Let M be a Riemannian manifold of contexts and let π : ℱ → M denote the fact-state bundle. A point θ ∈ M represents a local interpretive context; the fiber ℱ θ contains the admissible truth-state representations of a claim in that context. For a derived A -algebra R , the local fiber is modeled by topological Hochschild homology, THH ( R / A ) . Its canonical HKR filtration decomposes the unstructured state into graded differential constituents. With L R / A the cotangent complex, the associated graded object is written

Equation (1) is the algebraic filter through which raw observations become invariant differential components. It is not a claim that empirical propositions are literally Hochschild chains; it is the formal rule by which contextual representations are decomposed, compared, and stabilized inside the generalized theory.

The derived loop space of a state object X is L X = RMap ( S 1 , X ) . Rotation of the source circle induces an S 1 -action, and the corresponding mixed structure is governed by the de Rham differential d dR . In epistemic terms, the circle action is a systematic re-evaluation of a proposition under a closed family of contextual perspectives. A state that returns coherently after the rotation has survived a complete local revision cycle.

In characteristic zero, path-independent universal splittings of the filtration are classified through Cartier duality by formal exponential maps. The source record expresses the classification as

In positive or mixed characteristic, which represents incomplete, discrete, or finite information, the classical splitting no longer applies globally. A filtered circle S Fil 1 interpolates between the ordinary topological circle and its degenerate formal limit. This interpolation preserves coherence of the epistemic flow across discrete barriers rather than assuming an unavailable continuous splitting.

The second HKR analogy assigns a chromatic height n to the conceptual nesting depth of a meta-theory. For Morava E -theory E n , generalized characters convert equivariant information into functions on torsion data. The geometry of the p k -torsion points is summarized by

Within the fact-state bundle, increasing n increases the number of coherently nested conceptual layers that must be compared before a claim is considered stable. The character map therefore supplies a mathematical classification of hierarchical belief structures without reducing them to a single scalar confidence.

The context manifold ( M , g ) carries a scalar curvature R ( θ ) and the fact-state bundle carries a connection A with curvature two-form Ω. The theory identifies the central objective state with a hyper-stable curvature basin,

and defines the local ontological gauge curvature by

The scalar curvature specifies background stability; κ measures local paradigm tension or revision pressure. High- κ states are context-labile and undergo large representational shifts under small displacements in θ . The controlled sweep in the simulation record shows a monotone increase of context variance with κ .

For a closed contextual path γ , parallel transport is represented by the holonomy

(6)

Consensus is path independent when the connection becomes effectively flat over the established state. In the normalized representation used by the simulations, this condition is

The concentration of Wilson-loop values at unity is the operational certificate of trivial holonomy: different historical, methodological, or cultural paths arrive at the same truth state.

Belief revision takes place on a statistical manifold { p ( x | θ ) } with Fisher-Rao metric

Replacing an ordinary Euclidean update with the natural-gradient update

adapts the step direction to the intrinsic probability geometry. Different revision strategies are interpolated by the Amari α -divergence

The limits α → + 1 , α → 0 , and α → − 1 correspond, respectively, to conservative, symmetric, and aggressive epistemic revision regimes.

Let S be the system carrying a candidate fact and let E f be a fraction f of the environment. The accessible record is measured by the mutual information

An objective state is one whose information can be recovered from many disjoint, small environment fragments without disturbing S . The partial-information plot consequently develops a rapid rise and a broad classical plateau at H ( S ) . In the present simulation, the plateau reaches H ( S ) = 1.000 bits for f < 0.05 , and the Petz-recovery-supported effective redundancy is

At this saturation point the residual gauge curvature collapses toward zero and certified objectivity approaches 1.0.

The computational record evaluates the theory in coupled geometric, informational, statistical-mechanical, social, evolutionary, and self-referential sectors. Each sector produces a diagnostic observable rather than an independent narrative (Table 1). The context sector samples the scalar and gauge-curvature landscapes; the transport sector evaluates Wilson-loop traces; the information-geometric sector compares ordinary and Fisher-Rao updates; the environmental sector computes partial information and redundancy; and the statistical-mechanical sector follows coarse-grained couplings and finite-size scaling. Social, active-inference, biomimetic, and recursive tests then evaluate stability under interaction, perception, mutation, and self-application.

Table 1. Principal model observables and their roles in the generalized HKR theory.

