Formal Genetics: Proof, Computation and Evidence in Molecular Biology (Edition 11)

A student textbook in twelve sections. Each section develops one piece of molecular genetics three times over: as a mathematical argument the reader works through by hand, as a theorem checked by the Lean 4 kernel, and as a number produced by a solver or by analysis of a real dataset. Every section closes with an epistemic ledger separating what was measured, what was assumed, and what was proved.

This book has NOT been peer reviewed. It has had an internal review, which is described in its reproducibility appendix and is not a substitute. An earlier edition of this repository described work as “5-round Opus peer reviewed” on the basis of a function that made no model call; that description was withdrawn, and the appendix documents the withdrawal rather than quietly dropping it.

The organising case is a refutation. An earlier edition of this material taught that the cancer-associated polymerase variant POLE p.Pro286Arg acts by removing proofreading stages. Section 3 proves a no-go theorem showing no model of that kind reproduces the measured phenotype, and Section 4 gives the measurement that breaks it by a factor of 52. The mechanism is retracted in these pages rather than defended.

New in edition 5 — Section 12, Reticulate Evolution. Section 8 measured the topology of a single DNA sequence embedded by the chaos game, and Section 11 reported the negative result that followed. Section 12 replaces that test rather than only reporting its failure. Moving from one sequence to a population makes the null a theorem — a strict molecular clock on a clonal genealogy admits no unfilled cycle — so a violation becomes a certificate a kernel can check rather than a p-value against shuffles. The section then refutes its own first statistic: a clonal population with a relaxed clock and no recombination whatever violates the clock-level condition more strongly than a real recombinant does, so the tree-level four-point condition must carry the method, and a machine-checked proof that the two conditions nest is what licenses reading the difference between them. It closes on ten real viral populations, seven of which are reported blocked because a measured control shows the aligner alone would have produced the answer.

CORRECTION TO THE MATCHED-NULL FLOORS. The previous edition's matched-null floors were not matched. Each was tuned by bisecting a single realisation of a quantity that is chaotic in the mutation rate — the simulator draws its per-branch rate multipliers from the same random stream as its mutations, so changing the rate shifts every downstream multiplier — and the floor was then taken as a maximum over replicates whose realised divergences differed by up to an order of magnitude. Measured, the published nulls missed their populations' divergence by −49%, +11% and −12%; at fixed rate the simulator's divergence has a coefficient of variation near 50%, spanning 10–28× between extremes, so matching such a null in expectation is ill-posed however carefully it is implemented.

Re-deciding against a null matched per replicate — each replicate accepted only if its own realised divergence is within 10% of the population's — leaves every verdict and the 3/3 literature agreement intact, but moves two numbers in opposite directions. Coxsackievirus A6 clears 2.75× its floor rather than 1.24×, so the previous edition understated its own evidence. And the JC polyomavirus margin is four-fifths of its floor rather than the sixth previously reported: still correctly below, but a far closer call than published. Hepatitis B is unchanged at 4.50× with an identical rational floor. Two defects acted in opposite directions here — a non-converging bisection misses either way, while over-dispersion always inflates a maximum — so the net direction was a measurement, not an argument, and a prediction made before re-running it was wrong.

CORRECTION. The previous edition defined the Bandelt–Dress isolation index as half the minimum of a quadruple's two crossing pairing sums minus its own. That is not the isolation index: the definition takes the maximum, clamped below at zero. The two forms agree on a tree metric, where the crossing sums are equal, and differ exactly where the four-point condition fails — so the error was invisible on precisely the data where the condition holds, and the property used to choose between them (that the index is an internal edge length on a tree) is satisfied by both and so decided nothing. The previous edition is superseded.

