№ 5 — Analogue Cognition · BÄRŌ DYNAMICS

§ 01 · Purpose

A living scientific map, not an explainer

№ 5 is built to make a novel theory of consciousness — Robert Worden's Projective Wave Theory of Consciousness (PWTC) and the Requirement Equation — inspectable: every claim is rendered as a named equation pinned to the anatomical structure that implements it, sitting beside the established neuroscience scaffold the theory has to interoperate with (predictive coding, free energy, Hopfield CA3, HPA feedback, somatic markers, global-workspace ignition, metacognition, allostasis, communication policy, travelling waves, sleep-replay consolidation, drift-diffusion).

The goal is not pedagogy. The goal is to give scientists a single page where they can read the math, click the brain, watch the layers integrate, and decide whether the model's predictions are worth testing against real neural recordings.

§ 02 · Anatomical substrate

What the model is rendered on

The cortex you see is not stylised. It is the standard HCP cortical parcellation drawn in 3-D from the same coordinate system used in the connectomics literature.

Cortex

HCP-MMP1.0 multi-modal parcellation, 360 parcels (180 per hemisphere). Glasser, M. F., Coalson, T. S., Robinson, E. C., et al. (2016). A multi-modal parcellation of human cerebral cortex. Nature, 536, 171–178. Source: assets/cortex_glasser.json.

Subcortex

19 deep-brain structures (thalamus, hippocampus, amygdala, basal-ganglia complex, brainstem nuclei, cerebellum) rendered as discrete clickable meshes; each pins a causal chain. Source: assets/subcortex.json.

Connectome

White-matter edge sample drawn between parcel centroids, following the convention of the diffusion-MRI tractography literature. Source: assets/connectome_edges.json.

EEG bands

Canonical band-power traces (δ, θ, α, β, γ) drive the background phase + amplitude of the cortical wave layer. Source: assets/eeg_band_powers.json.

Worden RE trace

Pre-computed baseline of the Requirement Equation against free energy under the joint update rule, used to seed ω_W slider initialisation. Source: assets/traces/worden_re_baseline.json.

§ 03 · The falsifiable core (PWTC)

Worden's Projective Wave Theory of Consciousness — the five equations under test

These are the five equations whose survival of contact with real neural recordings determines whether PWTC stands. They are tagged PWTC throughout the side panel of the viewer.

P1 · Ⅰ
PWTC

Wave-weighted state transition (wave + state substrate)

$$P(s' \,|\, s, a) \;=\; \frac{\sum_k |\psi_k(s, a)|^2 \; T_k(s' \,|\, s, a)}{\sum_k |\psi_k(s, a)|^2}$$

Worden, R. (2026), The Projective Wave Theory of Consciousness, Frontiers in Psychology (10.3389/fpsyg.2026.1674983; preprint arXiv:2405.12071). Neural activity couples to a wave excitation that holds the analogue model of space. Discrete state s and wave amplitudes ψ_k co-evolve through a cycle-scheduled projection/collapse.

P11 · Ⅺ
PWTC
LIVE

Worden Requirement Equation + joint Friston/Worden update

$$R(s) \;=\; \sum_{i=1}^{N} w_i\,\bigl|\,g_i - p_i(s)\,\bigr| \\[6pt] \theta_{t+1} \;=\; \theta_t \,-\, (1-\omega_W)\,\eta\,\nabla_{\!\theta} F \,+\, \omega_W\,\eta\,\nabla_{\!\theta} E$$

Worden, R. (2024), The Requirement for Cognition, in an Equation, arXiv:2405.08601. Goals g_i are the system's requirements; p_i(s) its current performance; R(s) is minimised by adaptive behaviour. Runs as a live joint update beside Friston's free-energy gradient — the ω_W slider sets survival vs. fitness.

P12 · Ⅻ
PWTC

Projective wave in the thalamus + decoding

$$\psi(\mathbf{k},t) = \sum_{n=1}^{N} a_n\,\exp\!\bigl(i\,(\mathbf{k}_n \cdot \rho(\mathbf{r}) - \omega(\mathbf{k}_n)\,t)\bigr) \\[4pt] I(\mathbf{r}) \;=\; \bigl|\,\mathcal{F}^{-1}\{\psi\}(\mathbf{r})\,\bigr|^2$$

Worden, R. (2026), PWTC, Frontiers / arXiv:2405.12071. The thalamus holds an analogue model of 3-D space as a wave whose components have wave-vectors k corresponding to projectively-transformed positions ρ(r). The spatial form of conscious experience is the inverse Fourier transform of that wave.

