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Notation

Every symbol used on more than one of the theory pages is defined here, with the name it carries in the wdm code where there is one. A few letters have two jobs; the last section lists them.

Conventions

Mass in kilograms, energy in kilocalories, time in days; pounds and inches appear only at the input and output boundary. Subscript $k$ indexes days, and day $k$’s state is its morning state; subscript 0 marks a starting value ($T_0$, $W_0$). A hat denotes an estimate, a dot a time derivative, a star a target ($\phi^\star$, $W^\star$), a superscript $(i)$ a sigma point, a superscript $+$ a value just after a measurement update, and $^{\top}$ a transpose. A $\Delta$ before a symbol is a change or deviation ($\Delta C$, $\Delta I$); standing alone it is the distance to the target.

States

Symbol	Meaning	Units	Code name
$T$	Tissue mass: fat plus fat-free solids and structural water	kg	`tissue_kg`
$F$	Fat mass, a component of $T$	kg	`fat_kg`
$M$	Lean mass, $T - F$	kg	`lean_kg`
$G_c$	Glycogen-water compartment, deviation from its level at habitual intake	kg	`gly_water_kg`
$G_n$	Sodium-water compartment, deviation from its level at habitual intake	kg	`na_water_kg`
$A$	Adaptive thermogenesis ($\le 0$ in a deficit)	kcal/day	`adapt_kcal`
$W$	Total mass, $T + G_c + G_n$: what the scale weighs	kg	`scale_kg`
$S$	Stored energy	kcal
$t$, $s$	Continuous time; time within one day in the step map	days

Parameters

Symbol	Meaning	Units	Code name
$c_0$	Mass-independent expenditure (absorbs constant log bias)	kcal/day	`rmr_intercept`
$c_1$	Mass dependence of expenditure	kcal/day per kg	`rmr_slope`
$\beta_A$	Fraction of a sustained deficit offset by adaptation	dimensionless	`at_gain`
$\tau_A$	Adaptation time constant	days	`at_tau`
$k_G$	Glycogen-water gain	kg per g/day	`gly_gain`
$\tau_G$	Glycogen-water time constant	days	`gly_tau`
$k_{Na}$	Sodium-water gain	kg per mg/day	`na_gain`
$\tau_{Na}$	Sodium-water time constant	days	`na_tau`
$\kappa_P$	Shift in fat share per g/kg/day of protein above 1.6	per g/kg/day	`partition_kappa`
$\theta$	The nine parameters above, in natural units		`Params`
$a_w$	Waist offset; estimated only when waist readings exist	cm	`waist_offset_cm`
$b_w$	Waist slope: centimeters of waist per kilogram of fat	cm per kg	`waist_slope_cm_per_kg`

Priors for all of them are in the parameter table.

Derived model quantities

Symbol	Meaning	Units	Code name
$E$	Total expenditure, $c_0 + c_1 W + \mathrm{TEF} + X + A$	kcal/day	`total_expenditure`
$E_{\mathrm{base}}$	Expenditure excluding adaptation	kcal/day	`baseline_expenditure`
$\mathrm{TEF}$	Thermic effect of food	kcal/day	`thermic_effect`
$D$	Energy gap $E_{\mathrm{base}} - I$ driving adaptation; the same quantity is the controller’s deficit	kcal/day
$p$	Fat share of a tissue change, Forbes plus protein shift, clipped to $[0.50, 0.98]$	dimensionless	`fat_fraction`
$p_{\mathrm{Forbes}}$	Forbes fat share, $F/(F + 10.4)$	dimensionless	`forbes_fat_fraction`
$\rho_{\mathrm{eff}}$	Energy cost of a kilogram of tissue, $p\rho_F + (1 - p)\rho_L$	kcal/kg	`effective_energy_density`
$c_0 + c_1 W$	Maintenance energy at weight $W$ (no TEF, no adaptation)	kcal/day	`maintenance_kcal`, `maintenance_interval_kcal`
$I^\star$	Weight-stable intake, $(c_0 + c_1 W)/(1 - \phi)$ at thermic fraction $\phi = 0.10$	kcal/day	`weight_stable_intake_kcal`

