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Theorem · exact zero-or-unbounded stack-clock dichotomy

FPRD-T175

Letter projections have either an exact stack clock or unbounded clock skew

Exact statement

For a letter projection from a call-return-local alphabet, define the clock diameter of an observed word as the largest final stack-height difference among compatible hidden lifts. If no calls exist, the stack is never populated. Otherwise, the diameter is uniformly bounded exactly when each visible-letter fiber has one action type, in which case it is identically zero. If any fiber mixes action types, then for every n some observed word of length at most 2n has clock diameter at least n.

StatusSelf-contained elementary proof with exhaustive action-fiber audit
External reviewThe visibly pushdown action partition is classical. No external or specialist review of the projection-level dichotomy formulation is recorded.

Context

T174 proves effective selection when projection preserves one common stack clock. T175 shows that ordinary letter projections admit no projection-level hierarchy of finite clock-skew bounds between exact synchronization and unbounded divergence.

Hypotheses and scope

  • The hidden alphabet has a fixed call, return, and local partition.
  • The hidden-coordinate map is a memoryless letter projection with nonempty fibers.
  • Clock diameter ranges over all compatible hidden lifts, without source-language restrictions.

Proof or evidence

If an ambiguous fiber contains a call, repeating that visible letter separates call and noncall lifts by n. If it contains only local and return actions, first repeat any call-bearing visible letter n times, then repeat the ambiguous letter n times. One lift preserves height n while the other returns to zero.

Stack-clock dichotomy →

Verification notes

The checker exhausts all 399 nonempty action-fiber patterns on observed alphabets through size three. It verifies 52,084 zero-diameter words and 2,680 explicit linear-divergence witnesses through separation eight.

Limitations

  • A source language may forbid the hidden histories used by the unrestricted witness.
  • The invariant measures height disagreement rather than stack-symbol disagreement.
  • Context-dependent hidden-coordinate maps may have bounded nonzero lag.
  • The theorem does not decide causal selection in the unbounded tier.

Open work

Develop source-conditioned clock coherence, since no nontrivial bounded-skew tier exists at the unrestricted letter-projection level.