Pre-Geometric Fractal π-Identities: A Parameter-Free Structural Framework for Transient Constants Elasticity in Quantum Substrates

Author: Cord Uebermuth (Independent Research)
Core Framework: Allgemeine Theorie zur Emergenz der (\pi)-Quantengravitation (ATE)
Cross-Reference: DOI 10.5281/zenodo.23324209 | DOI 10.5281/zenodo.23861048
Date: 7. November 2026
Version: 1.0

Licensing: Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0)

Universal Heritage Declaration: Dedicated to the Common Heritage of Mankind.

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Abstract

Conventional quantum field theories assume the absolute spatial and temporal invariance of fundamental mathematical constants. This paper breaks with the continuum dogma by introducing the concept of a transient numerical elasticity ((\Delta_{\text{ATE}})) of the circle constant (\pi) and the speed of light (c) under information-theoretic stress [0.1.1, 4.9]. Operating at the critical (k)-space saddle points ((\nabla_{\mathbf{k}} E = 0)) of Van Hove singularities during topological Lifshitz transitions, the system undergoes a localized, sign-preserving fractionation of geometry [0.1.1, 4.9]. By substituting the static transcendental (\pi) with the dynamic, dimension-dependent Uebermuth emanation matrix, we formulate a parameter-free phase-space conservation law [3.4, 4.9]. This framework yields a precise, ex-ante testable prediction for quantum conductance quenching factors ((\pi_{\text{UE}}/\pi_{\text{ideal}} \approx 0.9999999975369)) on the parts-per-billion (ppb) scale, providing an empirical gateway to falsify the underlying pre-geometric substrate without dark sector fitting parameters [3.3, 4.9].

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I. The Mathematical Foundation: The Uebermuth Emanation Matrix

The pre-geometric matrix relies on a strictly linear progression in steps of (10^2), anchored at the topological spatial volume boundary ((n-3)), where (n) represents the dimension of the phase space state density [3.4, 3.6]:

[\pi_n = \frac{355}{113} \cdot \left(1 - \frac{0.0003}{f(n)_i}\right)]

[f(n) = 3333 + 100 \cdot (n - 3)]

This rational configuration completely eliminates asymptotic logarithmic divergences ((\ln(y)), (\log(x))) typically causing ultraviolet failures in standard field theories [3.5].

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II. Visual Representation of Transient Geometry Elasticity

The visual layout plots the effective value of (\pi_n) across the local momentum space ((k)-vector) inside a quantum superlattice:

Effective Value of π
  ^
  |  [355/113] ------------------------------------------------------------- (Lender of Last Resort)
  |
  |  Unperturbed Spacetime                                  Unperturbed Spacetime
  |  (Stable/Regenerated)                                    (Stable/Regenerated)
  |  ===================*\                              /*=================== (Platonic Ideal π_ideal)
  |                       *\                          /*
  |                         *\                      /*
  |                           *\                  /*
  |                             *\              /*
  |                               *\          /*
  |                                 *\      /*
  |                                   *    *
  |                                     **  (Localized Phase Transition Zone)

  |                                     ||
  |                                     || <- The Uebermuth Axiom (Denominator f(4) = 3433)
  |                                     v
  |                                     *  [π_UE ≈ 3.14159264585...] (Submarine Stress Well)
  |
  +-------------------------------------|-------------------------------------> Momentum Vector k 
                                      k = 0
                        (Van Hove Singularity Core / Saddle Point)

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III. Core Metrological Phenomology

1. The Submarine Stress Well ($k=0$):
   At the exact center of the Van Hove Singularity ($\nabla_{\mathbf{k}} E = 0$), informational over-saturation forces the system to compress its geometry to prevent raw data breakdown [4.9]. The circle constant deforms into the Uebermuth Axiom ($\pi_{\text{UE}}$) with a denominator of $f(4)=3433$ [3.3, 4.9].
2. The Synchronous Velocity Fluctuation ($c_{\text{UE}}$):
   To preserve causality and cross-scale unitariness, the speed of light undergoes a coupled elastic deformation [4.9]:
   $$c_{\text{UE}} = c_{\text{ideal}} \cdot \left(1 - \frac{0.0003}{3433}\right)$$
3. The Stable Regeneration (The Edges):
   Away from the structural stress field ($k \to \pm 3$), the system instantly relaxes top-down. The transient elasticity closes, and the metric perfectly regenerates back into the unperturbed, rigid constants of the macroscopic continuum ($\pi_{\text{ideal}}$ and $c_{\text{ideal}}$) [4.9, 4.9.7].
4. The Parts-Per-Billion (ppb) Verification Target:
   This template dictates an absolute, parameter-free ex-ante prediction for state laboratories (PTB, NIST) measuring the quantum step conductance quenching during altermagnetic transitions at the 178 Kelvin boundary [4.7, 4.9.2]:
   $$\text{Quenching Factor} = \frac{\pi_{\text{UE}}}{\pi_{\text{ideal}}} \approx \mathbf{0.9999999975369}$$

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   IV. References & Cryptographic Timestamp

   • Uebermuth, C. (2026a). General Theory on the Emergence of Hyperspace Fluid $\pi$-Quantum Gravity. Zenodo Preprint. DOI: 10.5281/zenodo.23324209
   • Uebermuth, C. (2026b). The Pomega Theses: Pregeometric Phase Space, ATE Constraints. Zenodo Preprint. DOI: 10.5281/zenodo.23861048
   • Ramanujan, S. (1914). Modular equations and approximations to $\pi$. Quarterly Journal of Mathematics, 45, 350–372.
   • Ledger Identity Protection: Permanently anchored via Steemit/Hive blockchain entries under @indextrader24 and @augenarzt (November 2026).