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Functional Stability Theory (FST)

Functional Stability Theory is a unified mathematical programme that identifies a single structural challenge — Functional Positivity under Gauge Constraint (Pattern A) — as the common substrate of open problems in number theory, mathematical physics, and cosmology.

The Three Masters

The programme rests on three foundation papers:

Master Title Role DOI
Zookeeper The Spectral Zookeeper RH proof via CCM microcluster closure 10.5281/zenodo.19673127
Zeta Zoo The Zeta Zoo — The Mathematical Side of FST Classification (SGE taxonomy, Boundary Theorem) 10.5281/zenodo.19673227
Spectrum Duality FST Spectrum Duality / RFEP Physical instantiation (Pattern A, DS1–DS3) 10.5281/zenodo.19162705

Domain Supplements

FST-Mathematics

Classified by the SGE taxonomy from the Zeta Zoo. These instantiate Pattern A on number-theoretic and algebraic structures. BSD, Hodge, and P vs NP are bridge species — they appear in both the mathematical and physical branches of the programme.

Paper Version Status Open Problem DOI
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

FST-Physics

Derive Pattern A + DS1–DS3 from Spectrum Duality. These instantiate the Dissipative Selection Principle on physical systems.

Paper Version Status Open Problem DOI
Turbulence v1.3 Journal-ready DFC1 empirical (only input) 10.5281/zenodo.19056813
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

FST-Cosmology

The cosmological branch of FST. The Dark Energy paper instantiates Pattern A on cosmological screening mechanisms (Hu–Sawicki f(R) gravity). A dedicated CRM paper re-examining the Cooperative Renormalization Model from the FST perspective is planned.

Paper Version Status Open Problem DOI
Dark Energy v1.6 Framework Note Hu–Sawicki parameters quantitatively open 10.5281/zenodo.19036235

FST-Biology

In development. See fst-biology/.

FST-Chemistry

Planned. See fst-chemistry/.

Proof Architecture

                        THREE MASTERS
     ┌──────────────────────┼──────────────────────┐
     │                      │                      │
  Zookeeper            Zeta Zoo            Spectrum Duality
  (RH proof)        (Classification)      (Pattern A, DS1-DS3)
     │                      │                      │
     │              FST-Mathematics          FST-Physics
     │                      │                      │
     │              ┌───────┼───────┐        ┌─────┼─────┐
     │              │       │       │        │     │     │
     │            BSD†    Hodge†  PvNP†      TU    YM    NS
     │                                                   │
     │                                      FST-Cosmology│
     │                                            │    NS-LDI
     │                                           DE
     │
  Status: PROVEN (unconditional, CCM route)

  † = bridge species (math + physics)

Detailed Proof Status

  PROVEN (rigorous)     CONDITIONAL (threshold)     OPEN (= research)
       │                       │                         │
      TU:                     YM:                      BSD†:
   DFC⇒NL'                Doeblin                   Rank ≥ 2
   (journal-ready)        α(β)>0                    Gross-Zagier
                              │
      DE:                    NS:                     Hodge†:
   Screening              G₂ (proj.)               Deligne's
   (validated)            G₃ (Gronwall)            question
                              │
     NS-LDI:                PvNP†:
   TLL+LDI                Uniformity
   (Lorenz ✓)              Bridge

  † = bridge species

Hierarchy

FST (Functional Stability Theory)
│
├── Masters
│   ├── Zookeeper         RH proof (CCM microcluster closure)
│   ├── Zeta Zoo          Mathematical classification (SGE taxonomy)
│   └── Spectrum Duality  Physical instantiation (RFEP, Pattern A)
│
├── FST-Mathematics       BSD, Hodge, P vs NP
├── FST-Physics           Turbulence, Yang–Mills, Navier–Stokes, NS-LDI
├── FST-Cosmology         Dark Energy
├── FST-Biology           (in development)
└── FST-Chemistry         (planned)

Chronological Development

2025/2026  CRM I–IV (dark energy)     RH "light" proof (even dominance)
           developed independently    developed independently
                 \                       /
                  \                     /
                   +---------+---------+
                             |
                   Recognition: both share the same
                   structural pattern (Pattern A)
                             |
                             v
                   RFEP formulated (general principle)
                             |
                   Several dead ends
                             |
                             v
                   Idea: classify zeta-type families
                   using techniques from the RH proof
                             |
                   Not enough — need deeper tools
                             |
                             v
        RH via Connes framework (CCM)
        microcluster closure → unconditional proof
                             |
                             v
                   Zeta Zoo opens: SGE taxonomy
                   classifies all zeta families
                             |
                             v
                   Reorganization under FST
                   (Three Masters + five domain branches)

Core Foundations — Principles and Naming

Level Name Acronym Meaning DOI
Programme Functional Stability Theory FST The programme name (umbrella over all) —
Principle Renormalized Free-Energy Principle RFEP The mathematical core principle 10.5281/zenodo.19162705
Pattern Pattern A: Functional Positivity under Gauge Constraint Pattern A The universal stability pattern 10.5281/zenodo.19162705

Independent Foundations

Name Role DOI
RH Even Dominance (v2.1) Independent RH proof, second route 10.5281/zenodo.19546593
CRM Cosmology (I-V) Independent dark energy model 10.5281/zenodo.18728935

These stand independently of FST. The RFEP was abstracted from them; they are not derived from it.

Numerical Validation Scripts (Domain Supplements)

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 FST-Physics: 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

Repository Structure

functional-stability-theory/
├── masters/                    Three foundation papers
│   ├── zookeeper/              RH proof (microcluster closure)
│   ├── zeta-zoo/               Classification (SGE taxonomy)
│   └── spectrum-duality/       Physical axioms (RFEP, Pattern A)
├── fst-mathematics/            Domain supplements — Mathematics
│   ├── bsd/                    Rank-1 positivity (reformulation)
│   ├── hodge/                  No-go + easy direction
│   └── p-vs-np/                Witness entropy gap (reformulation)
├── fst-physics/                Domain supplements — Physics
│   ├── turbulence/             K41 spectrum (unconditional)
│   ├── yang-mills/             Mass gap (conditional)
│   └── navier-stokes/          Regularity + NS-LDI (conditional)
├── fst-cosmology/              Domain supplements — Cosmology
│   └── dark-energy/            CRM screening (validated)
├── fst-biology/                Domain supplements — Biology (in development)
├── fst-chemistry/              Domain supplements — Chemistry (planned)
└── scripts/                    Numerical validation

Related Repositories

Author

Lukas Geiger -- ORCID: 0009-0005-7296-1534

License

CC-BY-4.0