Fixed constants

Symbol	Meaning	Value	Code name
$\rho_F$	Adipose energy density	9441 kcal/kg	`RHO_FAT`
$\rho_L$	Fat-free energy density	1816 kcal/kg	`RHO_LEAN`
$\rho_G$	Glycogen-complex energy density	1025 kcal/kg	`RHO_GLYCOGEN_COMPLEX`
$\alpha_P$	Thermic effect, protein	0.25	`TEF_PROTEIN`
$\alpha_C$	Thermic effect, carbohydrate	0.075	`TEF_CARB`
$\alpha_F$	Thermic effect, fat	0.02	`TEF_FAT`
$\alpha_A$	Thermic effect, alcohol	0.15	`TEF_ALCOHOL`
	Thermic effect when only total energy is logged	10% of intake	`TEF_MIXED_DIET_FRACTION`
	Forbes coefficient	10.4	`FORBES_COEFFICIENT`
	Reference protein intake for the partition shift	1.6 g/kg/day	`PROTEIN_REF_G_PER_KG`
	Floor on $F$ inside Forbes’ law	2 kg	`MIN_FAT_KG`
	Physical-activity level scaling the Mifflin-St Jeor prior	1.45	`DEFAULT_PAL`
	Fiber correction subtracted from logged energy	2 kcal/g	`KCAL_PER_G_FIBER_OVERCOUNT`
$\sigma_w$	Waist-reading noise standard deviation	1.5 cm	`WAIST_OBS_SD_CM`
$\sigma_I$	Spread of intake on an unlogged day	600 kcal/day (default)	`unlogged_intake_sd`
$s_k$	Uncertainty of an estimated day’s logged energy	a quarter of it	`intake_sd_kcal`

Inputs

Symbol	Meaning	Units	Code name
$I$	Metabolizable energy intake	kcal/day	`energy_kcal`
$P_g$	Protein	g/day	`protein_g`
$C$	Carbohydrate (in the controller, the grams prescribed)	g/day	`carb_g`
$\bar C$	Habitual (pre-diet) carbohydrate intake	g/day	`habitual_carb_g`, `carb_ref_g`
$\Delta C$	$C - \bar C$	g/day	`carb_dev_g`
$F_g$	Fat	g/day	`fat_g`
$A_g$	Alcohol	g/day	`alcohol_g`
$\mathrm{Na}$	Sodium	mg/day	`sodium_mg`
$\overline{\mathrm{Na}}$	Habitual (pre-diet) sodium intake	mg/day	`habitual_sodium_mg`, `sodium_ref_mg`
$\Delta\mathrm{Na}$	$\mathrm{Na} - \overline{\mathrm{Na}}$	mg/day	`sodium_dev_mg`
$X$	Logged exercise	kcal/day	`exercise_kcal`
$q$	Constant under-reporting fraction of logged intake	dimensionless
$u$, $u_k$	Daily input vector; the food of day $k$, eaten after the morning-$k$ weigh-in		`DailyInput`
$m_k$	1 if a food log exists for day $k$		`intake_known`
$o_k$	1 if a weigh-in exists for day $k$		`weight_observed`

The CSV columns behind these are described in Data.

Observations and noise

Symbol	Meaning	Units	Code name
$y_k$	Scale reading on the morning of day $k$	kg	`weight_kg` (CSV `weight_lb`)
$y^w_k$	Waist reading on day $k$	cm	`waist_cm` (CSV `waist_in` or `waist_cm`)
$H$	Observation row $(1, 0, 1, 1, 0)$		`H`
$v_k$, $R$	Scale noise and its variance, $R = \sigma_v^2$	kg, kg²	`obs_sd`
$v^w_k$	Waist-reading noise, standard deviation $\sigma_w$	cm
$w_k$, $Q_k$	Process noise and its covariance		`process_noise`
$\sigma_T, \sigma_F, \sigma_G, \sigma_N, \sigma_A$	Per-day process-noise standard deviations of the five states (defaults 0.08, 0.04, 0.15, 0.10 kg and 3.0 kcal/day)	state units	`tissue_sd`, `fat_sd`, `gly_sd`, `na_sd`, `adapt_sd`
$b$	Direction of the unlogged-intake inflation, $\rho_{\mathrm{eff}}^{-1}(1, p, 0, 0, 0)^{\top}$
$\nu$	Student-$t$ degrees of freedom, when that noise model is selected; the pipeline is Gaussian		`dof` (`--student-t`)

