Plain · open preprint
The Theory of Persistence
When a system passes through a constraint,
what disperses — what persists?
Everything says what it is by revealing what it no longer has. Reality exists in the cone of uncertainty between addition and multiplication.
Everything says what it is (its additive position) by revealing what it no longer has (its deferred factorisation — what was dispersed). Reality exists in the cone of uncertainty between addition and multiplication; the primes are the fixed points of that tension.
Everything says what it is (its additive position) by revealing what it no longer has (its deferred factorisation — what was dispersed). Reality exists in the cone of uncertainty between the additive structure (Σ ℤ) and the multiplicative structure (Π primes); the PT sieve identifies the fixed points, and the active backbone {2, 3, 5, 7, 11, 13} is its observable signature.
Persistence Theory (PT) starts from a simple idea: when a constraint acts, not everything disappears. One part disperses as entropy; another crosses the constraint and becomes structure. What persists is not what is strongest. It is what is most compatible with the logic of the filter. Persistence is not a victory. It is an agreement. Form is not decoration: it is the structure made visible by constraint. Observables measure that distribution.
Persistence Theory (PT) starts from a simple idea: under constraint, one part disperses as entropy while another persists as structure. What persists is not what is strongest. It is what is most compatible with the logic of the filter. Persistence is not a victory. It is an agreement. Form is not decoration: it is the structure made visible by constraint. That idea unfolds as a chain: informational conservation principle, sieve dynamics, crystallisation of 2, slide toward the reduced attractor μ* = 15, then derivation of observables with explicit epistemic status.
In one sentence : persisting does not mean remaining by inertia; it means surviving an admissible constraint without losing structural identity.
Technical reading : do not read this page as one global proof, but as an audit map. Theorems, bridges, physical derivations, validations, and predictions do not carry the same status.
Formal verification
30 foundational PT theorems, kernel-verified in Lean 4
The critical path T1 → T3 → s = ½ → T2 → L0 → T7 → W7-1
is formally verified by the Lean kernel with no sorry,
for a total of 187 modules (30 critical-path theorems in 8 modules + 179 secondary) and ~2,850 formal declarations.
Open the formalisation page →
Plain
The idea in four scenes
A constraint acts
The theory starts from a minimal situation: something is placed under constraint, and that constraint forces a separation.
Survivors appear
What does not cross the constraint disperses; what does cross it keeps a readable identity.
Measurement becomes possible
Observables are no longer fitted knobs: they measure the persistent part, the dispersed part, and their modes of coupling.
Continuous and discrete reconcile
General relativity (smooth spacetime) and quantum mechanics (discrete levels) looked incompatible. PT shows that they are two properties of the same arithmetic substrate — just as a triangle has both three vertices (discrete) and three angles (continuous), with neither preceding the other. The old opposition dissolves.
Standard
Read the theory as a chain, not as a catalogue
Order matters: informational conservation, sieve dynamics, attractor selection, numerical derivations, then epistemic audit. That thread keeps intuition, theorem, physical bridge, and validation separate.
Principle
Total distinction partitions into persistent structure and entropic dispersion.
02Sieve
The survivors of the sieve provide an exact laboratory for reading constraint.
03Attractor
After 2 crystallises, the cascade slides toward the reduced attractor μ* = 15.
04Derivations
Physical and chemical observables are derived without continuously fitted parameters, with explicit statuses.
05Validation
Each result receives a status: theorem, bridge, derivation, validation, or prediction.
Standard
One constraint, several registers
In mathematics, persistence appears in the sieve of Eratosthenes: primes are the irreducible survivors of the additive-multiplicative coupling, where the multiplicative constraint eliminates composites while additive progression reveals the dynamical trace of the survivors through prime gaps.
In PT, a prime is also a discrete resonance point of the sieve: it indexes a persistence mode. The cyclic phase associated with that prime gives an amplitude; compared with the persistence threshold, that amplitude fixes its status: boundary, active, echo, or super-echo.
In physics and chemistry, the question becomes riskier: are persistent structures sufficient to reconstruct constants, masses, geometry, shells, ionization energies, and electron affinities? The site separates what is proved, derived, validated, and still open.
The persistence points
Survivors, gaps, GFT, and theorems show how a constraint produces stable structures.
CosmogonyThe primordial cascade
The crystallisation of 2, the emergence of time, and μ* = 15 are presented as an instantaneous hierarchy.
PhysicsThe principle tested
Couplings, masses, angles, time, and geometry compare persistent structures with measurements.
ChemistryShell persistence
The sequence 2, 8, 8, 18, 18, 32, 32 follows from channels, spin doubling, and internal shells.
Technical
First audit handles
The technical depth of the index exposes what should be checked before judging the theory: informational conservation, the T0 → T6 chain, bridge status, and numerical reproducibility.
Conservation principle
Total informational capacity partitions into KL divergence and entropy: Hmax = DKL + H.
Demonstrative chain
Each step must be followed as a status transition: theorem, bridge, derivation, validation, or prediction.
Reproducibility
The companion scripts check the flagship calculations and expose points still dependent on a bridge.
Technical note
The formal framework comprises base bridge axioms (BA0–BA2) that identify the dynamical field with the gaps between consecutive primes, then derived bridges (BA3–BA5) promoted or constrained in the chain, and a cascade of seven theorems (T0 → T6) deriving the holonomy angles sin²(θ_p) = δ_p(2 − δ_p) and the anomalous dimensions γ_p, and theorem T5 establishing the reduced attractor μ* = 15 by exact rational exhaustion. The identity log₂(m) = D_KL(P‖U_m) + H(P) (GFT) is the fundamental principle of persistence: total informational capacity is conserved and partitions exactly into persistence and entropy.
Explore
T0 → T6, GFT, L0 — complete chain, zero drift across versions.
43 observables, mean error 0.30%, median 0.06%, zero fit.
Canonical registry: 45 companion script entries, 2,522/2,523 passing checks — flagship scripts runnable in the browser.
30 foundational theorems on the critical path T1 → T7 → W7-1 kernel-verified in Lean 4 + Mathlib (including W7-1 in both directions), plus 179 secondary modules (187 modules in total). Open the dedicated page →
PT sub-projects
Three public repos applying PT to specific domains.
PT Mathematics (PTM)
Five self-contained articles (M1–M5) on the mathematical foundations of PT — from the sieve of Eratosthenes to the reconstruction of physics.
Each article = one theorem + companion scripts that verify every statement numerically.
PT Physics (PTP)
Computational proof that PT works: 43 Standard Model observables reproduced from the derived symmetry s = 1/2.
Local HTML dashboard + reproducible companion scripts tracked by Appendix F.
PT Chemistry (PTC)
Computational chemistry engine derived entirely from PT. 0 adjustable parameters; constants descend from the symmetry s = 1/2 and the sieve structure.
Dissociation energies, MAE ≈ 2% (200+ molecules) — aromaticity, σ-bonds, NICS.
Cite this work
If you cite PT in a paper, thesis, or post, the recommended BibTeX entries are below. Each artefact has a permanent Zenodo DOI.
DOI: https://doi.org/10.5281/zenodo.18726591
@book{senez2026persistencemonograph,
author = {Senez, Yan},
title = {The Theory of Persistence: From the Sieve to the Standard Model},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.18726591},
url = {https://doi.org/10.5281/zenodo.18726591}
}Preprint — not peer-reviewed. Critical scrutiny is welcomed.