Finite-horizon residual tower

Exact statement

For a language L, Q_h is the finite set of response rows of prefixes on continuations of length at most h. Restriction gives surjections Q_(h+1) to Q_h, forming an inverse system. Reading one input symbol maps Q_(h+1) to Q_h by selecting the corresponding response subtree.

StatusDefined self-containedly and connected to residual derivatives

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Individual finite response tables do not explain how memory moves when a new symbol arrives. The residual tower records all horizons together and gives each input symbol a canonical shift map.

Definitions

Q_h contains the prefix response rows on all continuations of length at most h.
Restriction from horizon h+1 to h is a surjective bonding map.
A one-letter derivative consumes one horizon level by selecting the child response subtree.

Proof or evidence

The public note proves well-definedness, compatibility, compact inverse-limit completion, dense residual embedding, and the stabilization criterion formalized in FPRD-T156.

Residual tower →

Verification notes

The finite identities were checked on every binary language table through total word depth three.

Limitations

The inverse limit can contain ideal residuals not realized by any prefix.
The definition gives semantic transport but does not bound physical update cost in a machine representation.
Inverse-limit and profinite methods are classical; no novelty claim is made for that framework.

Open work

Use the inverse-limit shift as the semantic object for an encoding-independent theory of stationary transport presentations.