L_threshold_hist = L_threshold_eff * exp(-rho_B * B_overflow)


B_overflow = max(0, transfer_bandwidth - assimilation_bandwidth) / assimilation_bandwidth


B_gate_emotional = exp(-gamma_emotional * DeltaE_emotional_hist / kT_emotional_hist)


L_emotional = C_emotional,d * emotional_response_d(L_emotional_offload; theta_emotional,d) * lambda_phi^D_f * B_gate_emotional,d


L_emotional_offload = max(0, L_cog_raw - L_threshold_hist) * eta_offload_hist


DeltaE_emotional_hist = DeltaE_emotional_eff + chi_B * B_overflow


kT_emotional_hist = kT_emotional_eff / (1 + psi_B * B_overflow)


eta_offload_hist = eta_offload_eff * exp(-omega_B * B_overflow)


G_over(d) = 1 if L_cog_raw <= L_threshold_hist; else exp(-gamma_d * (L_cog_raw - L_threshold_hist) / kT_emotional_hist)


L_cog_raw(d,x) = C_d * response_family_d(x; theta_d) * lambda_phi^D_f * B_gate(d,constraints)


L_residual_stress = max(0, L_cog_raw - L_threshold_hist) * (1 - eta_offload_hist)


L_threshold = C_threshold * lambda_phi^D_f * B_gate_threshold


L_total = L_cog_eff + L_emotional + L_residual_stress


DeltaE_emotional_eff = DeltaE_emotional + chi_T * T_trauma


kT_emotional_eff = kT_emotional / (1 + psi_T * T_trauma)


eta_offload_eff = eta_offload * exp(-omega_T * T_trauma)


L_threshold_eff = L_threshold * exp(-rho_T * T_trauma)


{"heat_loss":"Q_i = max(0, L_info_i - L_threshold) * (1 - G_over_i)","magnetic_projection":"M_i = sigmoid(((chi_i H_i) + R_i - 0.5 C_loss_i) * G_over_i)","overflow_gate":"G_over_i = 1 if L_info_i <= L_threshold else exp(-1.25 * (L_info_i - L_threshold) / 0.9)","signal_load":"L_info_i = phi^D_f * (log(1 + 2 h_i) + MM(t_i;1,0.35) + (1 - r_i)^0.6)"}


{"byte_transition_rate":"domain agitation / susceptibility driver","capacity_overflow":"hysteresis heat-loss channel","entropy":"field demand / information pressure","repeated_4grams":"remanence / memory channel"}


Q_i = max(0, L_info_i - L_threshold) * (1 - G_over_i)


M_i = sigmoid(((chi_i H_i) + R_i - 0.5 C_loss_i) * G_over_i)


G_over_i = 1 if L_info_i <= L_threshold else exp(-1.25 * (L_info_i - L_threshold) / 0.9)


{"heat_loss":"Q_i = max(0, L_info_i - L_threshold) * (1 - G_over_i)","magnetic_projection":"M_i = sigmoid(((chi_i H_i) + R_i - 0.5 C_loss_i) * G_over_i)","overflow_gate":"G_over_i = 1 if L_info_i <= L_threshold else exp(-1.25 * (L_info_i - L_threshold) / 0.9)","signal_load":"L_info_i = phi^D_f * (log(1 + 2 h_i) + MM(t_i;1,0.35) + (1 - r_i)^0.6)"}


{"byte_transition_rate":"domain agitation / susceptibility driver","capacity_overflow":"hysteresis heat-loss channel","entropy":"field demand / information pressure","repeated_4grams":"remanence / memory channel"}


Q_i = max(0, L_info_i - L_threshold) * (1 - G_over_i)


M_i = sigmoid(((chi_i H_i) + R_i - 0.5 C_loss_i) * G_over_i)


G_over_i = 1 if L_info_i <= L_threshold else exp(-1.25 * (L_info_i - L_threshold) / 0.9)


compressed_total_bytes = payload_bytes + residual_bytes + witness_bytes + decoder_delta_bytes + container_bytes


Hutter-hard promotion additionally requires the total contest artifact for enwik9 to beat 109685197 bytes under the applicable prize rules


