A phenomenal field structured by 3-D projective geometry (ρ: ℝ³→ℙ³), carried by a Worden thalamic wave, bounded by a Markov blanket with an explicit η/s/a/μ partition, and updated by active inference under a coherence-gated policy.
Every quantity on screen is integrated from the equations in the next tab — nothing is decorative animation. The Kuramoto order parameter R is the hinge: it sets blanket strength, policy temperature, theta amplitude and gamma frequency simultaneously.
What you are looking at
Projective field — sphere radius displaced by ψ(k,t) evaluated at ρ(r); hue is intensity I(r)=|ψ|². This is the phenomenal field, not a decorative mesh.
Markov blanket — the shell. Cyan cubes are sensory states s, magenta cones are active states a. Radius tracks 0.2+0.8R.
External states η — 6 OU processes outside the blanket. Phenomenal objects (inside) are internal states μ, positioned through the projective divide.
Prediction error ε = o − ĝ(b), one bar per sensory channel, height = precision-weighted magnitude π⊙ε.
Six strata rings — Allen–Cahn gradient flow ∂ψᵢ/∂t = −τᵢ⁻¹δC/δψᵢ. Green/violet are the two wells of V(ψ); a locked colour is a stored memory.
Decode lattice (floating grid above) — the inverse-Fourier readout I(r)=|F⁻¹{ψ}(r)|², i.e. the spatial form of experience.
Kuramoto dial (below) — 12 oscillator phases θᵢ with the mean-field vector R·e^{iΨ}.
Provenance
Worden Spatial Cognition: A Wave Hypothesis (arXiv:2405.10112) supplies L-1/L-12/L-15/L-16. ACFM supplies the variational coherence principle, the stratified architecture, the homeostatic-divergence belief law and the row-stochastic distress chain. Friston supplies variational/expected free energy and the blanket partition.
Note: there is no "brain hypothesis" as such in the source corpus — the phrase occurs only as Bayesian brain hypothesis in citation lists. The formal claims implemented here are Worden's Wave Hypothesis and ACFM's central coherence hypothesis. The η/s/a/μ partition and its conditional-independence test are authored here: the corpus used "Markov blanket" only as a visual metaphor, with no partition equations.
I · Worden projective wave L-12 / L-15
ψ(k,t) = K-½ Σₙ aₙ(1+2ιₙ)·cos( kₙ·ρ(r) − ω(kₙ)t + φₙ )
I(r) = | F⁻¹{ψ}(r) |²
ρ: ℝ³→ℙ³ ρ(r) = r / max(¼, 1 + 𝒦·(r·d̂))
ω(k) = |k|·c λₙ = λ₀ + Δλ·n/(N−1)
ρ is a homography — a perspective divide in homogeneous coordinates. Because it is projective, straight world-lines map to straight image-lines; that invariance is Worden's reason the internal model must be projective rather than arbitrary. Toggle the invariance lines to see it hold. Imagination is ρ↦ρ∘g, g∈SE(3): the drifting eye direction d̂.
align = (k·d̂)/|k| ιₖ += 0.6·max(0,align) ι̇ₖ = −0.9ιₖ
Worden's Bayesian update from new sense data: a stimulus excites the k-band aligned with its direction, then that excitation decays.
II · Oscillatory kernel L-16
θ̇ᵢ = ωᵢ + (K/N) Σⱼ sin(θⱼ−θᵢ) + σ·ηᵢ(t)
R·e^{iΨ} = N⁻¹ Σⱼ e^{iθⱼ} R ∈ [0,1]
Ȧᵢ = (μ/2)·Aᵢ(1 − Aᵢ²/A₀²) + Dᵢ(t)
Gᵢ = Aᵢ·½(1 + cos(θᵢ − Ψ)) ∈ [0,A_max]
Phase gain: a module aligned with the mean field runs at full gain, an anti-aligned one (θᵢ=Ψ+π) is silenced. Timescale separation is the design invariant — τ_phase ≈ 1–10 steps, τ_A = 1/(2μ) ≈ 100 steps.
K_c = 2/(π·g_ω(0)) R ≈ √((K−K_c)/K_c) near threshold
Supercritical bifurcation. For a normal frequency spread g_ω(0)=1/(σ√2π), so K_c = 2σ√(2π)/π ≈ 10·σ_ω[Hz]. Both the measured R and this prediction are shown in Telemetry, and they agree just above K_c. Drop K below K_c and R collapses — and with it blanket strength, policy decisiveness and gamma frequency, all at once.
Two caveats on reading the measured and predicted R together, since they will not match exactly. K_c is the noiseless N→∞ mean-field result, whereas this kernel runs N=12 with phase noise σ. Finite N puts a floor of roughly N^{−1/2} ≈ 0.29 on the measured R, so below K_c it sits well above the predicted 0; above K_c the noise term drags it back under the predicted value. Neither gap is a broken bifurcation — set σ→0 and raise the k-band count to watch the two converge.