P15 · XV
PWTC

Perspectival imagination (wave transform under SE(3))

$$\rho:\ \mathbb{R}^3 \to \mathbb{P}^3 \quad\text{(projective)} \\[4pt] \psi(\mathbf{k},t)=\sum_n a_n\,e^{\,i(\mathbf{k}_n\cdot\rho(\mathbf r)-\omega_n t)} \\[4pt] \text{imagine: } \rho \mapsto \rho\circ g,\ \ g\in SE(3) \;\Rightarrow\; \text{lines}\to\text{lines}$$

Worden, R. (2024) Spatial Cognition: A Wave Hypothesis, arXiv:2405.10112. Imagination applies an imagined viewpoint change g to the thalamic wave so anticipation and planning run in the same projective space as perception. Projective invariance — straight world-lines map to straight image-lines — is Worden's reason the internal model must be projective rather than arbitrary.

P16 · XVI
PWTC

Cortical microcircuit — Van der Pol + Kuramoto neurons reading/writing the wave

$$x_i(t) = A_i(t)\,\cos\theta_i(t) \\[4pt] \dot\theta_i = \omega_i + \tfrac{K}{N}\sum_j \sin(\theta_j-\theta_i), \quad \dot A_i = \mu_i A_i\bigl(1-A_i^{2}\bigr) \\[4pt] \text{spike}_i \;\Leftrightarrow\; x_i \uparrow \theta_{\mathrm{thr}}$$

Worden, R. (2026) PWTC; single-neuron membranes as Van der Pol relaxation oscillators — Van der Pol & Van der Mark (1928); FitzHugh (1961); Nagumo (1962). Populations synchronise via Kuramoto coupling with order parameter R=|⟨e^iθ⟩|. Cortex is ~80% excitatory pyramidal / ~20% inhibitory — DeFelipe & Fariñas (1992); Markram et al. (2004). Dale's principle — Dale (1935); Eccles (1976). Gamma rhythms — Buzsáki & Wang (2012). Precision-from-phase — Feldman & Friston (2010).

Worden's claim is not that the cortex is the seat of consciousness; it is that the thalamic projective wave is, and the cortex is its sensor and actuator. The viewer makes this geometry literal: when the wave layer is active, the cortical mesh reads from and writes to it, and the perceptual decode is rendered as a separate field.

§ 04 · Mechanistic scaffold — 19 layer equations

The established neuroscience PWTC must interoperate with

Each row is one toggleable layer in the side panel. The LIVE tag marks layers whose update() actually steps the ODE / wave / accumulator each frame and writes to ctx.state; the others render an anatomical pulse driven by a precomputed trace. Layers tagged PWTC belong to Worden's core (§ 03 above, restated here in their full panel position).

P2 · Ⅱ

Phase coherence

$$R \;=\; \Bigl|\,\langle e^{i\theta}\rangle\,\Bigr| \qquad \pi_i \;=\; \pi_{\text{base}} + \alpha \cos\theta_i$$

Kuramoto (1975) order parameter; Friston & Feldman (2010) precision-weighted attention. Used in Worden's framework as the substrate over which the thalamic wave couples to neurons.

P3 · Ⅲ

Variational free energy (FEP)

$$F \;=\; \mathbb{E}_{q}\!\bigl[\log q(z) - \log p(o,z)\bigr] \;=\; \underbrace{\mathbb{E}_q[-\log p(o\,|\,z)]}_{\text{accuracy}} \,+\, \underbrace{\mathrm{KL}\!\bigl(q(z)\,\|\,p(z)\bigr)}_{\text{complexity}}$$

Friston (2010) Free Energy Principle. In PWTC, free-energy minimisation is what the cortex uses to maintain coupling with the thalamic wave.

