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 programme rests on five CoreCore foundation papers. All DOIs below are Concept-DOIs that always resolve to the latest version on Zenodo.
| Master | Title | Role | Concept-DOI |
|---|---|---|---|
| Zookeeper | The Spectral Zookeeper | RH proof via CCM microcluster closure | 10.5281/zenodo.19673126 |
| Zeta Zoo | The Zeta Zoo — The Mathematical Side of FST | Classification (SGE taxonomy, Boundary Theorem) | 10.5281/zenodo.19673226 |
| Spectrum Duality | FST Spectrum Duality / RFEP | Physical instantiation (Pattern A, DS1–DS3) | 10.5281/zenodo.19036190 |
| Atlas | Dirichlet Character Atlas | Micro-cartography (Galerkin diagnostics; negative method validation) | 10.5281/zenodo.19960809 |
| Selberg | NE-B Failure as Hilbert–Pólya Detection | SGE-YES validation (v2.0 universality on Selberg zeta) | 10.5281/zenodo.19962588 |
Atlas + Selberg form the method-validation pair: Atlas is the negative test (leading-order Galerkin diagnostics fall short for Dirichlet characters), Selberg is the positive test (v2.0 reproduces a classical operator-based result on Selberg zeta).
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 | Concept-DOI |
|---|---|---|---|---|
| BSD | v1.2 candidate | Reformulation; rank ≤ 1 verified | 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 |
Derive Pattern A + DS1–DS3 from Spectrum Duality. These instantiate the Dissipative Selection Principle on physical systems.
| Paper | Version | Status | Open Problem | Concept-DOI |
|---|---|---|---|---|
| Turbulence | v1.4 candidate | Journal-ready | DFC1 empirical (only input) | 10.5281/zenodo.19056813 |
| Yang–Mills | v2.2 | Conditional | Strong-coupling lattice gap; continuum transfer conditional | 10.5281/zenodo.19087433 |
| Navier–Stokes | v2.2 candidate | 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 |
The cosmological branch of FST. The Dark Energy paper instantiates Pattern A on cosmological screening mechanisms (Hu–Sawicki f(R) gravity).
| Paper | Version | Status | Open Problem | Concept-DOI |
|---|---|---|---|---|
| Dark Energy | v1.6 | Framework Note | Hu–Sawicki parameters quantitatively open | 10.5281/zenodo.19036235 |
In development. See fst-biology/ and the private companion repo research-line/fst-nash (protein folding via Nash equilibrium).
Planned. See fst-chemistry/.
| Term | Meaning |
|---|---|
| v2.0 | Method package developed in the RH programme (Trilogy v2.1, 10.5281/zenodo.19035640): reduces RH to even dominance of the Weil quadratic form QW_λ via four ingredients — Shift Parity Lemma, frontier-prime dominance, NE-A, NE-B. |
| NE-A | Non-existence theorem A. The Fourier multiplier of the prime shift operator A_λ on the critical line is non-positive — cannot serve as a Hilbert–Pólya operator. |
| NE-B | Non-existence theorem B. No universal symmetric operator commutes with all Shift-Parity difference matrices D_N(r) (computer-assisted proof for N ≤ 15). Together with NE-A this rules out the classical Hilbert–Pólya route — and is exactly why v2.0 is needed for Riemann. |
| SGE | Semigroup–Group Equivalence. Classification axis of the Zeta Zoo: HP-BL-YES (commuting operator exists, e.g. Selberg/Casimir), HP-BL-NO (commutant blocked, Riemann), HP-BL-OPEN (undecided, e.g. Prime-Hub). |
| Weil quadratic form QW_λ | Truncated explicit-formula quadratic form whose positivity controls zero locations. Universal across the zeta zoo; the operator behind it is family-dependent (and may be absent — see NE-B). |
| Hilbert–Pólya | Conjecture that the Riemann zeros are eigenvalues of a self-adjoint operator. v2.0 generalises this: where Hilbert–Pólya works (SGE-YES), v2.0 reproduces it; where it fails (NE-B / Riemann), v2.0 still applies. |
| Pattern A | Functional Positivity under a Gauge Constraint — the universal stability pattern of FST. |
| RFEP | Renormalized Free-Energy Principle. Mathematical core principle of FST; supplies DS1–DS3. |
| CCM | Connes–Consani–Moscovici. Fourier model for the Weil quadratic form used in the Zookeeper proof. |
| UCU | Universal Convexity Uniqueness lemma. Together with SGE and Weil, the trinity of meta-principles governing the zeta-type branch. |
FIVE MASTERS
┌────────────────┬────────────────┬─────────────────┬────────────────┐
│ │ │ │ │
Zookeeper Zeta Zoo Spectrum Duality Atlas Selberg
(RH proof) (Classification) (Pattern A, (Dirichlet, (NE-B
SGE / UCU / DS1–DS3, negative failure;
Weil QW_λ) RFEP) method test) SGE-YES
│ │ │ │ │
│ FST-Mathematics FST-Physics method-validation pair
│ │ │
│ ┌───────┼───────┐ ┌─────┼─────┐
│ │ │ │ │ │ │
│ BSD† Hodge† PvNP† TU YM NS
│ │
│ FST-Cosmology
│ │ NS-LDI
│ DE
│
Status: PROVEN (unconditional, CCM route)
† = bridge species (math + physics)
FST (Functional Stability Theory)
│
├── Masters
│ ├── Zookeeper RH proof (CCM microcluster closure)
│ ├── Zeta Zoo Mathematical classification (SGE taxonomy)
│ ├── Spectrum Duality Physical instantiation (RFEP, Pattern A)
│ ├── Atlas Micro-cartography of Dirichlet (negative method test)
│ └── Selberg SGE-YES method validation (positive)
│
├── 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)
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
Atlas (Dirichlet, negative) + Selberg (SGE-YES, positive)
method-validation pair completed
|
v
CoreCore expanded to FIVE Masters
(Zookeeper, Zeta Zoo, Spectrum Duality,
Atlas, Selberg) + domain supplements
| Level | Name | Acronym | Meaning | Concept-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.19036190 |
| Pattern | Pattern A: Functional Positivity under Gauge Constraint | Pattern A | The universal stability pattern | 10.5281/zenodo.19036190 |
| Name | Role | Concept-DOI |
|---|---|---|
| RH Even Dominance v2.1 (Trilogy, Part I-III) | Independent RH proof, second route | 10.5281/zenodo.19035640 |
| RH Direct Proof (Even Dominance) | Direct frontier-dominance route | 10.5281/zenodo.19764771 |
| 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.
