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Theorem · readable-cone capacity and sharp horizon radius

FPRD-T153

Finite-horizon scaffold responses have bounded readable-cone capacity

Exact statement

Fix a Loff--Moreira--Reis scaffolding automaton of degree d, distance k, working alphabet size g, and control set Q. For continuation horizon h, every response row is determined by the cut control state and the radius R=k+(h-1)(k-1) unfolded neighbourhood when k,h>=1, with radius zero when k=0 or h=0. If V_0=g+1 and V_(r+1)=1+(g+1)V_r^d, then every horizon-h cut has at most |Q|V_R distinct response rows. The radius R is uniformly sharp.

StatusSelf-contained proof with exhaustive finite structural audit
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

FPRD-T152 turns any upper bound on distinguishable cut presentations into a response-row bound. Native scaffold semantics supply such a bound because a finite continuation can pull information from only a bounded unfolded cone.

Hypotheses and scope

  • The machine is a fixed deterministic Loff--Moreira--Reis scaffolding automaton.
  • Transitions inspect only the radius-k unfolded neighbourhood and install missing, SELF, or old-top path descriptors of length at most k.
  • The continuation family contains words of length at most h.

Proof or evidence

A one-step pullback sends requested new radius r to old radius k+r-1. Induction gives R=k+(h-1)(k-1), and counting abstract unfolded views gives the row bound. Degree-one chains place a decisive bit exactly at depth R, proving uniform sharpness. The checker covers 13,056 old scaffolds and 4,253,184 one-step extensions, plus 24 sharpness witnesses.

Finite-horizon scaffold capacity →

Verification notes

The proof was checked for the h=0 and k=0 boundaries, SELF loops, invalid paths, the missing-versus-unlabelled radius-zero collapse, hidden physical aliasing, and reachability of the sharpness chains from ordinary prefixes.

Limitations

  • The theorem bounds a fixed automaton parameter tuple; it does not separate a language from all scaffolding automata or PEGs.
  • The count includes abstract views that may be unreachable in a particular automaton, so it is a uniform ceiling rather than an exact reachable-state count.
  • The bound concerns finite-horizon response rows and does not bound total archived memory.
  • No literature-priority or external-review claim is made.

Open work

Use FPRD-T154 to replace raw row cardinality by a structural invariant that survives larger fixed scaffold parameters.