P4 · Ⅳ

Kuramoto coupling with adaptive K

$$\frac{d\theta_i}{dt} \;=\; \omega_i \,+\, \frac{K}{N}\sum_{j=1}^{N}\sin(\theta_j-\theta_i) \qquad K_{t+1} = K_t + \eta\,\nabla_{\!K} F$$

Kuramoto (1975). Free-energy gradient adapts coupling K live. In Worden's framework, Kuramoto coupling maintains the thalamic wave but is not itself the seat of consciousness.

P5 · Ⅴ

Predictive-coding hierarchy

$$\varepsilon_\ell \;=\; \pi_\ell\bigl(\mu_\ell - g(\mu_{\ell+1})\bigr) \qquad \dot{\mu}_\ell \;=\; -\,\frac{\partial F}{\partial \mu_\ell}$$

Rao & Ballard (1999); Friston (2005). Bidirectional predictive coding. Worden argues these predictions are not the conscious content — they are how neurons read from and write to the projective wave.

P6 · Ⅵ

Hopfield CA3 (hippocampal autoassociator)

$$W_{ij} \;=\; \sum_{\mu=1}^{P} \xi_i^{\mu}\,\xi_j^{\mu} \qquad s_i(t+1) \;=\; \operatorname{sign}\!\Bigl(\sum_j W_{ij}\,s_j(t)\Bigr)$$

Hopfield (1982). CA3 content-addressable attractor; stores the priors that condition the wave's Bayesian update over remembered spatial scenes.

P7 · Ⅶ

HPA-axis distress (CRH → ACTH → cortisol negative feedback)

$$\frac{dC}{dt} \;=\; k\,A \,-\, w\,C \qquad \frac{dA}{dt} \;=\; \frac{p}{1+(C/K_d)^{\,n}} \,-\, q\,A$$

Glucocorticoid feedback — Sapolsky, Romero & Munck (2000), Endocrine Reviews. Amygdala-gated stress drive. The affective substrate that gates which sense-data the thalamic wave incorporates.

P8 · Ⅷ

Somatic markers (insular / vmPFC)

$$c_i \leftarrow c_i \cdot \bigl(1 - \omega \cdot \hat{m}_i^{\,-}\bigr) \qquad \hat{m}_i^{\,-} \;=\; \max\!\bigl(0,\, -\mathrm{soma}(\text{state})\bigr)$$

Damasio (1994) Somatic Marker Hypothesis. Negative mood deflates epistemic confidence; in PWTC, this is how the body signals which sense-data should be prioritised in the wave's Bayesian update.

P9 · Ⅸ

Thalamocortical ignition (global workspace)

$$i^* = \operatorname{argmax}_i\bigl(\pi_i \cdot a_i\bigr) \qquad a_{i^*} \leftarrow 1.5\,a_{i^*}, \;\; a_{j\neq i^*} \leftarrow 0.5\,a_j$$

Baars (1988); Dehaene & Naccache (2001) global workspace ignition. Worden places the thalamus at the centre of the ignition pattern: ignition is the thalamic wave being driven into coherent excitation by aligned neural input.

P10 · Ⅹ

aPFC metacognition

$$C_{\text{meta}} \;=\; \frac{1}{1 + |F|}\sum_{i=1}^{8} w_i\,c_i \qquad \eta_{\text{balance}} \;=\; \frac{\text{coherence}}{1 + |F|}$$

Fleming & Lau (2014) anterior PFC metacognitive arbitration. The metacognitive estimate gates how readily cortex commits to the current wave-derived spatial percept.

P13 · XIII
LIVE

Homeostatic divergence (allostasis)

$$\frac{db}{dt} \;=\; \alpha\,\nabla_{\!b} F(b,o) \,+\, \beta\,\bigl(o_t - g(b)\bigr) \,+\, \lambda\,\nabla_{\!b} H_D\!\bigl(q \,\|\, q^*\bigr)$$

H_D = KL(q ‖ q*) is the divergence from a low-entropy preferred distribution. Allostasis — the brain defends specific physiological set-points, not merely low surprise — Sterling & Eyer (1988); Sterling (2012).

