Appendix B: Units, Parameters, and Coefficients (Audit Table)#
TLDR#
Centralize units, parameters, and coefficients so thresholds and regularizers are auditable across chapters.
Use this when implementing: it prevents silent unit mismatches and makes “what is this constant?” a lookup, not a hunt.
B.1 Base Units (Information + Steps)#
We use a purely information-theoretic unit system:
Information / cost: nats (\(\mathrm{nat}\)).
Interaction time \(t\): discrete environment steps (\(\mathrm{step}\)).
Computation time \(s\): internal solver time (continuous; normalized units unless mapped to wall-clock).
Scale time \(\tau\): depth coordinate (dimensionless).
Memory time \(t'\): discrete past index (\(\mathrm{step}\), with \(t' < t\)).
Conventions:
Entropies \(H(\cdot)\), mutual information \(I(\cdot;\cdot)\), and divergences \(D_{\mathrm{KL}}\) are measured in \(\mathrm{nat}\).
Value/cost scalars (\(V\), \(F_t\), budgets, thresholds) are measured in \(\mathrm{nat}\).
Per-step rates (HJB terms, \(\Delta V\), any “cost rate”) are measured in \(\mathrm{nat/step}\) (interaction time).
Latent coordinates (\(z\), \(z_n\), \(z_{\mathrm{tex}}\), code embeddings \(e_k\)) are treated as normalized/dimensionless; any physical units should be absorbed into preprocessing and encoder normalization.
B.2 Parameter / Coefficient Units (by Role)#
Symbol |
Meaning (context) |
Units |
|---|---|---|
\(t\) |
interaction step index |
\(\mathrm{step}\) |
\(t'\) |
memory time index (\(t' < t\)) |
\(\mathrm{step}\) |
\(s\) |
computation time (internal solver) |
solver-time units (normalized) |
\(\tau\) |
scale time (depth) |
dimensionless |
\(\Delta t\) |
optional mapping from steps to wall-clock |
\(\mathrm{s/step}\) |
\(r_t\) |
reward/cost per step |
\(\mathrm{nat}\) |
\(\mathcal{R}\) |
reward 1-form (rate per length; paired with velocity gives nat/step) |
\(\mathrm{nat}/[\text{length}]\) |
\(A\) |
non-conservative component of the reward 1-form (\(\mathcal{R}=d\Phi + A\)) |
\(\mathrm{nat}/[\text{length}]\) |
\(V\) |
value / cost-to-go |
\(\mathrm{nat}\) |
\(\Delta V\) |
value change per step |
\(\mathrm{nat/step}\) |
\(\mathfrak{D}\) |
control-effort / regularization rate term |
\(\mathrm{nat/step}\) |
\(\lambda\) |
Lyapunov rate in \(\dot V\le -\lambda V\) (continuous-time form) |
\(s^{-1}\) |
\(\gamma\) |
discount factor (MaxEnt RL) |
dimensionless |
\(H\) |
horizon / planning depth |
\(\mathrm{step}\) |
\(T_c\) |
cognitive temperature / entropy-regularization coefficient (Definition 73) |
dimensionless |
\(\beta_{\text{ent}}\) |
exponential-family scale in \(\exp(-\beta_{\text{ent}} V)\) |
dimensionless |
\(\vartheta\) |
local conditioning proxy (Definition Definition 9) |
\([z]^2/\mathrm{nat}\) before normalisation; dimensionless after the stated \(\ell_0\) convention |
\(\beta_{\text{cpl}}\) |
coupling coefficient from \(\vartheta\) (Definition Definition 9) |
reciprocal units; dimensionless after the stated \(\ell_0\) convention |
\(\epsilon\) |
numeric stabilizer / threshold |
inherits compared quantity |
\(\eta\) |
step size / learning-rate symbol |
dimensionless (in normalized coordinates) |
\(\beta\) |
VQ-VAE commitment weight (Section 2.2b / 3.3) |
dimensionless |
\(\beta_n\) |
nuisance KL weight (structured residual) |
dimensionless |
\(\beta_{\mathrm{tex}}\) |
texture KL weight (reconstruction-only residual) |
dimensionless |
\(\beta_K\) |
macro codelength weight (rate term) |
dimensionless |
\(\lambda_{\text{use}}\) |
codebook usage regularizer weight |
dimensionless |
\(\lambda_{\text{*}}\) |
composite-loss weights (e.g. \(\lambda_{\text{shutter}},\lambda_{\text{ent}},\lambda_{\text{zeno}}\)) |
dimensionless |
\(\lambda,\mu,\nu\) |
VICReg component weights |
dimensionless |
\(\alpha,\beta_{\pi},\gamma,\delta\) |
scaling coefficients (Section 3.2) |
dimensionless |
\(V_{\text{max}},V_{\text{limit}},V_{\text{proxy}},V_{\text{true}},B_{\text{switch}}\) |
risk/cost budgets and thresholds |
\(\mathrm{nat}\) |
\(\lambda_{\text{in}},\lambda_{\text{mix}}\) |
grounding/mixing information rates |
\(\mathrm{nat/step}\) |
\(q_{k\to k'}\) |
macro transition probabilities |
dimensionless |
\(C_{\partial}\) |
boundary information capacity |
\(\mathrm{nat}\) |
\(I_{\text{bulk}}\) |
bulk information volume |
\(\mathrm{nat}\) |
\(\ell\) |
boundary resolution scale |
boundary-length units (chosen) |
\(\eta_\ell\) |
boundary area-per-nat at resolution \(\ell\) |