LB_route = payload_floor + residual_floor + witness_floor + decoder_delta_floor + container_floor + evaluator_cost_floor


proposal_score -> candidate_route_coordinate -> bounded_exact_route_metastate -> promotion_receipt


promote iff decoded_hash == source_hash and compressed_total_bytes < incumbent_bytes and ratio_schema is explicit and all witness, residual, decoder, and container bytes are counted


C = proposal_score(comp, phys, geom, scaling) or phi_HP = field + compression_gain + decoder/resource penalties


α_ij^(l) = softmax_j(LeakyReLU(a→^(l)^T [W^(l) h_i^(l) || W^(l) h_j^(l)]))


dY/dX = 2aX + b (where Y = aX² + bX + c)


ẍ_i + γẋ_i + ω_i²x_i + Σ_j κ_ij(x_i - x_j) = F(t)


C_ratio = U_size / C_size = 1.48× (achieved)


Shader(x,y,t,θ)→palette + GCL→LUT + SUBLEQ→compute


L_E(x) = BPB(x,w_prior) - BPB*(x) = (1/n)Σ_i log₂(P_w*(x_i)/P_wprior(x_i))


L_G(x,t) = Σ_{s=1}^{S} γ^s · ΔL_E(x_s,t+1) ≈ τ·L_E·log(S+1)/log(S_max+1)


L_R(x) = Σ_j c_j·1[f_j computed] + Σ_{l=1}^{D(x)} log₂|M_l|


L_M(x) = log₂|E| + α·1[hit] + β + λ·|E|/|E_max|


L_total = λI·l̂I + λE·l̂E - λG·l̂G + λR·l̂R + λM·l̂M (Σλ=1, λG≤λE)


η(x) = l̂I(x) / (l̂I + l̂E + l̂R + l̂M + ε)


η_net = (W_rec - W_erasure) / W_in; W_erasure ≥ k_B·T·ln2 per bit


Q = (E_flash + E_enthalpy + E_recovered - W_demon) / (E_work + E_loss) > 1.0


g = cos(Δθ·2π/16) · cos(Δφ·π/8) · (1 - 2|χ_i - χ_j|)


F_ij = w_ij · (a_j - a_i) · Δp_ij / |Δp_ij|


E(f) = -½ Σ_{i,j} w_ij · a_i · a_j + Σ_i V(a_i)


w_ij = cos(Δθ·22.5°)·cos(Δφ·22.5°)·cos(2πΔτ/16)·(1-2|Δχ|)·exp(-|Δp|²/2σ²)


F_ij^(τ) = w_ij^(τ) · (a_j - a_i) · sgn(τ_j - τ_i)


T_throat = {(i,j) | Φ_topo(i,j) >> Φ_metric(i,j)}


w_ij = w_p · w_π · w_τ · w_χ · w_topo · w_σ


d_N = path_length(γ_ij) + curvature_penalty(κ) + torsion_cost(T); d_T = d_E + λ_N·d_N


TrixalAxes = (thermal, work, irreversibility), each ∈ [0,1]; |axes| = √(th² + w² + ir²)


H = -Σ_b p(b) log₂ p(b), where p(b)=count(b)/len


score = (entropy_production + path_asymmetry + time_reversal_violation) / 3


L_thermo = Σ_i distance_i · (1 + irreversibility_i)


depth = entropy_production · ln(time_steps)


SHA256(axes || traj_hash || hardware_entropy || timing_jitter || process_nonce)


EM_risk = J^n · exp(E_a / (k_B · T)) / 10^{12}


CM_damage = (ΔT / ΔT_threshold)^m · 10^{-8}, m≈1.9


W_erasure ≥ k_B · T · ln(2) ≈ 2.87e-21 J/bit at 300K


dS/dt = power_dissipation / (k_B · T · ln 2) [bits/s]


stress(t) = stress_0 · e^{-t/300} + intensity · (1 - e^{-t/300})


RUL = MTBF / (AF · (1 + fatigue·0.01 + thermal_fatigue·0.1))


surprise = -ln(margin), margin = 1 - stress_magnitude; regret = max(0, stress - 0.5)


int_bits ≤ fp_mantissa_bits + 1 (signed) or ≤ fp_mantissa_bits + 1 (unsigned)


can_safely_narrow(src_bits, signed, dst_mantissa) → bool


fdiv.d=33, fdiv.s=19 → penalty=0.737; fadd.d=4, fadd.s=4 → penalty=0.0


ΔE = E_upper - E_lower; E_upper=hc/λ_min, E_lower=hc/λ_max


G = photons_per_e⁻ × n_wells; photons_per_e⁻ = ⌊E_electron / ΔE⌋


λ(T) = λ₀ + α · (T - T₀); α=5e-6 /K (GaAs/AlGaAs)


