Bounded Bayesian belief

### PDF page 21 entirely classical and information -theoretic, and they neither require nor preclude a quantum substrate. 6.5 Limitations First, the binary belief simplification is illustrative but restrictive. The full empirical content of the framework requires elicitation of both confidence (mean -like) and uncertainty (precision -like) information, which the binary case collapses. Second, the bridge hypothesis remains the framework's narrowest claim and most exposed point. Even with the operationalisation and controls in §4.1, gamma -band power's multiple functional correlates mean that a positive result is open to alternative interpretations until orthogonal controls are run. First experiments should target the single-agent bridge before the dyadic test. Third, the choice of coupling potential is principled but not forced. A negative result on the dyadic sinh prediction would not refute the rapidity framework; it would refute the specific coupling. Systematic model comparison — sinh vs tanh vs linear vs polynomial — is the more honest experimental design than a single confirmatory test of the sinh form. Fourth, alternative metabolic barriers (polynomial, exponential, biophysically detailed cost functions) yield qualitatively different saturation laws and merit dedicated investigation. The log barrier is the simplest convex divergent choice; whether the cortex implements it specifically, or only approximates it under typical operating regimes, is an open question. 7. Conclusion A finite agent updating bounded beliefs is best described in the coordinate where evidence adds and the metabolic envelope vanishes — the log -odds rapidity. By the bound ed composition theorem of [3], this coordinate exists uniquely, the boundary is unattainable from any finite composition, and the interior is the agent's mathematically forced state space. A pair of finite agents under joint attention is plausibly describe d by a coupling potential whose gradient produces a sinh alignment force, generating a specific nonlinear signature in inter-brain synchrony data. These claims are modest, mathematically clean, and testable. The framework describes how something bounded us es geometry, group structure, and environmental feedback to evolve coherently in a world it cannot fully resolve. It does not collapse consciousness into mathematics. It identifies one of the structures within which conscious belief operates, and it asks whether that structure leaves the empirical signature we predict. Appendix A. Variational Derivation of the Coupled Dynamics ### PDF page 22 Let λA, λB be the rapidity coordinates of two interacting agents. We define the joint information potential of the dyad as V(λA, λB) = −ΦAλA − ΦBλB + β cosh(λA − λB). Standard first-order gradient flow on V, consistent with the overdamped formulation of free-energy minimisation, yields directly dλA /dt = −∂V/∂λA = ΦA − β sinh(λA − λB), dλB /dt = −∂V/∂λB = ΦB + β sinh(λA − λB). This delivers the equations of the main text without introducing and eliminating inertial terms. Stability. Near Δλ = 0, sinh(Δλ) ≈ Δλ, and the relative dynamics linearise to d( Δλ)/dt ≈ −2βΔλ, exponential convergence with rate 2 β. Globally, the Lyapunov function U(Δλ) = cosh(Δλ) − 1 is non-negative, vanishes only at Δλ = 0, and satisfies d U/dt = −2 β sinh²(Δλ) ≤ 0 under zero differential drive. The aligned state is therefore globally attracting whenever ΦA = ΦB. Lyapunov stability of this continuous flow and the discrete Banach contraction of [3, Corollary 7.1] are different mathematical objects; the two results are structurally aligned in spirit but not in the technical sense that one would be a special case of the other. Appendix B. On Dimensionality and Curvature For the binary case, the agent's state space is the open interval (−1, 1), a one -dimensional manifold. A one-dimensional Riemannian manifold has no intrinsic curvature: the Riemann tensor vanishes identically, and any tw o metrics on it are related by a coordinate change. Statements about “the hyperbolic geometry of binary belief” should be understood as statements about the natural log -odds coordinate equipped with the metabolically -augmented Poincaré metric, not about intrinsic curvature of the underlying space. Genuine intrinsic negative curvature requires at least two dimensions, and arises naturally when the agent's belief is over a continuous environmental variable parameterised by both location and precision. The Gaussian location-scale family produces the Poincaré half-plane metric (§6.3) with constant negative curvature −1. The Bernoulli case in this paper inherits the form of its higher - dimensional parent by structural analogy on the rapidity coordinate. The classification theorem of [3, Corollary 3.4] guarantees that, up to scale and orientation, this 1D structure is unique: every smooth, associative, strictly monotone composition on a bounded interval is the same structure, with Einstein velocity addition and Bayesian binary belief composition both being instances of it. Appendix C. Continuous Belief Extension (Sketch) ---

