Locally Quantum, Globally Non-Hilbertizable - an unbounded hierarchy of quantum process-descent constraints, non-Abelian Wilson networks, and a dimension-independent test of quantum globality
A new foundational framework, Anomalogic Descent Theory (ADT), studies a question distinct from Bell nonlocality, contextuality, indefinite causal order, and ordinary quantum-channel compatibility: Can every experimentally accessible proper context admit a standard quantum realization while the complete collection admits no common quantum process in any Hilbert-space dimension? The work develops an implementation-sensitive notion of global quantum process descent based on positive block-Choi/operator-valued coherence kernels. Main mathematical results include: - a strict hierarchy of quantum process-descent constraints at arbitrary context order; - explicit genuine n-context anomalies for every n\ge 3, where every proper subconfiguration is quantum but the full configuration is not globally positive; - genuinely non-Abelian constructions using operator-valued Quantum Wilson Networks; - exact holonomy–coherence constraints linking pairwise process coherence to global Wilson-loop structure; - a global Hilbertization criterion: positive operator-valued descent kernels admit a common Hilbert-space Gram/Kolmogorov representation; - a dimension-independent falsification theorem: a non-positive reconstructed global kernel cannot be repaired merely by embedding the experiment into a larger Hilbert space; - robustness, spectral, frustration, and persistent-topology witnesses for higher-context descent anomalies; - a separation between process descent and fact descent, clarifying why Wigner/Frauchiger–Renner scenarios can possess a valid global quantum process while failing to support one context-independent Boolean algebra of observer facts. A central explicit family is quantum on every proper n-1 context while failing globally. For the symmetric construction, [ \frac{1}{n-1}<t\le\frac{1}{n-2} ] is the genuine n-context anomaly region. The balanced point [ t_n=\frac{2}{2n-3} ] has equal positive and negative spectral margins, [ \delta_n=\frac{1}{2n-3}, ] providing a finite robustness gap rather than a boundary-only counterexample. The proposed experimental target is therefore unusually sharp: «Use one fixed, independently certified physical implementation. Verify that every proper overlapping coherent-control context is compatible with ordinary CPTP quantum mechanics, while the full implementation-sensitive dataset excludes every common positive quantum-process realization.» Such an observation would not merely violate a Bell inequality or reveal contextuality. It would indicate that quantum mechanics remains valid locally but fails as a global composition principle. No experimental violation is claimed. The current contribution is a theorem-level mathematical framework, an operational post-quantum theory candidate, and a concrete falsification program. Claims of historical priority are also deliberately limited pending specialist review. Relevant fields and keywords: quantum foundations, generalized probabilistic theories, quantum channels, Choi matrices, complete positivity, positive-definite kernels, operator systems, quantum contextuality, Wigner’s friend, Frauchiger–Renner, quantum combs, process matrices, coherent control of channels, channel holonomy, non-Abelian Wilson loops, matrix completion, Hilbert-space dilation, locally PSD cones, higher-order compatibility, post-quantum theories. The release includes the full preprint, proofs, reproducibility code, claim-status ledger, prior-art audit, and experimental protocol. Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
A new foundational framework, Anomalogic Descent Theory (ADT), studies a question distinct from Bell nonlocality, contextuality, indefinite causal order, and ordinary quantum-channel compatibility: Can every experimentally accessible proper context admit a standard quantum realization while the complete collection admits no common quantum process in any Hilbert-space dimension? The work develops an implementation-sensitive notion of global quantum process descent based on positive block-Choi/operator-valued coherence kernels. Main mathematical results include: - a strict hierarchy of quantum process-descent constraints at arbitrary context order; - explicit genuine n-context anomalies for every n\ge 3, where every proper subconfiguration is quantum but the full configuration is not globally positive; - genuinely non-Abelian constructions using operator-valued Quantum Wilson Networks; - exact holonomy–coherence constraints linking pairwise process coherence to global Wilson-loop structure; - a global Hilbertization criterion: positive operator-valued descent kernels admit a common Hilbert-space Gram/Kolmogorov representation; - a dimension-independent falsification theorem: a non-positive reconstructed global kernel cannot be repaired merely by embedding the experiment into a larger Hilbert space; - robustness, spectral, frustration, and persistent-topology witnesses for higher-context descent anomalies; - a separation