PCM5 Generalized Mixed-Model Evidence Boundary

PCM5 extends the cichlid workflow into generalized mixed models and derived posterior quantities. Its governed records currently stop at source identity, claim decomposition, and executable contract wrappers. They do not contain a governed posterior result or numerical comparison.

What PCM5 Is Designed To Establish

PCM5 asks whether phylogenetic mixed-model evidence remains coherent when the response is no longer Gaussian and when reported quantities are derived from posterior draws. That creates three coupled proof problems:

Proof problem	Why it is scientifically material
generalized model identity	Poisson and multinomial families, links, baselines, offsets, and latent scales define different estimands
random-effect identity	phylogenetic, species, residual, and family-specific components require distinct covariance and variance meanings
derived posterior identity	signal ratios, probabilities, ancestral summaries, and plots must be computed from the same retained draw population

The nine dossiers partition those problems so a favorable low-level check cannot promote an unobserved derived claim. At the current boundary they show exactly which source analysis and transformations a future comparison must own; they supply no posterior observations and therefore no numerical evidence.

Derived Quantities Are Part Of The Analysis

PCM5's headline quantities are not columns emitted directly by a sampler. Probabilities, phylogenetic-signal ratios, variance shares, and ancestral responses are functions of linked posterior operands. Their scientific identity includes the parent model, chain and draw, input point or node, transformation, and denominator.

That dependency is why a printed posterior mean is insufficient. Nonlinear transformations and ratios must be applied within each admitted draw so posterior dependence and asymmetry survive. The derived draw population then receives its own summary and comparison rule; it cannot be reconstructed faithfully from separately rounded marginal summaries.

Do Not Start Closure From The Headline Plot

Suppose a source plot shows infection probability across a predictor. Its points are several evidence layers away from the model: category coding and observation denominator define the response; family and link define the linear-predictor scale; fixed and random effects define each draw; the inverse link creates a response-scale probability; and the plot selects predictor values and summarizes those derived draws.

Reproducing the curve shape does not establish correspondence at any parent layer. PCM5 must first retain aligned model and posterior draws, then create a draw-keyed probability ledger, then compare the summaries, and only afterward render the curve. The same dependency order applies to phylogenetic-signal ratios, variance partitions, and ancestral displays.

This is why PCM5’s current plot and derivation contracts are useful despite being not_comparable: they prevent a visually plausible endpoint from claiming evidence that its parent posterior does not yet supply.

flowchart LR
    inputs["Source scripts · data<br/>tree and workspace"]
    claims["Nine claims<br/>and manifests"]
    wrappers["Reference and Python<br/>contract wrappers"]
    missing["Zero governed<br/>primary outputs"]
    closure["Executed models · diagnostics<br/>observations · verdicts"]

    inputs --> claims --> wrappers --> missing
    missing -. required closure .-> closure

Record Inventory

Surface	Retained state	What remains absent
source	PCM5_PGLMM.R, PCM5_extra_code.R, data, tree, workspace, provenance	a governed invocation that captures source-model outputs
claim map	nine stable evidence and claim IDs	claim-level observed comparison rows
reference program	contract writer naming source locators and scripts	Poisson, multinomial, ancestral, and repeated-measure execution
Bijux program	bundle_contract_only writer with no build script	a matched native or adapter-backed scientific computation
result inventory	reviewer summaries and a manifest with zero primary outputs	retained posterior draws, diagnostics, parameters, and derived quantities
verdict	nine not_comparable records	numerical correspondence adjudication

Claim-Specific Closure Requirements

Bundle	Required governed evidence
evidence-001	exact repeated-observation workspace identity, species/tip mapping, and input comparison
evidence-002	draw-wise phylogenetic-signal numerator, denominator, scale, and posterior summary
evidence-003	stable node identities joined to ancestral posterior draws and intervals
evidence-004	plotted node/branch ledger linked to the exact analytical rows and tree
evidence-005	Poisson family, link, exposure or offset, priors, random effects, draws, and diagnostics
evidence-006	source multinomial coding, latent scale, category baseline, priors, and posterior summaries
evidence-007	draw-wise inverse-link transformation with response-scale intervals and comparison rules
evidence-008	distinct species and phylogenetic random effects under matched covariance and priors
evidence-009	draw-wise phylogenetic, species, residual, and family-specific variance partition

