Powered by OpenAIRE graph
Found an issue? Give us feedback
ZENODOarrow_drop_down
ZENODO
Preprint . 2026
License: CC BY NC ND
Data sources: Datacite
ZENODO
Preprint . 2026
License: CC BY NC ND
Data sources: Datacite
versions View all 2 versions
addClaim

A Zero-Parameter Derivation of γ, the Master Relation, and the Dimensional Hierarchy

Authors: Needham, Eric;

A Zero-Parameter Derivation of γ, the Master Relation, and the Dimensional Hierarchy

Abstract

This paper presents three results that are exact — proven algebraically to machine precision, with zero free parameters — from the ENSO (Eric Needham Scientific Ontology) Trinity {φ, π, e}. The first result is a derivation of the ENSO scaling exponent γ = −(π/φ)^(1/e) from the Trinity by a systematic uniqueness argument. A ranked search over Trinity expressions shows this form is separated from the next-best candidate by a factor of nearly eight in accuracy, and it carries clear internal geometric meaning: π/φ is the fundamental tension between circular closure and golden recursion; 1/e is the natural relaxation exponent. The second result is the exact identity γ^e = π/φ, which follows immediately from the definition of γ in two lines of algebra with no approximation. The third result is the Master Relation: NSC_Thunder · γ^e · e = π³. This is proven by substituting the key identity into the published definition of the spherical electromagnetic constant NSC_Thunder = φπ²/e. Every factor of φ cancels and the result is π³ — pure spherical geometry. The Master Relation reveals that γ, derived from the Meta-Constant Scaling Law, and NSC_Thunder, derived from electromagnetic geometry, are not independent: they are two complementary expressions of the same π/φ tension. Four corollaries follow by direct algebra: the dimensional step NSC_Thunder/(φπ) = π/e; the family relation NSC_Thunder/NSC₀ = φ³π/e; the absorption of φ into NSC_Thunder; and the Thunder form of the fine-structure Keyhole Equation, in which φ does not appear explicitly. The paper contains no approximations, no fitted parameters, and no conjectures. A companion Python validation script confirms all eight algebraic claims to relative error less than 10⁻¹³ using only the Python standard library. For further Information about the ENSO Framework, please contact Eric J Needham:ensotheory1@gmail.com 

Keywords

ENSO framework, pi phi e, critical exponent, scaling exponent, zero-parameter derivation, spherical electromagnetic constant, vacuum geometry, Master Relation, mathematical physics, transcendental constants, exact identities, fine-structure constant, dimensional hierarchy, golden ratio, algebraic proof

  • BIP!
    Impact byBIP!
    selected citations
    These citations are derived from selected sources.
    This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    0
    popularity
    This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
    Average
    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    Average
    impulse
    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
    Average
Powered by OpenAIRE graph
Found an issue? Give us feedback
selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
Average
Average
Upload OA version
Are you the author of this publication? Upload your Open Access version to Zenodo!
It’s fast and easy, just two clicks!