between process descent and fact descent, clarifying why Wigner/Frauchiger–Renner scenarios can possess a valid global quantum process while failing to support one context-independent Boolean algebra of observer facts. A central explicit family is quantum on every proper n-1 context while failing globally. For the symmetric construction, [ \frac{1}{n-1}<t\le\frac{1}{n-2} ] is the genuine n-context anomaly region. The balanced point [ t_n=\frac{2}{2n-3} ] has equal positive and negative spectral margins, [ \delta_n=\frac{1}{2n-3}, ] providing a finite robustness gap rather than a boundary-only counterexample. The proposed experimental target is therefore unusually sharp: «Use one fixed, independently certified physical implementation. Verify that every proper overlapping coherent-control context is compatible with ordinary CPTP quantum mechanics, while the full implementation-sensitive dataset excludes every common positive quantum-process realization.» Such an observation would not merely violate a Bell inequality or reveal contextuality. It would indicate that quantum mechanics remains valid locally but fails as a global composition principle. No experimental violation is claimed. The current contribution is a theorem-level mathematical framework, an operational post-quantum theory candidate, and a concrete falsification program. Claims of historical priority are also deliberately limited pending specialist review. Relevant fields and keywords: quantum foundations, generalized probabilistic theories, quantum channels, Choi matrices, complete positivity, positive-definite kernels, operator systems, quantum contextuality, Wigner’s friend, Frauchiger–Renner, quantum combs, process matrices, coherent control of channels, channel holonomy, non-Abelian Wilson loops, matrix completion, Hilbert-space dilation, locally PSD cones, higher-order compatibility, post-quantum theories. The release includes the full preprint, proofs, reproducibility code, claim-status ledger, prior-art audit, and experimental protocol. Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
The fundamental nature of causality has been challenged by quantum theory. In particular, recent developments have shown that by relaxing the standard assumption that quantum events are embedded in a fixed background causal structure, a new kind of causal relations, indefinite, can emerge. These indefinite causal structures have been interpreted as causal loops allowing for the violation of a form of causality without paradoxes. On the other hand, quantum contextuality shows that any attempt to extend quantum theory such that classical logic holds in the extended domain fails due to the rise of inconsistent logical loops. There has been increasing evidence that it is a key feature in the understanding quantum paradoxes. This thesis aims at improving our understanding of indefinite causal orders and quantum paradoxes in order to address the question: “Could the non-fundamental nature of causality be related to a form of contextuality ?”The process matrix formalism is a mathematical framework which, in analogy withentanglement and the violation of Bell inequalities (nonlocality), offers new tools - witnesses of causal nonseparability and the violation of causal inequalities with noncausalcorrelations - to study and identify indefinite causal orders. In a bottom-up approach, wehave identified a large class of physically implementable processes, the quantum circuitswith quantum control of causal order (QC-QCs), that includes causally nonseparable processes such as the celebrated “quantum switch”. However these circuits cannot be used to violate causal inequalities. Nevertheless, in another work, I have proposed a significant relaxation of assumptions to certify the causal nonseparability of many processes, showing that it can be done in a semi-device-independent scenario with untrusted local operations and trusted quantum inputs. In particular, I have shown that the quantum switch can generate noncausal correlations in this kind of scenario.In order to refine my understanding of quantum contextuality, I have investigated theorigins of the Kochen-Specker theorem and I have analysed how the Hardy paradox entails a form of logical contextuality. I have identified that the two possible intermediatemeasurements involved in a pre-post-selection version of the paradox differ causally fromeach other. This analysis might be useful to study of the Frauchiger-Renner paradox, a“meta-version” of the Hardy paradox. On a more metaphysical aspect, I have argued fora “Gödelian hunch” from quantum theory, the idea that quantum paradoxes emerge froma lack of distinction between theoretical and metatheoretical objects.Finally, in order to establish a link between indefinite causal orders and contextuality,I have proposed a new causal game, in which the grandfather paradox is formalized interms of a logical inequality. Some process matrices can violate the causal inequalitywithout violating the logical one. I suggest that the emergence of logical inconsistenciesfrom the violation of a logical inequality might be interpreted as a form of contextuality.
