Functional Stability Theory is a unified mathematical framework that connects thermodynamic stability, game-theoretic equilibrium selection, and renormalized energy functionals to derive rigorous results across multiple domains of mathematical physics and complexity theory.
The central object is a renormalized free-energy functional whose critical-point structure, combined with selection principles from evolutionary game theory, yields domain-specific theorems when instantiated with the appropriate state spaces and energy densities. The framework is governed by the Dissipative Selection Principle (DSP): among all critical points of the renormalized functional, physical evolution selects the unique dissipation-minimising attractor -- a meta-theorem that unifies the domain-specific results below.
- framework/ -- The Unified Renormalized Energy Framework (meta-level paper)
- domain-proofs/ -- Domain-specific instantiations and proofs
- applications/ -- Empirical applications (FST-I Particles, FST-II Chemistry, FST-III Biology)
- scripts/ -- Numerical validation scripts
- fst_references.bib -- Shared bibliography
| Paper | Version | Status | Open Problem | DOI |
|---|---|---|---|---|
| Turbulence | v1.3 | Journal-ready | DFC1 empirical (only input) | 10.5281/zenodo.19056813 |
| Dark Energy | v1.6 | Framework Note | Hu–Sawicki parameters quantitatively open | 10.5281/zenodo.19036235 |
| Yang–Mills | v2.1 | Conditional | Analytical proof of λ < 0 | 10.5281/zenodo.19087433 |
| Navier–Stokes | v2.1 | Conditional | Assumption G2 (projection regularity) | 10.5281/zenodo.19087449 |
| NS Log-Distance | v1.3 | Proof of Life ✓ | TLL for 3D NS analytically open | 10.5281/zenodo.19056807 |
| BSD | v1.1 | Reformulation | Higher Gross–Zagier (rank ≥ 2) | 10.5281/zenodo.19087443 |
| Hodge | v1.1 | No-Go Theorem | = Deligne's question (1982) | 10.5281/zenodo.19087439 |
| P vs NP | v1.2 | Reformulation | Uniformity Bridge | 10.5281/zenodo.19056809 |
| Framework (RFEP) | v1.6 | Meta-Theorem | Pattern A falsifiability clarified | 10.5281/zenodo.19036190 |
Framework (Pattern A)
+-- DS1-DS3 (Dissipative Selection)
+-- Second-Order Resolvent Dominance
|
+--------------+--------------+
| | |
PROVEN CONDITIONAL OPEN
(rigorous) (reduced to (= open
threshold axiom) research)
| | |
TU: YM: BSD:
DFC => NL' Doeblin a > 0 Rank >= 2
(journal) (Kingman l < 0) Gross-Zagier
|
DE: NS: Hodge:
Screening G2 (projection) Deligne's
(validated) G3 (Gronwall) question
|
NS-LDI: PvNP:
TLL + LDI Uniformity
(Lorenz OK) Bridge
The scripts/ directory contains computational validation scripts:
| Script | Paper | Description |
|---|---|---|
scripts/turbulence/compute_F_spectrum.py |
Turbulence | Verifies K41 as unique minimiser of F[E]; strict convexity test |
scripts/navier-stokes/compute_ds3_lorenz.py |
Navier-Stokes | DS3 stress test on Lorenz attractor; TV saturation |
scripts/navier-stokes/compute_tll_ldi_lorenz.py |
NS-LDI | Proof of Life: TLL+LDI on Lorenz attractor (5/5 tests passed) |
scripts/dark-energy/compute_w_vs_desi.py |
Dark Energy | w_eff(z) comparison with DESI constraints |
scripts/dark-energy/compute_w_mapping.py |
Dark Energy | Correct w_eff -> w_DE mapping + DESI grid scan |
scripts/dark-energy/compute_husawicki_mcmc.py |
Dark Energy | Hu-Sawicki f(R) MCMC fit against DESI+Planck+Cassini |
scripts/bsd/compute_height_saturation.py |
BSD | Height saturation test for quadratic twists |
scripts/hodge/compute_ghr_spectrum.py |
Hodge | GHR spectrum numerical verification |
2025-2026 RH Trilogy CRM I-IV
(self-contained) (self-contained)
| |
v v
+-----------+ +-----------+
| Riemann | | Cosmic |
| Hypothesis| | Recursion |
| Part I-III| | Model I-V |
+-----------+ +-----------+
| |
+----------+----------+
|
Abstraction / Generalisation
|
v
+---------------------+
| RFEP Framework |
| (Renormalized Free- |
| Energy Principle) |
| = Connecting Link |
+---------------------+
|
Instantiation / Application
|
+----------+----------+
| |
v v
+------------------+ +------------------+
| Domain Proofs | | Applications |
+------------------+ +------------------+
| NS, YM, TU, DE | | FST-I Thermo |
| Hodge, BSD, PNP | | FST-II Chemical |
+------------------+ | FST-III Biology |
+------------------+
|
under the umbrella name
|
v
+---------------------------+
| FST = Functional |
| Stability Theory |
| (programme name, came LAST)|
+---------------------------+
RH ──────────> RFEP Framework ──────────> FST Domain Proofs
(proven) (abstracted from RH) (instantiate RFEP)
|
CRM ──────────> RFEP Framework ──────────> FST Applications
(model) (confirmed by CRM) (validate RFEP empirically)
Arrows indicate logical dependency, not chronological order:
- RH stands on its own without RFEP/FST
- RFEP references RH as its "Reference Instantiation"
- FST references RFEP as its theoretical foundation
- Applications confirm RFEP predictions (not the other way around)
| Level | Name | Acronym | Meaning | DOI |
|---|---|---|---|---|
| Principle | Renormalized Free-Energy Principle | RFEP | The core mathematical principle | 10.5281/zenodo.19036190 |
| Pattern | Pattern A: Second-Order Resolvent Dominance | Pattern A | The universal stability pattern | 10.5281/zenodo.19036190 |
| Programme | Functional Stability Theory | FST | The programme name (umbrella over all) | -- |
| Foundation | Riemann Hypothesis Proof | RH | Self-contained proof, reference instantiation | 10.5281/zenodo.19035640 |
| Foundation | Cosmic Recursion Model | CRM | Self-contained model | 10.5281/zenodo.18728935 |
- rh-even-dominance -- Riemann Hypothesis: Even-dominance proof (foundation)
- crm-cosmology -- Curvature Relaxation Model (foundation)
- rfep-framework -- archived, now integrated here under
framework/
Lukas Geiger -- ORCID: 0009-0005-7296-1534
This work is licensed under CC-BY-4.0.