Mass Genesis: the particle mass spectrum as σ= 0 recognition-cost ground states on the recognition 3-cube

We derive the structure of the particle mass spectrum from a single functional equation. The Law of Logic, expressed as a reciprocal, normalized, composition-respecting, calibrated, continuous cost on positive ratios, forces a unique cost functional J(x) = ½(x + x⁻¹) − 1; self-similarity of the resulting discrete ledger forces the golden ratio φ; the ledger cadence is 2³ = 8; and the recognition substrate is the three-cube Q₃ = {0, 1}³. On that substrate a particle is the zero-cost (σ = 0) ground state of a recognition-cost functional, and we prove that a realized mass state exists unconditionally and is the unique σ = 0 ground state of its load in each scale orbit, with rest mass equal to a parameter-free φ-power prediction and with inertial, gravitational, and rest mass equal for the same reason. The carrier has now been sharpened: matter is the spin-1/2 quarter-turn summand of the eight-tick light register, that summand splits into left and right Weyl sectors as the ±i eigenspaces of J = P², conjugation swaps the sectors, and no nonzero neutral state is recognition-fixed. The sector of a pattern (lepton, up-type, down-type) is read from two cube labels, and the coupling structure that organizes the generation ladder is the boundary Poincaré duality of Q₃; the up/down orientation and the handedness orientation are both proved to be free binary choices rather than derivable facts of the tick-generated core. The charged generation torsion reduces to one cyclic rule with a single anomaly, and that anomaly, the up/down generation-spread scalar, is itself derived from the closure principle: a vertex-coupled carrier sits below the recognition loop's 1-cycle dimension, cannot host a closed loop intrinsically, and must borrow its Poincaré-dual face, producing exactly the observed step. Only the ℤ/2 up/down label stays conventional. A structural physical-stability predicate cuts the realized charged spectrum down to exactly nine fermion topologies with no fourth generation. The quark window assignment and the Majorana neutrino mass-squared ratio (a parameter-free 0.8% prediction) follow. Every claim tagged [THEOREM] is a machine-checked theorem in Lean 4; its kernel axiom footprint is contained in {propext, Classical.choice, Quot.sound}, with the sole exception of a few finite decision-procedure evaluations (the quark cube counts and the nine-row spectrum enumeration) that additionally invoke Lean's two compiler-evaluation axioms, flagged at each use; the foundational inputs, the data-selected choices, and the open questions are labeled as such throughout.