\([dA_G]/\mathrm{nat}\) |
\(\Lambda\) |
curvature/capacity offset constant (metric law) |
\([z]^{-2}\) |
\(\kappa\) |
coupling in \(R_{ij}-\tfrac12R G_{ij}+\Lambda G_{ij}=\kappa T_{ij}\) |
chosen so \(\kappa T_{ij}\) has units \([z]^{-2}\) |
\(\alpha_{\mathcal F}\) |
optional coefficient for the curl/Maxwell stress extension (Definition 51) |
\([z]^2\) under \([\mathcal F]=\mathrm{nat}/[z]^2\) |
\(U(z)\) |
hyperbolic information potential \(-d_{\mathbb{D}}(0,z)\) (Section 21.1) |
\(\mathrm{nat}\) |
\(T_c(\tau)\) |
generative temperature schedule (Section 21.2) |
dimensionless |
\(\kappa_T\) |
temperature annealing rate (Section 21.2) |
\(\tau^{-1}\) |
\(\phi_c\) |
Möbius translation \((-c)\oplus z\) moving \(c\) to the origin (Definition Definition 75) |
dimensionless (isometry) |
\(\lambda(z)\) |
conformal factor \(2/(1-\lvert z\rvert^2)\) (Section 21.4) |
\([z]^{-1}\) |
\(\sigma_{\text{tex}}\) |
base texture standard deviation (Section 21.4) |
\([z_{\text{tex}}]\) |
\(R_{\text{cutoff}}\) |
geometric stopping radius (Section 21.3) |
dimensionless |
\(\epsilon_{\text{conv}}\) |
convergence stopping threshold (Section 21.3) |
\(\mathrm{nat}/\tau\) |
\(S_{\mathrm{OM}}\) |
Onsager-Machlup stochastic action (Section 22.1) |
\(\mathrm{nat}\) |
\(\Phi_{\text{gen}}\) |
generative potential \(\alpha U + (1-\alpha)V_{\text{critic}}\) (Section 22.3) |
\(\mathrm{nat}\) |
\(\alpha\) |
generation-control interpolation parameter (Section 22.3) |
dimensionless |
\(\gamma\) |
friction coefficient in overdamped limit (Section 22.4) |
\(s^{-1}\) |
\(\beta_{\text{curl}}\) |
curl/Lorentz coupling coefficient (Section 22.1) |
dimensionless |
\(\lambda_{\text{jump}}\) |
Poisson jump intensity (Section 22.2) |
\(s^{-1}\) |
\(\eta\) |
multiplicative jump factor for mass (Section 22.2) |
dimensionless |
\(\mathcal{A}_y\) |
sub-atlas for class \(y\) (Definition 107) |
— |
\(V_y\) |
class-conditioned potential (Section 25.2) |
\(\mathrm{nat}\) |
\(\beta_{\text{class}}\) |
class temperature (inverse semantic diffusion) (Section 25.2) |
dimensionless |
\(\mathcal{B}_y\) |
attractor basin for class \(y\) (Section 25.2) |
— |
\(\gamma_{\text{sep}}\) |
class separation strength (Section 25.3) |
dimensionless |
\(\lambda_{i\to j}^{\text{sup}}\) |
class-modulated jump rate (Section 25.3) |
\(s^{-1}\) |
\(\mathcal{L}_{\text{purity}}\) |
chart purity loss \(H(Y\mid K)\) (Section 25.4) |
\(\mathrm{nat}\) |
\(\mathcal{L}_{\text{balance}}\) |
load balance loss (Section 25.4) |
\(\mathrm{nat}\) |
\(\mathcal{L}_{\text{metric}}\) |
metric contrastive loss (Section 25.4) |
\(\mathrm{nat}\) |
\(\mathcal{L}_{\text{route}}\) |
route alignment loss (Section 25.4) |
\(\mathrm{nat}\) |
\(\epsilon_{\text{purity}}\) |
purity threshold (Definition 107) |
dimensionless |
\(\tau_n\) |
per-sample nuisance surprisal fallback threshold for \(D_{\mathrm{KL}}(q(z_n\mid x)\Vert p(z_n))\) |
\(\mathrm{nat}\) |
\(\tau_{\mathrm{tex}}\) |
per-sample texture surprisal fallback threshold for \(D_{\mathrm{KL}}(q(z_{\mathrm{tex}}\mid x)\Vert p(z_{\mathrm{tex}}))\) |
\(\mathrm{nat}\) |
\(\tau_K\) |
per-sample macro surprisal fallback threshold for \(-\log p_\psi(K)\) |
\(\mathrm{nat}\) |
The surprisal thresholds are fallback triggers for the D.C (ungrounded) intervention. Calibrate each threshold as a high quantile of the corresponding surprisal on a held-out training-distribution set; they are not universal constants. The primary D.C signal remains Node 13’s \(I(X;K)>0\) check together with the operational coupling window.
B.3 Symbol Overload (Important)#
Some Greek letters are intentionally overloaded in different submodels:
\(\beta_{\text{ent}}\), \(\beta_{\text{cpl}}\), and \(\beta_{\text{curl}}\) are distinct subscripts for entropy, coupling, and curl terms; plain \(\beta\) is reserved for the VQ-VAE commitment weight and some contrastive-loss scales where noted.
\(\tau\) appears as (i) scale time (Section 1.3), (ii) the entropy-weight coefficient in Section 2.11.3, and (iii) the temperature in some contrastive losses; use local definitions.
\(\gamma\) appears as (i) discount factor, (ii) the World Model volatility scaling coefficient \(\gamma\) (Section 3.2), and (iii) friction coefficient in overdamped dynamics (Section 22.4).
\(\lambda\) appears as (i) Lyapunov rate (\(s^{-1}\)), (ii) generic loss weights (dimensionless), and (iii) other Lagrange multipliers (units stated locally).