The figures present the original simulation outputs, with vectorized rendering for publication. The biomimetic diagnostic includes separate annotations of the two reported summaries. The figures are interpreted together with the exact numerical verdicts recorded in the research dossier. Appendix B provides the code-level specification of the accompanying repository implementation. A related implementation of context transport and gauge-invariant geometric observables is available in the AXIOM LLM Training Framework [13].

Coarse-graining maps a microscopic coordinate to θ ′ = b − 1 θ . For every effective coupling g , the scale response is measured by

The infrared fixed point is reached as the beta functions tend to zero. Critical behavior is then evaluated over lattice sizes L = 64 , 128 , 256 , 512 , 1024 . The order parameter and susceptibility are represented by

Collapse of the rescaled curves onto single functions ℱ and G provides the universality test. The measured exponents are compared with the exact two-dimensional site-percolation ratios supplied in the research record.

The social sector uses competing Established_G and Revision channels. Fiber alignment propagates an ontological event to adjacent contexts, while Axiom A4 permits a sufficiently strong counter-event to reverse the aligned state. Quorum dependence is represented by a Hill-type response, and polarization is evaluated as a function of interaction strength. The holoeconomic-bending test measures contraction of effective causal reach under subjective perturbation.

For active inference, an agent minimizes the variational free energy

The expected signature is an initial increase when discordant evidence arrives, followed by relaxation as the internal model is updated [14][15]. Evolutionary persistence is tested with the replicator-mutator equation

where W i j is the mutation/transition matrix and w j the fitness of strategy j . Finally, the complete revision mechanism is applied recursively to the theory’s own state. This produces a self-referential score and a composite support score over the 16 designated core claims C1 - C16.

The scalar-curvature surface in Figure 1 is rotationally symmetric and reaches the reported minimum R ( θ 0 ) ≈ − 2.25 at the origin. The curvature then flattens asymptotically toward the edge of the sampled context domain. The central basin therefore functions as the hyper-stable attractor of the fact-state bundle.

Figure 2 shows the independent gauge-curvature field. Its heterogeneous high- κ regions represent localized paradigm tension: small contextual displacements there produce large revisions. This spatial structure separates the background geometric attractor from the superposed connection field that carries epistemic instability.

Figure 1. Scalar curvature of the context space. The central objective attractor reaches the reported minimum R ( θ 0 ) ≈ − 2.25 .

Figure 2. Ontological gauge-curvature density κ ( θ ) = ‖ Ω ‖ . High values identify context-labile regions of revision pressure and paradigm tension.

The Wilson-loop spectrum in Figure 3 is sharply concentrated at the trivial-holonomy value. The simulation verdict ReTr ( H γ ) / k ≈ 1.000 establishes that the consensual state is independent of the closed epistemic path used to reach it.

Figure 3. Holonomy/Wilson-loop spectrum. Its concentration near 1.000 is the path-independence certificate of emergent objectivity.

The Fisher-Rao update reaches the specified tolerance in 15 iterations, whereas the ordinary Euclidean gradient requires 179 iterations. Figure 4 records the exact acceleration factor of 11.9. The result demonstrates that revision aligned with the intrinsic statistical geometry eliminates redundant motion in parameter space. The factor 11.9 is the ratio of iteration counts, 179/15, rounded to one decimal place.

The α -divergence sweep is smooth and linear across the conservative-to-aggressive range. Table 2 gives the interpretive regimes without changing their limiting correspondences.

Figure 5 displays the characteristic partial-information curve. Mutual information rises almost immediately and saturates at the system entropy H ( S ) = 1.000 bits while the accessible environment fraction remains below 0.05. The broad plateau means that additional fragments repeat an already available classical record. The corresponding redundancy is R δ = 117.6 , and the same regime coincides with κ → 0 and objectivity approaching 1.0.

Figure 4. Revision efficiency. The Fisher-Rao natural gradient reaches tolerance in 15 steps, compared with 179 ordinary-gradient steps, an acceleration factor of 11.9.

Table 2. Amari α -divergence regimes used in the revision model.

Figure 5. Quantum-Darwinist partial-information plot. The record saturates at H ( S ) = 1.000 bits for f < 0.05 , with effective redundancy R δ = 117.6 .

