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.
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.
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.
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.
Official website: distinctiontheory.orgPublic portal for the start guide, papers, claim status, failure registry, prior-art boundary, and citation resources. Canonical GitHub repository:https://github.com/yiningwu-research/Distinction-Theory FDS-Q1 develops the quantum-record-boundary paper in the Q-series of Finite Distinction Systems (FDS) / Distinction Theory. It studies Wigner’s Friend scenarios as finite record-boundary promotion problems. The central claim is that a friend-relative operational record cannot be promoted into Wigner’s accessible fact algebra unless a physical record channel supplies sufficient cross-boundary mutual information. Q1 does not solve the measurement problem, derive the Born rule, replace decoherence theory, modify unitary quantum mechanics, or claim that consciousness causes collapse. It is not a collapse theory. It is a finite-record boundary theory: observers are treated as finite distinction-registers, and measurements are treated as stable record formation under finite capacity. The central quantitative object is the boundary-mismatch entropy MF|W = H2(ZF | ZW), where ZF is the friend’s accessible record class and ZW is Wigner’s accessible record class. A friend-relative record is Wigner-promotable to tolerance ε only when H2(ZF | ZW) ≤ ε, equivalently, I2(ZF; ZW) ≥ H2(ZF) - ε. Q1 carefully separates Wigner’s ignorance from physical coherence. Wigner’s ideal global coherent assignment requires isolation and reversible-control assumptions. Wigner’s operational state is instead obtained by tracing out inaccessible degrees of freedom. The friend’s internal record is represented as a stable record class relative to the friend’s accessible algebra, not as a claim of fundamental collapse. The paper introduces a concrete finite-record interface. An observer implements an operational measurement family MO = {EOz}, which outputs a finite classical record ZO. A POVM outcome counts as an FDS record only when it is stabilized in the observer’s register over a retention window [t, t + τ] with error probability at most δ. Thus Q1 treats record extraction as classical-register stabilization under finite accessibility, not as a fundamental projection or collapse trigger. Information-theoretic quantities are measured in bits: H2, I2, and C. When a physical entropy is associated with a record distribution, the conversion is S = kB ln 2 · H2. Q1 uses this convention because FDS tracks finite distinguishability in binary record units. Von Neumann entropy is used when the relevant object is a quantum reduced state rather than a classical record variable. The paper formulates several core results: a boundary-promotion criterion, a Wigner-friend boundary-separation proposition, a finite-access lower bound on friend-record reconstruction using data processing and Fano-style inequalities, and an objective-convergence condition through environmental redundancy. In the binary case, the reconstruction error satisfies a lower-bound relation involving the inverse of the binary entropy function on the monotone branch 0 ≤ Pe ≤ 1/2. Q1 also defines an objective availability threshold using environmental fragments. For fragments E1, ..., EN, the minimal fragment number is mε = min { m : I2(ZF; E1:m) ≥ H2(ZF) - ε }. A record becomes objectively available to a class of finite observers only when redundancy, observer capacity, and record stability are all sufficient. This connects Q1 to decoherence and quantum Darwinism while preserving the distinction between decoherence, internal record formation, and external record availability. The paper’s response to Frauchiger-Renner-style arguments is not a refutation of the theorem. Instead, Q1 provides a boundary-audit rule: a statement recorded in the friend’s accessible algebra cannot be inserted into Wigner’s accessible algebra unless boundary promotion succeeds. In slogan form: AF-fact does not imply AW-fact unless the physical promotion channel supplies sufficient mutual information. Q1 introduces a promotion strength ηF→W = I2(ZF; ZW) / H2(ZF), which organizes boundary-alignment regimes: no promotion, partial leakage, Wigner-promotable record, and full boundary alignment. Partial leakage licenses inference, not promotion. For quantum technology, Q1 distinguishes coherence preservation from record availability. It introduces the diagnostic record availability horizon τRAH = inf { t : I2(Zinternal; Zexternal(t)) ≥ H2(Zinternal) - ε }. This is a diagnostic time for external record availability, not a new fundamental timescale. It is intended to complement device metrics such as T1, T2, gate fidelity, readout fidelity, internal record stability, cross-boundary mutual information, record-erasure history, and reset or garbage-collection cost. The release includes deterministic normal-form demonstrations. They illustrate the record-boundary causal map, boundary-promotion condition, decoherence versus record availability, a three-bit toy model, the promotion lattice, and the relation map connecting FDS Core, T1, O1, Q1, P3/P4/P7, and Q2. These figures are conceptual normal forms, not quantum dynamical simulations and not empirical fits. This release includes the paper PDF, LaTeX source, reproducibility code, generated figures, CSV outputs, and model summary outputs.
