Cohort-aware population and historical observation — 20 Watts Was Enough

lusoris contributors

This note prevents aggregate counts and retrospective stories from silently standing in for lifecycle state, causality, or prospective prediction. It is a mathematical companion to the quantitative-history and demography audit and Candidate 014.

Stock and flow

For an accounting interval of duration $\Delta t$ seconds,

N_{t+\Delta t}=N_t+B_t-D_t+I_t-O_t,

where $N_t$ is the count of qualified active units at time $t$ , and $B_t$ , $D_t$ , $I_t$ , and $O_t$ are replication/entry, retirement/death, immigration, and outmigration counts during the same interval. A rate such as $b_t=B_t/(N_t\Delta t)$ has units s $^{-1}$ . The identity detects inconsistent bookkeeping; it neither explains the flows nor predicts the next interval.

Cohort-component transition

Let $\mathbf n_t$ be a vector of unit counts indexed by lifecycle stage, role, version cohort, or another declared partition. Then

\mathbf n_{t+\Delta t}=\mathbf L_t\mathbf n_t+\mathbf m_t,

where $\mathbf L_t$ is a transition matrix and $\mathbf m_t$ is the net migration vector in unit counts. Survival/transition entries in $\mathbf L_t$ are dimensionless probabilities per interval; replication entries are new units per source unit per interval. A constant matrix is a scenario assumption, not evidence that transition rates remain stationary.

Equal totals $\mathbf 1^\top\mathbf n_t$ can conceal different future paths because stage composition changes exposure, replication, failure, maintenance, and retirement. Every population-level result therefore retains the vector or a registered sufficient aggregation.

Age, period, and cohort are not freely separable

If lifecycle age $a$ , observation period $p$ , and entry cohort $c$ satisfy $c=p-a$ , then

g\!\left(\mathbb E[Y_{a,p}]\right) =\mu+\alpha_a+\beta_p+\gamma_c

is not uniquely identified in its unrestricted linear components. $Y$ uses a declared task unit and $g$ is a specified link. More samples or less noise do not remove the exact dependency. Constraints, priors, curvature, external variation, or mechanistic structure must be declared because they determine part of the decomposition.

Adoption curves do not identify influence

For adoption share $F(t)\in[0,1]$ ,

\frac{dF(t)}{dt}=(p+qF(t))(1-F(t)),

where $p$ and $q$ have units s $^{-1}$ . A close fit is compatible with multiple generative processes: independent exposure, common broadcast, command, homophilous selection, network influence, or mixtures. Identifying influence requires intervention or additional assumptions, not curve shape alone.

Selected observation

For latent event count $N_{\mathrm{latent}}$ and a simplified chain,

\mathbb E[N_{\mathrm{observed}}] =N_{\mathrm{latent}} p_{\mathrm{produce}}p_{\mathrm{survive}}p_{\mathrm{discover}} p_{\mathrm{retain}}p_{\mathrm{code}},

where every $p$ is a dimensionless conditional probability under a declared dependency order. The product is diagnostic, not an independence claim. If missingness depends on an unobserved value after conditioning on available data, unrestricted recovery is impossible without external data or additional selection assumptions.

Collapse and recovery remain vectors

A population or institution state is reported as

\mathbf r_t=(N_t,Q_t,A_t,S_t,G_t,H_t),

where $N_t$ is qualified unit count, $Q_t$ task/service quality, $A_t$ effective authority coverage, $S_t$ network/service connectivity, $G_t$ governance or maintenance capacity, and $H_t$ reserve/headroom. Every component uses its native declared unit. No scalar “collapse” or “recovery” score is formed without explicit authorized weights and sensitivity analysis.

Prospective gate

Retrospective explanation is separated from prediction by freezing model, features, coding, hyperparameters, data vintage, and evaluation before the held-out period, place, lineage, or regime becomes available. Rolling-origin, geographic, lineage, and live holdouts test different forms of transfer; random row splits are insufficient when adjacent rows share history.

flowchart LR
    E["Entry cohort · inherited version · exposure"] --> P["Population state by age · role · location"]
    P --> T["Replication · transition · migration · retirement"]
    T --> P
    P --> K["Production · survival · discovery · retention · coding"]
    K --> V["Versioned observed record + data vintage"]
    V --> I{"Identified quantity?"}
    I -->|"no"| A["Abstain · sensitivity range · alternate models"]
    I -->|"yes"| S["Scenario projection / causal estimate"]
    S --> F["Frozen temporal · place · lineage holdout"]
    F --> O["Observed outcome + calibration"]
    O --> T

Editable source: cohort-observation-contract.mmd.