Can a Dynamic Internal Field Govern a Transformer's Cognition? Certifiability, not Superiority, in Homeostatic Compute Control

Francisco M. Arrabal-Campos; Ignacio Fernandez; Francisco G. Montoya; Alfredo Alcayde

cs.AIAug 25, 2026

Can a Dynamic Internal Field Govern a Transformer's Cognition? Certifiability, not Superiority, in Homeostatic Compute Control

Authors: Francisco M. Arrabal-Campos, Ignacio Fernandez, Francisco G. Montoya, Alfredo Alcayde

Organizations: Department of Chemistry and Physics, Research Centre CIAIMBITAL Department of Engineering, Research Centre CIAIMBITAL University of Almería, 04120 Almería, Spain · Department of Engineering, Research Centre CIAIMBITAL University of Almería, 04120 Almería, Spain

Abstract

An intelligent system does not merely reason: it governs its own reasoning - how much to compute, when to stop, which module to activate. Can that role be played by a dynamic internal field - a low-dimensional homeostatic state with explicit physics and certified stability - that modulates cognition without performing it? Ours is a field on the module graph governed by a family of PDEs on the graph Laplacian, advancing with an adaptive-depth reasoner. We certify the stability of the integrator of the whole family - an integrator certificate, not a closed-loop one. New, and proved here: a discrete Schur-Cohn criterion for Verlet with velocity coupling, necessary and sufficient per latent root, with no commutation hypothesis. The answer is threefold: substance no, structure only in part, certifiability yes. The type of the field's physics is irrelevant for accuracy: wave, diffusion, gated mixtures and a 2D Navier-Stokes substrate tie. A twenty-seed preregistered deconfounding campaign bounds the structural claim: at equalized caps the second-order effect is strong in one family (+0.087 [+0.042, +0.132], t=4.0) but is not detected in the other (+0.014 [-0.013, +0.040], n.s.), so part of the original contrast was capacity, not order; and a matched-interface GRU is indistinguishable in the first and nominally exceeds the field in the second (-0.035 [-0.067, -0.002]). What distinguishes the field is not capability but that its one-step operator admits an exact runtime stability check - a difference of kind, not of existence: learned recurrences carry certificates too, sufficient and conservative ones. A kill-gate with a positive control finds no evidence for the field as evidence accumulator (Delta AUC +0.0007 [-0.0065, +0.0079] vs a 0.03 threshold). A dynamic internal field is a viable, certifiable compute governor, but not an enhancer of cognition: it modulates, it does not think.

Figures & tables

Figure 1: The Homeostatic Background Processor . An internal-state field $h$ (top) lives over the model’s module graph —backbone blocks, the adaptive-depth reasoner and the working memory (bottom)— coupled by the graph Laplacian ( $-c^{2}\nabla^{2}h$ ). Each module exposes an interoception signal $s_{i}$ (bottom-up) and receives modulation $m_{i}$ (top-down: halting threshold, block gain, memory gates; the interface also exposes a router-bias head, not consumed by the experiments reported here). The field evolves under a damped-wave PDE (one instance of the family of Sec. 3 ), one tick per reasoner iteration.

Figure 2: Intrinsic dynamics of the field (simulation of the solver of Sec. 3 ). An interoceptive impulse at node 1 induces a damped oscillation of the VEI (left, $\|h_{i}-h^{\ast}\|$ per node) that propagates to neighboring modules through the Laplacian coupling (right, node ${\times}$ tick map). It is the second-order regime that the analogy with coupled spins in NMR describes (a local impulse spreads and precesses); the Verlet scheme integrates it stably under the certificate of Prop. 1 .

	$S_{5}$ adjacent			$S_{5}$ 5-cycle $+$ transp.
Model	short	medium	long	short	medium	long
vanilla	$0.984$	$0.670$	$0.147$	$0.974$	$0.976$	$0.581$
gating	$0.985$	$0.761$	$0.218$	$0.974$	$0.980$	$0.725$
gating_wm	$0.983$	$0.747$	$0.197$	$0.974$	$0.982$	$0.714$
hbp_first	$0.987$	$0.776$	$0.245$	$0.974$	$0.985$	$0.730$
hbp_full	$0.986$	$0.771$	$0.248$	$0.974$	$0.984$	$0.741$