Rapidity Coordinates for Bounded Belief A Resource-Constrained Predictive Processing Framework with a Falsifiable Inter-Brain Synchrony Test Daniel John Murray Abstract We develop an information -geometric framework for belief updating in resource -bounded predictive agents, building on a recent structural result that any smooth, associative, strictly monotone binary operation on an open bounded interval admits a unique lin earising rapidity coordinate, with boundary unattainable from any finite composition of interior elements [Murray, 2026]. We apply this to binary predictive belief: the bias variable s = 2P − 1 ∈ (−1, 1) instantiates a bounded compositional structure under Bayesian evidence accumulation, and the log -odds coordinate λ = artanh(s) is its rapidity. Three independent motivations identify this coordinate: Bayesian additivity, symmetry around indifference, and compatibility with a metabolically - augmented information metric that takes the form of the one-dimensional Poincaré line element. A logarithmic metabolic barrier yields logistic dynamics ds/dt = (1 − s²)·Φ(t), equivalently dλ/dt = Φ(t), where Φ is an information -accumulation rate hypothesised to be linearly related to phase- locked gamma -band power. For two agents engaged in joint attention we postulate — by structural analogy with the two-dimensional hyperbolic case rather than by derivation in the one- dimensional setting — a cosh coupling potential whose gradient yields a sinh alignment force on the difference of rapidities. Under this postulated coupling, dual -EEG data should exhibit a pure estimate of the couplin g strength β, and distinguishable by Bayesian model comparison from linear, polynomial, and saturating alternatives. We are explicit throughout about what is derived versus what is hypothesised, give three pre -registered falsification criteria for the brid ge hypothesis and a decision tree mapping each empirical failure mode to its interpretation, and reproduce the foundational theorems in a self-contained appendix so the paper can be evaluated independently of [Murray, 2026]. Keywords: bounded composition; rapidity coordinate; predictive processing; free-energy principle; information geometry; hyperbolic embedding; inter-brain synchrony; Bayesian model comparison. Abstract

Rapidity Coordinates for Bounded Belief A Resource-Constrained Predictive Processing Framework with a Falsifiable Inter-Brain Synchrony Test Daniel John Murray Independent Researcher, Melbourne, Australia ORCID: 0009-0005-1794-5945 Abstract We develop an information -geometric framework for belief updating in resource -bounded predictive agents, building on a recent structural result that any smooth, associative, strictly monotone binary operation on an open bounded interval admits a unique linearising rapidity coordinate, and that the boundary of such an interval is unattainable from any finite composition of interior elements [3]. We apply this structural fact to the specific case of binary predictive belief, where Bayesian evidence accumulation on the bias variable s in (−1, 1) instantiates the bounded compositional structure and the log -odds coordinate λ = artanh( s) is its rapidity. Three independently motivated properties of this coordinate make it the right object of analysis: it is the unique coordinate in which Bayesian evidence accumulates additively; it is the proper -length consistent with a logarithmic resource cost. Together these motivate a variational framework in which a single agent's belief trajectory is described by s(t) = tanh(∫ Φ(t′) dt′), where Φ is an information-accumulation rate hypothesised to be linearly related to phase -locked gamma-band power. For two agents engaged in joint attention, we postulate — by structural analogy with the two-dimensional hyperbolic case rather than by deriva tion in the one -dimensional setting — a cosh coupling potential whose gradient yields a sinh alignment force on the difference of rapidities. Under this postulated coupling the framework produces a sharply testable signature in dual-EEG data: in the absenc e of differential stimulus drive, the rate of change of the relative rapidity between two observers should follow a pure negative sinh of the relative rapidity itself. The prediction is parameter-free in the operational sense that the coupling strength β is estimated on independent calibration trials and fixed for the test, not refit per condition; it is distinguishable by Bayesian model comparison from linear, polynomial, or saturating alternatives. We are explicit throughout about what the framework deriv es versus what it hypothesises, and we discuss the natural extension to continuous beliefs where a genuine two-dimensional hyperbolic plane arises from the Gaussian location -scale family. A self -contained statement and proof sketches of the Manuscript

