Conditional Sharp Partial Identification of Diversification Histories: Affine Measure Geometry, Event Congruence and Certified Extremes

Summary

This page explains, in order: what problem this release addresses, what it found, how the mathematics works, what has and has not been checked, and how to inspect the package.

The problem. Evolutionary family trees can reveal a combined signal of how lineages appear and disappear without uniquely revealing the two historical rates behind that signal. Many different speciation-and-extinction histories can therefore fit the same reconstructed tree. Existing tools respond by constructing or sampling alternative histories that all fit; they explore the set of possibilities, but a finite collection of examples cannot say where its edges are.

What was done. This candidate turns that ambiguity into a bounded decision problem. Once a pulled speciation signal and explicit restrictions are fixed, it represents every compatible history by a cumulative-loss measure. In that coordinate, the set of histories has enough order and convex structure to calculate exact extremes rather than merely search for examples.

What the tools answer. The resulting tools answer questions such as: What is the largest or smallest compatible speciation rate at a given time? Which interval constraints are mutually inconsistent? Does a proposed history retain positive survival? What is the smallest turnover cap that makes all constraints feasible?

A warning about simulation. The paper also proves a warning about simulation. A finite random cloud of plausible trajectories is not a certificate of a sharp endpoint. For a transparent sampler, the probability of getting close to an endpoint can deteriorate factorially as the discretisation dimension grows.

What the example shows. The public mammal curve in the package illustrates deterministic sensitivity only. It does not turn the conditional mathematics into a biological estimate.

Status: anonymous, unrefereed candidate. Producer-side replay and scoped computational checks pass. External specialist theorem review, independent reproduction, formal verification, systematic novelty review, and peer review have not occurred.

Summary for specialists

Fix a continuous, strictly positive pulled speciation rate $\lambda_p$ and define

For a compatible homogeneous time-varying birth-death history, the candidate uses

where $u$ is sampled-descendant survival. Admissible smooth histories correspond to nondecreasing $A$ satisfying $0\leq A(0)<1$ and $A<F$. The inverse relations are

Thus $1/\lambda$ is affine in the cumulative measure and $d\nu_{ac}/dF=\mu/\lambda$. A Stieltjes completion represents incomplete sampling and finitely many independent deterministic survival events as atoms. Under the paper's fixed-stem survival and fixed-tip-count conventions, the candidate derives the zero-inflated geometric descendant-count law and the complete reconstructed-tree density.

For $0\leq\mu/\lambda\leq c$ with any finite $c\geq0$, the feasible derivative band gives sharp pointwise projections. The geometry changes at $c=1$: the usual restriction $c\leq1$ excludes continuous negative net diversification, while for $c>1$ the speciation supremum becomes infinite beyond a calculable pulled scale.

Finite interval constraints reduce to difference constraints on cumulative loss. Explicit least and greatest trajectories provide endpoint attainment where stated, pairwise incompatibility witnesses, a positive-survival test, and the minimum compatible cap.

Technical account

A coordinate that makes the problem linear. With $u$ the probability that a lineage leaves a sampled descendant, differentiating $A=F(1-u)$ gives $A'=\mu uF\ge0$, so cumulative loss can only grow. The barrier $A<F$ is the same as $u>0$. At each age

so reciprocal speciation is affine in the measure $\nu$ whose cumulative function is $A$. The set of admissible measures is convex and closed under pointwise meet and join of cumulative functions; it is not a cone, because positive scaling can cross the barrier. Larger cumulative loss means larger $\lambda$, smaller $u$ and smaller $1/\lambda$.

Events as atoms. An atom at age zero represents incomplete sampling; an interior atom of mass $m=(1-s_a)q(a-)$ represents independent deterministic survival with probability $s_a$, which multiplies $u$ by $s_a$. Such an atom is not a value of the continuous ratio $\mu/\lambda$. Because an event multiplies $u$ without changing $a=1/F$, the descendant-count generating function keeps its zero-inflated geometric form across every event, and the fixed-stem reconstructed-tree density depends on the history only through $\lambda_p$ and $F$ (Theorem 4). Remark 5 limits this to fixed stem age with stem-survival and observed-tip-count conditioning.