Estimation

Symbol	Meaning	Dimension	Code name
$x$, $x_k$	Physiological state $(T, F, G_c, G_n, A)$ on the morning of day $k$	5	`State`
$\eta$	Unconstrained (standardized) metabolic parameters	9	`eta_mean`
$\eta_w$	Unconstrained waist calibration ($a_w$, $b_w$)	2	`waist_eta_mean`
$z$, $z_k$	Augmented state $(x_k, \eta)$, or $(x_k, \eta, \eta_w)$ with waist readings	14 or 16
$\sigma_\eta$	Per-day random-walk scale on $\eta$, in prior-sigma units (0.02)		`PARAM_DRIFT_PER_DAY`
$\hat x_{k \mid j}$, $P_{k \mid j}$	Mean and covariance of $x_k$ given $y_{1:j}$
$P_\eta$	Covariance of the parameter block, carried between episodes	$9 \times 9$	`eta_cov`
$e_k$, $S_k$	Innovation and its predicted variance		`innovation`, `innovation_var`
$z_k$	Standardized innovation, $e_k/\sqrt{S_k}$
$K_k$	Kalman gain
$F$, $F_k$	Transition matrix of a linear model; its Jacobian in the extended filter	$5 \times 5$
$J_k$, $C_k$	Smoother gain; sigma-point cross-covariance used in its place		`cross_cov`
$\ell$, $b$	Local linear trend model: weight level, daily slope
$\sigma_\ell, \sigma_b, \sigma_v$	Local-linear-trend noise standard deviations: level, slope, observation		`level_sd`, `slope_sd`, `obs_sd`
$n$	Dimension of the belief being sigma-pointed
$L$	Cholesky factor of the belief covariance	$n \times n$
$\alpha$, $\beta$, $\kappa$	Unscented-transform tuning: spread, prior-knowledge correction, secondary spread (0.3, 2, 0)		`SigmaPointSpec`
$\lambda$	Derived scaling, $\alpha^2(n + \kappa) - n$
$\mathcal{X}^{(i)}$, $\mathcal{Y}^{(i)}$	Sigma points; their images under the step map
$W_m^{(i)}$, $W_c^{(i)}$	Sigma-point weights for the mean and covariance
$\mathcal{Z}^{(i)}$	Sigma points mapped through the waist observation
$f$, $g$	One-day step map; the continuous-time vector field it integrates		`step`, `rate`
$c$, $s$	Center and scale of a parameter’s reparameterization map		`Transform.center`, `Transform.scale`
$w$, $\Delta t$, $\tau_{\mathrm{forget}}$	Prior weight when relaxing across a gap; the gap in days; the relaxation time constant (730 days)		`FORGET_TAU_DAYS`
$n_\sigma$	Gross-error gate threshold, in predicted standard deviations (4)		`gate_sigma`
$R_k^{\mathrm{eff}}$	Observation variance after gating or Student-$t$ reweighting	kg²
$\omega$	Latent precision multiplier of the Student-$t$ scale mixture
$\varepsilon$	Intake error on an unlogged day	kcal/day
$h$	Forecast horizon in the backtest (default 1 and 7 days)	days	`horizons`
$j$	Origin day in the backtest; conditioning index in $\hat x_{k \mid j}$

