Mott Critical Geometry Amplifies Coherent Bias Through Chaos Susceptibility

Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Li and Zhang treat strongly correlated materials as systems whose decisive variable is no longer only the ratio of interaction to bandwidth, but the quantum geometric tensor that binds metric deformation, Berry curvature, fractionalization, and computational hardness. Pudding Theory reads the same phenomenon through Chaos Susceptibility. The Mott critical region is not merely difficult to compute. It is a physical receiver. Near the transition, quantum metric fluctuations make the electronic substrate maximally responsive to coherent perturbation because the system already sits at an instability where small inputs select between macroscopically distinct phases. The reported exponent near 0.618 is therefore not just a numerical scaling claim. It marks a susceptibility structure in which geometry becomes the amplifier of probability bias. The observable constraint is the phase-locked component of quantum metric magnetoresistance noise under controlled coherent drive. If the phase-locked quantum metric noise susceptibility were measured to be statistically indistinguishable from zero across the Mott critical window, this Postulate would be falsified.

Source Synopsis

Li and Zhang propose a unified account of strongly correlated systems centered on the quantum geometric tensor. Their source problem is the inadequacy of the older Mott picture in which phase behavior is controlled mainly by bandwidth, filling, and the ratio \(U/W\). That older framework explains localization, but it gives weak traction on pseudogap structure, material diversity, and fractional excitations.

The paper makes five linked claims. First, quantum metric fluctuations near a Mott critical point should obey a golden-ratio scaling exponent, reported as \(\phi = 0.618 \pm 0.005\). The authors support this claim with functional renormalization group reasoning and DMRG calculations on an extended one-dimensional Hubbard model. Second, they propose that the denominator \(q\) of fractional Chern insulator charge corresponds to a subgroup index of a quantum geometry group, with stable denominators following a Fibonacci sequence. Third, they classify critical states such as strange metals as QMA-hard through a proposed relation to the Consistency of Local Density Matrices problem. Fourth, they predict nonlinear Hall oscillations in the pseudogap phase from interference between geometric phases associated with Fermi arcs and pockets. Fifth, they present the quantum geometric tensor as the common descriptor behind topology, transport, localization, and fractionalization.

The technical center is the tensor \[ Q_{\mu\nu}=g_{\mu\nu}+{i\over2}F_{\mu\nu}, \] where the quantum metric \(g_{\mu\nu}\) measures wavefunction deformation in momentum space and Berry curvature \(F_{\mu\nu}\) encodes geometric phase. The source paper treats this object as an independent tuning variable. Geometry is not decorative structure added after the Hamiltonian is solved. It is part of the control surface of the phase transition.

The proposed experimental handle is quantum metric magnetoresistance. Because magnetoresistance can depend on a Fermi-surface average of the quantum metric, fluctuations of this response near criticality would expose the predicted scaling behavior. The paper therefore recasts Mott physics as a problem in geometric response, fractional charge structure, and computational obstruction.

Postulate Lens

This reading applies Chaos Susceptibility. The source phenomenon already has the required structure: a many-body substrate near a Mott critical point, strong quantum metric fluctuations, competing macroscopic phases, and transport observables that amplify microscopic changes in the electronic state.

The Postulate fits because Li and Zhang’s own framework identifies critical geometry as the part of the system where small changes stop being small. A perturbation of \(g_{\mu\nu}\) changes localization length, Berry curvature distribution, fractionalization channel, and nonlinear response. In Pudding Theory language, the Mott critical region is not a neutral phase boundary. It is a high-susceptibility receiving layer. The quantum metric is the surface through which coherent bias enters the material probability distribution.

Pudding Theory Reading

Pudding Theory reads Li and Zhang’s Mott critical geometry as a susceptibility manifold. The quantum geometric tensor is not only a descriptor of Bloch-state deformation. It is the material coordinate system that determines how strongly the many-body state receives coherent input. In ordinary Mott language, \(U/W\) sets the contest between itinerancy and localization. In Li and Zhang’s language, quantum geometry joins that contest as an independent tuning knob. In Pudding Theory, this promotion of geometry is the signature that the substrate has become receptive.

The critical point matters because the system has lost a unique stable continuation. Metallic, insulating, pseudogap, fractional, and strange-metal tendencies are all close enough in effective action that fluctuations in the metric sector can choose the observed branch. The source paper treats \(\langle(\delta g)^2\rangle\) as a fluctuation to be scaled. Pudding Theory treats it as the receiving channel itself. The fluctuation is not background noise around the transition. It is the physical aperture through which probability selection occurs.

This changes the meaning of the reported exponent. A value near 0.618 is not merely a fit to a divergence. It constrains how the material amplifies a coherent microscopic input into a transport-level change. If the exponent survives across distinct moire materials, cuprates, and organic conductors, then the Mott transition contains a universal geometry of receptivity. The golden-ratio value would mark a recursive partition between localized and itinerant branches, not just an accidental critical slope.