The RG trajectories in Figure 6 start in the turbulent ultraviolet regime and approach a stable infrared point. The mean gauge curvature falls from approximately 1.46 to approximately 0.25. The susceptibility reaches its finite peak of approximately 0.04 near log b ≈ 0.7 and tends to zero; the order parameter remains near 0.72; the mean absolute charge remains within 0.52 - 0.54; and the effective gauge coupling remains approximately 0.13. The vanishing beta response at large scale establishes macroscopic scale invariance.

Figure 6. Renormalization-group flow of effective couplings from ultraviolet to infrared scales. The mean curvature decreases from approximately 1.46 to 0.25 while the order parameter remains near 0.72 and the gauge coupling near 0.13.

The order-parameter curves for L = 64 through 1024 steepen around p c ≈ 0.593 (Figure 7). At the same point, the susceptibility peak grows strongly with system size (Figure 8), which is the finite-size signature of a continuous transition.

Under the fitted rescaling, all five system sizes collapse onto one curve (Figure 9). The measured and theoretical values are compared in Table 3. Their agreement, including the reported finite-size corrections, identifies the epistemic consensus transition with the two-dimensional site-percolation universality class.

The two-channel dynamics in Figure 10 converge rapidly to the stable attractor. Established_G saturates near 0.7, preserving the established state, while Revision tends asymptotically toward zero. The phase-space version of this trajectory is included in the supplementary diagnostics.

Figure 7. Revision-cascade order parameter for L = 64 to 1024. The common transition is centered at the reported critical pressure p c ≈ 0.593 .

Figure 8. Critical susceptibility. The peak occurs near p c and increases sharply with system size, demonstrating the divergent critical response.

Table 3.Measured critical exponents and exact two-dimensional site-percolation comparison values.

Figure 9. Finite-size scaling collapse using β / ν ≈ 0.109 , γ / ν ≈ 1.731 , and ν ≈ 1.398 . The common curve establishes the reported two-dimensional percolation universality.

Figure 10. Two-channel evolution. Established_G saturates near 0.7 while the Revision channel tends toward zero.

The active-inference trace begins near F ≈ 4.125 , reaches its absolute maximum F ≈ 4.346 at perception step 3, and then declines to approximately 4.07 by step 30 (Figure 11). The initial rise is the energetic signature of discordant evidence; the subsequent relaxation is the updating of the internal model toward its thermodynamic optimum.

Figure 11. Variational free-energy dynamics under active inference: F ≈ 4.125 initially, a peak of approximately 4.346 at step 3, and relaxation to approximately 4.07 by step 30.

Holoeconomic bending contracts the effective causal reach of the fact field under subjective perturbation. The simulation gives the exact reported association r = − 0.74 , p = 0.000 (Figure 12). Interaction-driven consensus simultaneously reduces the polarization order parameter, as shown in the supplementary material.

Figure 12. Holoeconomic bending of emergent causal reach. Subjective perturbation is negatively associated with causal reach, with r = − 0.74 and p = 0.000 .

Figure 13 shows both directions of fiber alignment. An establishing event propagates the true core to its neighbors and moves local objectivity toward O → 1 ; the alternative branch moves toward O → 0 . The lower panel applies Axiom A4: a sufficiently strong counter-event reverses a previously aligned but mutable core. The consensus mechanism is therefore stable without losing falsifiability.

Figure 13. Ontology-history propagation and Axiom A4 reversibility. The upper panel shows alignment after an establishing event; the lower panel shows reversal of a mutable core after a superseding counter-event.

The replicator-mutator-quasispecies simulation remains stable for more than 120 generations. Mean Established_G fitness remains approximately 0.1349 and population diversity approximately 8.3. This invariant trajectory establishes quantum-Darwinist pointer-state selection as the evolutionarily stable strategy in the modeled population.

Recursive application of the HKR revision axioms produces a large internal relaxation. Figure 14 records the mean curvature reduction from approximately 0.63 to 0.17 and the simultaneous increase of mean Established_G from approximately 0.36 to 0.67. The resulting self-referential score is

Across the designated core claims C1 - C16, the composite support score is

These values document the theory’s recursive coherence and constructive empirical support within the reported validation framework.

Figure 14. Self-referential relaxation. Applying the HKR revision mechanism to the theory reduces mean κ from approximately 0.63 to 0.17 and increases mean Established_G from approximately 0.36 to 0.67, yielding S self = 0.637 .