(sec-appendix-b-units-parameters-and-coefficients)= # {ref}`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. (sec-appendix-b-base-units)= ## 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. (sec-appendix-b-parameter-coefficient-units)= ## 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 ({prf:ref}`def-cognitive-temperature`) | dimensionless | | $\beta_{\text{ent}}$ | exponential-family scale in $\exp(-\beta_{\text{ent}} V)$ | dimensionless | | $\vartheta$ | local conditioning proxy (Definition {prf:ref}`def-local-conditioning-scale`) | $[z]^2/\mathrm{nat}$ before normalisation; dimensionless after the stated $\ell_0$ convention | | $\beta_{\text{cpl}}$ | coupling coefficient from $\vartheta$ (Definition {prf:ref}`def-local-conditioning-scale`) | 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 ({ref}`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 ({ref}`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 ({prf:ref}`def-extended-risk-tensor`) | $[z]^2$ under $[\mathcal F]=\mathrm{nat}/[z]^2$ | | $U(z)$ | hyperbolic information potential $-d_{\mathbb{D}}(0,z)$ ({ref}`Section 21.1 `) | $\mathrm{nat}$ | | $T_c(\tau)$ | generative temperature schedule ({ref}`Section 21.2 `) | dimensionless | | $\kappa_T$ | temperature annealing rate ({ref}`Section 21.2 `) | $\tau^{-1}$ | | $\phi_c$ | Möbius translation $(-c)\oplus z$ moving $c$ to the origin (Definition {prf:ref}`def-mobius-translation`) | dimensionless (isometry) | | $\lambda(z)$ | conformal factor $2/(1-\lvert z\rvert^2)$ ({ref}`Section 21.4 `) | $[z]^{-1}$ | | $\sigma_{\text{tex}}$ | base texture standard deviation ({ref}`Section 21.4 `) | $[z_{\text{tex}}]$ | | $R_{\text{cutoff}}$ | geometric stopping radius ({ref}`Section 21.3 `) | dimensionless | | $\epsilon_{\text{conv}}$ | convergence stopping threshold ({ref}`Section 21.3 `) | $\mathrm{nat}/\tau$ | | $S_{\mathrm{OM}}$ | Onsager-Machlup stochastic action ({ref}`Section 22.1 `) | $\mathrm{nat}$ | | $\Phi_{\text{gen}}$ | generative potential $\alpha U + (1-\alpha)V_{\text{critic}}$ ({ref}`Section 22.3 `) | $\mathrm{nat}$ | | $\alpha$ | generation-control interpolation parameter ({ref}`Section 22.3 `) | dimensionless | | $\gamma$ | friction coefficient in overdamped limit ({ref}`Section 22.4 `) | $s^{-1}$ | | $\beta_{\text{curl}}$ | curl/Lorentz coupling coefficient ({ref}`Section 22.1 `) | dimensionless | | $\lambda_{\text{jump}}$ | Poisson jump intensity ({ref}`Section 22.2 `) | $s^{-1}$ | | $\eta$ | multiplicative jump factor for mass ({ref}`Section 22.2 `) | dimensionless | | $\mathcal{A}_y$ | sub-atlas for class $y$ ({prf:ref}`def-semantic-partition`) | — | | $V_y$ | class-conditioned potential ({ref}`Section 25.2 `) | $\mathrm{nat}$ | | $\beta_{\text{class}}$ | class temperature (inverse semantic diffusion) ({ref}`Section 25.2 `) | dimensionless | | $\mathcal{B}_y$ | attractor basin for class $y$ ({ref}`Section 25.2 `) | — | | $\gamma_{\text{sep}}$ | class separation strength ({ref}`Section 25.3 `) | dimensionless | | $\lambda_{i\to j}^{\text{sup}}$ | class-modulated jump rate ({ref}`Section 25.3 `) | $s^{-1}$ | | $\mathcal{L}_{\text{purity}}$ | chart purity loss $H(Y\mid K)$ ({ref}`Section 25.4 `) | $\mathrm{nat}$ | | $\mathcal{L}_{\text{balance}}$ | load balance loss ({ref}`Section 25.4 `) | $\mathrm{nat}$ | | $\mathcal{L}_{\text{metric}}$ | metric contrastive loss ({ref}`Section 25.4 `) | $\mathrm{nat}$ | | $\mathcal{L}_{\text{route}}$ | route alignment loss ({ref}`Section 25.4 `) | $\mathrm{nat}$ | | $\epsilon_{\text{purity}}$ | purity threshold ({prf:ref}`def-semantic-partition`) | 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. (sec-appendix-b-symbol-overload)= ## 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 ({ref}`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 $\gamma$ ({ref}`Section 3.2 `), and (iii) friction coefficient in overdamped dynamics ({ref}`Section 22.4 `). - $\lambda$ appears as (i) Lyapunov rate ($s^{-1}$), (ii) generic loss weights (dimensionless), and (iii) other Lagrange multipliers (units stated locally).