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 eliminatesarbitrary empirical constants and fine-tuned parameters from foundational mathematicsand physics. Rather than treating topological invariants as structures imported ontonumbers, 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 theMajorana fixed point, we derive the generalized Collatz map Ca,b(n).By analyzing the 2-adic valuation distribution and uniform residue classes modulopowers of two, we identify an expected logarithmic dissipation per parity block EX =ln(a/4). Aggregation over a mathematically forced minimal control layer m = 4 yieldsstable renormalization bounds. Applying the Dalvi Dictact—the principle of local-toglobal topological completion—forces a unique integer base B = ⌊(4/a)16⌋, yielding thetranscendental primordial invariant ∆ = 4 ln 99 ≈ 18.3804794 for the a = 3 system Substituting ∆ into the canonical form of Srinivasa Ramanujan’s 1914 modular seriesgenerates 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 asprimitive—by reformulating general relativity via the parameter-free Mahapatra–HilbertAction SMH. The scale hierarchy problem is natively resolved via the Primordial ActionProduct Law (PAPL), which scales gravitational forces down to the electroweak scale bya 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 emergentarena to yield link invariants (µ = 1). The paper includes fully deterministic, parameterfree Python verifications yielding π to machine precision, establishing a falsifiable physicalprediction of an axion topological resonance at fa = 1.528 GHz.