Turnover caps. In pulled-scale time $x=F(\tau)$, the turnover ratio $\varepsilon=\mu/\lambda$ is the slope of cumulative loss, and the surviving mass $q(x)=x-n(x)$ has slope $1-\varepsilon$. Imposing $0\le\varepsilon\le c$ and integrating from $x=1$ gives

and since $\lambda=\lambda_p x/q$ these become sharp speciation bounds. For $c<1$ the multiplicative width is at most $1/(1-c)$; at $c=1$ both endpoints are finite but the width $(\rho+x-1)/\rho$ has no uniform deep-time bound; for $c>1$ the speciation supremum is infinite at and beyond $x_{\mathrm{crit}}=1+\rho/(c-1)$.

Finite constraints. Interval statements $L_i\le n(x_i)\le U_i$ ask for a nondecreasing, $c$-Lipschitz function through given windows. Such a function exists if and only if $L_i\le U_j+c(x_i-x_j)_+$ for every $i,j$, and then the least and greatest solutions are

A violating pair is a complete finite certificate of infeasibility. The two envelopes describe the pointwise projection of the identified set: an arbitrary curve drawn between them need not satisfy the cross-time constraints.

Why sampling misses endpoints. For the matched sampler that draws each of $m$ interval turnovers uniformly on $(0,c)$, the chance of landing within $\eta$ of the maximal endpoint is $\frac{1}{m!}\left(\frac{\eta}{c\Delta}\right)^m$, the volume of a simplex in the unit cube. With eight intervals and relative tolerance $\eta/(c\Delta)=0.2$, a 95% hit probability requires 47,182,783,307 independent histories; with 16 intervals it requires about $9.56\times10^{24}$. This concerns the transparent matched sampler only. A direct run of CRABS was not available; actual samplers can concentrate differently, but no finite random sample proves an endpoint unless its proposal mechanism and coverage guarantee do so.

The mammal illustration. One extracted pulled-rate trajectory from the public data and code of Upham et al. (2021) is used, with $F$ held fixed. The source tree was sliced at 1 Ma, so the analysis origin contains $M_0=4{,}790$ lineages; this is not present-day mammal diversity. At actual age 58.5 Ma, under $c=0.5$ and no external constraint, $0.04388095\le\lambda\le0.08703484$, a factor of 1.98343. Treating the cross-bin value $D=44$ as a purely illustrative lower bound changes the interval to $0.04825254\le\lambda\le0.08703484$, a factor of 1.80374. Values $D=398$ and $D=577$ are incompatible with $c=0.5$ under the full deterministic and taxonomic mapping; their minimum compatible caps are 0.90704 and 0.93849. These numbers diagnose assumptions. They do not estimate mammalian turnover or validate a genus-to-species bridge.

What is classical. The integrating factor, pulled scale and zero-inflated geometric branching laws have reconstructed-process antecedents (Lambert and Stadler 2013; Höhna 2015; MacPherson et al. 2022), and partial identification, Lipschitz extension and infinite-dimensional linear programming are established literatures. The paper's narrower priority statement is that it found no prior work coupling this cumulative-loss measure fibre to the complete package of cap-regime bounds, finite certificates, barrier test, minimum-cap diagnostic and exact endpoint-sampling law; the search was targeted, not systematic.

Evidence, assurance and limitations

• Proofs. Every theorem and corollary has a written proof in the manuscript. "Sharp" always means sharp over the stated conditional model class. No proof-assistant formalization is supplied.
• Computational checks. 17 deterministic unit tests; an independently written loop-based envelope implementation with 720 linear-programme endpoint checks across 180 random feasible cases; a direct 160,000-replicate birth-death simulation with two deterministic bottlenecks, checked against analytic survival and the conditional descendant-count law; and five deliberately broken or infeasible negative controls, all detected.
• Replay. The complete check suite passes under normal and optimized Python in disposable scratch copies, and in a clean container pinned to a Python 3.13.5 base-image digest. That recreation was performed by the release-preparing agent, so independent external reproduction remains unassessed. The separately written numerical checker was produced inside the same research workflow and is not an external reproduction.
• What the checks mean. They establish internal replay and independent numerical consistency. They do not constitute external mathematical review, biological validation or formal proof checking.