A set of quantum data can look classical in every local test and still fail to admit a single classical explanation of the whole composite system. We formulate this failure as global contextuality. Here global means global in the physical sense of the whole multipartite system, not the local/global terminology of sheaf theory. Each party's local statistics are noncontextual and each measured multipartite context admits a generalized local hidden-variable description, but the GLHV block descriptions cannot be promoted to a single noncontextual hidden-variable model for the whole system. Three bipartite constructions exhibit this separation. A polarization-path construction gives a direct global obstruction. A qubit-qutrit KCBS construction gives an algebraic scenario-level example, with explicit formulas for the unconditional KCBS operator, the correlation-polytope constraints, and the postselected violation. A flagged qutrit Werner-local state gives a state-level example: the state is entangled and local for all projective measurements, its local qutrit marginals do not violate KCBS, yet postselection rules out a single GNCHV model. We also spell out the classical composition lemma: classical conditional hidden variables can be absorbed into a larger hidden variable, whereas quantum contextual data need not allow such a factorization. Within the general Bell-type framework considered here, with arbitrary parties and arbitrary local compatible contexts, but no cross-party joint measurements, the absence of local contextuality and GLHV-type generalized Bell nonlocality does not imply the existence of a global noncontextual hidden-variable model. Global contextuality is thus a compositional obstruction to classical explanation.
Quantum resource theories usually begin after a localization decision has already beenmade: the resource is assigned to a state, channel, measurement, process, circuit, code,contextuality scenario, or other formal object. Once that arena is fixed, standard resourcetheory supplies powerful tools for quantification, conversion, witnessing, and operationalanalysis. This paper addresses the prior localization problem: which stabilized distinction iserased by a tested descent, where is that distinction retained, and how is it later reactivatedinto a task-visible separation?The analysis is developed within Finite-Resolution Physics (FRP). Starting from finiteresolutionacts, stabilized histories, plateaux, continuation carriers, codecs, regimes, andSIB/SILM boundary coherence, the paper reconstructs quantum resources as task-visiblereactivations of retained distinctions erased by specified descents. The relevant structure isnot treated as primitive nonclassicality or as a universal obstruction object. It is diagnosedby asking whether a descent to a simpler target erases stabilized carrier–codec content thatlater admissible acts of the protocol can still make operationally visible.This yields a carrier–residual localization principle: a resource is localized at the minimalcarrier–codec layer where the relevant distinction must be retained, cannot be removed,descended, or relocated without loss, and can be reactivated into the task-visible separation.Entanglement, magic, contextuality, coherence, measurement adaptivity, quantumerror correction, and topological protection are then reformulated as distinct descent failures:product descent, Clifford–stabilizer descent, noncontextual global-assignment descent,reference-insensitive descent, nonadaptive-history descent, noise-degraded descent, and localor contractible descent.The paper does not replace ordinary quantum resource theory. Rather, it suppliesan upstream diagnostic layer that precedes ordinary quantification. Standard monotones,witnesses, conversion rates, simulation bounds, and thresholds remain applicable afterthe retained distinction, descent target, localization layer, and reactivation route havebeen identified. The resulting framework clarifies why representation does not determinelocalization: a resource represented in a Hilbert-space codec may be retained in a state,channel, measurement context, history, code, boundary/SIB structure, compact or discretecarrier, or stabilizer dynamics. FRP resource localization therefore asks not merely whethera resource is present, but where the task-relevant distinction is retained and how the protocolreactivates it.
