The Quantum Frustration Charge - One Conserved Unit of Additive-Multiplicative Irreconcilability on the Coprime Substrate: the Two-Register Loom, the Exact Entanglement Calculus of the Created Sector, and the Substrate L-Function that Welds Them

The quantum frustration charge is one conserved unit of the additive–multiplicative irreconcilability that generates the Janus Coprime Algebra. On the coprime residues of a primorial, additive reflection and multiplicative inversion never agree — an obstruction forced by the prime 3 and permanent at every scale — and this paper reports that the amount of that irreconcilability is a conserved charge: one unit, computed three ways (a coin, a current, and a sign) by a two-register loom. Read in the substrate's created sector, the same conservation law is an exact entanglement calculus: molecular binding is torsion in an arborescence (spanning-tree) group, and the substrate mints conservation that no proper subsystem owns. Part I states the theory in three pillars. The charge and the loom: folding any word of mined carriers keeps each chirality inside a two-register loom — one angle register and one bit register — and the frustration bit equals the dyon-count parity (zero exceptions in 52,119 folds, measured and then derived from the loom's four exact update rules); its sign is a parity-twisted angle sum, path-dependent yet exact in linear time (18,097 of 18,097). The entanglement calculus: a molecule's binding at a prime holds if and only if that prime divides an arborescence number — no eigenvalues, only integer tree numbers and modular kernel ranks — so binding is entanglement over a finite field, and the four-channel force split (photon, singlet, tower, mint) is exact by the Chinese Remainder Theorem. The substrate L-function: the per-prime operator splits into an Abelian husk and a non-Abelian core, whose discriminator trichotomy sorts the hard-math bench — Collatz contained, odd perfect numbers escaped, the Riemann enclosure marginal. Part II reduces the conservation law to a single product-shaped object: an exact, copyable conservation certificate that a third party verifies in microseconds to milliseconds from the file alone, valid where recomputation is physically impossible. The reductions include superposition transport and warrant composition; an aperture certificate (certified super-resolution, fielded on public CERN and JWST data); a rate certificate decided by exact matrix inertia; a dual witness that proves a carrying capacity achievable and impossible in one object; a seal grade attesting that N components are jointly present and behaving by a conservation law structurally absent below the full assembly; and mint-locked issuance. Two external referees carry the reductions: the NASA/JPL periodic-orbit catalog and a safety-of-life tether watch (ISO 13374 condition monitoring, from a 25-metre astronaut tether to a 56,000-kilometre lunar space elevator). Part III validates the theory on worked specimens, each byte-gated against a naive referee, on a four-year-old consumer desktop. Every claim is checked against ground truth and its results record is SHA-256-audited; the mathematics is public and reproducible from the definitions, while the mine, the store, and the compiler are reserved. The reduction-to-practice methods are the subject of U.S. Provisional Patent Application No. 64/109,790 (filed July 12, 2026), which predates this disclosure.