Item 34: Does the selection-punishing menu bite the ported tower ⟹ Value chain?

The source line refutes the hard-selector tower ⟹ Value route on a selection-punishing menu — every option worth nothing exactly when it is the one chosen — where the tower holds and Value fails, so a scope condition on Value's own menu quantifier is necessary. It separately reports a δ-hedged soft route as punishment-robust, at a stated cost: proved modulo a feature-introspection step it files as open, with the robustness observation flagged same-session, unvetted and not machine-checked.

Workspace.Deference.Contrib.InheritedAlgebra.value_asymptotic (lean/Workspace/Deference/Contrib/InheritedAlgebra.lean) is also a soft construction — a vanishing-gap mixture over the menu, with hSoft as the softmax-gap step of its chain — and is therefore not the refuted hard-selector route. But it is not identical on its face to the source's hedged construction either: that one is a ramp over the options within 2δ of the top quote at a fixed δ, and the port carries δ → 0. Resemblance is not identification, and neither the survival nor the failure of the port's hypotheses on the punishment family has been established.

Instantiate the port's full hypothesis package on that family — at a fixed positive gap and at a gap shrinking with the day index — and determine whether the package is jointly satisfiable there and what conclusion it actually supports. Two outcomes, both successes:

(A) The port survives. Ship an inhabitation witness, a precise statement of why the port lies on the surviving soft side, and any additional hypothesis the instantiation turns out to require. (B) The port fails or goes vacuous. Ship the counterexample as a necessity witness, name the exact hypothesis or combination that fails, and give the weakest defensible restriction restoring a nonvacuous statement.

Do not prejudge which hypothesis fails, and do not define success by importing the source line's own scope condition as a definition. Whether that condition is the right restriction for this port is part of what the item asks.

Deliverable shape: witness-checked or enumeration-verified over the declared finite menu, or lean-proved for the inhabitation term if (A); either way a named verdict per hypothesis. Acceptance check: python3 -m checkers.run accepts the registered entry; or the lean gate is green and the inhabitation term typechecks.

Context: lean/Workspace/Deference/Contrib/InheritedAlgebra.lean, the value_asymptotic block and its ported source projects/deference/note-dump-2026-06-27/lean/LeanDeference.lean — the superseded June tree, cited because provenance records where the port was made from, not because it states the current result; projects/deference/note-dump-2026-08-11/wiki/total-trust-implies-value.md §"Necessity of the scope condition" for the menu, its arithmetic, and the hard-selector failure; projects/deference/note-dump-2026-08-11/wiki/soft-self-endorsement.md for the hedged construction, its robustness claim and the grade that claim carries; projects/deference/rounds/2026-08-12-corpus-reconciliation/RECONCILIATION.md §1. A solution ships: the instantiation and the per-hypothesis verdict, in whichever of the two outcomes it reaches. Either outcome is worth having — a bounded theorem with a witness and an unbounded one with a witness are both better than the present reading.