Explanation of physics_complexity_implies_limits

(1) What the declaration says in plain English

The theorem physics_complexity_implies_limits states that whenever the physics complexity structure derived from the RS ledger holds, the corresponding computation limits from the ledger also hold. It is an identity: the two propositions are definitionally the same, so the implication is immediate.

(2) Why it matters in Recognition Science

RS grounds physical complexity in the convex J-cost minimization on ledgers (unique minimum at ratio 1, linear verification, exponential growth via φ-rungs). This theorem shows that those physical properties directly force computational constraints, linking the ledger-based physics to limits on what algorithms can achieve. It is part of the IC-005 certificate establishing that RS physics is computationally tractable in specific ways (O(1) ground state, O(N) balance checks, EXPTIME for high rungs).

(3) How to read the formal statement

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

• The hypothesis h assumes physics_complexity_from_ledger.
• The conclusion is computation_limits_from_ledger.
• The proof is just h (reflexivity).
• Note that physics_complexity_from_ledger is defined as computation_limits_from_ledger, making the theorem a definitional bridge rather than a deep inference.

(4) Visible dependencies or certificates in the supplied source

The theorem depends on physics_complexity_structure, which itself is computation_limits_structure. It sits after the J-cost core results: jcost_nonneg, jcost_unique_minimum, jcost_squared_form, verification_equivalence, and phi_rung_complexity_unbounded. The module ends with the string certificate ic005_certificate summarizing all IC-005 results (including this implication) as derived with no sorry.

(5) What this declaration does not prove

It does not establish the concrete content of the computation limits (e.g., specific classes such as P or EXPTIME), which reside in the imported ComputationLimitsStructure. It does not prove the full complexity classification or any empirical predictions; those appear only as the non-theorem list rs_complexity_classes and the summary string. No new physical or computational bounds are derived here.