Closure Dependency Graph

The evidence records form a scientific dependency graph. Input identity feeds every model; posterior diagnostics qualify every derived quantity; tree and node identity connect ancestral values to their visualization:

flowchart LR
    workspace["evidence-001<br/>workspace and population"]
    signal["evidence-002<br/>Gaussian signal"]
    ancestral["evidence-003<br/>ancestral effects"]
    plot["evidence-004<br/>tree-linked plot"]
    poisson["evidence-005<br/>Poisson model"]
    infection["evidence-006<br/>infection model"]
    transform["evidence-007<br/>probability transforms"]
    repeated["evidence-008<br/>repeated effects"]
    partition["evidence-009<br/>variance partition"]

    workspace --> signal
    workspace --> ancestral --> plot
    workspace --> poisson
    workspace --> infection --> transform
    workspace --> repeated --> partition
    infection --> repeated

Closing a downstream record requires immutable links to the exact upstream artifacts it consumed. Similar column names, the latest local file, or a shared execution directory are not sufficient provenance.

Model-Family Identity

A comparison record identifies the observation unit, response family, link, category baseline, offset or exposure, fixed effects, random effects, phylogenetic covariance, residual convention, priors, chain policy, retained parameters, and transformation formulas. Those identities are evaluated before numerical tolerances.

Poisson, multinomial, and Gaussian quantities cannot share a variance or response-scale interpretation by name alone. Species and phylogenetic random effects remain distinct. Node and branch outputs resolve against the exact tree revision rather than an incidental row order.

Observation And Diagnostic Ledger

A future lane needs one denominator across reference execution, comparison execution, chain diagnostics, parameter observations, transformations, and exclusions. Each observation identifies its claim, parameter or node, source and Bijux values or distributions, rule, tolerance, outcome, and mismatch reason.

Non-convergence and absent parameters remain visible. A missing comparable execution retains not_comparable; an unexplained observed disagreement can support mismatch_unexplained; agreement supports a matched verdict only after the full eligible observation set is adjudicated.

Same-Draw Algebra Is Part Of The Estimand

Nonlinear posterior quantities must be computed from operands belonging to the same model, chain, and retained draw. In general:

mean(inverse_link(eta_draw)) != inverse_link(mean(eta_draw))

mean(A_draw / (A_draw + B_draw))
    != mean(A_draw) / (mean(A_draw) + mean(B_draw))

The difference is scientific, not merely numerical: the draw-wise calculation preserves posterior dependence and nonlinear uncertainty. A valid child ledger therefore retains:

Child field	Required parent identity
model and parameter keys	exact family, link, fixed/random terms, priors, covariance and scale
chain and draw key	one admitted posterior state shared by every operand
predictor/category/node key	the response-scale target or structural object being derived
expression identity	versioned operands, constants, inverse link, denominator and domain rules
child state	finite, missing, refused, or excluded with the inherited parent reason

Pairing operands by row position after chains are filtered independently is not same-draw evidence. Neither is reconstructing a ratio from published marginal summaries. Those outputs may be retained as separate descriptive records, but they cannot close the registered derived-posterior claim.