P14 · XIV

Communication policy (active-inference over speech acts)

$$G(\pi) \;=\; \mathbb{E}_{q}\!\bigl[\ln q(o,\theta\,|\,\pi) - \ln p(o,\theta\,|\,C)\bigr] \\[4pt] \pi^* \;=\; \operatorname{softmax}\!\Bigl(-\,\tfrac{G(\pi)}{\tau}\Bigr), \quad \pi \in \{\text{withhold}, \text{qualify}, \text{meta}, \text{speak}\}$$

Active-inference policy selection — Friston et al. (2017). Silence is a first-class outcome. Built on Worden's Requirement Equation: the system speaks only when speaking reduces R for the receiver.

P17 · XVII

Cortical travelling waves

$$\phi(\mathbf r,t)=\mathbf k\cdot\mathbf r-\omega t,\quad v_\phi=\omega/|\mathbf k|\approx 0.1\text{–}0.8\,\mathrm{m/s} \\[4pt] a(\mathbf r,t)=\sum_{m} A_m\cos\!\bigl(\mathbf k_m\cdot\mathbf r-\omega_m t\bigr)$$

Muller, Chavane, Reynolds & Sejnowski (2018), Nat. Rev. Neurosci. Waves sweep cortex at 0.1–0.8 m/s during perception, memory, and sleep. In PWTC the cortical wave reads the thalamic projective wave.

P18 · XVIII

Sleep & dreaming (sharp-wave-ripple replay)

$$\text{UP/DOWN slow oscillation} \sim 0.8\,\mathrm{Hz} \ \text{gates sharp-wave ripples} \\[4pt] \text{hippocampus} \rightarrow \text{neocortex replay (consolidation)}$$

Wilson & McNaughton (1994); Buzsáki (2015); Diekelmann & Born (2010). Offline, the projective wave runs on remembered scenes — a dream.

P19 · XIX
LIVE

Drift-diffusion + sequential Bayesian belief update

$$\dot x \;=\; v(t)\,dt + \sigma\,dW, \qquad \text{decide when } |x|\ge a \\[4pt] q_{t+1}(s) \;\propto\; q_{t}(s)\,p(o_{t+1}\,|\,s)\,p(s)$$

Ratcliff (1978), Psychological Review drift-diffusion model; Mathys et al. (2011) hierarchical Gaussian filter; Gold & Shadlen (2007) LIP/FEF/DLPFC correlates. Steps the SDE each frame (Wiener increment σ√dt·z), reads drift from ctx.state.pred_error, and applies a Bayesian update to a four-bin posterior q(s) on each boundary crossing. Closes the largest gap in the perception chain — there is now an explicit accumulating belief variable behind the wave decode.

§ 05 · Six causal chains (pinned by clicking the structure)

Each chain walks four anatomical stages from raw input to emergent expression

Click the named structure in the viewer to pin the chain in the side panel. Each stage names the anatomical site, the equation that runs at that hop, and the citation that grounds it.

Chain 1 · Conscious perception

pinned by clicking THALAMUS

V1 · primary sensory cortex

$p(o,s)=p(o\|s)\,p(s)$

Generative model — Friston (2010) FEP.

Association cortex

$F=\mathbb E_q[\log q(s)-\log p(o,s)]$

Variational free energy — Friston (2010).

Thalamus · PWTC

$\psi(\mathbf k,t)=\sum_n a_n e^{i(\mathbf k_n\cdot\rho(\mathbf r)-\omega_n t)}$

Worden PWTC, arXiv:2405.12071 §6. LGN, MGN, VPL/VPM host the projective wave.

Global access · percept

$I(\mathbf r)=|\mathcal F^{-1}\{\psi\}(\mathbf r)|^2$

Decoding rule — Worden; broadcast — Dehaene & Changeux (2011).

Chain 2 · Episodic memory

pinned by clicking HIPPOCAMPUS

Neocortex · experience pattern

$\xi^\mu\in\{-1,+1\}^N$

Distributed activity pattern, conveyed via entorhinal cortex.

CA3 · autoassociator

$W_{ij}=\sum_\mu \xi_i^\mu \xi_j^\mu$

Treves & Rolls (1994); Hopfield (1982).

SWR reinstatement

$\text{SWR: hippocampus}\to\text{neocortex}$

Sharp-wave-ripple replay — Buzsáki (2015).

Recollection

$\hat\xi\approx\xi^\mu$

Pattern completion → conscious recollection.

Chain 3 · Affective salience

pinned by clicking AMYGDALA

Thalamic low road

$o_t\to\text{amygdala (subcortical)}$

Subcortical fast route — LeDoux (1996).