| Script | Paper | Description |
|---|---|---|
masters/zookeeper/scripts/ |
Zookeeper | CCM microcluster-closure reproducibility pipeline; compact outputs in masters/zookeeper/results/ |
scripts/turbulence/compute_F_spectrum.py |
Turbulence | K41 as unique minimiser of F[E]; strict convexity test |
scripts/turbulence/compute_goy_shell_dfc.py |
Turbulence | Sabra/GOY shell-model DFC1/DFC2 verification and result plot |
scripts/yang-mills/compute_dobrushin_su2.py |
Yang-Mills | SU(2) lattice Dobrushin influence scan and gap plot |
scripts/yang-mills/compute_birkhoff_rg.py |
Yang-Mills | Birkhoff contraction scan for hierarchical RG steps |
scripts/navier-stokes/compute_ds3_lorenz.py |
Navier-Stokes | DS3 stress test on Lorenz attractor; TV saturation |
scripts/navier-stokes/compute_bv_selection.py |
Navier-Stokes | Balanced-viscosity selection test on the Lorenz attractor |
scripts/navier-stokes/compute_bv_multi_attractor.py |
Navier-Stokes | BV-selection stress test on Lorenz, Roessler, and Chen attractors |
scripts/navier-stokes/compute_mu_reach.py |
Navier-Stokes | Measure-theoretic reach scan on Lorenz and KS attractors |
scripts/navier-stokes/compute_tll_ldi_lorenz.py |
NS-LDI | Proof of Life: TLL+LDI on Lorenz attractor (5/5 tests) |
scripts/navier-stokes/compute_tll_ldi_ks.py |
NS-LDI | TLL+LDI diagnostics and grid refinement on the KS attractor |
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/bsd/compute_bsd_verification.py |
BSD | BSD formula sanity checks for selected LMFDB curves |
scripts/bsd/compute_rank2_lmfdb.py |
BSD | Rank-2 regulator positivity sample and plot |
scripts/hodge/compute_ghr_spectrum.py |
Hodge | GHR spectrum numerical verification |
scripts/hodge/compute_voisin_test.py |
Hodge | Voisin-style negative-control stress test |
scripts/p-vs-np/compute_sat_entropy.py |
P vs NP | SAT slice-entropy experiment and result plot at the 3-SAT phase transition |
scripts/zeta-zoo/dedekind_ne_b_test.py |
Zeta Zoo | Dedekind Q(sqrt(-5)) NE-B analog probe |
scripts/zeta-zoo/ihara_petersen_sge_test.py |
Zeta Zoo | Ihara/Petersen SGE YES-side test |
scripts/zeta-zoo/sge_control_experiment.py |
Zeta Zoo | SGE YES/NO discriminating control experiment |
masters/atlas/scripts/ |
Atlas | Galerkin computation pipeline (35 scripts: basis, κ-grid, asymptotic scans, χ-specific tests) |
functional-stability-theory/
├── masters/ Five CoreCore foundation papers
│ ├── zookeeper/ RH proof (microcluster closure)
│ ├── zeta-zoo/ Classification (SGE taxonomy)
│ ├── spectrum-duality/ Physical axioms (RFEP, Pattern A)
│ ├── atlas/ Dirichlet micro-cartography (negative method test)
│ └── selberg/ NE-B failure as HP detection (SGE-YES validation)
├── 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 (per-paper subdirectories)
- rh-even-dominance — Riemann Hypothesis: Even-dominance proof (foundation)
- crm-cosmology — Cooperative Renormalization Model (foundation)
- fst-nash — Protein folding via Nash equilibrium (FST-Biology, private)
Lukas Geiger — ORCID: 0009-0005-7296-1534