t3_to_canonical_empty_ledger_bridge_holds
plain-language theorem explainer
Given T3 (ledger forced from cost symmetry), the canonical empty ledger is a balanced witness, so the legacy existential "there exists a balanced ledger" holds by naming that witness. Forcing-chain assemblers cite this bridge when wiring T3 into the complete chain. The proof is a short term construction that packages two facts from LedgerForcing.
Claim. Assume the ledger is forced: the empty consistent floor entry has zero recognition cost, empty joins are neutral on floor states, and empty joins are cost-neutral. Then the canonical empty ledger is balanced, and there exists a balanced ledger (namely that empty ledger).
background
The module UnifiedForcingChain aims to show that T0–T8 are forced from the cost foundation (Recognition Composition Law, normalization, calibration), starting from an absolute floor. In that ladder, T3 is the ledger step: at the pre-analytic floor the ledger is the additive bookkeeping of recognition work, with a zero-cost empty consistent entry and cost-additive independent joins; the reciprocal scalar ledger is treated as an analytic refinement.
The bridge structure packages what T3 already implies for the ledger API used downstream. It does not invent a free choice of balanced ledger. It names the canonical witness LedgerForcing.empty_ledger, records that this witness is balanced, and exposes the legacy existential surface $\exists L,,\mathrm{balanced}(L)$ that older chain code expects.
T3 itself is the cost-symmetry step in the primer chain (ledger forced by $J(x)=J(1/x)$). This declaration only certifies the empty-ledger bridge once a T3 hypothesis is in hand.
proof idea
Term-mode structure inhabitant, not a tactic script. Under hypothesis $h_3 :$ ledger forced, fill the two bridge fields directly:
empty_ledger_balancedis definitionally the upstream fact that the canonical empty ledger is balanced (LedgerForcing.empty_ledger_balanced).balanced_exists_legacyis the pair $\langle$ empty ledger, that balance proof $\rangle$, discharging the existential.
No further case analysis or cost algebra is performed here; the work lives in LedgerForcing.
why it matters
This is the T3 → canonical empty-ledger bridge in the complete inevitability chain. Downstream, complete_forcing_chain threads T−1 through T8 by composing such bridges; without a named balanced ledger witness, later recognition and $\varphi$-forcing steps lack a concrete ledger object.
In the module’s stronger claim, T3 is forced by cost symmetry rather than postulated. The bridge closes the bookkeeping gap between that Prop and the LedgerForcing API (empty ledger as neutral zero-cost entry). It does not touch T5 J-uniqueness, T6 $\varphi$, T7 eight-tick, or T8 $D=3$; those sit later in the same chain. Doc-comment is minimal: T3 supplies the canonical empty-ledger bridge. That is exactly the role it plays for chain assembly.
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