η = (0.5 + window_bonus) · spacing_eff · (1 - stress_penalty); clamped to [0,1]


Windows: (3,5), (8,12), (16,20) μm; transmission=1.0 inside, exp(-dist·0.5) outside


ν = 10⁴/λ [cm⁻¹]; DFB: ±7.5 cm⁻¹, EC: ±200 cm⁻¹; λ_min=10⁴/ν_max, λ_max=10⁴/ν_min


MI_pred = Σ_i (w_i · MI_i · S_i) / Σ_i (w_i · S_i); w_i = 1/(d_i + ε)


surprise = log(1 + |MI_actual - MI_predicted|)


d(z₁,z₂) = √Σ_i w_i · ((z₁_i - z₂_i) / s_i)²


alignment = ∇g_i · ∇g_j / (‖∇g_i‖ · ‖∇g_j‖)


ds² = g_tt·dθ² + 2·g_tp·dθ·dφ + g_pp·dφ²


Gamma^k_ij = ½·g^kl·(∂_i g_jl + ∂_j g_il - ∂_l g_ij)


a^θ = -(Gamma^θ_tt·v_θ² + 2·Gamma^θ_tp·v_θ·v_φ + Gamma^θ_pp·v_φ²); v' = v + a·dt; x' = x + v'·dt


u = 2·tan(θ/2)·cos(φ), v = 2·tan(θ/2)·sin(φ); θ = 2·atan(√(u²+v²)/2), φ = atan2(v,u); select when |cos(θ)| < 0.01


D+D→D, L+L→L, D+L→W(COLLAPSE→W); chiralityToTernary: D→Active, L→Active, W→Latent


φ = (τ mod T)/T, T=942/1000; ternary: φ<1/3→Q, φ<2/3→A, else L; φ' = (φ+1/4) mod 1 if surprise>threshold


Same acceleration as M85 + Verlet velocity update + chart transition check


a_μ^theory = a_μ^experiment     within 0.5σ


Ψ_observed(x, t) = ∫ K_θ(x - x') Ψ_unified(x', t) dx'


g_n(θ_obs) = g_0 · sin(θ_obs)^{1/n} · cos(θ_obs)^{1 - 1/n}


Q_n = (1/2π) ∮_C ∇_n φ · dn = m/n   for m ∈ ℤ


ds² = -dθ²/ω(θ)² + a(θ)² [dr²/(1-kr²) + r² dΩ²] + ℓ_P² dθ² Γ(θ)


A(r) = 2π (cosh(r) - 1) ≈ π · exp(r)    for r >> 1


n_s = 1 - 2/(1 + θ_max/θ_recombination) ≈ 0.965


E_dissipated ≥ k_B T · ln(2)     per bit erased


Δx · Δp ≥ ℏ/2


Ψ_total = Ψ_A + Ψ_B = A · exp(i ω_Ψ θ) · [exp(i k_Ψ x_A) + exp(i k_Ψ x_B)]


|Ψ_total|² = 2|A|² · [1 + cos(k_Ψ (x_A - x_B))]


E = ℏ ω = ω_Ψ     (in natural units ℏ = 1)


Phenotype(x, t) = Ψ_E [ Genotype(x) × Regulatory_State(t) ]


Residual(n) = Ψ_decode [ Basis, Context(n) ] XOR Byte(n)


H_Ψ(data) = -Σ_n p(n) log_2 p_Ψ(n) ≤ H_uniform(data) = 8 bits/byte


G = (V, E) represents a quantum state |ψ⟩ through a
collection of local tensors {Tv }v∈V , one for each vertex


injec= CD and Hv ∼
tive if this map has trivial kernel


d ≥ D|N (v)| , which generically holds after coarse-graining
when D = O(1)


e = (v, n) of
the graph, the tensor Tv ⋆ T̄v of the norm network defines
a superoperator on the virtual space from any set of legs
to its complement Eq


