A shared stack clock makes fixed-budget causal selection effective

Exact statement

Let A be deterministic visibly pushdown and let pi preserve the call, return, and local partition. For every fixed finite family of synchronous causal selectors, the language of accepted observations missed by the family is visibly pushdown: all source and selected stacks zip into one product stack. Coverage is therefore decidable. For every fixed selector budget (m,s), a computable finite horizon B_(A,pi)(m,s) decides global feasibility and bounds obstruction radius.

StatusSelf-contained product-stack proof with classical VPL calibration and bounded exhaustive audit

External reviewThe synchronized-stack closure mechanism and visibly pushdown decision properties are established by Alur and Madhusudan. No specialist review of the FPRD selector application is recorded.

Context

T173 makes regular hidden sources effective. T174 extends effective compactness to unbounded stacks when projection preserves the shared push-pop-local clock, isolating action erasure rather than stack memory itself as the source of the hard multi-history product.

Hypotheses and scope

A is recognized by a deterministic visibly pushdown automaton.
The letter projection preserves the call, return, and local classes.
Selector count and selector state size are fixed finite bounds.

Proof or evidence

All compatible lifts perform the same stack action at each observed position. Their stacks have equal height and can be zipped coordinatewise into one VPA stack. The resulting VPA recognizes exactly the observations with some accepting hidden lift but no accepting selected lift. VPA emptiness and shortest-witness computation then make each budget effective.

Synchronized stack-clock theorem →

Verification notes

The saved checker audits a restricted fixture family of 4,096 deterministic two-state VPA tables, eight action-preserving selectors, 36 selector families, and 40 observed words; it does not cover all deterministic VPAs or compute the general horizon. It verifies 5,898,240 direct-versus-zipped language cases across 15,040,512 synchronized input steps.

Limitations

The construction assumes a deterministic source VPA.
Projection must preserve the visible stack-action partition.
The theorem gives an algorithmic horizon but no closed-form size bound.
It does not apply to the forgotten-action lift from FPRD-T168, where histories can use independent stack clocks.

Open work

Use T175's clock dichotomy to study intermediate tiers restricted by source language, selector outputs, bounded phase structure, or contextual maps.