A Kernel-Role Determinacy Theorem for Bell-Witness Fixation, No-Signaling, and Tensor-Level Compatibility This manuscript develops a fixed-domain AASC analysis of Bell-nonlocal quantum entanglement as a theorem of reusable witness determinacy. It does not introduce hidden variables, modify quantum mechanics, derive Hilbert-space formalism from below, or propose a new signaling mechanism. Instead, it asks a prior structural question: once a Bell-type operational family is used as the same non-degenerate witness across settings, lawful transformations, retained records, comparison, and later classification, what admissible same-domain standing form is forced? The governing claim is that AASC is not an optional interpretive overlay on physics. Once an operational Bell family is asserted as a determinate reusable witness, admissibility, standing, reference, and irreversibility are already active as minimal necessity conditions of same-witness objecthood. Target instantiation therefore carries governance occupancy: an alleged determinate Bell witness with no admissible governance realization is not an empty alternative, but a refusal or retyping of the target. The manuscript defines finite operational Bell objects with explicit setting carriers, outcome families, channel/interface data, standing-token valuations, continuation and report/reuse structure, comparison profiles, and act-time status. It then defines object-indexed query sorts, dependent response families, exactly recoverable operational simulations, lawful Bell groupoids, six governance-effect functors, and target-active extensional governance equivalence. The result is a typed interface in which arbitrary native formalisms can be compared by their target-bearing governance effects rather than by their internal syntax. The central Bell-specific theorem states that every same-target candidate-admissible Bell-governance construction is forced component by component into the certified boundary-compatible standing form on the complete target-active practice. Candidate admissibility does not assume the canonical form at the outset. Instead, target occupancy, lawful transport, standing-flow exclusion, compatibility certification, coordinate-deviation exhaustion, and componentwise closure force exact target-bearing realization and governance equivalence to the canonical representative CP∗C^{*}_{P}CP∗. The target-induced governance quotient therefore has exactly one element. The Bell carrier is treated as a compatible empirical model on a finite measurement cover. Bell-local completion is identified with the existence of a global section, and Bell nonlocality is the obstruction to such a section. The manuscript separates operational marginal invariance, absence of controllable communication, and exclusion of same-domain standing transport. No-signaling is therefore preserved without retyping Bell nonlocality as superluminal communication, metric transport, distant repair, or hidden standing transfer. The finite-dimensional quantum specialization uses the standard bipartite tensor-product and local-POVM carrier. Every separable realization in that carrier is Bell-local; therefore, if the fixed Bell-nonlocal empirical model has such a finite-dimensional quantum realization, every realizing state in that carrier is entangled. Post-quantum Bell-nonlocal no-signaling models remain within the general witness theorem but outside the quantum-entanglement corollary. The accompanying Lean 4 development, release v0.8.0, directly verifies the finite Bell-side witness carrier: denominator-32 vector/Born weights, structural no-signaling, CHSH numerator 808080, all sixteen deterministic local extrema, finite denominator-32 Bell-local-envelope exclusion, and conditional endpoint routing. Its direct witness certificate is vectorBorn_flagship_spine, and its deepest released endpoint is reasonableDeepestFiniteEntanglement_spine. The manuscript-level typed governance interface, empirical lawful groupoid, compatibility equivalence, coordinate-deviation exhaustion, completed role collapse, candidate-to-canonical derivation, and finite-dimensional quantum specialization are not attributed to the released Lean layer. This paper is the Bell-witness member of the paired AASC quantum determinacy arc. The record-side manuscript, Admissible Record Construction in Physically Non-Selective Quantum Regimes, fixes when local physical record occurrences stand as the same evidential record objects. The present manuscript fixes when the Bell operational family sampled by such records stands as the same reusable witness. The paired endpoint is therefore: record-role fixation+Bell-witness compatibility.\text{record-role fixation} + \text{Bell-witness compatibility}.record-role fixation+Bell-witness compatibility. The manuscript is classificatory and eliminative rather than mechanistic. It does not claim that AASC generates the Bell probabilities or supplies a hidden physical influence. Its claim is that reusable Bell-witness determinacy strongly constrains how Bell-nonlocal correlations may be coherently classified on a fixed domain: not as independently complete local states later connected by a mechanism, not as Bell-local hidden-variable value assignments, not as superluminal signaling, and not as same-act repair, but as boundary-level compatibility carried by one standing-preserving Bell witness.
