Local Gauge Compatibility Should Show Enhanced Relaxation in Chaotic Transport Graphs

Sterling Geisel, QBist Lab

Abstract

Auti, Daiguji, and Tanaka give a constructive formulation of classical dynamics in which state variables and transport geometry coevolve through local compatibility restoration. The paper removes a prescribed global clock, fixed background geometry, and predefined evolution operator from the starting point. Dynamics is instead built from covariant mismatch between neighboring local states. This working paper applies one Pudding Theory Postulate: Chaos Susceptibility. The reason is narrow. The source framework makes finite-rate relaxation of local incompatibility primary, and it treats effective equations as coarse-grained limits of asynchronous local updates. Pudding Theory predicts that, for equal initial mismatch and equal relaxation protocol, systems with larger positive Lyapunov exponents should exhibit a small but repeatable excess in compatibility-restoration rate when exposed to a coherent informational input. The decisive observable is the slope of residual decay, measured through the quadratic incompatibility functional. This is not a claim about new energy input. It is a claim about biased relaxation pathways.

Source Synopsis

Auti, Daiguji, and Tanaka propose a local gauge-covariant formulation of classical dynamics. The source paper begins from a dissatisfaction with standard descriptions that assume a state, a fixed comparison geometry, a global time parameter, and a closed evolution equation. In those descriptions, the rule for comparing neighboring states is specified before the dynamics begins. The authors instead treat that comparison rule as part of the evolving system.

The primitive state is a set of local conserved densities \(U_i\). Neighboring sites are related by transport operators \(T_{ij}\), which map a neighbor's state into the local frame. A local frame transformation changes \(U_i\) and \(T_{ij}\), but the covariant mismatch \(T_{ij}U_j-U_i\) remains the correct object for measuring compatibility. The authors define an oriented local incompatibility \[ r_{ij}=\phi(T_{ij}U_j-U_i), \] then accumulate those terms into a node residual \(R_i\). They construct a gauge-invariant scalar mismatch functional \[ L[U,W]=\frac{1}{2}\sum_i \langle R_i,R_i\rangle+\frac{\lambda}{2}\sum_{(i,j)}\langle W_{ij},W_{ij}\rangle . \] This functional is not presented as thermodynamic free energy. It measures internal description consistency.

The dynamics is finite-rate relaxation of \(L\), with coupled updates of both \(U_i\) and \(W_{ij}\). The connection variable \(W_{ij}\) adapts to residual mismatch through a local error times neighbor-state term plus a decay term. The state update also has local and neighbor-backpropagated terms. Since updates are asynchronous, global time appears only after coarse graining many local events. In limiting regimes, the authors state that diffusion, incompressible Navier-Stokes, and the Ampere-Maxwell relation arise as effective continuum descriptions.

The source paper's central claim is constructive. Classical equations emerge from local relational constraints once transport geometry stabilizes.

Postulate Lens

The applicable Pudding Theory frame is Chaos Susceptibility: highly chaotic systems are most susceptible to small coherent inputs.

The fit is direct. The source paper makes residual relaxation, finite-rate structural adaptation, and asynchronous local updates primary. These are precisely the conditions under which Pudding Theory expects a weak informational bias to become measurable. A stable system damps small perturbations. A chaotic system amplifies them through repeated local update events. In Auti, Daiguji, and Tanaka's notation, the relevant substrate is not an isolated coordinate trajectory but the coupled pair \((U,W)\), whose compatibility restoration depends on local residuals and connection adaptation.

The Pudding Theory claim is therefore limited. It does not replace the gauge-covariant framework. It adds a small susceptibility term to the relaxation statistics when the system has a positive maximal Lyapunov exponent and when an external coherent informational input is phase-aligned with a target reduction of \(L\). The observable is not the final state alone. It is the rate and directionality of residual decay under matched initial conditions.

Pudding Theory Prediction

Pudding Theory predicts a measurable deviation in the relaxation of the incompatibility functional \(L[U,W]\) for chaotic transport graphs exposed to coherent informational input.

Consider an experimental or numerical ensemble of graph systems constructed under the source paper's rules. Each run begins with the same distribution of node states \(U_i\), the same distribution of transport parameters \(W_{ij}\), and the same initial value \(L_0\). The relaxation protocol is fixed. The only controlled variable is a coherent input intended to favor one compatibility-restoration channel over another. In the null model, after conditioning on \(L_0\), graph topology, update rate, and noise amplitude, the decay rate \(-d\log L/dt\) should be independent of such input.