Table 1: Accuracy by difficulty (in-distribution, mean over

n{=}3

seeds; per-seed values and paired contrasts in the repository).

	gating_wm (WM)	hbp_first (diffusive)	hbp_full (wave)
$S_{5}$ adjacent	$0.002\pm 0.087$	$0.036\pm 0.090$	$0.143\pm 0.134$
vs WM	–	$t{=}0.77$ , $p{=}0.46$	$\mathbf{t{=}2.68}$ , $p{=}0.025$
$S_{5}$ 5-cycle $+$ transp.	$0.213\pm 0.068$	$0.185\pm 0.077$	$0.268\pm 0.085$
vs WM	–	$t{=}{-}1.42$ , $p{=}0.19$	$\mathbf{t{=}2.62}$ , $p{=}0.028$

Table 2: Pure-OOD compute adaptivity, protocol v3 (learnable physics),

n{=}10

fresh seeds (

\pm

std). Fisher-

z

paired contrasts against the gating_wm control. Boldface marks uncorrected

p<0.05

: neither contrast survives Holm-4 individually, and the order contrast these numbers support is bounded by the deconfounding of Sec. 5.4 .

Figure 3: OOD compute adaptivity by variant and generator set ( $n{=}10$ , $\pm$ SEM), protocol v3 (learnable physics). The hbp_first arm runs at its original caps: the twenty-seed replication of Sec. 5.4 shows that the 5-cycle gap displayed here is confounded with capacity and vanishes once the caps are equalized.

Figure 4: The deconfounding campaign of this section (twenty fresh seeds, $20$ – $39$ ). Left: out-of-distribution compute adaptivity per arm and generator family; the first-order arm now runs at caps equalized to the second-order ones, which the unconditionally contractive implicit kernel of Proposition 2 makes safe. Right: the same evidence as seed-paired differences from the second-order field, which is where the verdict actually lives. The order effect clears zero only in adjacent ; the matched-interface GRU governor ties the field there and its interval falls entirely below zero in cycle_transp . Error bars are $\pm$ SEM (left) and $95\%$ CI (right).

Adaptive Computation

Adaptive computation allocates model depth, tokens, attention, or sensing effort according to input difficulty and resource constraints.

See all 101 papers

Aug 14, 2026

Algorithmic Information Dynamics of Learning: A Certified, Differentiable Complexity Controller for Grokking

Algorithmic Information Dynamics (AID) studies systems by perturbing them and measuring changes in algorithmic complexity, but its usual estimator, the Block Decomposition Method, is piecewise constant, restricting the calculus to finite differences. We use

K^{\mathrm{CDM}}_{\mathrm{s}F}

, a certified, differentiable estimator, to bring the calculus into learning dynamics: grokking, where a complexity order parameter is known but has not been made to act. As a transient loss kick, the estimator becomes a controller that accelerates grokking in Levin's description-length--versus-time sense, within a data-dependent Occam boundary whose finite-size trend,

f_c\sim\ln p/p

, is consistent with a coupon-collector interpretation. Ablations show that a complexity gate matches a train-loss gate in rescuing failing seeds with

27\%

less intervention; among the tested signals, only map complexity marks the transition's completion; the certified prior and the per-parameter

\nabla K

attribution are both fungible (a uniform-prior sensor makes bit-identical gate decisions, and random supports match

\nabla K

-selected ones above a sparsity threshold); and direct field perturbation shows a nucleation-like response to the Occam field (no linear regime is resolved over the probed amplitudes, so these measurements do not justify a fluctuation--dissipation surrogate), with a finite-field response growing by orders of magnitude toward the phase-transition. These measurements account for the empirically tuned staircase: bang--bang pulses, stall-fired and released on yield, whose iteration plausibly builds the response it exploits. The kick transfers to sparse parity and to a transformer; a sustained weight-space loss fails. The algorithmic estimator's distinct contribution is timing (when to fire and when to release), not attribution.