Derived-Quantity Ledger

Quantity	Draw-wise source expression	Identity that must be retained
Gaussian phylogenetic signal	treetip / (treetip + units)	model, chain, draw, covariance scaling, and both variance columns
infection phylogenetic signal	treetip / (treetip + units + pi^2 / 3)	multinomial2 family, latent-scale convention, chain, draw, and variance columns
response probability	inverse_logit(intercept + slope * predictor)	chain, draw, predictor value and units, coefficient keys, and category coding
repeated phylogenetic share	treetip / (species + treetip + units)	repeated-record population and all three same-draw variance columns
repeated species share	species / (species + treetip + units)	the same denominator and draw identity as the phylogenetic share
ancestral response	intercept plus named tip or node effect	exact tree digest, node identity, chain, draw, and summary convention

The ledger is part of the comparison contract, not explanatory decoration. A governed artifact should retain the input columns or their content identities, the expression version, the output scale, and the resulting observation key.

Treat Every Derived Quantity As A Child Evidence Record

flowchart LR
    parent["Parent posterior draw<br/>model · chain · state"]
    operands["Named operands<br/>with scale and units"]
    rule["Versioned expression<br/>and domain checks"]
    child["Derived draw<br/>with parent identity"]
    summary["Summary, interval,<br/>and comparison verdict"]

    parent --> operands --> rule --> child --> summary

The child record must retain enough identity to join back to one parent draw. Computing a ratio from separately summarized numerator and denominator posteriors destroys their dependence and is not equivalent to summarizing the draw-wise ratio.

Parent condition	Required child state
parent chain or draw is excluded by the declared diagnostic rule	excluded with the same parent reason
a required operand is absent	missing, with the operand key recorded
denominator is invalid under the expression contract	refused or non-finite under an explicit domain rule
category, node, or predictor identity cannot be resolved	not_comparable at the derived-observation layer
parent and transform identities align	eligible for summary and numerical adjudication

A rendered probability, signal fraction, or ancestral label cannot become the primary evidence owner. It remains a projection of the child draw ledger and must carry the same exclusions and denominator.

Adjudicate Each Scale Separately

PCM5 derived quantities cross latent, variance, probability, count, and tree display scales. Agreement on one scale cannot be projected onto another without retaining the draw-wise transformation and its denominator.

flowchart LR
    draws["Governed posterior draws<br/>chain and state identity"]
    latent["Latent-scale parameters<br/>and variance components"]
    transform["Declared draw-wise<br/>transformation"]
    response["Probability, count, signal,<br/>or ancestral quantity"]
    presentation["Summary table,<br/>interval or tree display"]

    draws --> latent --> transform --> response --> presentation

Create a distinct observation for the latent parameter, transformed draw, posterior summary, and rendered projection when a claim traverses those layers. The bundle can match at one layer and remain not_comparable or disagree at another. A matching posterior mean cannot repair a different category baseline, link, residual convention, tree-node identity, or same-draw variance denominator.

Transformation failures, non-finite draws, absent categories or nodes, and discarded chains remain counted at the layer where they occur. Summarizing only transformable draws without publishing that reduced denominator is not a complete comparison.

What Execution Would Change The Verdict State

A scientific execution changes a bundle only when it produces governed primary outputs and an observation ledger, not merely a successful process exit. The reference and Bijux lanes must declare the same analytical identity, preserve raw or losslessly normalized posterior records, publish chain diagnostics, and account for every expected parameter, node, transform, and exclusion.

The comparison then applies claim-specific structural and numerical rules. If the comparable population is absent, the verdict remains not_comparable; if observed comparable values disagree without an accepted explanation, the record can become mismatch_unexplained; only complete adjudicated agreement can support a matched verdict.

Safe Present-Tense Claims

• PCM5 source scripts, data identities, provenance, and nine claim boundaries are checked in and reviewable.
• All nine bundle manifests record not_comparable and zero governed primary outputs.
• The runtime contains narrower comparative, Bayesian, ancestral, and visualization capabilities, but these dossiers do not establish their equivalence to the PCM5 MCMCglmm models.
• No checked-in PCM5 result currently supports posterior signal, ancestral, count, infection, probability-scale, repeated-effect, or variance-partition correspondence.

Do not describe PCM5 as a reproduced PGLMM study until executed outputs, diagnostics, comparison observations, and claim-level adjudication are retained.