Amygdala · aversive learning

$\Delta V=\alpha\beta(\lambda-\sum V)$

Rescorla–Wagner associative value; LeDoux (2000).

Salience network · precision

$\pi_i\uparrow\Rightarrow \varepsilon_i=\pi_i(\mu_i-g(\mu_{i+1}))$

Anterior insula / dACC — Menon & Uddin (2010).

Autonomic + felt response

$\text{CeA}\to\text{hypothalamus / brainstem}$

Somatic-marker — Damasio (1996).

Chain 4 · Action selection

pinned by clicking BASAL GANGLIA

Cortex · candidate actions

$\{a_1,a_2,\dots,a_n\}$

Corticostriatal motor plans (M1, premotor, PFC).

Striatum · DA-gated

$\delta_t=r_t+\gamma V(s_{t+1})-V(s_t)$

DA reward-prediction error — Schultz (1997). D1/D2 Go/NoGo.

Pallidal output · disinhibition

$i^*=\operatorname{argmax}_i(\pi_i a_i)$

GPi/SNr — Mink (1996); Redgrave (1999).

Motor thalamus → M1

$u_t=a_{i^*}$

Released as movement.

Chain 5 · Arousal & precision

pinned by clicking BRAINSTEM

Reticular activating system

$a(t)=\text{reticular arousal}$

ARAS — Moruzzi & Magoun (1949).

Neuromodulatory nuclei · gain

$\pi=\pi_0 e^{\gamma a}$

LC gain — Aston-Jones & Cohen (2005); NA, 5-HT, DA, ACh.

Thalamocortical modulation

$K,\pi \propto a$

Intralaminar thalamic nuclei — Steriade (2000).

Global brain state

$\text{wake}\leftrightarrow\text{NREM}\leftrightarrow\text{REM}$

Sleep-state switch — Saper et al. (2005); VLPO, orexin.

Chain 6 · Forward model

pinned by clicking CEREBELLUM

Motor cortex · efference copy

$u_t \text{ (command + efference copy)}$

Efference copy — von Holst & Mittelstaedt (1950).

Cerebellum · forward model

$\hat o_{t+1}=f_w(x_t,u_t)$

Internal forward model — Wolpert (1998); Ito (2008). PF–Purkinje LTD.

Cerebello-thalamo-cortical loop

$\Delta u = -\eta(o-\hat o)$

Deep cerebellar nuclei → ventrolateral thalamus → cortex.

Coordinated action

$\text{smooth, well-timed behaviour}$

Emergent motor coordination.

§ 06 · Chamber Ⅱ — the Projective Consciousness Model

An exploratory model of coupled dynamics

Chamber Ⅰ pins equations to anatomy. Chamber Ⅱ explores coupled wave, oscillator, inference and pattern dynamics in PCM v5 (numerical revision pcm-v5-r2). Eight equation groups use a shared numerical core with fixed model-time steps, not one update per rendered frame. The η/s/a/μ partition and its coupling rules are project-authored; this combination is not a validated implementation of consciousness or of any single source theory.

The Kuramoto order parameter R measures phase alignment of twelve simulated oscillators. Authored mappings couple it to boundary strength, policy-weight sharpness and rhythm read-outs. Changing coupling K tests these chosen relationships; K_c is a continuum reference, not a guaranteed transition for twelve noisy oscillators. R is not PCI, a consciousness measurement, or a clinical diagnosis.

C1 · Ⅰ
PWTC
LIVE

Finite wave synthesis and a clamped perspective map

$$\psi(\mathbf r,t)=N_{\mathrm{band}}^{-1/2}\sum_n a_n\bigl(1+2\iota_n\bigr)\cos\!\bigl(\mathbf k_n\cdot\rho(\mathbf r)-|\mathbf k_n|ct+\varphi_n\bigr) \\[6pt] \rho(\mathbf r)=\frac{\mathbf r}{\max\bigl(\tfrac14,\;1+\mathcal K\,(\mathbf r\cdot\hat{\mathbf d})\bigr)}$$

Worden (2024) arXiv:2405.10112; Worden (2026) PWTC. The field is finite cosine-mode synthesis, not a reconstruction from brain data or a direct image of beliefs. The perspective divide is projective only where its denominator is unclamped; crossing the clamp can bend a test line, so a small collinearity residual is a local diagnostic, not proof of global homography. Model viewpoint drift is separate from the display camera. Stimuli directly excite direction-aligned wave bands; this authored branch is not a derived Bayesian posterior.

