djvu Coherent Mathematics: From Incomplete Self-Continuation to Mathematical Structure

What if mathematics does not precede physics — but emerges from it?

Coherent Mathematics (CoMath V7) is a self-contained foundational framework in which mathematical structure is not postulated but derived. Starting from a single survival principle:

Axiom 0: Coherence is not assumed — coherence is what survives.

Mathematical structure is not given — it is what survives under physical selection.

At the heart of CoMath lies a single dynamical principle: incomplete self-closure. No structure closes perfectly onto itself — the residual gap between a system and its own closure is not a defect but the generative force behind arithmetic, geometry, and logic. Mathematics does not rest on completed foundations. It moves because closure is never final.

The theory constructs arithmetic, fractal trigonometry, emergent logic, and the closure of coherence operators from physical selection dynamics alone — without importing classical mathematics as a prerequisite.

• Emergent zero ∅_f: the fractal void as the unique fixed point of recursive non-survival — not assumed, but derived as the limit of vanishing coherence
• Emergent infinity ∞_f: unbounded coherence extension as a fractal spiral limit — replacing the classical point-at-infinity with a dynamic recoherence divergence
• Derivation of the Fibonacci sequence and golden ratio φ as fixed points of recursive coherence
• Fractal trigonometric functions sin_f, cos_f, tan_f satisfying an exact Pythagorean identity
• Operative closure of the coherence space C_f via eigenvalue problem ℒ_R[C_f]μ_f = λ_f μ_f
• Emergent conjunction ∧_f constructed without classical logic import
• Parameter-free recoherence survival amplitude 𝒜_f(θ) = cos²(θ_f/2) · φ^(−D_f(1−cosθ_f)/2) connecting Axiom 0, the Born rule, and fractal damping
• Weinberg angle candidate sin²θ_W = 3/13 as a geometry-emergent conjecture
• Emergent disjunction ∨_f and the open frontier: the sole remaining closure step — identified, bounded, and formulated as a constructive conjecture

CoMath V7 is the rigorous mathematical companion to Fractal Uncertainty Theory (FUT V10). While FUT derives the fine-structure constant α ≈ 1/137, spatial dimensionality, and the four fundamental forces from recoherence dynamics, CoMath provides the foundational layer: it shows that the mathematical structures FUT relies on are themselves not assumed but emergent.

Together they form a physics-first reconstruction of mathematics and physics — reversing three centuries of convention in which mathematics was the unquestioned foundation of physical theory.

This book is written for researchers and theorists in foundations of mathematics, mathematical physics, and philosophy of science — and for anyone willing to ask the question that CoMath takes seriously:

No prior knowledge of category theory, set theory, or formal logic is required — only the willingness to follow a structure that builds itself from nothing but the question: what survives?

What must mathematics look like if reality — not convention — is its source?

If convention is not its source, where does it come from and what does this mean?

Jens Deutschmann is an independent researcher based Germany (ORCID: 0009-0007-1431-4149). His work is published via Zenodo (community: closure-first-physics), Amazon KDP, and GitHub.

CoMath V7 is part of a two-volume foundational program. The companion volume Fractal Uncertainty Theory (FUT) is available separately on Amazon KDP.