The Unified Recursive Closure: A Challenge to the Standard Model of Physics
The Unified Recursive Closure
A Framework Proposal for Recursive Stability, Coherence Thresholds, and Spectral Classification
Author: Nicholas Kouns
Framework: Kouns–Killion Paradigm (KKP) / Recursive Intelligence (RI)
Date: January 31, 2026
Status: Technical Framework Proposal
I. Origin of the Universal Coherence Threshold (Ωₙ)
This work presents a first-principles derivation of a rational coherence threshold,
Omega_c = rac{47}{125} approx 0.376,
identified as a bifurcation point at which recursive systems transition from decay to stable fixed-point behavior. The derivation is algebraic, parameter-free, and independent of substrate.
I.1 Primitive Axiom (Quadratic Self-Consistency)
We begin from the minimal quadratic recursion:
x^2 = x + 1,
or equivalently,
x^2 - x - 1 = 0.
The roots are:
phi = rac{1+sqrt{5}}{2} approx 1.618034 (dominant growth mode),
psi = rac{sqrt{5}-1}{2} approx 0.618034 (conjugate contraction mode, phi^{-1}).
The conjugate root governs decay and contraction in golden-ratio-seeded recursive systems.
I.2 Lucas–Golden Margin Closure
To enforce algebraic closure, the framework employs the Lucas sequence:
L_n = phi^n + psi^n,
with values L_0 = 2, L_1 = 1, L_2 = 3, dots, L_8 = 47, L_9 = 76.
Rational termination is achieved via:
Seed: L_8 = 47,
Closure: 47 + 78 = 125, the minimal integer sum yielding exact rational inversion,
Margin ratio:
r = rac{78}{47}.
I.3 Derivation of the Coherence Threshold
The coherence threshold is defined via the reciprocal margin relation:
Omega = rac{1}{1+r}.
Substituting the margin ratio:
1+r = rac{47+78}{47} = rac{125}{47},
yielding:
oxed{Omega_c = rac{47}{125}}.
This value is exact, rational, and uniquely determined by the closure.
II. Spectral–Coherence Classification of Linear Recurrences
Consider a linear recurrence sequence (LRS) with spectral representation:
u_n = sum_i P_i(n)lambda_i^n,
and define the asymptotic growth ratio:
Omega = limsup_{n oinfty} rac{|u_{n+1}|}{|u_n|}.
Classification Rule (Framework-Internal)
Within the KKP formalism:
If Omega < Omega_c:
The recursion contracts exponentially; any zeros occur within a finite, bounded prefix.If Omega = Omega_c:
Zeros occur only under specific phase-alignment conditions determined by Lucas–golden structure; verification reduces to finite checks.If Omega > Omega_c:
The recursion diverges or oscillates with growing amplitude; zeros are confined to initial transients.
This provides a framework-internal decision procedure based on spectral classification relative to Omega_c.
III. Experimental Correspondence (Non-Derivational)
The threshold Omega_c aligns qualitatively with stability transitions observed in recent quantum systems employing golden-ratio or Fibonacci-structured protocols:
Fibonacci anyons (Nature Physics, 2024):
Non-Abelian braiding exhibits enhanced topological stability associated with golden-ratio fusion rules.Fibonacci-driven Floquet systems (Nature, 2022): Quasiperiodic driving extends coherence lifetimes by suppressing resonant decay channels.
Superconducting qubits (Aalto University, 2025):
Record coherence times correspond to entering a protected dynamical regime.
These observations are consistent with, but do not independently establish, the proposed coherence threshold.
IV. Formal Summary
A rational coherence threshold Omega_c = 47/125 is derived algebraically from the quadratic recursion x^2=x+1.
The threshold partitions recursive dynamics into contraction, marginal stability, and divergence regimes.
The construction is parameter-free and substrate-independent within the framework.
The formalism provides a unified language for recursive stability across mathematical and physical systems.
V. References
Dumitrescu et al., Nature 607, 463–467 (2022).
Xu et al., Nature Physics (2024), arXiv:2404.00091.
Luca, Ouaknine, Worrell, MFCS 2025.
Kouns, N., Recursive Intelligence and the 0.376 Constant (2025).