C2 · Ⅱ
PWTC
LIVE

Kuramoto phase × Van der Pol amplitude kernel, with a continuum reference

$$\dot\theta_i=\omega_i+\frac{K}{N}\sum_j \sin(\theta_j-\theta_i)+\sigma\,\eta_i(t), \qquad \dot A_i=\frac{\mu}{2}A_i\!\left(1-\frac{A_i^{2}}{A_0^{2}}\right) \\[6pt] R\,e^{i\Psi}=\frac{1}{N}\sum_j e^{i\theta_j}, \qquad K_c=\frac{2}{\pi\,g_\omega(0)}\approx 10\,\sigma_\omega[\mathrm{Hz}], \qquad R\approx\sqrt{\frac{K-K_c}{K_c}}$$

Kuramoto (1975); Van der Pol & Van der Mark (1928). Twelve phase oscillators are distinct from the Fourier wave bands. Module gain G_i = A_i·½(1+cos(θ_i−Ψ)) attenuates an anti-aligned module. The critical coupling and near-threshold scaling are ideal continuum references, not exact transition laws for this finite noisy system. An incoherent ensemble can still have nonzero R of order N^−1/2; this is a typical finite-size scale, not a hard lower bound or a stability guarantee.

C3 · Ⅲ
LIVE

Theta and gamma as constructed read-outs of one ensemble

$$A_\theta = 3R, \qquad A_\gamma = 0.3\,\mathrm{Var}(\theta)+1.5K, \qquad A_\gamma^{\text{eff}} = A_\gamma\bigl(1+c_{\text{PAC}}\sin\varphi_\theta\bigr)$$

Buzsáki & Wang (2012) gamma mechanisms; Canolty et al. (2006), Science theta-gamma phase-amplitude coupling. Derived mode maps ensemble coherence, phase dispersion and coupling to rhythm amplitudes, with theta-phase modulation of the gamma envelope. These are constructed model read-outs, not independent neuronal oscillators or calibrated EEG measurements. Turning derived amplitudes off selects the manual values.

C4 · Ⅳ
LIVE

Markov blanket partition and a conditional-dependence proxy

$$\text{states}=\eta\ \cup\ \underbrace{s\ \cup\ a}_{b}\ \cup\ \mu, \qquad \text{hypothesis:}\ \ \mu \perp \eta \mid b \\[6pt] s=\gamma_s W\eta+\text{noise}, \qquad \dot\eta=-\eta/\tau_\eta+\sigma\,dW+\gamma_a V a \\[6pt] z=\bigl(b(t),\ \mu(t-\Delta t)\bigr)\in\mathbb R^{11}, \qquad \hat I_G\bigl(\bar\mu;\bar\eta \mid z\bigr)\approx-\tfrac12\ln\!\bigl(1-\rho^{2}_{\bar\mu\bar\eta\mid z}\bigr)$$

Pearl (1988); Friston (2013) Life as we know it; Kirchhoff et al. (2018). The authored partition has 6 external, 4 sensory, 3 active and 4 internal states. The ordinary route is η → s → μ → a → η; leak ℓ adds direct coupling outside it. The diagnostic transforms a regularized, exponentially weighted partial correlation between scalar mean internal and external states into a Gaussian conditional-dependence proxy, in nats. It conditions on current sensory and active vectors plus the previous internal vector. It needs history and can be unavailable or noisy. A small estimate is not proof of full-state conditional independence, a blanket-integrity percentage, or a clinical boundary measure.