Luan Ozelim, Abicumaran Uthamacumaran, Hector Zenil

Oxford Immune Algorithmics, Oxford University Innovation & London Institute for Healthcare Engineering, U.K · Department of Biomedical Computing, School of Biomedical Engineering and Imaging Sciences & King’s Institute for AI, King’s College London, U.K

Neural Network Training Dynamics Grokking Kolmogorov Complexity

Ask PDF

Aug 23, 2026

Where Cognition Lives: Dissecting Emergent from Computed Function in a Minimal Complete Cognitive Architecture

A cognitive architecture is more than the module that reasons: it must also decide how long to think and what deserves the effort. We built a minimal but complete system - a recurrent reasoner with adaptive halting, a homeostatic control field, and a value module - and asked of each part: does this function emerge from gradient descent, or must it be computed? Competence emerges. Stopping appears to emerge too, and to be worth more than everything decidable in advance, but that appearance is instrumentation: payoff at matched mean compute climbs from 0.467 (uniform) through 0.546 (difficulty) to 0.698 (ex-ante value), and the further climb to 0.921 (posterior self-observation) does not survive audit. PonderNet-style halting returns a halting-weighted mixture of hidden states while forced-depth baselines return one, and the language head is trained on the mixture alone; equalizing the readout annihilates the apparent advantage of native execution (residual +0.000 [0.000, 0.000]). Value does not emerge: trained couplings capture zero of a payoff an explicit allocator captures completely (+0.151, routing correlation +0.79), so the second-order decisions that pay must be computed, at least where value is orthogonal to content, as here by construction. On a frozen LLM actuator the same instruments show self-consistency voting to be a measured bound (+0.0236 [+0.0150, +0.0326]) and inter-sample agreement nearly worthless as a stopping signal, its mass concentrating on wrong answers. Every null we assert carries a mechanism and a positive control, and the protocol is part of the contribution. Executing our own falsifiable prediction, value under commitment pays +0.1312 [+0.1124, +0.1502] in a cliff-cost family, some seven times the smooth-family estimate - not because the cliff shifts information ex ante, but because it multiplies the attainable range fivefold (5.1x [3.4, 8.2]).

Francisco M. Arrabal-Campos, Francisco G. Montoya, Alfredo Alcayde +1

Department of Chemistry and Physics, Research Centre CIAIMBITAL Department of Engineering, Research Centre CIAIMBITAL University of Almería, 04120 Almería, Spain

Cognitive Architectures for AI Agents Computational Cognitive Modeling Adaptive Computation Inference-Time Compute Allocation Metacognition in Language Models

Ask PDF

Apr 30, 2026

Mechanized Foundations of Structural Governance: Machine-Checked Proofs for Governed Intelligence

We present five results in the theory of structural governance for cognitive workflow systems. Three are mechanized in Coq 8.19 using the Interaction Trees library with parameterized coinduction; two are proved on paper with explicit reductions. The Coinductive Safety Predicate (gov_safe) is a coinductive property that captures governance safety for infinite program behaviors, indexed by a boolean permission flag that is provably false for ungoverned I/O and true for governed interpretations (mechanized). The Governance Invariance Theorem establishes that governance is uniform across the meta-recursive tower: governance at level n+1 reduces to governance at level n by definitional equality of the type (mechanized). The Sufficiency Theorem proves that four atomic primitives (code, reason, memory, call) are expressively complete for any discrete intelligent system, formalized as compositional closure of a Kleisli category (mechanized). The Alternating Normal Form provides a canonical decomposition of any machine into alternating code and effect layers, with a confluent rewriting system (paper proof). The Necessity Theorem proves via explicit reduction to Rice's theorem that an architecturally opaque component (the reason primitive) is mathematically necessary for problems requiring semantic judgment (paper proof). A sixth contribution connects the abstract model to the deployed runtime: the Verified Interpreter Specification formalizes the BEAM runtime's trust, capability, and hash chain logic in Coq, then tests the running system against this specification using property-based testing with over 70,000 randomly generated directive sequences and zero disagreements. The mechanization comprises approximately 12,000 lines across 36 modules with 454 theorems and zero admitted lemmas.

Alan L. McCann

Mashin, Inc.

Property-Based Testing Formal Verification AI Agent Governance

Ask PDF

Abstract

Adaptive Computation

Explore similar work

Algorithmic Information Dynamics of Learning: A Certified, Differentiable Complexity Controller for Grokking

Where Cognition Lives: Dissecting Emergent from Computed Function in a Minimal Complete Cognitive Architecture

Mechanized Foundations of Structural Governance: Machine-Checked Proofs for Governed Intelligence