Explanation of physics_complexity_implies_limits

(1) Plain English

The declaration asserts that the physics complexity structure (derived from ledger J-cost minimization) directly entails the computation limits from the ledger. Because the two propositions are defined to be identical, the implication holds by direct substitution.

(2) Why it matters in Recognition Science

Recognition Science grounds physics in convex J-cost minimization on ledgers. This theorem ensures the resulting physical model inherits explicit computational bounds: ground-state verification is linear-time (O(N)) while φ-rung states grow exponentially. It therefore places RS physics in a concrete complexity regime without external assumptions.

(3) How to read the formal statement

The Lean statement is:

theorem physics_complexity_implies_limits (h : physics_complexity_from_ledger) :
    computation_limits_from_ledger := h

In ordinary language: given any proof h that physics_complexity_from_ledger holds, conclude that computation_limits_from_ledger holds. The proof is simply the hypothesis h itself. This works because physics_complexity_from_ledger is defined by equality to computation_limits_from_ledger.

(4) Visible dependencies or certificates in the supplied source

The theorem sits after physics_complexity_structure, which proves physics_complexity_from_ledger by invoking the imported structure. It is supported by the J-cost core: jcost_unique_minimum (unique ground state at x=1), verification_equivalence (balance ↔ total J-cost = 0, O(N) check), and phi_rung_complexity_unbounded (exponential rung growth). The module certificate ic005_certificate summarizes these results.

(5) What this declaration does not prove

It does not establish the content of the computation limits themselves. It supplies no new proof of J-cost convexity or ledger balance; those appear in the preceding local theorems. It likewise does not classify concrete complexity classes (P, EXPTIME, NP-hard) beyond the informal list in rs_complexity_classes.