Z = log ZBP +
ϕ(W)ZW ,
(7)
connected W

where a cluster is collection of loops with multiplicities,
W = {(ℓ1 , α1 ), (ℓ2 , α2 ),


i ≥ 1 is its multiplicity in the cluster


ZW = i Zℓαii and the
P
weight of a cluster is |W| := i αi |ℓi | where |ℓ| for a loop
ℓ ∈ L denotes the number of edges in ℓ


Z = ⟨ψ|ψ⟩ to 1/ poly(N ) multiplicative
error
(ii) local observables ⟨OA ⟩ = ⟨ψ|OA |ψ⟩/⟨ψ|ψ⟩ to
1/ poly(N ) multiplicative error, given ⟨OA ⟩ ̸= 0
(iii) correlation functions ⟨OA OB ⟩ − ⟨OA ⟩ ⟨OB ⟩ to
Õ(1/ poly(N )) additive error
where A, B ⊂ V are disjoint local regions


e = (v, n) with d(v, A) =
r, we have

∥µ′⋆,⃗e − µ⋆,⃗e ∥1 = O e−r/ξ∗
(10)

5
Where 1/ξ∗ = O(log ε∗ /ε)


W = Wnear ∪ Wfar , where clusters in
Wnear are distance at most a cutoff Rth away from B


t < r and
starts increasing thereafter, owing to the strict lightcone
in the message-passing dynamics [see SM for a proof]


t = r−1, leading
to

∥µ⋆,⃗e − µ′⋆,e ∥1 = O e−r/ξ∗
(13)
establishing locality of the BP fixed-points


i=1

For p = ∞ we have,
∥x∥∞ := max |xi |
1≤i≤m

(S3)

For an operator A ∈ B(H), we define the operator Schatten norm in terms of its singular values σ(A) as,
∥A∥p := ∥σ(A)∥p

(S4)

for any p ∈ [1, ∞]


p ≤ q ≤ ∞ and any operator A, we have,
∥A∥p ≥ ∥A∥q

(S5)

∥A∥1 ≥ ∥A∥2 ≥ ∥A∥∞

(S6)

In particular,

10
2


p ≤ q ≤ ∞ and any non-zero operator A,
1

1

∥A∥p ≤ rank(A) p − q ∥A∥q

(S7)

p
rank(A)∥A∥∞

(S8)

∥Ax∥2
= σmax (A)
x̸=0 ∥x∥2

(S9)

In particular,
∥A∥2 ≤
We also note that, the ∞−norm obeys,
∥A∥∞ = sup

we will denote ∥ · ∥∞ by just ∥ · ∥ for convenience


B = Ac , reshape the tensor into a linear map
T : HA → HB
by grouping the indices in A and B into multi-indices a = (ia1 ,


q = 1 we have,

| tr A† B | ≤ ∥A∥p ∥B∥q

(S12)

Graphs For a graph G = (V, E), we define some useful notation


e = {v, w} ∈ E, we will refer to it’s directed versions ⃗e = (v, w) and ←
e = (w, v)


e = {v, w}, we define, for any u ∈ V ,
d(e, u) := min{d(v, u), d(w, u)}
and similarly, for any A ⊂ V ,
d(e, A) := min d(e, u)


li=1 be finite-dimensional Hilbert spaces (the l virtual legs), and
let Hphys be the physical Hilbert space


i=1 Hi −→ Hphys from

We perform a singular value decomposition
T = V ΣU † ∈ Cnp ×nv ,

(S14)

where np ≥ nv by injectivity, where U ∈ Cnv ×nv and V :∈ Cnp ×nv satisfy U † U = 1nv (unitarity) and V † V = 1nv
(isometry), and Σ = diag(λe ) ∈ Cnv ×nv collects the singular values


e = 1,
e

e

(S15)

