Biological Calculus Theory: Toward the Mathematical Language of Living Systems A foundational proposal, conditional theorem, nucleotide-level pilot, second- organism consistency test, and qualified FtsZ candidate application

BIOLOGICAL CALCULUS THEORY | VERSION 4 Biological Calculus Theory: Toward the Mathematical Language of Living Systems A foundational proposal, conditional theorem, nucleotide-level pilot, second- organism consistency test, and qualified FtsZ candidate application Author: Gerald M. Goll ORCID number is 0009-0001-9148-9221 - 10.5281/zenodo.21937838 Affiliation: Independent Researcher, East Brunswick, New Jersey, United States Version: Version 4.0, 14 August 2026 Status: Conceptual and computational manuscript. Not peer reviewed. License: Creative Commons Attribution 4.0 International Revision V4: addition of Escherichia coli FtsZ84 (G105S) as a rigorously bounded candidate application; formalization of a condition- dependent division-output model; addition of a provenance ledger distinguishing measured, database-derived, calculated, fitted, inferred, and assumed quantities. Abstract Biology possesses a molecular alphabet but not yet a complete mathematical language that carries a specified nucleotide change through successive biological levels to a defined functional outcome. This paper proposes Biological Calculus Theory as a staged research program for constructing that language. Its first objective is deliberately limited: derive a reasonably accurate algorithm for predicting the effect of a minimal nucleotide change on one measurable function in a simple, completely sequenced organism. Biological levels are represented by composable deterministic maps and stochastic kernels whose parameters are learned from intervention data. A foundational conditional theorem states how a mutation-induced distribution propagates through such a composition, when an outcome is necessary or impossible, and when it remains probabilistic. A reproducible pilot enumerates every single-nucleotide substitution in the essential phosphoglycerate kinase coding sequence of JCVI-syn3A. Of 3,645 substitutions, 772 are synonymous, 2,700 missense, 155 premature-stop, 9 start- Goll | 1BIOLOGICAL CALCULUS THEORY | VERSION 4 codon changes, 8 stop-loss, and 1 stop-retaining. Two substitutions at coding position 9 convert TAC into TAA or TAG and deterministically reduce the encoded product from 404 amino acids to 2. A retrospective temporal-holdout test in Mycoplasma genitalium G37 examines the G2057A 23S ribosomal mutation. Using mechanistic evidence available before a 2025 experimental report, the framework predicted disruption of a canonical ribosomal pair, an azithromycin minimum inhibitory concentration increase of at least four-fold, and preserved drug-free growth. The later experiment reported an eight-fold azithromycin increase, increases for two additional macrolides, and no detectable drug-free growth defect. This is a consistency demonstration rather than prospective validation. Version 4 also identifies Escherichia coli FtsZ84 (G105S) as a candidate conditional application because published work provides quantitative molecular measurements and temperature- and context-dependent division phenotypes. Those observations do not yet supply a completed FtsZ algorithm; instead, they demonstrate why temperature, time, genetic background, protein abundance, medium, and division-system modifiers must enter the model explicitly. Absolute accuracy, universal organism prediction, cross-species extrapolation, and ancestral reconstruction are not claimed. The immediate empirical test is whether a fitted base-level algorithm predicts withheld mutations with useful calibration and accuracy exceeding declared simpler baselines.