Functional Stability Theory III: Biological Stability and Nash Frustration
Abstract
A conjectural framework for biological organization spanning molecular to ecosystem scales, organized around thermodynamic game theory and the Maximum Entropy Production Principle (MEPP). The paper examines whether many biological stability problems can be modeled as dissipative structures whose attractors admit Nash-style fixed-point descriptions once players, strategies and payoffs are specified at the relevant scale. The framework integrates England's dissipative adaptation theory, Friston's Free Energy Principle, Kauffman's Boolean network dynamics and classical evolutionary game theory into a working hierarchy. As a concrete application, we introduce Nash frustration for protein structures and report a preliminary, self-parametrized single-protein consistency check against NMR chemical shift perturbation data (Spearman rho_S = 0.44, p = 0.033, n = 24), not an externally calibrated predictor. The normalized residue ranking is eta-independent in the checked local scan window; absolute rho(J)-based scales remain open calibration targets. These exploratory outputs are not new biological validation. Series position FST-III is the biology / evolutionary stability companion (Level 1c) of the FST application series. The programmatic umbrella is the FST Hub (Concept-DOI 10.5281/zenodo.20130499). Sibling application companions: FST-I Particles, FST-II Chemistry, FST-IV Cosmology. Code & sources GitHub: research-line/functional-stability-theory / applications/fst-iii-biological. Corrective scope, version 1.7 (October 2026). The historical name Nash frustration denotes a step-dependent Hessian/gradient-surrogate diagnostic, not a Nash-equilibrium or kinetic-stability certificate. The synchronous derivative I−ηH differs from the sequential approximate-response routine. Local unilateral optimality, global best responses, stationarity, and algorithmic stability must be checked separately. In the cited MFG setting, existence and monotonicity-based uniqueness have distinct hypotheses and do not establish convergence or biological validity for the protein model. Zusammenfassung der Korrektur. Der Hesse-basierte Gradientensurrogatscore zertifiziert weder Nash-Status noch kinetische Stabilität. Existenz, Eindeutigkeit und algorithmische Konvergenz sind getrennt nachzuweisen. Historische Proteinbefunde bleiben explorativ; native HP35-Stationarität, tatsächliche Sweep-Ableitung, proteinspezifische PDE-Regularität und vollständige Claim-/Quellenprüfung bleiben offen. Changes in Version 1.7 (October 2026) Corrects the R119 Nash/spectral inference: I−ηH is the synchronous gradient-surrogate derivative, not the sequential approximate-response Jacobian. Expansion or proximity to the unit circle does not decide local/global Nash status. Separates own residue blocks, stationarity, local versus global unilateral tests, step overshoot, and the >1 / ≥1 boundary convention. Historical numerical outputs are retained without recomputation or upgraded predictive claims. Separates MFG existence, monotonicity-based uniqueness, and algorithmic convergence; removes unproved attraction/repulsion, native-fold and prion shortcuts. Protein-specific PDE assumptions remain unchecked. Synchronizes corrective DE/EN text and primary-source citations, repairs the overlapping hierarchy diagram, and checks Unicode umlauts and PDF layout. Ships separate English and German PDFs. No new combined PDF is built; older versions retain their historical combined files. This release repairs selected published statements. It does not certify the whole paper, native HP35 biology, the previous proof proposals, or journal readiness. Changes in Version 1.6 (July 2026) QUANT-III-2b non-identifiability lemma: added mathematical proof that per-protein max-normalized scores s_{p,i} = r_{p,i} / M_p cannot identify raw scale M_p or a universal threshold T across proteins; absolute scoring remains a calibration target. Negative diagnostic gate: continuous linear mapping of HP35 scores to BMRB 4428 CSP values yields negative leave-one-out Q^2 = -0.176, confirming that Spearman rank correlation (rho_S = 0.44) remains logically separate from continuous absolute prediction. Cross-protein threshold control: cross-application of Youden thresholds between HP35 and Protein G yields low-sensitivity performance (sens. 0.21--0.42), confirming that functional label thresholds do not transfer across proteins without matched holdouts. German claim hygiene & DE/EN sync: German