agent is, accordingly, a theorem about the algebraic structure of composition rather than a softer claim about resource constraints. The presen t paper applies this structural foundation to predictive belief and derives the dynamics that follow when the natural flow is driven by an evidence rate. This paper makes three contributions on that foundation. First, we identify three independent motivations for the rapidity coordinate in the predictive-processing setting — Bayesian additivity, symmetry around indifference, and compatibility with a metabolically -augmented information metric. Second, we show that a logarithmic metabolic barrier transforms t he information metric on the bias interval into the one -dimensional Poincaré metric, in which the rapidity is the proper

length, and that the resulting dynamics in the bias coordinate are exactly the one-parameter flow predicted by the bounded composition theorem, driven at evidence rate Φ. Third, we postulate a specific coupling potential between two agents whose gradient produces a sinh alignment force, yielding a falsifiable inter-brain prediction. We are careful, throughout, about what is derived and wh at is hypothesised. The existence and uniqueness of the rapidity coordinate, the boundary unattainability, and the structural classification are mathematical theorems established in [3]. The metabolic barrier's role in producing the Poincaré metric is a th ermodynamic argument from a chosen convex cost; the barrier itself is a modelling choice. The mapping from information rate to gamma power is a bridge hypothesis, not a theorem. The sinh coupling is the gradient of a postulated potential motivated by analogy with hyperbolic embeddings; in the 1D binary case it is not derived from intrinsic geometry. The paper's intended contribution is a coherent, testable application of an established structural theorem to predictive belief, not a closed proof of necessity for everything it discusses. We do not claim that consciousness is geometry. We claim that bounded agents — finite biological systems that must update beliefs about an environment they share with other finite agents — naturally evolve along coordinates wi th the structure we describe. The phenomenology of conscious belief, the kinematics of confidence updating, and the dynamics of joint attention are consistent with, and predictively constrained by, this coordinate structure. Whether and how phenomenal experience supervenes on these dynamics is a question we set aside. 2. The Bounded Belief Interval and Its Natural Coordinate 2.1 Setup Consider an agent tracking a binary environmental variable through a subjective probability P in Theorem 4.1(ii)], the boundary cannot be reached by any finite composition of interior elements. The coordinate we choose for the interior must respect three further constraints derived from the predictive-processing setting. 2.2 Three Constraints on the Coordinate

In log-odds, evidence simply adds; in bias, evidence adds with a multiplicative envelope. The coordinate change is not deep mathematics — it is a substitution — but it makes visible the structure that bounded predictive agents naturally inhabit. 2.5 The Integrated Rapidity Expression Integrating dλ/dt = Φ(t) from indifference at t = 0 gives, identically, s(t) = tanh (∫0t Φ(t′) dt′).

