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The Fractal Logic of Phi-adic Recursion

Milan Rosko

TL;DR

The Fractal Logic of $\Phi$-adic Recursion develops a fully constructive bridge between logical inference and additive recurrence by embedding Modus Ponens within Zeckendorf-based arithmetic and a $\Phi$-scaled geometric framework. It introduces a carryless Zeckendorf pairing, a geometric Iterant that encodes proofs as constrained Fibonacci-aligned configurations, and a $\Delta_0$-based witness predicate $W(a,b)$ whose existence is primitive recursive. These components are re-embedded in Tarski geometry and related to Diophantine representations via MRDP, yielding a unified operator that classifies tautologies by witnessability and semantic status while preserving a constructive, finite verification regime. The approach yields a fractal proof spine governed by $\Phi$, with explicit complexity bounds for verification and a conservative extension over elementary arithmetic, highlighting a deep connection between proof theory, combinatorics, and geometry. The work offers a novel constructive semantics for logical validity and practical pathways to finitely verifiable proof systems rooted in Fibonacci structure and geometric interpretation.

Abstract

Our central observation is that unbounded additive recurrence establishes a homomorphism between $\mathbb{N}$ and Modus Ponens in a constructive sense. By finding sums of nonconsecutive Fibonacci indices, each inference step corresponds to a geometric constraint whose verification requires $O(M(\log n))$ bit-operations. Logical entailment can be interpreted constructively as arc-closures under $Φ$-scaling, offering a bridge between additive combinatorics, proof theory, and symbolic computation.

The Fractal Logic of Phi-adic Recursion

TL;DR

The Fractal Logic of -adic Recursion develops a fully constructive bridge between logical inference and additive recurrence by embedding Modus Ponens within Zeckendorf-based arithmetic and a -scaled geometric framework. It introduces a carryless Zeckendorf pairing, a geometric Iterant that encodes proofs as constrained Fibonacci-aligned configurations, and a -based witness predicate whose existence is primitive recursive. These components are re-embedded in Tarski geometry and related to Diophantine representations via MRDP, yielding a unified operator that classifies tautologies by witnessability and semantic status while preserving a constructive, finite verification regime. The approach yields a fractal proof spine governed by , with explicit complexity bounds for verification and a conservative extension over elementary arithmetic, highlighting a deep connection between proof theory, combinatorics, and geometry. The work offers a novel constructive semantics for logical validity and practical pathways to finitely verifiable proof systems rooted in Fibonacci structure and geometric interpretation.

Abstract

Our central observation is that unbounded additive recurrence establishes a homomorphism between and Modus Ponens in a constructive sense. By finding sums of nonconsecutive Fibonacci indices, each inference step corresponds to a geometric constraint whose verification requires bit-operations. Logical entailment can be interpreted constructively as arc-closures under -scaling, offering a bridge between additive combinatorics, proof theory, and symbolic computation.

Paper Structure

This paper contains 25 sections, 11 theorems, 53 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hilbert skeleton
  4. Fibonacci arithmetic
  5. Recurrence-Inference Resonance
  6. Carryless Pairing
  7. Definition and basic properties
  8. Computational form and intuition
  9. Geometric Iterant
  10. Abstraction
  11. Head-index verification constraints
  12. MP-chain classification
  13. Witness formulation and alignment search
  14. Geometric re-embedding in Tarski geometry
  15. Fractal Logic
  16. ...and 10 more sections

Key Result

Lemma 3.3

The mapping $\pi_{\mathrm{CL}}:\mathbb N^2\to\mathbb N$ is injective. Moreover, its inverse $\pi_{\mathrm{CL}}^{-1}$—recovering $(x,y)$ from $\pi_{\mathrm{CL}}(x,y)$—is primitive recursive, definable by bounded subtraction (search problem) and index partitioning.

Figures (4)

  • Figure 1: The wheelchart reveals by polar alignment: For any three consecutive positive Fibonacci numbers $F_n,F_{n+1},F_{n+2}$, the arcs of the south pair $(F_n,F_{n+1})$ complete the circle with $F_{n+2}$; the north pair $(F_{n+1},F_{n+2})$ similarly aligns with the full circumference; whereas the northwest pair $(F_n,F_{n+2})$ necessarily varies, converging toward the ratio $[1-1/\Phi]:[1/\Phi]$. When the middle term $F_{n+1}$ is removed from any sum, uniqueness emerges: for each tuple there is a designated number $n\in\mathbb N^+$ preserving symmetry.
  • Figure 2: The Iterant as a geometric abstraction: three consecutive Fibonacci numbers $\{5,8,13\}$ appear as projections of Farey arcs $\{\phi_a,\phi_b,\phi_c\}$ on nested circles scaled by $\Phi$ and $\bar{\Phi}$. The construction acts as an Oracle verifying $F_n + F_{n+1} = F_{n+2}$. The configuration then tests $e = F_{n+4}$ via $F_{n+4} - F_{n+2} = F_{n+3}$: alignment confirms recurrence; deviation detects arithmetic failure. Fractal nesting yields recursive verification, with Tarski Geometry$\mathcal{G}$ serving as a $\Sigma_1$ generator and $\Delta_0$ arithmetic providing the verifying trail. This realizes a constructive analogue of a Turing--Lovelace mechanism shaped by the nonlinear invariant $\log_\Phi 2$.
  • Figure 3: Dyadic partitioning. Each propositional connective subdivides the "Boolean" truth space into nested semicircular arcs over $[0,1]$, indexed by powers of 2. Shortcuts are obstructed because dyadic fractions are redundant (1:2, 2:4, 4:8, etc.), preventing unique witness identification. The uniform scaling by factors of $1/2$ provides no invariant to exploit.
  • Figure 4: Replacing dyadic subdivision with Farey Arcsfarey1816 transforms the search space: consecutive reduced fractions $a/b$ and $c/d$ connect through their mediant $(a+c)/(b+d)$, generating the Stern–Brocot tree stern1858. Gaps shrink non-uniformly, and each rational appears exactly once, enabling unique witness recovery by greedy descent.

Theorems & Definitions (32)

  • Remark
  • Definition 2.1: Axiom schemata and rule
  • Definition 2.2: Zeckendorf representation
  • Remark
  • Definition 3.1: Carry
  • Definition 3.2: Untyped Carryless Pairing
  • Lemma 3.3: Carryless Injectivity
  • Remark
  • Example 3.4: Explicit Computation
  • Corollary 3.5: Realizable Arithmetic Pairing
  • ...and 22 more