C5 · Ⅴ
LIVE

Weighted-error + KL score and homeostatic belief updating

$$F=\underbrace{\tfrac12\sum_j \pi_j\varepsilon_j^{2}}_{\text{weighted error}}+\underbrace{\ln n-H[q]}_{\mathrm{KL}(q\,\|\,\mathrm{uniform})}, \qquad H_D=\mathrm{KL}\bigl(q\,\|\,q^{*}\bigr) \\[6pt] \pi_j=\bigl(\pi_{\text{base}}+\pi_\alpha\cos\theta_j\bigr)\frac{A_j}{A_0}, \qquad \text{schematic belief drive}=\mathrm{lr}(1-c)\bigl[L^{\!\top}(\Pi\varepsilon)+\text{topdown}\bigr]-\mathrm{lr}\,\lambda_{HD}\nabla_{\!b} H_D \\[6pt] V=\alpha F+\lambda H_D \quad\text{(diagnostic score, not a proven Lyapunov function)}$$

Friston (2010); precision from phase — Feldman & Friston (2010); low-entropy setpoint — Sterling (2012). F is an implemented weighted-error-plus-KL score, not a fully derived variational-free-energy model or subjective surprise. Phase- and amplitude-gated precision weights sensory error; changing precision can change F without changing that error. Observations o = softmax(s/T_o) use sensory states on the ordinary partitioned route. Confidence c = 1 − H[q]/ln n gates the drive; posterior gain and bounded, timestep-scaled belief updates are numerical choices. The H_D gradient holds q* fixed within an update and recomputes the target between updates. V and dV/dt are diagnostics, not a whole-model Lyapunov guarantee.

C6 · Ⅵ
LIVE

Heuristic policy weights, coherence-coupled sharpness, dual-gated commitment

$$w=\operatorname{softmax}\bigl(-\beta\,\mathcal G_{\text{heuristic}}\bigr), \qquad \beta=\max(0.05,\beta_0 R), \qquad 0\le R\le1 \\[6pt] \text{commit requires}\quad G_{\text{act}}\ge\gamma_A \ \ \wedge \ \ \tfrac12\bigl(1+\cos(\theta_{\text{belief}}-\theta_{\text{act}})\bigr)\ge\gamma_\varphi$$

Inspired by active-inference policy selection — Friston et al. (2017). The four policies are withhold, qualify, meta and speak, as in P14. Their scores are authored heuristics, not expected free energies from future-state rollouts or a complete expected-free-energy planner. For fixed scores, increasing β sharpens the weights; R is bounded by 1. The score offsets favour speak, so shifting weights need not change the highest-ranked policy. Commitment also requires amplitude, pairwise phase alignment and the event interval; R alone is not the gate. Rings mark numerical events, not demonstrated psychological acts.

C7 · Ⅶ
LIVE

Six-stratum pattern flow and conditional energy descent

$$\mathcal C=\sum_{i,j}\bigl[\tfrac D2(\psi_{i,j+1}-\psi_{i,j})^{2}+\lambda V_h(\psi_{i,j})\bigr], \qquad V_h(\psi)=\tfrac14\bigl(\psi^{2}-1\bigr)^{2}-h\psi \\[6pt] \left.\dot\psi_{i,j}\right|_{\text{unforced}}=\tau_i^{-1}\bigl[D\Delta_d\psi_{i,j}-\lambda V_h'(\psi_{i,j})\bigr], \qquad i\in\{\text{mech},\text{mem},\text{belief},\text{proj},\text{act},\text{meta}\}$$

Allen–Cahn-style pattern dynamics — Allen & Cahn (1979). Six periodic fields have distinct relaxation times; the energy uses diffusion coefficient D and the same tilted double-well potential as the reaction, with unit grid spacing and discrete Laplacian Δ_d. Descent applies only to the unforced subsystem with fixed D, λ and h, no external injection, and a suitable stable timestep and discretization. It is not a guarantee for the driven full simulation. A parameter quench can change the energy function itself. Persistent domains are a memory analogy, not demonstrated encoding or recall.

C8 · Ⅷ
LIVE

Distress chain, ε-control, and a small-gain diagnostic

$$D_{t+1}=D_t M \quad (M\ \text{tridiagonal, row-stochastic}), \qquad \dot\varepsilon=-k_d(1+\sigma)\,\varepsilon+k_p\bigl(1-\text{perf}\bigr) \\[6pt] \text{candidate small-gain bound:}\quad A_{\max}\cdot\max_i L_i\cdot\bigl\|\partial\Phi/\partial X\bigr\| < 1$$

Small-gain reasoning — Zames (1966). Distress runs as a row-stochastic model chain; ε-control combines stress-scaled decay with a performance-dependent exploration drive. The small-gain expression requires valid gain and Lipschitz bounds for the subsystem under analysis. A displayed product or phase-gated attenuation does not prove contraction of the stochastic, driven full model. The conditional energy-descent statement in C7 has its own restricted assumptions; neither diagnostic supplies a whole-model stability guarantee.

