels upper-bound the residual freedom and make rigidity refutable at a finite level, as shown in Theorem J, without claiming that the limit is attained; and a compensated transport family based on odds-ratio-preserving Plackett transports, for which the two-qubit layer, the record-family seal lift in closed form, and the full-support three-qubit extension on compact sets are proved. This family satisfies every axiom instance generated by record-chain protocols from two-qubit initial configurations at every depth, as established in Theorem M prime. Consequently, no capacity-three rigidity proof can proceed through record chains alone. Full capacity-three admissibility reduces to a single explicitly stated Extension Conjecture; under that conjecture, Open Problem G three has a positive answer and the minimum threshold N zero of one and the pair one, two equals four. Depth-two and depth-three record probes pass at machine precision, and an eta-scan Pareto front reaches the axis numerically. Part II specifies a reproducible artefact in which every major claim is either proved, reduced to a finite check, or implemented as a property test or golden regression over a rich class of instances, with each item’s status—Theorem, conditional result, conjecture, or probe—labelled explicitly.
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.
The Enter calculus [EC] derives the Born rule and projective update from five operational requirements: composability, no-signaling, nullity, continuity, and records. This paper equips the calculus with a strictly linear resource geometry and develops its finite-resource meta-theory as a single coherent framework. An update grammar with four primitive moves makes every cut-indexed causal potential a conservation law, as stated in Theorem E, and derives scalar monotonicity rather than assuming it. A latent-active split turns monogamy of causal change into a typing invariant and reconciles token activation with monotone coarsening, as shown in Theorem F. Resourcerelative admissibility is downward closed along runs, as stated in Lemma 4.2; hence forcing is upward monotone in capacity and the forcing threshold is well defined. The minimal register cost of the decohered fine-graining construction is N_F^0(s, m⃗) = s + ceiling(log₂ M) + ceiling(log₂(max_k m_k)), as proved in Theorem H. The resulting realizable rational net strictly refines the dyadic mesh and improves the rigidity bound at capacity 4 from the vacuous value δ = 1 to δ⋆ = 1/6, as shown in Theorem C⋆. The central new result is a rigidity theorem: a multiplicative functional equation extracted from the sequential cocycle with one qubit of headroom forces every admissible strategy whose update diagonal is Lüders-faithful to agree exactly with the Born strategy on every sub-capacity configuration, for every capacity N ≥ 3, as established in Theorem I. The proof is domain-explicit: a witness state is constructed with specified amplitudes, normalisation, and Lüders weights; the cocycle is derived on its exact open domain of validity; an overlapping-interval argument reduces the relation to a function of the product; and the resulting continuous Pexider equation is solved from first principles. Consequently, all residual finite-capacity freedom is confined to full-capacity configurations and to diagonally unfaithful update slack, as stated in Corollary K. Open Problem 17.2 of [EC] is therefore equivalent to a diagonally unfaithful instance of Open Problem G at capacity 3, as shown in Corollary L, while an LP hierarchy converts the remaining question into a computable limit, as established in Theorem J. An exact Choi-rank ledger assigns costs to geometric instruments and identifies decoherence rate with capacity flux, as shown in Theorem D′. Part II specifies a reproducible computational artifact in the style of the Enter-calculus repository: every theorem is either executed as a derivation or enforced as a runtime invariant, and every open problem is represented by a numerical probe.
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 1-2 of this series derived Born weights conditionally from the block’s Riemannian geometry: the canonical volume determines the Hilbert structure, the induced projective geometry determines the ray measure, and Gleason noncontextuality fixes the trace assignment. What remained was the confirmation problem: in a block ontology every compatible history and every compatible observer-instance exists, so probability must be a measure over realised record-relative viewpoints, not a chance of becoming actual. This paper builds the record-theoretic and statistical structure that closes that gap, up to two explicitly priced commitments. The central claim, calibrated. In a non-splitting block ontology, perfect record cells inherit the Born measure and its exact conditional calculus. Under a persistent-prefix distinguishability condition (Assumption PF, stated prominently), interference remains uniformly controlled through arbitrarily deep finite record refinements. The resulting static record filtration obeys martingale concentration without independence assumptions, and — conditional on the explicitly priced viewpoint-typicality principle (Commitment TY) — Born conditional probabilities are exponentially confirmed against persistently different rival policies, including adaptive history-dependent rivals. The supporting structure: an exact perfect-record projector calculus; a closed sequential interference bound whose index is refinement depth, not flowing time; an explicit block-compatible recorder family (translated Gaussian fragments, and intrinsically, normalised heat-kernel packets on the block’s own S3) for which the record-overlap exponents and minimum redundancy thresholds are derived in closed form — the recorder family itself is not yet derived from block dynamics; and a canonical Gram-polar construction that associates to any imperfect record family, at each fixed depth, an exact Boolean hierarchy of projectors within explicitly bounded distance of the raw records — a finite-depth canonical Booleanisation, with cross-depth projective consistency an open problem. The coherent event weight Q is used only comparatively: it is not an additive probability measure for imperfect records, and the paper states the resulting trade-off explicitly. Two commitments carry the non-mathematical load and are priced, not hidden. Commitment OL: observer continuity is represented by record-functional equivalence across compatible continuations. Commitment TY: rational expectation over the record-equivalent continuations follows the canonical conditional measure. Given TY, the block-derived Born measure is internally self-confirming on record-typical viewpoints; without it, the mathematics fixes the measure but not its first-person authority. No collapse, branch creation, privileged actuality, pointer-basis selection, or theory of consciousness is used. Finite-resolution results on the block’s S3 truncation ladder (exact spectral-leakage identities, a conditional Sobolev rate, the Gleason dimension threshold) are collected in Appendix B.