T3_To_CanonicalEmptyLedger_Bridge
plain-language theorem explainer
Bridge certificate from T3 (ledger forced) to a named canonical witness: the empty ledger is balanced, and the legacy existential "some balanced ledger exists" follows from that witness. Anyone assembling the complete T0–T8 forcing chain cites it to pin T3's bookkeeping surface to LedgerForcing.empty_ledger rather than an arbitrary balanced L. It is a Prop-valued structure (definitional interface), discharged by a one-line constructor theorem.
Claim. Given a T3 certificate that the recognition ledger is forced, a bridge asserts two facts: (i) the canonical empty ledger is balanced, and (ii) there exists at least one balanced ledger (the legacy existential surface). The intended witness for both is the empty ledger of the ledger-forcing module.
background
The Unified Forcing Chain module claims every level T-1 through T8 is forced from the cost foundation (Recognition Composition Law plus normalization and calibration). T3 is the ledger step: cost symmetry $J(x)=J(1/x)$ forces an additive bookkeeping structure for recognition work, with a neutral empty entry and cost-additive independent joins.
In LedgerForcing, a ledger is balanced when its event list is balanced; balanced is that predicate on ledgers. The empty ledger is the canonical zero-event object, and empty_ledger_balanced records that it satisfies balance. T3 itself is stated at the pre-analytic floor (empty consistent entry has zero cost; empty join is neutral), with the reciprocal scalar ledger as a later analytic refinement.
This bridge is the naming layer between that T3 certificate and the concrete canonical witness used downstream: it does not re-prove ledger forcing, it packages the empty-ledger universal property next to the older existential surface $\exists L,,\mathrm{balanced}(L)$.
proof idea
No proof body: the declaration is a structure (Prop-valued certificate) with two fields. Field one demands LedgerForcing.balanced of LedgerForcing.empty_ledger. Field two is the legacy existential, witnessed by that same empty ledger.
Discharge is external: t3_to_canonical_empty_ledger_bridge_holds builds an instance by setting empty_ledger_balanced := LedgerForcing.empty_ledger_balanced and packing ⟨empty_ledger, empty_ledger_balanced⟩ for the existential. A Subsingleton instance on the bridge (allEq by rfl) records that certificates are propositionally unique for fixed T3.
why it matters
In the forcing chain, T3 is "Ledger ← Cost symmetry." Without a canonical witness, T3 only yields an existential balanced ledger; physics bookkeeping needs a named neutral object. This bridge fixes that object as the empty ledger and keeps the legacy existential for older call sites.
It is consumed by CompleteForcingChain, the top-level structure that packages T-1 through T8 plus the quarter-turn, Hamiltonian, projective, and measurement layers. The companion theorem t3_to_canonical_empty_ledger_bridge_holds is the actual inhabitant used when assembling that chain. Landmark link: T3 sits between discreteness (T2) and recognition/observables (T4), before unique $J$ (T5) and $\varphi$ (T6).
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.