abstract, introduction, Pattern-A items and cosmology handoff aligned with the English lead version; German text extraction preserves real umlauts. PDF design-check & table layout: English, German and combined PDFs rebuilt; table column widths, line wrapping and ragged-right L-specifiers aligned across bilingual tables. Bilingual section-parity completion (August 2026): eight discussion subsections (punctuated equilibrium, stability vs. maximization, MEPP, Nash frustration beyond the Hessian, consolidated limitations, claim status, falsifiable predictions, “is biology just physics?”) and the synthesis matrix translated and integrated into the German edition; AlphaFold-era discussion extended with recent deep-learning limitations and additional references (including Kuzu 2025); full BibTeX key parity between both language versions. English typography pass (August 2026): non-breaking spaces before all citations, native UTF-8 text layer, table of contents added; English, German and combined PDFs rebuilt (39/35/74 pages). No claim upgrade: no new empirical validation or mathematical strengthening; unnormalized multi-protein holdouts and head-to-head controls remain future validation gates. Version 1.3 (May 2026) Strict-review claim guardrails: Abstract, German translation, computational protocol, Cross-Protein section, claim-status table and metadata were revised to mark the framework as conjectural and the HP35/NMR result as a preliminary single-protein consistency check. Eta interpretation clarified: normalized residue rankings are eta-independent only in the checked local scan window with an unchanged unstable active set; absolute score scales and rho(J)-based thresholds remain calibration targets. Coordinate-metric caveat added: residue-level Nash frustration is identified as a local Euclidean (phi, psi)-chart diagnostic; an intrinsic or mass-weighted variant remains future work. English-style calibration: Abstract, introduction, Pattern-A/RH analogies, FEP--MEPP bridge and MFG scope conditions were further calibrated to candidate, heuristic and structural-analogy language. Citation metadata maintenance: Maynard Smith--Price citation anchoring, Ferreiro et al. labels, the Hamilton Part-I title and DOI metadata for 20 classical references were corrected or completed. Unicode text-map maintenance: EN/DE sources now use lmodern plus explicit glyph-to-Unicode mapping so German PDF text extraction preserves real umlauts and ß. 2XWR provenance correction: the active paper table now reports the 2XWR structure used here as 199 modeled residues. The table uses the best-response prediction source (`results\2XWR` / server log); the extended-result JSON reports a separate Nash-Stability metric and is documented separately. Technical PDF maintenance: German PDF metadata were regenerated to remove pdfinfo mojibake; the combined bilingual PDF was regenerated with explicit merge metadata. Design check: Frontmatter, tables, formulas, links, bibliography and closing pages were visually spot-checked; no layout or content issue was found. Files rebuilt: EN, DE and combined PDFs rebuilt from the active v3 source set. Version 1.2 (May 2026) Source-check correction: the villin/BMRB reference now uses the correct Vardar et al. DOI 10.1006/jmbi.1999.3321; the previously listed DOI 10.1006/jmbi.1999.3363 belonged to an unrelated Holliday-junction paper. BMRB provenance clarified: BMRB 4428 is identified as the HP67/1QQV villin-headpiece NMR dataset, mapped to the C-terminal HP35/1YRF subdomain for the benchmark; BMRB DOI 10.13018/BMR4428 added. Self-citation metadata updated: FST-I, FST-II, RH Landscape, RH Even Dominance and RFEP/Spectrum Duality references now use stable Zenodo Concept-DOIs. Files rebuilt: EN, DE and combined v3 PDFs generated after the source check. Version 1.1 (May 2026) Hub-link corrected: "FST-IV Overview" references replaced by "FST Hub" (programmatic umbrella). The Hub paper holds Concept-DOI 10.5281/zenodo.20130499 and is no-numeral; FST-IV is now the cosmological collector slot. GitHub source link added. Related Identifier: FST-DE reference relabelled to "FST-IV Cosmology / FST-DE". Version 1.0 (May 2026) Initial release as part of the FST series. Status: ~7.5/10 readiness; remaining open items per referee moderate-revision verdict: absolute score/threshold calibration (QUANT-III-2 via BMRB 4428 + McKnight 1996 protection factors), cosmology-section relocation, TP53 mutational analysis with calibrated score/threshold logic, falsifiable predictions with quantitative thresholds, MEPP-as-heuristic framing.