This is a calculus identity once one accepts the definition of Φ as the rate of change of the rapidity. The equation has no empirical content on its own; the empirical content of the framework is carried entirely by the bridge hypothesis discussed in §4.1, which makes Φ measurable. The structural content — that the integrated rapidity remains strictly inside (−1, 1) for every fi nite cumulative drive — is the continuous-form shadow of the bounded composition theorem [3, Theorem 4.1(ii)]. The qualifier matters: the theorem proves boundary unattainability for finite compositions, and the continuous analogue holds for finite cumulati ve drives. An infinite cumulative drive (∫Φ → ∞) 3.1 Joint Free Energy and the Postulated Coupling Two agents A and B attending to the same stimulus each have a belief si and a rapidity λi = artanh(si). They form predictions of one another's states through social signals (gaze, vocalisation, posture) — a setting analysed in the active-inference literature on joint attention and inter-subjective alignment [10] — and a coupling term in the joint variational free energy penalises predictive divergence. We write the dyad's joint potential schematically as V(λA, λB) = −ΦA · λA − ΦB · λB + β · cosh(λA − λB), where the linear terms represent each agent's independent sensory drive and the cosh term represents the alignment potential under joint attention, with strength β > 0. The coupling potential is postulated. In the full two -dimensional Gaussian extension (§6.3),

The sinh has two notable features: it is linear in Δλ for small differences (small disagreements relax linearly, as one would expect from any reasonable coupling) and exponential for la rge differences (large disagreements relax much faster than any polynomial coupling predicts). The exponential tail is the empirical signature we propose to test. 4. Empirical Predictions and Experimental Protocol 4.1 The Bridge Hypothesis: Φ and Gamma Power The framework's empirical contact with neural data is the proposal that the information - accumulation rate Φ is linearly related to a measurable neural quantity: Φ(t) = k · Pγ(t), where Pγ(t) is the phase -locked gamma -band power in task -relevant channels and k is a saccadic suppression), and any specific identification with information accumulation requires careful operationalisation and control. Operationalisation. We propose the following pre-registered pipeline for the bridge hypothesis to be a well -defined empirical claim. Band: 30–80 Hz, with the specific window committed in advance per experimental modality. Channels: task-relevant channels identified by ICA on a held-out training set, or by anatomical priors (occipito -parietal for visual discriminatio n tasks).

discrete trial-level rapidity increment Δλtrial on the integrated normalised gamma power over the same trial, on a held-out training set per subject per session. What would falsify the bridge. The hypothesis is that k is stable within a subject and session under the pre-registered pipeline. We specify three concrete falsification criteria : (i) k varies by more than a pre -registered threshold (we suggest a coefficient of variation > 0.3) across matched conditions within a session; (ii) the relationship between Δλtrial and integrated gamma is significantly non -linear in a generalised additive model with the same number of effective parameters; (iii) an alternative neural measure (alpha desynchronisation, beta power, theta - gamma coupling, P300 amplitude) shows a tighter linear relationship with Δλtrial on the same data, indicating that gamma is not the privileged correlate. Why linearity? The bridge hypothesis posits a linear relationship between Φ and gamma power, not merely a positive correlation. This is a strong claim and deserves explicit justification. The Communication-Through-Coherence proposal [8] suggests that gamma -synchrony gates the efficacy of neural communication; the more pre-synaptic groups are coherent with their target, the more bits of evidence pass per unit time. To first order in coupling strength, this gives a linear motivating intuition, not a derivation. A non-linear bridge — a power-law with exponent ≠ 1, or a saturating relationship — would still be consistent with the structural framework but would change the empirical predictions of §4.2 quantitatively. The linearity assumption is par t of the empirical bet, not a structural commitment. EMG contamination in dyadic settings. Dual-EEG protocols involving social interaction face a specific severity of gamma-band contamination: facial micro-movement, vocalisation, and head- stabilisation mu scle activity all generate broadband high -frequency signal that overlaps the