Control

Symbol	Meaning	Units	Code name
$W_k$, $W_H$	Predicted scale weight on plan day $k$; at the horizon’s end	kg	`predicted_weight_kg`
$T_k$, $T_H$	Predicted tissue mass on plan day $k$; at the horizon’s end (what the rate limit and the objective score)	kg	`predicted_tissue_kg`
$W_0$, $T_0$	Scale weight and tissue at the start of the plan	kg	`start_weight_kg`
$W^\star$	Target scale weight	kg	`target_weight_lb` (in lb)
$r_k$, $r_H$	Reference path toward the goal; its value at the horizon’s end	kg	`reference_trajectory`
$J$	Controller objective
$\Delta I_k$	Day-to-day change in prescribed intake	kcal/day
$D_k$	Deficit, expenditure without adaptation minus prescribed intake	kcal/day
$\mathbf{d}$	Decision vector: energy, protein per kg and fat share for each of the $B$ blocks	$3B$
$\tilde d$; $d_{\min}$, $d_{\max}$	A decision scaled to $[0, 1]$ across its bounds; the bounds
$B$, $b$	Number of decision blocks; block index		`n_blocks`
$H$	Horizon length; as a subscript, its last day	days	`horizon_days`
$I_{\min}$, $I_{\max}$	Bounds on prescribed energy	kcal/day	`min_kcal`, `max_kcal`
$P_{\min}$	Protein floor, per kg of the starting weight	g/kg/day	`min_protein_g_per_kg`
$C_b$, $C_{\min}$	Carbohydrate in block $b$; the carbohydrate floor	g/day	`min_carb_g`
$\phi_F$	Fat share of energy, a decision	dimensionless	`min_fat_fraction`, `max_fat_fraction`
$\phi$, $\phi^\star$	Energy share of the targeted macronutrient; its target	dimensionless	`target_carb_fraction`, `target_fat_fraction`
$\bar\phi$	Midpoint of the fat-share bounds, the default target	dimensionless
$\gamma$	Permitted weekly loss as a fraction of tissue, applied daily as $T_{k-1} - T_k \le \gamma T_k/7$	per week	`max_loss_frac_per_week`
$\mathbf{g}$	Inequality constraints $\mathbf{g} \ge 0$: one rate row per day, one carbohydrate row per block
$\Phi$	Weighted total constraint violation, used by the fallback
$\mu$	Escalating penalty multiplier on $\Phi$; zero for SLSQP		`penalty_scale`
$w_T, w_t, w_p, w_s, w_e, w_c$	Cost weights: terminal, tracking, progress, smoothness, effort, carbohydrate aversion		`CostWeights` fields
$w_\phi$	Macro-share weight: 0.1 as a tie-breaker, 100 for a stated target		`NEUTRAL_FAT_FRAC_WEIGHT`, `FAT_TARGET_WEIGHT`, `CARB_TARGET_WEIGHT`
$\Delta$, $n_{\min}$	Distance from the current scale weight to the target; fewest days to cover it at the maximum rate	kg, days

Symbols with two jobs

Within the section that defines it, every symbol above has one meaning. These letters also have a second job; a subscript, a typeface or the context separates them.

Symbol	Uses
$\alpha$, $\beta$, $\kappa$	Bare: unscented-transform tuning. Subscripted ($\alpha_P$, $\beta_A$, $\kappa_P$, …): thermic-effect and model coefficients.
$b$	Local-linear-trend slope; unlogged-intake direction; block index in the controller. $b_w$: waist slope.
$s$	Time within a day; the sex constant in Mifflin-St Jeor; a reparameterization scale.
$z$	$z_k$ as a vector: augmented state. $z_k$ as a scalar: standardized innovation.
$g$	Italic: the continuous-time vector field. Bold $\mathbf{g}$: the controller’s constraints.
$F$	Fat mass; transition matrix $F$, $F_k$ in the generic filter equations; $F_g$: fat grams eaten.
$H$	Observation row in the filter; horizon length in the controller.
$C$	Carbohydrate grams; $C_k$: smoother cross-covariance.
$W$	Total mass; $W_m^{(i)}$, $W_c^{(i)}$: sigma-point weights.
$P$	Covariance; $P_g$: protein grams.
$w$	$w_k$: process noise; bare $w$: prior weight across a gap; $w_T, \dots, w_\phi$: cost weights.
$\sigma$	Subscripted: a standard deviation. As a function: the logistic sigmoid in the logit map.
$c$, $J$	$c_0$, $c_1$: expenditure coefficients; bare $c$: a reparameterization center. $J$: controller objective; $J_k$: smoother gain.
$\phi$	The thermic fraction in the weight-stable intake formula; the controller’s macro energy share.
$j$	Backtest origin day; conditioning index in $\hat x_{k \mid j}$.

$D$, the energy gap that drives adaptation, and the controller’s deficit are the same physical quantity, not a reuse.

