meta_language_distinguishes
plain-language theorem explainer
The ambient logic distinguishes at least two propositions: there exist P and Q with P unequal to Q. Anyone tracking the T-1 self-bootstrap limit cites this as the meta-language half of the meta/object separation. The proof is a one-line wrapper of the True/False witness already proved upstream.
Claim. There exist propositions $P$ and $Q$ such that $P \neq Q$.
background
The module records a sharp limit of the T-1 self-bootstrap route: the formal language can tell propositions apart, yet that meta-level fact does not force every inhabited object carrier to be non-singleton. The unit type is the counterexample carrier.
Upstream, meta_language_distinguishes_props already states the same existence claim and proves it by exhibiting True and False as unequal propositions. The present declaration simply re-exports that fact inside the MetaDoesNotForceObject module so the certificate can name a local lemma.
No Recognition-cost, phi-ladder, or J-functional structure is used here; the content is pure propositional logic inside Lean's meta-language.
proof idea
One-line term wrapper: the proof is exactly the upstream theorem meta_language_distinguishes_props. That lemma constructs the witness pair True and False, then derives a contradiction from any assumed equality by transporting True.intro along the equality and eliminating False. No additional tactics or local lemmas are introduced.
why it matters
This is the first field of the meta/object separation certificate metaDoesNotForceObjectCert, which packages meta-level distinguishability together with the negative claim that meta distinction does not force object-level distinction on every inhabited carrier.
In the Recognition foundation stack it marks the boundary of the T-1 self-bootstrap: language-level non-triviality is secured, but object carriers (in particular the unit carrier) need not inherit a two-point distinction. Downstream work that wants a theorem-backed certificate of this separation cites the cert, which in turn cites this lemma for the positive meta half.
It does not advance the forcing chain T0–T8, the RCL, or dimensional claims; it is a foundational hygiene result about what the bootstrap may and may not force.
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