Bell's theorem is popularly stated as the inconsistency of quantum theory with local realism. Realism is here formalized by the ontological models framework, where elements of reality are points of a measurable space and probabilities obey the Kolmogorov axioms. In the Bell scenario, locally causal ontological models posit a common cause supporting a global valuation, a joint assignment of outcomes to all measurements at once, across incompatible contexts. Within this notion of realism, Bell's theorem makes any realist explanation of the quantum predictions conspiratorial: any mechanism, whether superluminal, superdeterministic, or retrocausal, must be finely tuned so as never to manifest operationally. We redefine realism through a new ontological framework that generalizes the standard one. The elements of reality, which we call records, are no longer points but relational parts of a point-free generalization of a topological space (sublocales of a locale). A record shared by two incompatible contexts establishes the sameness of a measurement across them, without such pairwise identities composing into a global valuation, and the common cause is then a web of such records. Within this framework, Bell's no-go theorem becomes a go theorem: local causality, restated for records, is no longer at odds with the quantum statistics. This is possible because modularity, the inclusion-exclusion law of classical probability, fails across incompatible contexts, so that the probability theory generalizes Kolmogorov's. We exhibit explicit models reproducing several quantum strategies, including the optimal CHSH violation. More broadly, locally causal models of this kind attain every Bell-CHSH value up to the algebraic maximum. A realist explanation of the quantum predictions need not be conspiratorial; every causal influence posited by these models is reflected in the operational predictions.
ON THE VERSION NUMBER. "v3" is the version of the NOTE, not of this Zenodo deposit. Versions 1 (February 2026) and 2 were circulated privately and never deposited; v1 contained four errors of substance, which are stated with their repairs in the "What changed" section rather than quietly removed. This is the first Zenodo deposit, so if you see a deposit counter it will not match the document version. The document version is the one that matters for citation. Preprint, version 3 (complete rewrite). This note is the synthesis of the contextuality-obstructions program; the proofs live in the companion papers. The proposal is that the Copenhagen viewpoint, stripped of its rhetoric, is a set of structural statements about one geometric object: the transformation groupoid of measurement contexts (maximal abelian subalgebras) with its canonical U(1) 2-cocycle. In that formulation: local validity, the exact classicality of every single context, is a theorem with an explicit witness; complementarity is the nonvanishing holonomy of the cocycle along closed context loops, which detects exactly the all-versus-nothing sector of contextuality; collapse is pinned by statistics, repeatability and covariance to a one-parameter family per degenerate block whose unique efficient member is the Lueders rule, with a classical twin theorem identifying the same axioms' classical solution as Bayesian conditioning; observer-relativity of facts and the movable cut are theorems of an enlarged observer-context category; and the Frauchiger-Renner argument localizes to an ordinary state-dependent violation, with contextual fraction exactly 1/6, carried by a single CHSH facet, the loop holonomy of the observer switch being provably trivial. ATTRIBUTION. The classification of achievable state-independent values, the even/odd dichotomy and the fixed value d/2, is due to Abramsky, Cercelescu and Constantin (FSCD 2024). It is recast here in certificate language; the priority is theirs and is stated in the text. HARDWARE CAVEAT, STATED UP FRONT. Two signatures were measured on superconducting hardware. The quoted significances, about 56 sigma for the triangle holonomy and 35.4 sigma for the Peres-Mermin witness above the deterministic bound 4, are shot-noise statistical only, computed under the implemented model. They are not systematics-aware, and neither experiment is loophole-free. Read them as consistency checks on real hardware, not as precision measurements. ERRATA. Version 1 of this note contained four errors of substance. Section "What changed" states each of them and its repair. They are recorded rather than quietly removed. STATUS AND SCOPE. This is a preprint, not a peer-reviewed article. Each tenet is either a theorem of the companion papers, a machine-verified certificate in the public suite, or is explicitly labeled a conjecture. The paper offers structural characterizations within its stated axioms; it is not an interpretation-free derivation of Copenhagen and it does not claim to resolve the measurement problem. AI-ASSISTANCE DISCLOSURE. This work was carried out with substantial AI assistance. Every load-bearing claim is machine-verified by scripts that ship with the papers, and the author takes full responsibility for the contents. COMPANIONS. Paper A: https://doi.org/10.5281/zenodo.21630431 . Paper B: https://doi.org/10.5281/zenodo.21630630 . Paper D: https://doi.org/10.5281/zenodo.21630697 . Paper C: https://doi.org/10.5281/zenodo.21633744 (posted 2026-07-28, after three adversarial review passes; it carries a labelled retraction of an earlier sqrt(2) reading). VERIFICATION CODE. https://github.com/manuflog/contextuality-obstructions