The corrected statement is stronger. For sorted pairing sums s1 ≤ s2 ≤ s3 the three splits' indices are (s3−s1)/2, (s3−s2)/2 and 0, so the certificate's margin is twice one index — the middle one — and the four-point condition is exactly its vanishing. The reading is the payoff: a tree metric has exactly one split with positive index, a certificate makes a second positive, and any two distinct splits of a four-set are incompatible. So a quartet certificate is the split-decomposition condition for a split system to fail to be a tree, not an analogue of it — checked as an identity, with quadruples carrying two positive indices and quadruples violating the four-point condition both numbering 47,802 across the grid. The measurement that followed survives unchanged: the supported-split index is identical to five decimal places on a clonal population and on both reticulate populations built from it, so it is blind to the reticulation and tracks divergence instead. The supported index carries the tree; the index a tree forces to zero carries the reticulation.

What edition 9 claimed, now corrected. The closure of Section 12's topological line is replaced by a positive statement about what the certificate was already measuring. Phylogenetic combinatorics assigns each split of a quadruple a Bandelt–Dress isolation index — on a tree metric, the supported split's index is the internal edge's length — and between the two non-minimal splits that index gap is exactly half the pairing-sum gap. So the certificate's margin is twice an isolation-index gap, and the four-point condition is precisely the non-positivity of that gap; both are machine-checked, and neither proof needs a tree, a complex or a metric space. Measuring the split decomposition's own headline quantity then shows where the signal is not: the largest supported-split index is identical to five decimal places on the clonal population and on both reticulate populations built from it, tracking divergence instead. The supported index carries the tree; the gap carries the reticulation.

What edition 8 added. Section 12's ledger now records the closure of its topological line on measured grounds. Re-run at sixty-four taxa, the persistent-homology statistic fires on the two populations containing no reticulation — gap-blocked and saturated — while missing a single recombinant entirely, and no cheap filtration change fixes it: correcting for saturation makes it worse, and restricting to globally comparable columns removes the gap false positive completely (confirming it was the missing-data confound reappearing as topology) but leaves the miss. The section's topological content is therefore the certificate itself, the same theorem evaluated per quadruple, which is what makes complete deletion within the subset available at all.

What edition 7 added. Section 12's comparison table is corrected — the earlier edition judged this book's own statistic against a mismatched null and so understated its specificity — and its ledger now reports two results found since, both against the section's own interest: a published-breakpoint test on SARS-CoV-2 recombinants that held on one population and failed on another, with margins of two and four sites that cannot carry a verdict; and a persistent-homology experiment that is inconclusive because its null produced no cycle in 495 of 495 replicates. The lesson both taught is in the ledger: check that a null varies before believing a p-value derived from it.

What edition 6 added. Section 12 now contains the comparison it previously listed as missing: against the established test that shares its no-clock null, the pairwise homoplasy index. On the five populations where both can be measured the two agree exactly — both specific on the strict-clock and the relaxed-clock null, both detecting all three reticulate populations at every seed. The incumbent is not beaten on power, so the case for a certificate is made for its discipline and its exhibitable witness instead, and the section says so.

What edition 5 corrected in its own apparatus. A review of the surrounding literature, run after the work was done, found four errors in it and they are fixed in the open: the band-edge test used to certify the aligner was not a certificate and had really failed on this project's own data, which inflated a headline count threefold; Buneman's theorem is restated with its actual hypotheses (a nonnegative dissimilarity map, not a metric, because the triangle inequality is a consequence of the four-point condition); a negative control was demoted because recombination in that family is documented rather than merely unexcluded; and the simulator's indel parameters are labelled chosen rather than calibrated, no published viral indel rate having been found.

Contents. Reading a proof about a cell · kinetic proofreading · a no-go theorem · the POLE refutation · mismatch repair · mutation accumulation · variant landscapes · topological genomics · DNA topology · from model to clinic · gaps closed · reticulate evolution. With worked examples, exercises and solutions, a reproducibility appendix naming the command behind every number, a research-directions appendix, and an annotated reference list in which three entries are marked no verified DOI rather than given a plausible-looking string.

Bundle. The PDF, the complete LaTeX sources and build script, the Lean 4 modules behind Section 12's theorems, and the raw JSON output and data provenance behind its numbers — so that the claims can be checked or disputed without trusting this description of them.

Repository: SocrateAI-Scientific-Genetic, release v1.8.0.