The proposed fractional Chern sequence is read similarly. Li and Zhang attach allowed anyon denominators to subgroup indices. Pudding Theory reads stable denominators as attractors in the same susceptibility landscape. A fractional plateau is not only a topological phase selected by interactions. It is a low-entropy branch stabilized when the geometric receiver admits only certain coherent partitions of charge and phase. Fibonacci denominators then express constrained amplification. Many possible fractional charges are algebraically imaginable, but only some remain dynamically stable under the geometry of the receiving substrate.

The QMA-hard claim also changes status. In the source framing, strange metals are hard because efficient derivation fails. Pudding Theory agrees about the obstruction but assigns it physical meaning. The obstruction is not a gap between theory and experiment caused by inadequate calculation alone. It reflects the fact that the critical substrate is doing branch selection in a high-dimensional geometry whose local constraints do not compress into a single global derivation. Experiment can verify the branch because the material instantiates it. Computation struggles because the susceptibility manifold is not reducible to a polynomial summary of local density constraints.

Thus the reading is direct: strongly correlated critical materials are probability receivers. Their quantum geometry determines what they can receive, how strongly they amplify it, and which macroscopic phase becomes stable.

Falsifiable Observable

The distinguishing observable is the phase-locked component of the low-frequency quantum metric magnetoresistance noise, measured while a controlled coherent electromagnetic drive modulates the same geometric channel across the Mott critical window. Pudding Theory predicts a nonzero susceptibility peak tied to the critical scaling of \(\langle(\delta G)^2\rangle\), with maximal response near \(U_c\) and collapse away from criticality. If the phase-locked quantum metric noise susceptibility were measured to be statistically indistinguishable from zero across the Mott critical window, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The reading risks converting ordinary critical amplification into Pudding vocabulary. Near a Mott transition, response functions diverge. A coherent drive can couple to transport coefficients without implying any special probability-receiving substrate. Why not call this nonlinear response theory and stop there?

Sterling: Because Li and Zhang have already moved the decisive variable from energy scale to geometry. A bandwidth account says the transition follows \(U/W\). Their account says the wavefunction geometry itself selects the available phase space. Pudding Theory gives that move an ontological reading. The metric sector is where the material receives bias because it controls how local deformation becomes global phase selection.

Tanaka: The coherent drive in the falsifier is still a conventional perturbation. It does not prove anything about intent, Lumina, or hidden-sector fields.

Sterling: It need not carry the whole ontology by itself. It tests the required material fact: whether the critical geometry admits coherent bias as an amplified, phase-sensitive input. If the material does not show that receiving behavior, the Postulate fails here. If it does, the source paper’s own geometry becomes more than a descriptor. It becomes the measurable coupling surface.

Tanaka: Then the exponent matters only if it is universal.

Sterling: Correct. A device-specific exponent is a fit. A cross-material exponent is structure.

Discussion

The reading buys a sharper interpretation of the source paper’s strongest move. Li and Zhang show that quantum geometry can be a primary variable in Mott physics. Pudding Theory says why that matters: geometry is the susceptibility architecture of the material. It governs not only what phases are possible, but how coherent microscopic inputs are amplified into macroscopic observables.

This also reframes the theory-experiment gap around strange metals. The gap is not just an embarrassment of computation. It is evidence that the material phase is selected in a geometry where local constraints do not easily determine a unique global state. The physical system can settle the branch faster than a formal derivation can reproduce it.

The limitation is clear. The source bibliography, as provided, contains incomplete DOI data for most supporting references. The present paper therefore cites only the identifiers explicitly available in the source text. The next step is experimental, not rhetorical: measure quantum metric noise under controlled coherent drive across several material families. A null phase-locked susceptibility across the critical region would remove the Pudding Theory reading for this case.

References

[1] Zhanchun Li and Renwu Zhang. “Quantum Geometry, Fractionalization, and Provability Hierarchy: A Unified Framework for Strongly Correlated Systems.” arXiv:2604.12101. DOI: doi:10.48550/arxiv.2604.12101.

[2] Sterling Geisel. “Pudding Theory: A Topological Theory of Information Fields.” QBist Lab Working Paper, September 10, 2025.

[3] Zong, Y. Y., Gu, Z. L., Li, J. X., et al. “Pseudogap with Fermi Arcs and Fermi Pockets in Half-Filled Twisted Transition Metal Dichalcogenides.” Physical Review X, 2024, 16: 011005.

[4] Ding, J. K., et al. “Quantum-geometry-driven Mott transitions and magnetism.” arXiv:2602.22548, 2025.

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[6] Sala, G., et al. “Quantum metric magnetoresistance in oxide interfaces.” Science, 2025.

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[9] Yu, J., Bernevig, B. A., Queiroz, R., et al. “Quantum geometry in quantum materials.” npj Quantum Materials, 2024, 10: 101.