distinction_forces_T0

This module opens the closure path from a supplied distinction witness to the early forcing spine (Phase 1 of T−1→T8). Given $h:\exists x,y:K,,x\neq y$, the upstream construction TMinus1ForcedFromDistinction builds the forced observable quotient and proves it equivalent to Bool. The point is to thread T0 through that distinction-generated floor rather than through unthreaded global Boolean facts.

A cost function on a configuration space is a nonnegative map $C$ satisfying dichotomy (zero cost iff the configuration is consistent) and independent additivity (cost adds over joins of configurations that share no predicates). The lemma cost_pos_iff_inconsistent restates dichotomy as positivity of cost exactly on inconsistent states.

Here the configuration space is the forced quotient of $h$, equipped with the transported recognition cost obtained by pulling back the Boolean T−1→T0 cost along the forced quotient–Bool equivalence. The structure T0_FromDistinction packages the Boolean character of that quotient together with the full recognition-work surface on it.

Tactic-mode construction of the five remaining fields of T0_FromDistinction (the Boolean equivalence is supplied as a witness in the signature).

• recognition_work is the preexisting certificate forcedQuotient_recognition_work_constraint h.
• consistency_cheap rewrites via the transport identity forcedQuotientRecognitionCost_transport, simplifies through the Bool equivalence, and closes by reflexivity (empty Bool config has zero cost).
• contradiction_expensive applies CostFunction.cost_pos_iff_inconsistent to the transported cost, converting inconsistency of $\Gamma$ into positive cost.
• logic_emergent is the dichotomy axiom of forcedQuotientRecognitionCost h.
• additive_indep is the additivity axiom of the same transported cost, applied to independent pairs.

This is the first concrete forcing step from a raw distinction to the Recognition spine: T0 on the quotient the distinction itself generates. Downstream, distinction_forces_T1 is a one-line lift of this result, and distinction_forces_T3 chains it with the T2 step. The spine package distinction_forces_T0_spine records the Boolean quotient witness among the T−1-to-T4 data forced by any distinction.

In the broader framework this anchors Phase 1 of the T−1→T8 closure: once T0 sits on the distinction-generated floor, later steps (J-uniqueness at T5, $\varphi$ at T6, the eight-tick octave at T7, $D=3$ at T8) can be routed through transported structure rather than global Bool facts. It does not yet force those later landmarks; it only installs the recognition-work surface that the rest of the chain rides on.