FPRD LabShare an approach

← Claims

Theorem and contextual lower bound

FPRD-T84

Output-minimal anchors retain a linear-bit archive

Exact statement

A fixed deterministic three-sweep reverse-copy transducer over ({0,1,\#}) has two named output-minimal one-cell anchors of one fixed effect/component type and trace output (#). For every K≥4K\ge4, legal right insertion into the inter-anchor slot gives a contextual quotient for operatornameSmallPerK=⋃p=1KPer⁡poperatorname{SmallPer}_K=\bigcup_{p=1}^K\operatorname{Per}_p with at least (2^{K-3}) and at most (|M_K|) classes. Thus the exact Boolean small-period archive still requires (Theta(K)) bits.

StatusProved and internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The lower bound survives after both productive anchors, their rooted types, and their one-letter outputs have been fixed.

Hypotheses and scope

  • The quotient is for one explicit marked right-insertion action, not for an unmarked effect semigroup.
  • K is an external observation cap and the predicate concerns the full extended word produced amid the named anchors.

Proof or evidence

Fresh-marker words (W_z=##z#) have a doubled marker skeleton with no period below K. Inserting the context derived from z makes (W_{z'}W_z) have a period through K exactly when (z'=z), separating all (2^{K-3}) payloads.

Output-minimal anchor archive →

Verification notes

Endpoint conventions, anchor order, output-minimality, insertion orientation, Fine–Wilf use, and all ordered codeword pairs through K=8 were checked independently.

Limitations

  • This is not a lower bound for the raw between-anchor output alone or for a general marked-path category.
  • It does not eliminate quantification over successful runs and inversions.
  • For one fixed transducer, its internally computed cap is fixed; the theorem alone is not a device-size complexity lower bound.

Open work

Determine the complexity of exposing the relevant marked record from a successful run, rather than compressing the record after exposure.

Notes

This is a lower bound for one concrete marked-record right action, not a general marked-path category or an unmarked effect quotient. SmallPer_K is the extended inversion-word period-existence observation used to formalize Theorem 3.6 P2 and stated explicitly in Lemma 9.1. The result does not lower-bound the raw between-anchor factor alone or the stronger Proposition 7.6 conjunction requiring the period to divide both unit anchor lengths. K is an external observation cap; for one fixed transducer its computed B(T) is fixed, so this is not by itself a transducer-size complexity lower bound. No external novelty claim is made.