Behavioural Profiles Are Not Heritable Sequences: a Refuted Transfer of Phylogenetic Recombination Detection to Genetic Programming

Evolutionary program synthesis in the AlphaEvolve family applies mutation and selection to populations of programs, usually with a crossover operator whose contribution is asserted rather than measured. Phylogenetics has an exact instrument for that question: a quartet certificate refutes the four-point condition on four taxa with an integer comparison a proof kernel can recheck, and a recombinant genealogy violates that condition where a clonal one does not. This paper proposed transferring the instrument, tested the transfer, and reports its failure together with the reason. The transfer looked clean. A candidate program's behavioural profile — its outcome at each of T probes, drawn from {correct, incorrect, raised, non-finite} with a gap symbol for an inadmissible probe — is a four-state sequence with missing data, exactly the model a nucleotide alignment assumes. A programs×probes matrix is therefore an alignment and the machinery runs unmodified. The null comes for free and is correct rather than convenient: a population varied only by mutation from single parents is a tree by construction. On synthetic profiles it appeared to work, separating clonal from recombinant arms by margin across a usable window of mutation rates, with both extremes unusable for opposite reasons (a degenerate null below, saturation above). On real program populations it fails. Expression trees over a closed grammar, evolved against a rational target, every individual executed at 600 probes. Three pre-registered controls fail in the same direction: the recombinant arm exceeds the clonal floor in 0 of 24 runs, 0 of 23 restricted to comparable best fitness, and 0 of 22 restricted to comparable realised behavioural spread (22 of the 24 were comparable, so this is not a matching artifact). Stratifying all 49 populations by spread explains everything: above a spread of 0.05, 4 of 4 clonal populations carry certificates and so does 1 of 1 recombinant; below 0.01, none of 29 does in either arm. Behavioural spread accounts for certificate presence and arm membership adds nothing — the clonal arm, which cannot recombine, carries them at the same rate. Crossover also bought no fitness: clonal median best error 0.5707 against recombinant 0.6028, best individual identical in both arms. The diagnosis indicts the proposal, not the implementation. The four-point condition detects non-tree-like structure in a distance matrix; it detects recombination only when characters are independently heritable units, because recombination's signature is mosaic ancestry along the sequence. A probe outcome is not inherited — it is computed from the whole program — so two unrelated point mutations routinely alter overlapping probe sets and manufacture the same pattern. The synthetic validation was circular: its profiles were generated by mutating profile strings position by position, which builds site-wise heritability in by construction, so it confirmed that the method detects recombination in site-wise-heritable data — never in doubt — while appearing to validate an application where that property fails. What survives. The proof-carrying fitness ladder, whose upper rungs are Lean 4 obligations read through #print axioms so that no empirical gain buys a rung and sorry cannot pass, is untouched by this result. Three measurement guards earned their place by catching this rather than ratifying it: that a null must be able to vary, that a floor must be shown converged, and that matching must be on the realised nuisance rather than the nominal budget — the last of which produced the control without which the failure would plausibly have been dismissed. Dataset. 49 evolved program populations (25 clonal, 24 recombinant across four crossover shares), each with its behavioural profile matrix in FASTA, violating-quadruple counts, exact margins, realised spread, distinct-profile counts, best fitness, and the best program rendered as an expression. Useful to anyone asking whether a crossover operator contributes, independently of the method refuted here. Transferable negative: before carrying a phylogenetic method to a non-biological population, ask what it assumes about heritability of characters, not merely about the alphabet. Matching the data model was necessary and nowhere near sufficient. All figures are produced by committed code and stored artifacts; every one of 26 quantitative claims was machine-verified against its artifact before deposit. Not peer reviewed.