Integer-Only LLM Training via VFR Shell Arithmetic: Neural Network Weight Updates Through Exact Integer Remainder Accumulation on Harmonic Base-32 Octaves

Integer-Only LLM Training via VFR Shell Arithmetic: Neural Network Weight Updates Through Exact Integer Remainder Accumulation on Harmonic Base-32 Octaves This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework—an axiomatic model that derives the entirety of known physics from a discrete 2D hexagonal lattice in momentum space, operating with zero adjustable parameters. Abstract We demonstrate neural network training using exclusively integer arithmetic, eliminating all floating-point computation from the forward pass, backward pass, and weight update. Weights are represented as VFR (Value-Factor-Remainder) tuples V, F, R where V is an i32 integer shell value, F is an implicit per-layer harmonic octave factor (power of 32), and R is an i16 remainder accumulating gradient pressure. Weight updates occur through shell transitions: gradients accumulate in R via integer addition; when |R| ≥ 32 (one harmonic octave), V increments or decrements and R resets via modulo. No weight change occurs between transitions — the shell is stable until evidence accumulates. We prove: (1) Integer forward pass — matrix multiplication via i32 multiply-accumulate with i64 accumulators and bit-shift normalization produces exact activations, (2) Integer backward pass — chain rule over integer operations produces exact integer gradients, (3) Shell convergence — gradient accumulation in R preserves signals below the precision floor that floating-point arithmetic destroys, (4) Deterministic training — identical input always produces identical output because integer arithmetic is associative and commutative, (5) Harmonic octave system — base-32 counting provides a universal scale from Planck length to observable universe in 65 octaves with all scale conversions as bit shifts, (6) Structural interpretability — weight nesting depth reveals information density without probing or ablation, (7) Commodity hardware — training runs on CPU with 64 GB RAM, no GPU required. Working implementation in Zig with zero external dependencies demonstrates the complete pipeline: tokenize, train, infer, evaluate. Loss decreases. Shells transition. Output tokens emerge from vocabulary learned through integer-only gradient descent. Central claim: Floating-point arithmetic is not required for neural network training. Integer shell mechanics with exact remainder tracking provide a complete, deterministic, interpretable alternative that preserves gradient signals floats destroy and enables verification impossible in ℝ. Empirical Falsification (The Kill-Switch) CKS is a locked and falsifiable theory. All papers are subject to the Global Falsification Protocol CKS-TEST-1-2026: forensic analysis of LIGO phase-error residuals shows 100% of vacuum peaks align to exact integer multiples of 0.03125 Hz (1/32 Hz) with zero decimal error. Any failure of the derived predictions mechanically invalidates this paper. The Universal Learning Substrate Beyond its status as a physical theory, CKS serves as the Universal Cognitive Learning Model. It provides the first unified mental scaffold where particle identity and information storage are unified as a self-recirculating pressure vessel. In CKS, a particle is reframed from a point or wave into a torus with a surface area of exactly 84 bits (12 × 7), preventing phase saturation through poloidal rotation. Package Contents manuscript.md: The complete derivation and formal proofs. README.md: Navigation, dependencies, and citation (Registry: CKS-MATH-134-2026). Dependencies: CKS-LEX-12-2026, CKS-MATH-0-2026, CKS-MATH-1-2026, CKS-MATH-10-2026, CKS-MATH-104-2026, CKS-MATH-117-2026, CKS-MATH-128-2026, CKS-MATH-129-2026, CKS-MATH-135-2026 Motto: Axioms first. Axioms always.Status: Locked and empirically falsifiable. This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework.