Finite-Sample Signal Uncertainty and Sharp Partial Identification of Diversification Histories

Plain-English summary

A reconstructed family tree does not preserve a unique history of speciation and extinction. Many different histories can produce the same observable signal. This creates a practical risk: one fitted curve or a large finite cloud can look decisive even when other histories supported by the same tree reverse the conclusion.

This candidate supplies a finite-sample safeguard against that false confidence. Under one exact fixed-stem, stem-survival, homogeneous time-varying model, it builds a confidence set for the entire pulled scale $F$ from the observed number of tips and unordered node ages. It then carries every signal in that set through the affine fibre and reports one of three conclusions for a registered target:

1. certified incompatible for every signal in the confidence set;
2. compatible throughout the band; or
3. unresolved because the decision changes across the band.

This matters because it distinguishes conclusions that survive finite-sample uncertainty from conclusions that only look secure on one estimated signal. In the frozen 22-tip example, the method certifies one turnover cap as incompatible across the whole confidence-containing set. For another cap, it withdraws a plug-in "compatible" answer and reports unresolved. The method therefore changes the kind of scientific conclusion that the evidence permits, not merely the width of an error bar.

The separate CRABS benchmark produced two results that must be read together:

• every one of the 240 returned primary clouds missed at least one sharp endpoint beyond tolerance, with 60 additional structural censors; and
• exact certification changed 532 of 3,840 registered decision statuses, or 13.854%, below the preregistered 20% H4 gate.

The first result is a concrete warning: in this benchmark, a finite random cloud was not a certificate of the sharp limits. The second limits the breadth of the claim: the evidence does not show that exact certification is already a must-have or broadly useful complement to CRABS. Failing that ambitious utility gate does not erase the finite-sample method or the endpoint diagnostic; it states honestly how much decision impact the registered benchmark measured.

Status: anonymous, unrefereed theorem-and-method candidate published with notes. Producer-side replay passes locally and on public Linux CI. No unaffiliated reproduction, external specialist review, proof-assistant formalisation, editorial peer review, empirical validation, or broad CRABS utility is claimed.

The statistical object

Let $T$ be the fixed stem age. Write the pulled scale as

Conditional on the stem lineage leaving sampled descendants, the tip count $N$ has a geometric law with parameter

Conditional on $N$, the $N-1$ unordered internal node ages have a known distribution whose cumulative probability is a monotone transformation of $F$. These two facts separate the finite-sample problem into two exact pieces:

• invert the geometric count tails to obtain an interval for $F(T)$; and
• use a Dvoretzky--Kiefer--Wolfowitz--Massart band for the transformed node-age distribution.

The candidate combines the two pieces by Bonferroni propagation. Under the declared fixed-stem law, the resulting set contains the full pulled-scale trajectory with at least the stated nominal probability.

This guarantee is conditional and model-specific. It is not a confidence set for an estimated topology, uncertain node dates, a smoothed derivative, a lineage-dependent process, or a misspecified biological model.

From a signal band to an identified set

For a fixed signal $F$, the compatible homogeneous histories are represented by a nondecreasing cumulative-loss coordinate $A$ below the survival barrier:

The corresponding rates are

This coordinate turns many restrictions and targets into affine or monotone questions in $A$. For each admissible signal in the confidence set, the code computes the conditional affine fibre and the sharp target range. The union of those fibres contains the true conditional identified set whenever the signal set contains the true $F$.

The decision rule deliberately avoids a plug-in yes/no answer:

The third state is substantive. It prevents an answer based on one estimated curve from being reported as robust when nearby signals reverse it.

Frozen synthetic decision

The worked example is synthetic; it is not a fit to an empirical clade. The observed tree has 22 tips. At the 95% joint nominal level, the count component alone gives

After the node-age band and affine constraints are propagated:

• a registered turnover cap of 0.70 is certified incompatible under the stated normalized deterministic-diversity constraint; and
• a separate interior target that is compatible on the plug-in signal becomes unresolved over the full confidence set.

The second result is the main practical warning. Plug-in compatibility can be an artefact of ignoring uncertainty in the signal that defines the fibre.

Sanity-check coverage

The frozen 20,000-replicate simulation reports:

This simulation checks the implementation in one declared setting. It does not create the coverage theorem; the analytic count inversion, DKW--Massart bound and Bonferroni argument do that work.

What the CRABS benchmark found

The separate comparison was prospectively frozen before its high-cost Stage 2 cells were evaluated. It contained 1,100 registered cells and preserved every result or structural-censor status in a sealed ledger.

The revised H2 accounting separates returned clouds from structural censors:

• all 240 returned primary rejection clouds missed at least one sharp endpoint beyond the frozen tolerance; and
• 60 additional primary cells were structurally censored and are not described as returned-cloud endpoint misses.

That supports a narrow conclusion: a finite random cloud is not an extremum certificate in the registered benchmark.

The preregistered H4 gate asked a different question: how often does exact certification change a decision status? Across 3,840 clustered queries, 532 statuses changed:

The threshold was 20%, so H4 failed. Of the changes, 502 withdrew a prior call to unresolved and 30 moved a below-threshold call to above-threshold. The package reports the full transition and stratum tables rather than treating the clustered queries as independent replicates.

Accordingly, this release does not claim that the affine method is a must-have or essential complement to CRABS, a replacement for CRABS, or broadly superior. It supports the narrower conclusion that finite random clouds are not extremum certificates in this registered benchmark, while the measured decision impact fell short of its preregistered target.