K ← clip(K + κ·s, 0, K_max), s ∝ −F
Free-energy-adapted coupling, the loop intended to make the kernel autonomous rather than driven. Read literally it is a one-way decay, not a homeostat: F = accuracy + complexity is a sum of non-negative terms, so s = −F ≤ 0 always and K falls monotonically to 0 with no restoring term to bring it back. Left on, it walks the system below K_c and decoheres it within about 20 s. It ships switched off for that reason — turn it on to watch the decoherence, and note that a genuine homeostat needs a setpoint, s ∝ (F* − F).
III · Theta / gamma read-out
A_θ = 3R f_θ = ω₀/2π
A_γ = 0.3·Var(θ) + 1.5·K f_γ = 30 + 20K + 10R
Theta and gamma are not independent oscillators — they are read-outs of the one Kuramoto ensemble. Theta amplitude tracks coherence; gamma amplitude tracks phase dispersion. Untick "derive" to override manually.
A_γ^eff = A_γ·(1 + c_PAC·sin φ_θ)
Phase–amplitude coupling: gamma envelope modulated by theta phase.
IV · Markov blanket authored
states = η (external) ∪ s (sensory) ∪ a (active) ∪ μ (internal)
blanket b = s ∪ a
claim: μ ⊥ η | b
s = γ_s·W·η + noise η̇ = −η/τ_η + σdW + γ_a·V·a
The only legitimate route is the cycle η → s → μ → a → η. Sensory gain γ_s is the original "permeability".
leak ℓ > 0 ⇒ b += ℓ·η̄ , η += ℓ·b̄ (breaks CI)
z = ( b(t), μ(t−1) ) ∈ ℝ¹¹ b = s ∪ a (7-dim)
ρ_{μη|z} = (c_{μη} − c_{zμ}ᵀΣ_{zz}⁻¹c_{zη}) / √(r_μ·r_η)
r_μ = v_μ − c_{zμ}ᵀΣ_{zz}⁻¹c_{zμ} (residual variances)
I(μ;η|z) = −½·ln(1 − ρ²) nats
A genuine test, not a metaphor: Gaussian conditional mutual information from an exponentially-weighted partial correlation, conditioned on the whole 7-dimensional blanket b = s ∪ a and the internal state's own immediate past μ(t−1) — together, μ(t)'s full set of Markov parents. Conditioning on a scalar summary, or on b alone, leaves residual correlation and reports a violation that isn't there. With ℓ=0 this sits at ≈0 and the blanket is a real blanket. Raise ℓ and watch it climb — the boundary stops being Markov.
S_blanket = clamp(0.2 + 0.8R, 0.1, 1.0)
scale = clamp(1 + (θ_def + γ_def)·S, 0.2, 2.8)
V · Free energy & belief
F = E_q[−log p(o|z)] + KL(q(z)‖p(z))
= ½Σⱼπⱼεⱼ² + (ln n − H[q])
└─ accuracy ─┘ └─ complexity ─┘
q = softmax(γ_b·b) q* = softmax(γ_b·b/T), 0<T<1, γ_b=4
H_D(q‖q*) = Σ q ln(q/q*)
∇_b H_D = q ⊙ ( (ln q − ln q*) − H_D·𝟙 )
The posterior gain γ_b is added here, not taken from the corpus. The belief law below ends in tanh and clips b to [−1,1], so a plain softmax(b) caps at q_max ≈ 0.63 and confidence can never exceed ≈0.23 however strong the evidence. Reading the posterior at an inverse temperature restores the full simplex without altering the law itself.
o = softmax(s/T_o), T_o = 0.10 ĝ = L·q
Observations are formed from the sensory states alone — the blanket is the only inbound channel, so nothing in the inference may read η directly. L has unit column sums, so ĝ and o live on the same simplex.
Homeostatic divergence: the distance from the current posterior to a sharpened target. Its closed-form gradient enters the belief law directly.
πⱼ = (π_base + π_α·cos θⱼ)·(Aⱼ/A₀)
ε = o − ĝ(b), ĝ = L·q
Δb = lr(1−c)·[Lᵀ(Π ε) + topdown] − lr·λ_HD·∇_b H_D
b ← clip( tanh(b + Δb), −1, 1 )
The (1−c) factor is confidence gating: updates are small when confident, large when uncertain. c = 1 − H[q]/ln n.
V(b) = αF + λH_D is a strict Lyapunov function ⇒ dV/dt ≤ 0
Telemetry reports the measured sign of dV/dt. Transient positives after a stimulus are expected; the running mean should stay ≤ 0.
