Multiplicity Knot Theory v3.0: Prime-Weighted Braid Invariants and a Cryptographic Commitment Prototype

This document presents v3. 0 of Multiplicity Theory's knot-theoretic framework, a substantial reframing of the v2. 1 preprint that addresses critical issues identified through independent audit. KEY CHANGES FROM V2. 1: The previous version claimed a parameter-free topological invariant P (K). Audit revealed gaps in the Markov invariance proof, incomplete derivations of constants c₀ = ln (10) and z = 1/ (2cos1), and miscategorization of the representation (projective rather than true). This v3. 0 reframes all three as explicit conjectures supported by numerical evidence. TECHNICAL CORE: We define a prime-colored braid category with strand-dependent R-matrices R₏, ₐ = (Oₚ ⊗ Oq) Rₛtd (Oₚ† ⊗ Oq†). The Yang-Baxter equation holds projectively (Conjecture 2. 1). A prime-weighted functional Z (K) = Tr (ρ (βK) WK) uses a modified trace. Truncated prime averages c₀ (X) and z (X) have conjectural limits (Conjectures 3. 1–4. 1). The protection functional P (K; X) = exp (c₀ (X) c (K) ) |Z (K) |^z (X) is valid for finite X. CRYPTOGRAPHIC APPLICATION: We sketch a Multiplicity-Based Commitment (MBC) scheme with security model and Python prototype for 4-strand, 12-bit commitments. Binding is heuristic, tied to encoding injectivity and invariant stability. STATUS: This is a research program document, not a finished theorem. Previously overclaimed results are demoted to falsifiable conjectures with finite-X formulations suitable for experimental validation.