Response Upgrade Well-Foundedness

Abstract

Finite T2-compliant response traces cannot retry forever: a sufficiently long same-stimulus trace must stop or change class, and blind retries are decidable.

Theorem 1.1 (T2 response traces stop or change class).

$$\begin{array}{r}\forall S,C,R:Type,\\ [\mathrm{Fintype}\left(S\right)],[\mathrm{Fintype}\left(C\right)],[\mathrm{Fintype}\left(R\right)],\\ [\mathrm{DecidableEq}\left(S\right)],[\mathrm{DecidableEq}\left(C\right)],[\mathrm{DecidableEq}\left(R\right)],\\ trace:\mathrm{List}\left(\mathrm{ResponseEvent}\left(S,C,R\right)\right),\\ stimulus:S,responseClass:C,\\ T2Compliant\left(trace\right)\Rightarrow \forall event:\mathrm{ResponseEvent}\left(S,C,R\right),event\in trace\Rightarrow \mathrm{stimulus}\left(event\right)=stimulus\Rightarrow \mathrm{card}\left(R\right)<\mathrm{length}\left(trace\right)\Rightarrow \exists event,event\in trace\wedge (\mathrm{stopped}\left(event\right)=true\vee \mathrm{responseClass}\left(event\right)\ne responseClass).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/OperationalTuition/ResponseUpgradeWellFounded.t2_response_upgrade_well_founded (✓ std3). ∎

Source. Repository-derived.

Commentary.

A response event records a stimulus, response class, response value, and an explicit stop bit. T2 compliance is a structural predicate requiring the nonterminal responses in each finite stimulus/class slice to be duplicate-free.

The finite response alphabet bounds every duplicate-free list by its Fintype cardinality. If a same-stimulus trace exceeds that bound, the compliant trace therefore contains a stopping event or an event whose response class has changed.

Theorem 1.2 (Blind-retry T2 violations are decidable).

$$\begin{array}{r}\forall S,C,R:Type,\\ [\mathrm{Fintype}\left(S\right)],[\mathrm{Fintype}\left(C\right)],[\mathrm{DecidableEq}\left(S\right)],[\mathrm{DecidableEq}\left(C\right)],[\mathrm{DecidableEq}\left(R\right)],\\ trace:\mathrm{List}\left(\mathrm{ResponseEvent}\left(S,C,R\right)\right),\\ t2ViolationDecision\left(trace\right)=true⟺\neg T2Compliant\left(trace\right).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/OperationalTuition/ResponseUpgradeWellFounded.t2_violation_decidable (✓ std3). ∎

Source. Repository-derived.

Commentary.

All carriers and the finite trace are explicit, so the universal T2 predicate has a decision procedure. The Boolean classifier returns true exactly for its negation.

A two-event trace repeating the sole response in a one-element alphabet is a compiled blind-retry witness and is classified as a violation.

References

• Truth anchor: D5/S3/ConceptDynamics/OperationalTuition/ResponseUpgradeWellFounded.t2_response_upgrade_well_founded
• Truth anchor: D5/S3/ConceptDynamics/OperationalTuition/ResponseUpgradeWellFounded.t2_violation_decidable