no-response blocks to rule out motor preparation as the driver; and concurrent EOG/EMG monitoring with explicit artefact rejection to rule out saccade and muscle contamination of high- frequency power. Crucially, we specify in advance the pattern that would falsify a generic-arousal account: k calibrated on orthogonal-engagement trials (where attention is directed away from the stimulus) should be statistica lly indistinguishable from zero, while k calibrated on the standard task should be positive. A flat positive k across both conditions indicates that gamma is tracking arousal rather than information accumulation specifically. 4.2 Single-Agent Prediction In a binary perceptual discrimination task with continuous confidence reporting and concurrent EEG under the protocol of §4.1, the framework predicts d/dt [ artanh(s(t)) ] = k · Pγ(t). After calibration on the training set, the relationship is parameter-free across trials and conditions on the held -out test set. Failure of this relationship on test data, or failure of any of the three falsification criteria in §4.1, falsifies the bridge hypothesis. 4.3 Dyadic Test For two participants jointly viewing the same ambiguous stimulus and reporting confidence continuously, we measure sA(t), sB(t) via behavioural rating and PγA(t), PγB(t) via dual EEG. We define equalised epochs operationally by stimulus-level criteria — windows in which the displayed response of Φ to a transient stimulus change to decay below a pre -registered threshold). This stimulus-level definition is non-circular: we do not use Φ itself to identify the epochs in which Φ is to be tested. During such epochs the framework predicts d(Δλ)/dt = −2β · sinh(Δλ), where β is estimated from the alignment rate in independent calibration trials with unambiguous stimuli, fit by linear regression in the small-disagreement regime where sinh ≈ identity. The shape of d(Δλ)/dt as a function of Δλ — pooled across many equalised epochs and participants — should be a pure negative sinh curve. 4.4 Distinguishing the Coupling Form Three competitor coupling forms make qualitatively different predictions for the same observable:

on noise estimates that do not yet exist. What follows is a discriminability intuition, not a sample- size argument, and it should be replaced by a proper analysis once pilot data are available. Distinguishing the sinh from the tanh in the | Δλ| ∈ [1.5, 2.5] regime, where the two predictions differ by a factor of approximately 3–4 in the magnitude of d(Δλ)/dt, at a Bayes factor of 10:1 with realistic trial-level noise plausibly requires of the order of 30–50 dyads, each contributing several hundred equalised-epoch samples across the | Δλ| ≥ 1.5 regime. A pilot study with ~10 dyads should be sufficient to estimate noise levels and effect sizes for a proper power calculation. Pre- rich space of EEG analyses. 4.6 Failure Modes and What Each Implies Because the dyadic test sits on top of two layers of modelling (the bridge hypothesis for Φ; the postulated cosh coupling), a negative result is multiply interpretable. We make the inference structure explicit: • Calibration of k fails (k unstable or coefficient-of-variation > 0.3 across matched conditions): the bridge hypothesis is refuted. No dyadic test is interpretable without a stable bridge. • Calibration of k succeeds on standard task but is also non-zero on orthogonal- engagement controls: gamma tracks arousal/attention rather than information accumulation. The bridge is refuted in its specific form, though a corrected bridge based

as KL-divergence-based coupling become candidates. • Bridge survives, dyadic response is sinh in the tails: the postulated coupling is supported. This is the framework's positive prediction. The single-agent bridge test is therefore the gating experiment. The dyadic test should not be attempted until the bridge has been independently calibrated and survived its three falsification criteria. 5. Figures Four figures carry the visual argument of the paper. Figures 1 and 2 illustrate the conceptual core (bounded interior, rapidity stretching, saturating envelope). Figure 3 shows the discriminating