When initially disjoint observers later consult physical records, under what specified conditions should their readouts agree? This paper gives a sufficient, access-relative certification within standard quantum mechanics. This certification requires stable, repeatably readable labels, low accessible sensitivity to coherence between the corresponding outcome sectors, and, for distributed consultation, carriers encoding a common label. Sequential nondemolition readouts of a single carrier with orthonormal record labels produce memory registers whose product record-basis outcomes match with Born probability one, independently of coherence in the carrier. For disjoint carriers, if their accessible joint state is close in trace distance to the corresponding marginal of an ideal spectrum broadcast structure, disagreement is bounded by that distance and by the two local discrimination errors. In a two-outcome model, the relevant coherence belongs to the branch-bearing system-record state rather than the record-only marginal; defining dephasing and off-diagonal observables on that same space yields an interference bound. These results concern record statistics, not interpretation-neutral identity of first-person experiences or observer-independent reality. Applied to Extended Wigner’s Friend scenarios, they support ideal ask-the-friend agreement. That fixed-setting equality supplies neither the setting-indexed joint model required by Absoluteness of Observed Events in Local Friendliness nor Brukner’s common Boolean representation, and it does not entail Frauchiger and Renner’s distinct quantum certainty, inter-agent-consistency, and single-valuedness assumptions. The no-go results remain valid conditional consequences of those premises.
Context: This manuscript serves as the foundational quantum mechanics capstone for the DAGI (Directed Acyclic Graph Interpretation) research program at Whytics. It provides the computable, hardware-verified kinematic engine required to complete Relational Quantum Mechanics (RQM) and resolve macroscopic measurement paradoxes. Abstract: The tension between unitary evolution and irreversible wavefunction collapse is acutely encapsulated in the Wigner's Friend paradox and extended no-go theorems. While relational frameworks (e.g., Relational Quantum Mechanics) posit that physical states are inherently observer-dependent, they have historically lacked a computable kinematic mechanism to determine exactly \emph{when} and \emph{for whom} a measurement becomes an irreversible fact. Here, we introduce \emph{Observable-Selective Branching} within the Directed Acyclic Graph Interpretation (DAGI). We propose that wavefunction collapse is not a universal foliation of spacetime, but a topological phase transition governed by the mereological complexity of the observer. Using M"obius inversion on Boolean subset lattices, we define an observable's "informational cross-section" by its irreducible synergistic support (interaction order k). A macroscopic record induces irreversible branching relative to an observable if and only if the record intersects that topological support. Crucially, we test this topological stratification experimentally on IBM superconducting processors (\texttt{ibm_fez}, \texttt{ibm_marrakesh}). Under strictly pulse-matched controls, we demonstrate a "split outcome": a spatially separated physical record produces a bounded/null effect on a highly localized k=1 probe (an operational clock), confirming unitary isolation for the simple observer. Simultaneously, the identical record drastically suppresses a distributed k≥3 GHZ-class coherence witness, confirming collapse for the complex observer. Finally, we demonstrate that applying a coherent quantum-eraser control actively un-branches the graph, restoring global coherence. These hardware results provide a physical mechanism for relational quantum mechanics, demonstrating that the objective, universally shared "global stage" of classical reality is simply the thermodynamic limit of observable-selective branching. Key Highlights: Resolution of Wigner's Friend: Replaces the philosophical ambiguity of the measurement problem with a strict, computable topological boundary based on Möbius interaction order (k). Observable-Selective Collapse: Demonstrates mathematically and empirically that wavefunction collapse is not a universal scalar event, but a relational graph-cut bounded by the observer's informational cross-section. Hardware Validation: Translates the Wigner's Friend paradox into a concrete, schedule-matched validation suite on IBM superconducting hardware, proving that low-complexity probes bypass records while high-complexity probes suffer irreversible collapse. Topological Healing: Proves via coherent erasure (ERASE) that measurement backaction is a programmable, informational structure rather than irreversible thermodynamic hardware damage. Bridge to Quantum Darwinism: Explains the emergence of an objective macroscopic reality as the thermodynamic limit of observable-selective branching for dense, macroscopic (k≈1023) entities.
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.