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Definition

FPRD-D20

Generalized-Dyck Hankel profiles

Exact statement

At level N, a generalized-Dyck Hankel profile records the residual rows visible to continuations of length at most N, the square prefix-versus-continuation window with both sides of length exactly N, their field and Boolean-semiring factor ranks, and the deterministic one-pass configuration interface at the midpoint of length-2N inputs.

StatusPrecisely stated with rank, row, and midpoint conventions separated
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The profile connects residual semantics, Hankel-style matrices, and one-pass memory without treating their numerical invariants as interchangeable.

Definitions

  • For k≥1k\ge1 bracket types and ℓ≥0\ell\ge0 neutral symbols, a live residual is indexed by its unmatched opener stack s∈[k]∗s\in[k]^*; malformed prefixes enter the zero residual K⊥K_\bot.
  • The bounded residual window tests continuations of every length through N and includes one representative of each visible residual row.
  • The midpoint window is a literal square matrix indexed by prefixes and continuations of length exactly N.
  • Boolean rank means factor rank over the Boolean semiring, not rank over a field.

Hypotheses and scope

  • At least one matched bracket type is present.
  • The observation horizon N is finite.
  • The midpoint machine interface is deterministic, one pass, exact, and restricted to total length 2N.

Proof or evidence

The specification fixes the two different windows and the meanings of their row, rank, and configuration counts. The proof record then derives each count from typed-stack residuals and canonical closing continuations.

Generalized-Dyck Hankel profile →

Verification notes

The definition was checked against the M23 proof and closure audit. The reachable all-zero failure row is included in state and row counts when appropriate but contributes no nonzero linear dimension.

Limitations

  • Bounded-horizon residual rows and exact-length midpoint rows are different objects.
  • Matrix rank alone does not count a reachable zero row.
  • The definition does not assign randomized or nondeterministic memory cost.

Open work

Add a level-two example showing the bounded window, exact midpoint window, zero row, and identity minor.

Notes

Rank counts nonzero linear row dimensions; deterministic midpoint width also counts the reachable all-zero failure row. Boolean rank always means Boolean-semiring factor rank in this milestone.