hyperbolic sinh (purple, solid), linear Euclidean (teal, dashed), and saturating tanh (coral, dotted), all with the same coupling strength β. The three curves a gree near zero and diverge dramatically at the tails: at |Δλ| = 2 they predict d(Δλ)/dt of approximately −7.25β, −4β, and −1.93β respectively. The shaded region indicates the expected experimental coverage |Δλ| ≤ 2.5; the experimental design must reach into the |Δλ| ≥ 1.5 regime for the model comparison to be decisive. Figure 4. Dyadic protocol for the sinh discriminating test. Two participants jointly view a shared ambiguous stimulus while reporting continuous confidence (sA, sB) on a rating dial with concurrent EEG (PγA, PγB). The pre-registered preprocessing pipeline of §4.1 yields trial-level Δλ and integrated gamma power, which are then tested for the predicted sinh relationship during equalised epochs. 6. Discussion 6.1 What the Framework Claims, and What It Doesn't This paper proposes a coordinate system, a metabolic-barrier model, a bridge hypothesis, and a coupling potential, anchored on the structural theorem of [3]. It claims: 1. Bayesian evidence accumulation on the binary belief interval is an instance of an admissible bounded compositional structure. By [3, Theorem 3.1, Theorem 3.2, and Corollary 3.4], its unique linearising coordinate is the rapidity λ = artanh( s), and the

itself a modelling assumption; alternative cost functions yield different saturation laws but do not change interior occupancy. 3. The bridge hypothesis — that Φ is linearly related to phase-locked gamma power under a pre-registered pipeline — is the empirical bet of the single-agent framework. If a stable calibration constant k does not exist under that pipeline, or if an alternative neural measure fits better, the bridge fails. 4. The dyadic sinh coupling is a postulated coupling potential motivated by structural analogy with the 2D hyperbolic case. It is not derived from intrinsic 1D geometry. Its falsification disconfirms the specific coupling form, not the broader framework. It does not claim: • That consciousness reduces to geometry or kinematics. • That the geometry of binary belief is intrinsically hyperbolic in the curvature sense (a • That the form of the coupling potential is mathematically forced from the binary case. • That microtubules, gravitational self-energy, or any quantum mechanism is required. We separate these registers deliberately. The mathematics and the predictions are what the paper is for; the philosophy is what the paper is about. 6.2 Between Something and Nothing The phrase between something and nothing can now be given a precise mathematical reading. The bounded composition theorem [3, Theorem 4.1] establishes that under three minimal axioms

representing unbounded structure. The framework does not explain consciousness; it identifies one of the structures within which conscious belief operates, and it asks whether that structure leaves the empirical signature we predict. 6.3 Genuine Hyperbolic Geometry from Continuous Beliefs The binary case is the simplest illustration. The framework extends naturally to continuous beliefs by replacing the Bernoulli with the univariate Gaussian location -scale family parameterised by mean μ and a precision-related coordinate σ. The Fisher-Rao metric on this family is exactly of how the 1D case projects from the 2D dynamics, is in preparation. 6.4 Relation to Other Frameworks The framework is consistent with, and extends, predictive processing and the free -energy principle [1, 2]. It rests on the bounded composition theorem of [3], draws on the established information geometry of statistical manifolds [7], on the empirical literature documenting hyperbolic structure in neural representations [4, 5, 6], and on the literature on hyperbolic embeddings in machine learning [5]. It does not depend on any specific account of phenomenal

framework requires elicitation of both confidence (mean -like) and uncertainty (precision -like) information, which the binary case collapses. Second, the bridge hypothesis remains the framework's narrowest claim and most exposed point. Even with the operationalisation and controls in §4.1, gamma -band power's multiple functional correlates mean that a positive result is open to alternative interpretations until orthogonal controls are run. First experiments should target the single-agent bridge before the dyadic test. Third, the choice of coupling potential is principled but not forced. A negative result on the dyadic sinh prediction would not refute the rapidity framework; it would refute the specific coupling. Systematic model comparison — sinh vs tanh vs linear vs polynomial — is the more honest experimental design than a single confirmatory test of the sinh form. Fourth, alternative metabolic barriers (polynomial, exponential, biophysically detailed cost functions) yield qualitatively different saturation laws and merit dedicated investigation. The log environmental feedback to evolve coherently in a world it cannot fully resolve. It does not collapse consciousness into mathematics. It identifies one of the structures within which conscious belief operates, and it asks whether that structure leaves the empirical signature we predict. Appendix A. Variational Derivation of the Coupled Dynamics