Fundamental Laws of Biology

Fundamental Laws of Biology

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.

Definitions and notation.

A mathematical law for a biological model is a relation between specified state variables, parameters and their evolution, accompanied by a domain of validity. Universality would mean applicability across a declared class of biological systems. The challenge asks whether principles with explanatory and predictive roles comparable to physical laws can be identified; the system class and standard of validation remain to be formulated.

Catalogue wording retained for traceability.

Identify governing principles for biological systems, analogous to physical laws.

Scope and source qualification.

A programme, not a quantified conjecture: neither the class of biological systems nor a terminal success criterion is fixed. The historical DARPA mirror was unavailable during this check.

Short English statement

Find mathematical principles that govern broad classes of biological systems.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Proved mean-rate monotonicity for fixed-rate replicator dynamics and explicitly disproved its extension to arbitrary common environmental variation. No universal biological law is established.

Sources and literature check.

Read the supplied complete definition, catalogue wording, scope qualifications and source list for this canonical ID. The original-source citations are inherited as supplied references; no fresh primary-source or novelty search was performed for this attempt. No current open-status claim is inferred from this calculation.

Testing a candidate universal monotonicity law.

Let $x_i(t)>0$, $1\le i\le m$, be frequencies with $\sum_i x_i=1$, and assign fixed reproductive rates $a_i\in\mathbb R$. The replicator equations are $\dot x_i=x_i(a_i-\bar a)$, where $\bar a=\sum_i x_i a_i$. Summing proves preservation of total frequency; the explicit solution is $x_i(t)=x_i(0)e^{a_it}/\sum_jx_j(0)e^{a_jt}$, so positivity is preserved. Direct differentiation gives \[\dot{\bar a}=\sum_i x_i a_i^2-\bar a^2=\sum_i x_i(a_i-\bar a)^2\ge0.\] Thus average reproductive rate increases in this sharply specified constant-environment model, and equality occurs precisely when all currently supported types have equal rates.

An explicit failure beyond that class.

Permit a common environmental decline $a_i(t)=b_i-t$. The common $-t$ cancels from all frequency equations, so frequency trajectories are exactly those generated by the constants $b_i$. Nevertheless \[\bar a(t)=\sum_i x_i(t)b_i-t,\qquad \dot{\bar a}=\operatorname{Var}_{x(t)}(b)-1.\] Choosing all $b_i=0$ gives constant frequencies and $\bar a(t)=-t$, strictly decreasing. Even choosing unequal rates in an interval of length less than one makes the variance at most one quarter, by centering at the interval midpoint, so the average rate still decreases. The failure is not caused by extinction of types or a boundary singularity.

Programme-level obstruction.

A frequency law alone is invariant under adding the same time-dependent rate to every type; absolute mean growth is not. Consequently one cannot infer a universal mean-fitness increase principle from frequency selection dynamics without fixing an environmental convention and admissible coupling. This countercheck narrows a natural candidate law but does not identify the requested general biological principles. The system class and validation standard, left open by the supplied source, remain essential missing inputs; the unavailable historical mirror has not been independently recovered.

Verification and adversarial audit.

Checked simplex preservation, positive explicit solution, equality case and a smooth interior counterexample; the counterexample concerns a proposed extension, not the broad programme itself. This was an LLM adversarial audit, not independent human verification; no theorem prover was used. Prompts A then B applied to this ID: Checked simplex preservation, positive explicit solution, equality case and a smooth interior counterexample; the counterexample concerns a proposed extension, not the broad programme itself. Scope remains limited: Define a biologically meaningful model class and environmental normalization with predictive validation beyond the toy selection equation.

Coordinator checked the variance identity, explicit common-environment counterexample and bounded-range variance estimate. Model-dependent fitness monotonicity is not a universal biological law. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.

Python 3 and local shell used to read supplied files and serialize this manuscript; mathematics derived symbolically in the text, with no numerical experiment or formal proof assistant. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.

Final status and first remaining gap.

UNRESOLVED. Define a biologically meaningful model class and environmental normalization with predictive validation beyond the toy selection equation.

Completion Estimate

COMPLETION: 0% The calculation is an elementary benchmark or conditional reduction; it establishes no claimed new portion of the complete source target.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath, record 75 (DARPA-023). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 DARPA DSO, Mathematical Challenges, BAA 07-68 (catalogue attribution: 2007), Challenge 023. Historical source mirror; unavailable in this check. https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html.