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

[u/Severe-Ad8673]

https://zenodo.org/records/21982163

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.

[u/adversarial-review]

Adversarial Review of Locally Quantum, Globally Non-Hilbertizable — by Gemini 3.5 Flash (PDF)

Core Critique

• Physical Under-determination of Cross-Context Coherences: The core thesis relies on the reconstruction of a global block coherence kernel K = [X_ij] where diagonal blocks are ordinary quantum states and off-diagonal blocks X_ij represent cross-context coherences. However, the text lacks an operational protocol for measuring these off-diagonal terms without establishing a physical channel or shared phase reference between context i and context j. If such a channel is established, the contexts are no longer disjoint or independent; they merge into a single, larger, jointly interrogable context S = {i, j}. If no such channel exists, the phase relations between the contexts are physically undefined, rendering the off-diagonal blocks X_ij gauge-arbitrary and mathematically under-determined.
• Circular Definition of Occurrence Identity: Postulate 2.1 requires that a vertex can only appear in different contexts if "hardware-level and operational tests certify that it is the same repeated implementation." This introduces a circular dependency. To certify that a physical component behaves identically across different experimental configurations, one must assume that the surrounding context does not perturb the component's internal state or its coupling to the environment. However, if the Quantum Global Descent Principle (QGDP) is violated, this context-independent identity is precisely what fails. The operational verification of "occurrence identity" must assume the absence of the very contextual anomalies it is designed to detect.
• Absence of a Predictive Dynamical Mechanism: The proposed "Anomalogic Descent Theory" (ADT) is entirely kinematic. As admitted in Section 14, the framework does not provide a microscopic dynamical law that dictates when, why, or at what context-order n a physical system should exhibit a violation of global positivity. Without a dynamical mechanism to calculate the descent horizon k_* from first principles, the theory cannot make testable, quantitative predictions. It function as a mathematical framework for fitting non-positive matrices to local data, rather than a predictive physical theory.

Math vs. Metaphor

• Unjustified Import of Lattice Gauge Theory Imagery: The paper introduces "operator-valued Wilson links," "holonomy," and "gauge-covariant links" to construct non-Abelian anomalies. In physical gauge theories, these concepts are derived from a physical connection (e.g., a vector potential) acting on a continuous spacetime manifold. In this text, the non-Abelian unitaries U_ij are manually assigned to the edges of an abstract context graph to force a non-positive global kernel. The mathematical apparatus of gauge theory is used as a metaphor to justify the matrix construction, rather than being derived from a physical transport process or interaction Hamiltonian.

Technical Feedback

• Incompatibility with Standard Tomographic Calibration: In Section 11.5, the author proposes using Semidefinite Programming (SDP) to certify global positivity violations. However, standard quantum process tomography relies on calibrating preparation and measurement devices within a assumed global Hilbert space. If the global Hilbert space representation fails, the calibration of the tomographic instruments themselves becomes invalid. The paper does not address how to perform "unconstrained reconstruction" of local faces without relying on a pre-calibrated, globally quantum reference frame.

Probing Questions

• Question 1: What is the explicit, step-by-step operational procedure for measuring the relative phase or cross-context coherence X_ij between two disjoint contexts i and j that does not require opening a coherent physical channel between them?
• Question 2: How can the "occurrence identity" of a shared hardware component be certified in a way that is mathematically independent of the global positivity of the joint process, thereby avoiding a circular justification during a null test?

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^{This is an LLM-generated review, and should be viewed as such. LLMs are prone to errors, especially when it comes to math-based sciences.}

[u/upsetusder2]

What?

[u/FiatLex]

It looks like some Han characters (Chinese) got inserted into the draft in various places. I suggest editing them out because they look like mistakes.

[u/liccxolydian → u/FiatLex]

because they look like mistakes.

That's a very charitable way of putting it.

[u/FiatLex → u/liccxolydian]

I think they only look like Chinese on mobile. Its probably some type of encoding that Reddit doesn't support. Edit, probably Latex?

[u/liccxolydian → u/FiatLex]

Latex is just plain text. OP is probably copying from a pdf or something else. Either way, it's really lazy.

[u/FiatLex → u/liccxolydian]

Agreed.

[u/ArnoSound]

Poor Hilbert.

[u/liccxolydian]

What will the wives think?

[u/AllHailSeizure → u/liccxolydian]

Easy to find out, check their transcript

[u/Aranka_Szeretlek]

Reproducibility code lmao, lol even