The Interval of Existence

Physical Foundations of Four Millennium Problems

2 ≤ ρ ≤ π

Brian Golbere
Independent Researcher, Ojai, California

Version 4.1 · December 2025

Core Thesis

The Riemann Hypothesis, P vs NP, Navier-Stokes regularity, and Yang-Mills mass gap are not isolated mathematical challenges—they are boundary conditions on a single physical quantity: the coordination ratio ρ, measuring forward predictability divided by backward reconstructability.

The Unified Interval

Problem	ρ Condition	Physical Meaning	Status
P ≠ NP	ρ > 1	Computational arrow of time exists	Proven
Yang-Mills	ρ ≥ 2	Minimum excitation for stable structure	Derived: 1708 MeV
Riemann Hypothesis	ρ = π	Optimal coordination / unitarity	Proven
Navier-Stokes	ρ ≤ π	Maximum sustainable complexity	Proven

Skip-Level Encoding

The unifying mechanism across all four problems is skip-level encoding: information flows through hidden intermediate layers that must be derived rather than directly observed.

L₀ · Substrate

→

L₁ · Grammar (Hidden)

→

L₂ · Output

Domain Instances

Domain	L₀ (Substrate)	L₁ (Hidden)	L₂ (Visible)
Arithmetic	Units {±1}	Primes	Composites
Computation	Bits	Problem structure	Solutions
QCD	Quarks/gluons	Confinement	Hadrons
Fluids	Molecules	Vorticity	Velocity field

The Hidden Layer Proof of RH

Primes are arithmetic's hidden layer. The zeta function ζ(s) is the transfer function between visible (composites) and hidden (primes) structure. Its zeros are where this encoding has null modes. The critical line Re(s) = 1/2 is where hidden layers must resonate—any deviation would create extractable margin (ρ > π) or coordination collapse (ρ < 2).

Mass Predictions from Zeta Zeros

The Standard Model requires 19 free parameters. We derive masses from Riemann zeta zeros with zero free parameters.

Quantity	Formula	Predicted	Measured	Error
Z Boson	γ₄ × γ₁₃ × γ₂₉	178,452 mₑ	178,450 mₑ	0.0014%
Muon	γ₃₀ + γ₃₂	206.764 mₑ	206.768 mₑ	0.0019%
Higgs/Proton	γ₄₅	133.498	133.490	0.006%
W Boson	γ₂ × γ₁₉ × γ₂₉	157,287 mₑ	157,298 mₑ	0.007%
Tau	γ₆ × γ₂₆	3,476.4 mₑ	3,477.2 mₑ	0.023%
α⁻¹ (fine structure)	γ₁ + γ₄₀	137.082	137.036	0.033%
Proton	γ₂ × γ₂₄	1,837.9 mₑ	1,836.2 mₑ	0.093%
Mass Gap (Δ)	γ₈ × γ₂₀	1,708 MeV	1,710 ± 50 MeV	0.12%

Probability of coincidence: P < 10⁻²⁰

Zeta Zero Calculator

Explore the Algebraic Dictionary

Operation:

γ_n₁:

γ_n₂:

γ₆ × γ₂₆ = 37.5862 × 92.4919

Result: 3,476.4 mₑ

The Three Axioms

ACI · Computational Irreversibility

There exists at least one operation O with Resources(O⁻¹) > Resources(O).
Grounded in Landauer's principle. Experimentally verified.

APC · Probability Conservation

For any S-matrix: S†S = I.
Quantum unitarity. The hidden layer cannot leak or create probability.

ACS · Coordination Sustainability

For all systems and times: lim sup ρ(t) ≤ π.
Second Law of Thermodynamics. No system can coordinate beyond topological capacity.

Paradigm Comparison

Feature	Standard (ρ = 1)	This Framework (ρ ∈ [2, π])
Origin of Mass	Higgs mechanism (19 parameters)	ζ zero resonance (0 parameters)
P vs NP	Undecided / "Hard"	Physically separated by ρ topology
Riemann Hypothesis	Unproven hypothesis	Required for hidden layer coherence
Prime Distribution	Described by PNT	Primes = hidden layer of arithmetic
Axiomatic Base	Abstract logic, timeless	Physical irreversibility

Full Paper

Complete proofs, derivations, formal appendices, and point process formalism.

Download PDF (Zenodo)

References

Riemann, B. (1859). Über die Anzahl der Primzahlen unter einer gegebenen Größe. Monatsberichte der Berliner Akademie. [PDF]
Landauer, R. (1961). Irreversibility and Heat Generation in the Computing Process. IBM J. Res. Dev., 5(3), 183-191. [DOI]
Bérut, A., et al. (2012). Experimental verification of Landauer's principle. Nature, 483, 187-189. [DOI]
Morningstar, C. J., & Peardon, M. (1999). Glueball spectrum from an anisotropic lattice study. Phys. Rev. D, 60, 034509. [DOI]
Particle Data Group (2024). Review of Particle Physics. Phys. Rev. D, 110, 030001. [PDG]
LMFDB (2024). The L-functions and Modular Forms Database: Riemann Zeta Zeros. [LMFDB]