Strict Theory Extension on a Lawful Continuous Cantor Shell

We study a twenty-state stochastic substrate in which six coupled mechanisms (operator rewrite, admissibility gating, timescale adaptation, lens selection, packaging, and a budget ledger) rewrite a Markov kernel at every step. We describe it at two levels: a growth theory, which records how selector-weighted products of the evolving kernels grow, and a completion theory, which records the unique fixed point of an evolve–forget–reinstate map built from the current kernel and lens. We prove four results. First, the unmodified update law admits a nonempty forward-invariant controlled shell. Its admissible futures are indexed by a Cantor space of kernel parameters, and each of the six mechanisms changes the dynamics somewhere on it. Second, on this shell the multiplicative growth pressure exists for every parameter s ≥ 0, is Lipschitz and strictly decreasing, equals log 20 at s = 0, and has a unique zero in (1/3, 1). Third, two explicit shell states agree on every specified growth observation (every horizon, every parameter, both selector observables) and yet have different completion objects, so the completion theory is a strict extension of the growth theory. Fourth, the same pair shows that growth is blind to this extra structure: from the second step on, the two states have identical history matrices. Hence strict extension does not force a conditional pressure gap. On the same shell, one admissible reference law gives an identically zero gap, while another gives a certified positive gap. Lean 4 checks the twenty-state witness, the matrix identities and the abstract pressure lemmas. The all-time shell inequalities are certified by exact rational interval arithmetic. The correspondence with the implemented update formulas and the construction of the reference laws are analytic. This is version 3 of the preprint (v1: 31 March 2026; v2: 19 April 2026). It is a substantive mathematical revision; Appendix B of the paper records the version history. The Lean 4 development, exact rational interval certificates, and source code are available at https://github.com/ioannist/six-birds-cantor.