VI · Expected free energy & commitment
𝒢(π) = Σₖ[ E[KL(q(sₖ|π)‖q(sₖ|oₖ,π))] − E[ln p(oₖ|C)] ]
└──── epistemic ────┘ └── pragmatic ──┘
𝒢_meta = 0.6 − 0.3·C
𝒢_qualify= 0.4 − 0.2·Comm
𝒢_speak = 0.2 − 0.4·Comm
𝒢_withhold=0.5 − 0.2·U
Comm = C(1−U)R
π(a|s) = softmax(−β·𝒢), β = β₀·R
Coherence-coupled temperature. Proposition 3: ∂H[π]/∂R ≤ 0 — policy entropy falls monotonically as coherence rises. Uniform as R→0, greedy argmin 𝒢 as R→∞. This is why decisiveness is not a separate parameter.
commit ⇔ G_act ≥ γ_A AND align(θ_belief,θ_act) ≥ γ_φ
align(θ,θ') = ½(1 + cos(θ−θ'))
Dual-gated: amplitude alone is not enough, phase must agree. Each commitment fires a ring from the active states outward.
VII · Variational coherence principle
C[ψ] = ∫ ( |∇ψ|² + λV(ψ) ) dx V(ψ) = ¼(ψ²−1)²
∂ψ/∂t = −δC/δψ = ∇²ψ − λV'(ψ), V'(ψ)=ψ(ψ²−1)
stationary (memories): ∇²ψ = λV'(ψ) dC/dt ≤ 0
|∇ψ|² penalises rapid spatial variation → diffusive integration. V encodes drives and salience → selective organisation. Pure diffusion (λ=0) smooths all structure away and stores nothing; no diffusion fragments. Only the combination yields stable distributed attractors.
∂ψᵢ/∂t = −τᵢ⁻¹·δC/δψᵢ , i ∈ {mech, mem, belief, proj, act, meta}
Multi-scale stratification. Proposition One: a system minimising adaptive coherence across multiple temporal scales spontaneously differentiates into partially independent dynamical strata. τ = (0.05, 0.50, 0.20, 0.10, 0.08, 2.00) s.
VIII · Distress, ε-control, Lipschitz
M tridiagonal row-stochastic: M_ii=0.60, M_{i,i±1}=0.20
D_{t+1} = D_t·M λ_m = p_s + 2p_d·cos(πm/N)
Ḋ = αD + βC + γS + ξW + ε(M·D)
α=.10 β=.20 γ=.15 ξ=.10 ε=.05
ε̇ = −k_d(1+σ)·ε + k_p(1 − perf), perf = R
σ > 0.7 ⇒ ε ← ε·γ_d (exploit) σ < 0.3 ⇒ ε ← ε/γ_d (explore)
Both corpus terms are present: stress-scaled decay and performance feedback. Decay alone pins ε at its floor whenever distress stays above the calm threshold, so the feedback term is what gives exploration a working range.
contractive ⇔ A_max · maxᵢLᵢ · ‖∂Φ/∂X‖ < 1
Small-gain theorem on cross-stratum transfer Xᵢ^out = Gᵢ·hᵢ(Xᵢ^in). Because gain vanishes as a module desynchronises (Gᵢ→0 as θᵢ−Ψ→π), a desynchronised or low-confidence module cannot dominate downstream dynamics. The gauge turns red if the product exceeds 1.
H(t) = w₁R − w₂Σσ_ab − w₃H_ent(X) system health
Oscillatory kernel
order parameter R–
mean phase Ψ–
coupling K–
critical K_c–
R predicted √((K−K_c)/K_c)–
phase variance Var(θ)–
coupling energy E_c–
mean amplitude Ā–
Theta / gamma read-out
A_θ / f_θ–
A_γ / f_γ–
Projective wave
ψ at thalamic centroid–
wave energy E = √(K⁻¹Σa²)–
Σ impulse ι (Bayesian)–
eye direction d̂–
max foreshortening 1+𝒦·depth–
projective invariance residual–
Markov blanket
strength S = 0.2+0.8R–
‖s‖ sensory / ‖a‖ active–
‖η‖ external–
partial corr ρ_{μη|b}–
I(μ;η|b) nats–
blanket integrity–
Active inference
free energy F–
· accuracy ½Σπε²–
· complexity KL(q‖p)–
H_D(q‖q*)–
posterior entropy H[q]–
confidence c–
‖ε‖ prediction error–
mean precision π̄–
Lyapunov V = αF+λH_D–
⟨dV/dt⟩ (must be ≤ 0)–
beliefs b–
posterior q–
Policy · expected free energy
β = β₀·R–
selected policy π*–
policy entropy H[π]–
𝒢 (withhold/qual/meta/speak)–
gate G_act ≥ γ_A–
gate align ≥ γ_φ–
commitments fired–
Coherence field · strata
C[ψ] total–
⟨dC/dt⟩ (must be ≤ 0)–
stratum ⟨|ψ|⟩ (mech→meta)–
domain walls / 576 cells–
Distress · ε · stability
distress σ = mean(D)–
chain λ₁ (2nd eigenvalue)–
mixing time O(N²/p_d)–
exploration ε–
Lipschitz A·L·‖∂Φ/∂X‖–
system health H(t)–
Worden efficiency E=C+K–
fps / steps–