We call δ the injectivity parameter


Lc = dim(HLc )1L ,
Ka,L→Lc Ka,L→L
(S18)
c = dim(HL )1Lc ,
a

a

which follow from the fact that U is a unitary


Lc =
[Ū a ](i′L ),(iLc ) [U a ](iL ),(iLc ) =
δiL ,i′L = dim(HLc )1L
a

a,iLc

iLc

Moreover, when T is δ−injective with δ = 1 (i


G = (V, E) with vertex set V and edge set E


G = (V, E) with uniform virtual
dimension D and physical dimension dp is specified for each v ∈ V by a tensor
O
Tv :
H(n,v) → Hphys ∼
= Cdp ,
n∈N (v)

where N (v) denotes the set of neighbors of v, and each virtual space H(n,v) ∼
= CD


e = (v, w), we define a bond-space H⃗e ∼
= CD


e = (v, w) is a positive
operator µ⃗e ∈ Pos(H⃗e ) representing the message from v to w


e > 0 such that
PEPS if for each directed edge ⃗e ∈ E,
f⃗e (µ⋆ ) = λ⃗e µ⃗e


Z = ⟨ψ|ψ⟩
can be formally described as a Taylor series in terms of ‘loops’ on the network as described below


Z = ZBP 1 +
Zl 

(S21)

Γ⊂L
l∈Γ
Γ finite, compatible

where the sum runs over all finite sets Γ of mutually compatible loops


i=1

We call a cluster W connected if the interaction graph GW is connected, meaning there is a path between any two
vertices in the interaction graph


GW =
Pk
(VW , EW ) has |VW | = i=1 αi vertices, with loop ℓi corresponding to αi vertices


Z = log ZBP +
ϕ(W)ZW ,

(S25)

connected W

where the sum runs over all connected clusters W


c > c0 := log(2e∆) + 12 such that
|Zl | ≤ e−c|l|

(S27)

then, the series for log Z converges absolutely


Fm = log ZBP +

X

ϕ(W)ZW ,

(S28)

connected W
|W|≤m

is bounded by
|log Z − Fm | ≤ N e−d(m+1)
where d = c − c0 , ∆ is the degree of the graph, and N is the number of vertices


T = ⟨ψ|ψ⟩ and
A
T = ⟨ψ|OA |ψ⟩ respectively after suitable BP normalization


Z = ⟨ψ|ψ⟩ and Z A = ⟨ψ|OA |ψ⟩
for a local observable with ⟨OA ⟩ ̸= 0


m = ⟨OA ⟩BP · exp




conn W←LA







supp(W)∩A̸=∅
|W|≤m

leads to a relative error δm = | ⟨OA ⟩ − ⟨OA ⟩m |/| ⟨OA ⟩ | bounded by

δm ≤ O |A|e−(c−c0 )(m+1)

(S33)

where d = c − c0 = O(1)


A = ∂λ log ZλA |λ=0 = ⟨OA ⟩BP +
ϕW Z W
αl
− ⟨OA ⟩BP


degree ≥ 2 everywhere except in all regions Ai


c ≤ O e−d(A,B)/ξ
for a finite correlation length ξ ≤ O((1/(c − c0 ))) and d(A, B) being the graph distance


Z = ⟨ψ|ψ⟩ can be computed to 1/ poly(N ) multiplicative error in poly(N ) time
(ii) local observables ⟨OA ⟩ = ⟨ψ|OA |ψ⟩/⟨ψ|ψ⟩ with ⟨OA ⟩ ̸= 0 can be computed to multiplicative error ϵ in poly(1/ϵ)
time
(iii) 2-point correlation functions can be computed to additive error Õ(1/ poly(N )) in poly(N ) time

18
Proof


m ≤ N e−d(m+1)

(S37)

Hence, to ensure log Z − F̃m < O(1/ poly(N )) we require m = Ω(log N )


m
≤ O e−d(m+1) ≤ ϵ
⟨OB ⟩

(S38)

which can be ensured given m = O(log 1/ϵ), again computable in poly 1/ϵ time


m = d(A, B) + O(log 1/ϵ), leading to an error O m2 (|A| + |B|)e−dm , which is
(note that (|A| + |B|) = O(1),
!

2

1
1
2
−d[d(A,B)+log 1/ϵ]
O d(A, B) log
·e
= O log N ·
= Õ(1/ poly(N ))
(S39)
ϵ
poly(N )
since we already have d(A, B) = O(log N ) and we choose ϵ = 1/ poly(N )


a ≥ δ 2 > 0 and

†
a Ka Ka = D

P

(S43)

a

1,
X

λ2a Ka† Ka ⪰ δ 2

X

a

Ka† Ka = δ 2 D 1,

a

hence
Tr f (X) ≥ δ 2 D Tr(X) = δ 2 D