recognition_from_balanced_floor_ledger
plain-language theorem explainer
A balanced empty Boolean ledger (cost of the empty state is zero) yields a nonempty recognition witness on Bool. Anyone proving T4 from T2 discreteness and T3 ledger balance cites this extraction. The proof is a one-line projection of the balanced-floor recognition package onto its recognizer field.
Claim. If the Boolean recognition-work cost vanishes at the empty state, $C(\mathsf{false})=0$, then $\mathrm{Recognize}(\mathsf{Bool},\mathsf{Bool})$ is nonempty: there exists a recognition witness on the two-point floor.
background
This module is the public T-1 through T8 forcing spine. T2 forces two-state discreteness of the floor, T3 forces additive ledger bookkeeping with a balanced empty state, and T4 asserts a recognition witness on that discrete floor.
The Boolean recognition-work cost assigns $C(\mathsf{false})=0$ and $C(\mathsf{true})=1$. Balance of the empty ledger is exactly $C(\mathsf{false})=0$. Upstream, the balanced-floor recognition package records that hypothesis and builds the canonical pre-analytic event: the empty consistent state recognizing itself via the pair $\langle\mathsf{false},\mathsf{false}\rangle$.
Recognize is the Recognition-Science witness type for one carrier recognizing another. Here both carriers are Bool, the normalized two-point floor.
proof idea
Term-mode one-liner. Given hbalanced : C(false) = 0, apply the upstream constructor balanced_floor_recognition hbalanced, which returns a BalancedFloorRecognition record whose recognition field is already Nonempty (Recognize Bool Bool) (witnessed by ⟨false, false⟩). Project that field. No further algebra.
why it matters
Closes the T2/T3 → T4 step on the public forcing spine. Downstream, t4_corollary_of_t2_t3 sets floor_recognition by feeding T3's empty_balanced into this theorem, and t2_t3_to_t4_bridge_holds packages the same witness into the bridge certificate. The UnifiedForcingChain twin states the same fact: a balanced empty Boolean ledger supplies the minimal recognition witness.
Without this extraction, T4 would sit as a free sibling rather than a corollary of discreteness plus ledger balance. It is the explicit tie between T3 bookkeeping and the T4 recognition event that later feeds T5 J-uniqueness and the rest of the chain through T8.
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