The Ontological Hierarchy of Mathematics and Physics: First-Principle Derivation of Topological Invariants and Geometric Scale from the Generalized Collatz Map

This paper demonstrates a strict, unidirectional ontological hierarchy that eliminates
arbitrary empirical constants and fine-tuned parameters from foundational mathematics
and physics. Rather than treating topological invariants as structures imported onto
numbers, or geometric parameters like π as primitive values, we construct a deterministic,
self-constrained dynamic engine originating from the absolute bedrock of number theory.
Starting from three primitive, triadic coefficients (a, b, n) operating around zero as the
Majorana fixed point, we derive the generalized Collatz map Ca,b(n).
By analyzing the 2-adic valuation distribution and uniform residue classes modulo
powers of two, we identify an expected logarithmic dissipation per parity block E[X] =
ln(a/4). Aggregation over a mathematically forced minimal control layer m = 4 yields
stable renormalization bounds. Applying the Dalvi Dictact—the principle of local-toglobal topological completion—forces a unique integer base B = ⌊(4/a)
16⌋, yielding the
transcendental primordial invariant ∆ = 4 ln 99 ≈ 18.3804794 for the a = 3 system                                                                        Substituting ∆ into the canonical form of Srinivasa Ramanujan’s 1914 modular series
generates the exact numerical value of the geometric scale π without circularity (∆ → π).
We resolve the historical category error of the 1915 Göttingen school—which treated π as
primitive—by reformulating general relativity via the parameter-free Mahapatra–Hilbert
Action SMH. The scale hierarchy problem is natively resolved via the Primordial Action
Product Law (PAPL), which scales gravitational forces down to the electroweak scale by
a factor of 99−20 ≈ 1.35 × 10−40
.
Finally, we bridge this space to arithmetic topology by nesting Morishita’s Galois-tomanifold mapping, demonstrating how discrete primes are knotted within this emergent
arena to yield link invariants (µ = 1). The paper includes fully deterministic, parameterfree Python verifications yielding π to machine precision, establishing a falsifiable physical
prediction of an axion topological resonance at fa = 1.528 GHz.