Symbol	Meaning	Dimension	Code name
\(x\), \(x_k\)	Physiological state \((T, F, G_c, G_n, A)\) on the morning of day \(k\)	5	`State`
\(\eta\)	Unconstrained (standardized) metabolic parameters	9	`eta_mean`
\(\eta_w\)	Unconstrained waist calibration (\(a_w\), \(b_w\))	2	`waist_eta_mean`
\(z\), \(z_k\)	Augmented state \((x_k, \eta)\), or \((x_k, \eta, \eta_w)\) with waist readings	14 or 16
\(\sigma_\eta\)	Per-day random-walk scale on \(\eta\), in prior-sigma units (0.02)		`PARAM_DRIFT_PER_DAY`
\(\hat x_{k \mid j}\), \(P_{k \mid j}\)	Mean and covariance of \(x_k\) given \(y_{1:j}\)
\(P_\eta\)	Covariance of the parameter block, carried between episodes	\(9 \times 9\)	`eta_cov`
\(e_k\), \(S_k\)	Innovation and its predicted variance		`innovation`, `innovation_var`
\(z_k\)	Standardized innovation, \(e_k/\sqrt{S_k}\)
\(K_k\)	Kalman gain
\(F\), \(F_k\)	Transition matrix of a linear model; its Jacobian in the extended filter	\(5 \times 5\)
\(J_k\), \(C_k\)	Smoother gain; sigma-point cross-covariance used in its place		`cross_cov`
\(\ell\), \(b\)	Local linear trend model: weight level, daily slope
\(\sigma_\ell, \sigma_b, \sigma_v\)	Local-linear-trend noise standard deviations: level, slope, observation		`level_sd`, `slope_sd`, `obs_sd`
\(n\)	Dimension of the belief being sigma-pointed
\(L\)	Cholesky factor of the belief covariance	\(n \times n\)
\(\alpha\), \(\beta\), \(\kappa\)	Unscented-transform tuning: spread, prior-knowledge correction, secondary spread (0.3, 2, 0)		`SigmaPointSpec`
\(\lambda\)	Derived scaling, \(\alpha^2(n + \kappa) - n\)
\(\mathcal{X}^{(i)}\), \(\mathcal{Y}^{(i)}\)	Sigma points; their images under the step map
\(W_m^{(i)}\), \(W_c^{(i)}\)	Sigma-point weights for the mean and covariance
\(\mathcal{Z}^{(i)}\)	Sigma points mapped through the waist observation
\(f\), \(g\)	One-day step map; the continuous-time vector field it integrates		`step`, `rate`
\(c\), \(s\)	Center and scale of a parameter’s reparameterization map		`Transform.center`, `Transform.scale`
\(w\), \(\Delta t\), \(\tau_{\mathrm{forget}}\)	Prior weight when relaxing across a gap; the gap in days; the relaxation time constant (730 days)		`FORGET_TAU_DAYS`
\(n_\sigma\)	Gross-error gate threshold, in predicted standard deviations (4)		`gate_sigma`
\(R_k^{\mathrm{eff}}\)	Observation variance after gating or Student-\(t\) reweighting	kg²
\(\omega\)	Latent precision multiplier of the Student-\(t\) scale mixture
\(\varepsilon\)	Intake error on an unlogged day	kcal/day
\(h\)	Forecast horizon in the backtest (default 1 and 7 days)	days	`horizons`
\(j\)	Origin day in the backtest; conditioning index in \(\hat x_{k \mid j}\)

Symbol	Uses
\(\alpha\), \(\beta\), \(\kappa\)	Bare: unscented-transform tuning. Subscripted (\(\alpha_P\), \(\beta_A\), \(\kappa_P\), …): thermic-effect and model coefficients.
\(b\)	Local-linear-trend slope; unlogged-intake direction; block index in the controller. \(b_w\): waist slope.
\(s\)	Time within a day; the sex constant in Mifflin-St Jeor; a reparameterization scale.
\(z\)	\(z_k\) as a vector: augmented state. \(z_k\) as a scalar: standardized innovation.
\(g\)	Italic: the continuous-time vector field. Bold \(\mathbf{g}\): the controller’s constraints.
\(F\)	Fat mass; transition matrix \(F\), \(F_k\) in the generic filter equations; \(F_g\): fat grams eaten.
\(H\)	Observation row in the filter; horizon length in the controller.
\(C\)	Carbohydrate grams; \(C_k\): smoother cross-covariance.
\(W\)	Total mass; \(W_m^{(i)}\), \(W_c^{(i)}\): sigma-point weights.
\(P\)	Covariance; \(P_g\): protein grams.
\(w\)	\(w_k\): process noise; bare \(w\): prior weight across a gap; \(w_T, \dots, w_\phi\): cost weights.
\(\sigma\)	Subscripted: a standard deviation. As a function: the logistic sigmoid in the logit map.
\(c\), \(J\)	\(c_0\), \(c_1\): expenditure coefficients; bare \(c\): a reparameterization center. \(J\): controller objective; \(J_k\): smoother gain.
\(\phi\)	The thermic fraction in the weight-stable intake formula; the controller’s macro energy share.
\(j\)	Backtest origin day; conditioning index in \(\hat x_{k \mid j}\).