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_{p,q} = (O_p ⊗ O_q) R_std (O_p† ⊗ O_q†). The Yang-Baxter equation holds projectively (Conjecture 2.1). A prime-weighted functional Z(K) = Tr(ρ(β_K) W_K) 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.