Question: can bang PROVE things? Does that require "logic as a programmable feature"? (Raised via: "if a conversion to the abstract shape is provided, shouldn't two instances satisfying that shape be substitutable when only the shape is required? — you lose specific info but retain the generic structure.")

The answer is TWO levels:

substitutability / uniformity   FREE from PARAMETRICITY — the type IS the proof, no new feature   [bang has this NOW]
arbitrary propositions          needs DEPENDENT TYPES (Curry-Howard: propositions-as-types)        [Q31 far end]

The operator's insight IS a real theorem — Reynolds' ABSTRACTION THEOREM (parametricity / representation independence; Wadler's "theorems for free"):

a generic client that uses ONLY the abstract shape is PARAMETRIC — it cannot inspect the concrete type,
  only call the interface's operations
  ⟹ it CANNOT distinguish two instances satisfying the interface
  ⟹ they are SUBSTITUTABLE   ("lose specific info, retain generic structure" = the forgetful map to the interface)

PRECISIFICATION (owed for correctness): it is NOT the conversion ALONE that proves it — it is parametricity (the client uses only the interface) PLUS the conversion being a LAWFUL HOMOMORPHISM (it preserves the interface's operations + laws). The conversion witnesses "this concrete thing IS an instance of the shape"; parametricity guarantees the client respects only the shape. A law-BREAKING conversion would let the client observe the difference. Given both, substitutability is the theorem.

bang already EARNS these proofs. Everything from bite-0 through #55 is PARAMETRIC polymorphism, so the abstraction theorem applies: a generic bang function proves its own UNIFORMITY. map : (a→b) → List a → List b cannot inspect a, so it MUST treat every element uniformly — a proven property carried by the type, no proof written. That is "logic as a programmable feature" in latent form: the type system discharges substitutability for free. (This is the type-level twin of Q41 type isomorphism — an iso is a proof of type equality; parametricity is a proof of client-indistinguishability.)

Full proving = CURRY-HOWARD. To prove ARBITRARY propositions, climb to dependent types (Q31's Nat → refinement → dependent road): a PROPOSITION is a TYPE, a PROOF is a PROGRAM of that type. Then a well-typed bang program IS a proof, and bang can prove things about ITSELF. It fits the thesis: a proof becomes a first-class DESCRIPTION (of why a proposition holds), forced with $ like everything else. Today the KERNEL is proven in Lean (the metalanguage); Curry-Howard would give bang's OWN type system that power (the object language becomes a prover).

parametricity   (HAVE)   free theorems: uniformity · substitutability-through-an-interface — the type is the proof
refinement      (Q31)    types carry constraints ({n : Nat // n < 10}) — proofs ABOUT values
dependent       (Q31)    propositions-as-types — a program IS a proof — bang becomes a prover

Blocked on: nothing for the parametricity level (it's a metatheorem about the existing type system — could be DOCUMENTED/exploited as free theorems now); the full-proving level needs dependent types (Q31, post-v1, research). Performance/soundness of the kernel is unaffected (this is object-language proving, orthogonal to the Lean-verified core).

Revisit signal: free theorems are wanted as a TOOL (e.g. a parametricity-derived optimization or a substitutability check); OR dependent types are taken up (Q31) — then propositions-as-types makes bang a prover; OR Q41 type isomorphism gets a propositional-equality form (an iso as a proof term). Ties Q31 dependent types (the road to full proving), Q41 type isomorphism (proof of type equality), ADR-0075 (parametric polymorphism = the source of free theorems), ADR-0081 (generic data — the parametric values).