Compatible cut fields assemble into languages

Exact statement

Diagonally compatible literal cut fields are exactly the cut fields of unique languages. Their combined bounded-future quotients are the finite observations of the reachable residual orbit, their section maps are exposure-one derivatives, and the two iterated prefix-reach and future-observation constructions recover respectively the residual orbit and its bounded-observation completion, with exact actions and readout.

StatusProved and internally audited

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

Context

The theorem explains when a collection of exact cut tables really comes from one language and how its finite views join into the usual residual dynamic.

Definitions

Prefix reach limits which residuals have appeared; future observation limits how deeply those residuals are tested.
The comparison map sends each literal residual $u^{-1}L$ into its compatible thread of bounded observations $(\rho_N^L(u))_{N\ge0}$ .

Hypotheses and scope

All literal cut cells satisfy the named diagonal section law.
The finite quotients retain compatible horizon bonds, epsilon output, and shifted letter actions.

Proof or evidence

Repeatedly moving named suffix letters across the diagonal determines membership on the zero-future axis and hence one unique language. Exact rows form blocks of the joint bounded observation. Taking future depth first reconstructs each reached residual; then increasing prefix reach gives the literal residual orbit. Reversing the finite-image order gives the completed orbit imported from FPRD-T37.

Assembly theorem →

Verification notes

The M26 proof and closure audit separate unique-language assembly, marginal-to-joint identification, action typing, both limit orders, density, and the imported completion theorem. The independent challenge reports 44,400 checks and fourteen named mutation separators.

Limitations

The theorem imports the general completion result from FPRD-T37 rather than reproving or reclaiming it.
An infinite residual orbit need not be closed, and infinitude alone says nothing about whether the comparison map is onto.

Open work

Compare the two iterated constructions with standard profinite and residual-automaton presentations.

Notes

This is the new marginal-to-joint identification and comparison map. It imports M22 completion rather than reclaiming inverse-limit reconstruction as a new general theorem.