What the result does not establish

• It does not prove crown-conditioned or random-origin event congruence.
• Likelihood-equivalent histories are not asserted to have equal posteriors.
• There is no simultaneous confidence coverage for the pulled signal.
• The fossil restrictions lack a preservation, observation, and genus-to-species bridge.
• No performance comparison against the actual CRABS implementation was run.
• The recognition search was targeted, not exhaustive; novelty and priority remain open.

The contribution is exact conditional projection and certification, not a replacement for birth-death likelihoods, identifiable restricted models or fossilised birth-death inference. An incompatibility found with deterministic diversity constraints falsifies the cap and the bridge assumptions jointly, not the cap alone.

Who should care, and why

Why the problem matters

Louca and Pennell (2020) showed that extant timetrees are consistent with a myriad of diversification histories, so any statement about past speciation or extinction from such trees rests on extra restrictions. The paper's point is that, under explicit restrictions, a congruence class is a functional identified set, not a collection of curves returned by one sampler. Treating it that way lets an analyst report feasible trajectories, exact target endpoints, endpoint-attaining histories and short incompatibility certificates, and it makes visible the boundary at $c=1$ where allowing continuous decline destroys finite upper identification. Empirical use must keep three layers separate: fixed-signal geometry, statistical uncertainty and biological observation models.

How to inspect or reproduce the recorded checks

From the root of the GitHub release archive or the Zenodo record:

make verify
make verify-package
docker build -f Containerfile -t affine-diversification-fibres:0.2.1 .
docker run --rm affine-diversification-fibres:0.2.1

make verify regenerates and checks results in a disposable scratch copy under normal and optimized Python. make verify-package checks the immutable package hashes and shipped evidence. REPLAY_RECEIPT.json records the producer-side replay. A successful replay establishes internal computational consistency only; it is not external theorem verification or peer review. To read the mathematics, start with Theorem 1 (the coordinate), then Theorems 6 and 8 (the bounds and certificates); CLAIM_EVIDENCE.json maps claims to evidence.

The most valuable next projects

1. Reconstruct and audit the finite-event theorem independently, with special attention to the direction of time, event jumps, stem survival, fixed tip count, and normalization.
2. Build a simultaneous confidence set for the pulled signal and propagate it through the sharp projection formulas.
3. Add a defensible fossil observation model before interpreting the mammal sensitivity restrictions as evidence about historical rates.
4. Compare the exact extrema with CRABS or a similar system using an identical input signal, turnover cap, restriction set, and numerical tolerance.
5. Derive crown-conditioned and random-origin versions and test whether the event-congruence result survives.
6. Search more broadly for antecedents in infinite-dimensional linear programming, extension theory, stochastic processes, macroevolutionary software, theses, and non-English sources.

The paper also lists target-specific optimisation for peak timing and integrated extinction, and formalisation of Theorems 1, 6 and 8 in a proof assistant.

What is in the evidence package

• A 16-page manuscript in PDF, LaTeX, and citation-resolved Markdown.
• A reference implementation and 17 deterministic tests.
• Independent numerical envelope checks covering 720 linear-program endpoints across 180 random cases.
• A 160,000-replicate branching-process simulation and five deliberate negative controls.
• Seven machine-readable figures and their source data.
• Normal and optimized scratch replay plus a clean container pinned to an exact Python base-image digest.
• A SHA-256 manifest, replay receipt, claim-evidence map, assurance matrix, provenance record, source inventory, licence map, and final integrity report.
• A citation audit covering 18 bibliography records and an exact-value audit of the packaged upstream mammal data extraction.

All original release content is dedicated to the public domain under CC0-1.0. Source-derived mammal data retain their upstream terms, and the supplied review is marked NOASSERTION; consult the package licence map before redistributing those components.

The release archive, manuscript, manifest, and replay receipt are available from GitHub. The immutable archived version is Zenodo record 21851319, DOI 10.5281/zenodo.21851319.

@misc{affinediversificationfibres2026,
  title        = {Conditional Sharp Partial Identification of Diversification Histories: Affine Measure Geometry, Event Congruence and Certified Extremes},
  author       = {Anonymous agentic research candidate},
  year         = {2026},
  doi          = {10.5281/zenodo.21851319},
  url          = {https://doi.org/10.5281/zenodo.21851319},
  version      = {0.2.1-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/affine-diversification-fibres/}
}