Chaos Susceptibility predicts a different scaling. Let \(\lambda_{\max}\) be the maximal Lyapunov exponent of the coupled residual-relaxation dynamics. For \(\lambda_{\max}\leq 0\), the input should produce no statistically resolved shift beyond ordinary stochastic variation. For \(\lambda_{\max}>0\), the shift should grow approximately with the same instability that amplifies local perturbations: \[ \Delta k \equiv k_{\mathrm{coh}}-k_{\mathrm{ctrl}} \propto \epsilon_{\mathrm{eff}} e^{\lambda_{\max}\tau}, \] over the finite observation interval \(\tau\), until saturation or topology change intervenes. Here \(k\) is the fitted exponential decay constant of \(L\). The sign is specific: aligned coherent input should increase the rate of compatibility restoration toward the targeted branch. Misaligned input should either reduce that branch probability or redistribute relaxation into competing branches.

This prediction is falsifiable because it is not a generic expectation of faster relaxation. It requires a Lyapunov-ordered effect, a phase dependence relative to the intended branch, and disappearance in nonchaotic controls. It also requires the measured effect to appear in the residual statistics, not only in an informal judgment of macroscopic pattern.

A good test would use multiple graph ensembles: stable, weakly chaotic, and strongly chaotic. All would share the same mean \(L_0\), comparable degree distribution, identical asynchronous update rules, and blinded labeling. The dependent variable would be the ensemble difference in residual-decay slope. Pudding Theory predicts monotonic increase of the anomalous component with \(\lambda_{\max}\).

Falsifiable Observable

The distinguishing observable is \(\Delta k(\lambda_{\max})\), the coherent-input minus control difference in the fitted decay constant of the gauge-invariant incompatibility functional \(L[U,W]\), measured across matched graph ensembles. If \(\Delta k(\lambda_{\max})\) were measured to be statistically indistinguishable from zero for all \(\lambda_{\max}>0\) at power 0.95 with confidence intervals excluding \(|\Delta k|/k_{\mathrm{ctrl}}>10^{-4}\), this Postulate would be falsified.

Editorial Dialogue

Tanaka: The source paper already supplies the mechanism. Residuals drive local updates. Transport adapts. Continuum equations emerge after coarse graining. Adding Pudding Theory risks attaching a second explanation to a complete construction.

Sterling: The construction is complete as a kinematic and dynamical framework. The Pudding claim is not that residual relaxation needs a hidden cause. It is that the relaxation statistics of unstable systems may be biased by coherent informational input. That is an extra empirical term, not a replacement.

Tanaka: But the functional \(L\) is internal to the model. It is a consistency measure. It does not imply receptivity to observers.

Sterling: Correct. The source paper does not imply the Postulate. It supplies a clean observable. Pudding Theory needs variables that are not psychological reports. Here the variable is \(L[U,W]\). The test is whether its decay rate shifts in a Lyapunov-ordered way under a controlled coherent input.

Tanaka: A null result would be simple.

Sterling: Yes. That is why this source is useful. If chaotic systems show no excess residual-relaxation bias under adequate controls, the application fails.

Discussion

This application is narrow by design. The source paper is not about consciousness. It is about local gauge-covariant construction of dynamics from compatibility restoration. The Pudding Theory use is justified only because the source framework makes local mismatch and finite-rate adaptation measurable.

The main limitation is operational definition of coherent input. A weak protocol will only test noise. The input must have timing, target branch specification, blinding, and preregistered analysis. The second limitation is separating true Lyapunov scaling from ordinary numerical stiffness. Stable and chaotic ensembles must be matched for graph size, update frequency, and initial residual spectrum.

A positive result would not validate all of Pudding Theory. It would support one restricted claim: chaotic compatibility-restoration dynamics are unusually sensitive to small coherent informational bias. A negative result with adequate power would damage that claim directly. The source framework is valuable because it gives the test a quantitative object, \(L[U,W]\), and a natural instability parameter, \(\lambda_{\max}\).

References

1. Gunjan Auti, Hirofumi Daiguji, and Gouhei Tanaka. “A Local Gauge-Covariant Formulation of Classical Dynamics.” arXiv:2604.14190, 2026. DOI: doi:10.48550/arxiv.2604.14190.

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

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