These two findings are compatible. CRABS clouds can miss mathematical endpoints, while exact endpoint recovery changes fewer registered decisions than the broad-utility hypothesis predicted.

Why the failed gate matters

The release began with an ambition to become an essential complement to CRABS. The evidence does not support that language. The failed H4 gate is therefore a load-bearing negative result, not a footnote.

The candidate may support more targeted future work:

• identifying regimes where a cloud is especially likely to miss a decision-relevant boundary;
• replacing broad pooled utility claims with registered transition-specific targets; and
• using exact certification when the cost of a false compatibility or incompatibility call is high.

It does not establish general superiority, replacement value, broad community acceptance, or that every CRABS analysis needs an affine certificate.

Evidence and replay boundary

The release package contains:

• a 23-page PDF and accessible Markdown companion;
• exact count inversion, DKW propagation and affine decision code;
• 21 deterministic tests under ordinary and optimized Python;
• 720 endpoint comparisons across 180 feasible cases using a separately structured linear-programming oracle;
• a 160,000-replicate branching-process simulation;
• five semantic negative controls, all detected;
• the frozen 20,000-replicate signal-band sanity check;
• the complete 1,100-cell CRABS comparison and a 1,900-entry sealed result manifest;
• exact H2/H4 summaries and transition tables;
• source, citation, licence, provenance, status and assurance records;
• a 5,033-entry successor release manifest; and
• public GitHub Actions replay, including a pinned Linux-container job.

The quick successor gate is:

PYTHONPATH=. python3 verification/verify_route_a_candidate.py

It verifies the protected historical receipts, the sealed Stage 2 ledger, the successor manifest, schemas, DOI identity, fixed results, figure alt text, PDF, component licensing, publication boundary, and ordinary and optimized tests.

These are producer-side checks. A passing manifest establishes byte identity; a passing program establishes the encoded checks. Neither establishes that the probability model, scientific interpretation, or source-to-code bridge is correct.

What is not established

• No unaffiliated stochastic-process or phylogenetics specialist has reviewed the fixed-stem conditioning, count law, node-age factorization or coverage proof.
• No unaffiliated group has rerun the immutable package or written an independent implementation.
• No proof assistant has checked the theorem chain.
• The confidence set does not cover topology estimation, node dating, smoothing, model selection, derivative recovery or misspecification.
• The one-dimensional fibre excludes lineage-, state-, trait-, clade- and diversity-dependent processes.
• Fossil preservation, observation and taxonomic-scale uncertainty are not modeled.
• The worked finite-sample decision is synthetic and is not empirical validation.
• The recognition search is structured but not exhaustive; novelty and priority remain partial or unassessed.
• Independent rights review is absent beyond the component-level licence map.
• The failed H4 gate does not support broad CRABS utility.

Relationship to the predecessor

The 0.2.1 affine-diversification release established the conditional affine cumulative-loss representation, sharp fixed-signal target bounds, finite infeasibility certificates and an exact endpoint-sampling diagnostic.

This successor adds:

1. an exact finite-sample fixed-stem confidence set for the pulled scale;
2. propagation of signal uncertainty through the affine fibre;
3. three-valued robust decisions;
4. the complete frozen affine--CRABS comparison; and
5. a corrected interpretation that preserves the failed H4 gate.

The old release and DOI remain immutable. Reuse of its framework is not independent confirmation, and the successor does not retroactively add finite-sample coverage to the earlier archived object.

Who should read what

The most valuable next projects

1. Commission a focused external process-theory review of the fixed-stem law and simultaneous-coverage argument.
2. Build an independently authored implementation in another language or statistical stack and compare normalized outputs from the immutable tag.
3. Extend the observation boundary to topology, dating, smoothing and misspecification without differentiating an unsupported confidence band.
4. Add a fossil preservation and observation model before making empirical claims involving fossil restrictions.
5. Design a new preregistered comparison around specific decision transitions or high-cost regimes instead of reviving the failed broad H4 claim.
6. Develop crown-conditioned, random-origin and heterogeneous-process counterparts as separate, newly reviewed research objects.

What is in the public package

• The exact tagged source archive and DOI-bearing 23-page PDF.
• RELEASE_MANIFEST.sha256 with 5,033 sealed release paths.
• SHA256SUMS for the public GitHub and Zenodo assets.
• Machine-readable status, assurance, claims, sources, provenance, licences, environment and AI index.
• The complete frozen CRABS ledger, receipts and derived H2/H4 tables.
• The Stage 3-prime review and Stage 4.5 integrity report.

The scholarly creator is Anonymous. Operational maintenance and publication roles are recorded in the machine-readable provenance. Original prose, figures, structured records and project-created data are dedicated under CC0 1.0; original code is MIT-licensed; third-party exceptions retain the terms in the component licence map.

The immutable candidate is available from the GitHub prerelease. The archival version is Zenodo record 22041054, DOI 10.5281/zenodo.22041054.

@misc{finitesampleaffinediversification2026,
  title        = {Finite-Sample Signal Uncertainty and Sharp Partial Identification of Diversification Histories},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22041054},
  url          = {https://doi.org/10.5281/zenodo.22041054},
  version      = {0.3.0-candidate-r2},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/finite-sample-affine-diversification/}
}