Provenance note

Wave and projective ideas draw on Worden and Rudrauf; the pattern-field analogy draws on ACFM. The explicit η/s/a/μ partition, numerical couplings, inference score, heuristic policies and interventions are this project's combination. Evidence for individual ingredients does not validate the combined model. The application's Methods & data separates empirical findings, theoretical accounts and authored choices.

Where to start

Start with the guided introduction, or open the full instrument. Change one control at a time: lower K and compare R with the heuristic policy weights. Raise leak ℓ and inspect the conditional-dependence proxy after it has enough history. Inject a stimulus and compare score F, prediction error and the excited wave bands; no single response establishes learning or stability.

Render stack

Three.js r161, vendored — vendor/three/build/three.module.js. program-5-intro.html and program-5-pcm.html load shared ES modules for the numerical core, experiment session, content, interface, rendering and shaders. These are not self-contained single-file pages. No runtime CDN or build step is added.

Chamber Ⅰ answers "where in the brain would this live". Chamber Ⅱ tests what follows from these numerical choices — not whether the combined theory is biologically validated.

§ 07 · Transparency

Live integration vs. display-only

Each layer tile in the viewer carries a LIVE or DISPLAY badge. The distinction is structural, not cosmetic:

Live integrated

The layer's update() steps the ODE / wave / stochastic accumulator each frame and writes new state into ctx.state that other layers can read. P11 (Requirement Equation joint update), P13 (homeostatic divergence), and P19 (drift-diffusion + sequential Bayes) are live. P11's ω_W slider is wired to the integrator and changes the trajectory in real time.

Chamber Ⅱ

Shared ES modules advance PCM v5 in fixed model-time steps, independently of rendered frames. State measurements, heuristic scores and conditional diagnostics are distinct from display mappings: wave relief, colour, shell opacity and event rings encode model values or events, not physiological measurements. The partition, inference score, policy rules and numerical couplings are authored choices. Their limits are documented in C1–C8 and the application's Methods & data.

Display

The layer renders an anatomically-correct pulse driven by a pre-computed trace or by other live layers it reads. The equation shown is the source equation, not a runtime simulation. This is the honest default for layers where the full simulation is computationally infeasible in the browser (e.g. P12's full thalamic wave field) or where the equation is a literature scaffold rather than novel-theory core.

The badge exists so a reader cannot mistake a literature-reference equation for evidence that the equation has been simulated.

§ 08 · Methods

How the page is rendered and what it cites

Render stack

Three.js (WebGL2) for the cortex + subcortex meshes in chamber Ⅰ and for the projective field, decode lattice and strata rings in chamber Ⅱ — vendored at r161, no CDN; KaTeX 0.16.11 for in-panel and in-page equation rendering; vanilla ES modules for the layer / chain definitions. No build step.

Primary citations

Worden (2024) arXiv:2405.08601, arXiv:2405.10112, arXiv:2405.12071; Worden (2026) Frontiers in Psychology 10.3389/fpsyg.2026.1674983; Friston (2010); Glasser et al. (2016); Hopfield (1982); Ratcliff (1978); Buzsáki (2015); Muller et al. (2018); Kuramoto (1975); Allen & Cahn (1979); Pearl (1988); Friston et al. (2017). Full citation list inline in §§ 03–06 above.

§ 09 · Acknowledgment

On whose shoulders this stands

This page rests on the work of others. The Projective Wave Theory of Consciousness and the Requirement Equation are Robert Worden's; the Free Energy Principle and the active-inference framework are Karl Friston's. I am deeply grateful to both — and to Robert Worden in particular for the generosity of putting forward a novel, falsifiable theory of consciousness in a form that other people can pick up, render, integrate, and test. The synthesis itself is mine: the choices of which equations to include, which to omit, which to integrate live and which to render as anatomical illustration, and how to wire them together into a single inspectable surface. Any error of attribution, of interpretation, of implementation, or of judgment that the reader finds here is mine alone.