Forced, Purchased, or Seed: A Verified Classification of Physical Law from the x = ¬x Substrate

We report a complete classification experiment on a single hypothesis: every law of known physics is exactly one of (a) forced, a consequence of the self-negating substrate x = ¬x plus consistency; (b) purchased, additional structure admissible at a stated price; or (c) seed, an initial-condition choice. Nothing is a fourth thing. The experiment is carried by two artifacts held to a common ledger: a modular Python simulation of the substrate (a single-cone split-step quantum walk with a Gaussian free-fermion correlation engine, plus quantized gauge-link automata), and a Lean 4 development (a 32-file axiom kernel and eight derived theorems) that machine-checks the forced results. Every implemented rule carries a status tag, every measured claim carries a stated tolerance tied to a regression test, and every proof is audited for its axiom dependencies. At the close of the experiment no structural unknowns remain: quantum mechanics, Dirac matter, three spatial dimensions, the light cone, the metric law, the Einstein equation, Newton’s constant, the gauge Gauss laws, the Standard Model hypercharges, the anomaly law, and interior undecidability are forced or measured with zero free parameters; the gauge groups, the Higgs sector, the continuum, and the cosmological constant assembly are purchased at explicit prices; the initial asymmetry and the generation count are seeds. The cosmological-constant assembly yields the holographic form ρ = 3ℏc/(8πl2 PL2) with ρpred/ρobs = 1.19 and a falsifiable prediction w0 = −0.885, evolving, currently under live observational test. The classification terminates in a level invariant undecidability theorem: a complete description of a universe does not certify its own completeness or its own ground.