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Definition

FPRD-D29

Literal-local Fold-and-Look model

Exact statement

A literal-local Fold-and-Look presentation uses a connected strip with finitely many input-controlled feature sites, an input-independent sequence of all-layers simple folds, no cutting, punching, or unfolding after the input is encoded, and one final through-hole observation. The standing geometric hypotheses are clean alignment, no crease through a punch, interiority, uniform overlay, and positive punch radius.

StatusPrecisely stated after correction of the zero-radius boundary case
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

This is a prepared-substrate specialization of one-dimensional fold-and-punch computation. Input bits determine which preassigned sites are open; the fold program itself is fixed before the input is known.

Definitions

  • The address map φ:P→R\varphi:P\to\mathbb R sends a material point to its final coordinate, and Φ(y)=φ−1(y)\Phi(y)=\varphi^{-1}(y) is the complete layer fibre above yy.
  • PsitesP_{\mathrm{sites}} counts potential input-dependent sites, DD maximum layer depth, SS effective variable-bearing fibres, FF folds, Λ\Lambda material length, and δ\delta minimum site separation.

Hypotheses and scope

  • Clean alignment (GP): two site images coincide or are separated by more than twice the punch radius.
  • No crease through a punch (NC): every punch interval lies within one affine piece of the address map.
  • Interiority (IN): every punch image lies strictly inside the final footprint.
  • Uniform overlay (UF): the set of maximal affine pieces covering a point is constant across each punch image.
  • The punch radius satisfies R>0R>0.

Proof or evidence

The source memo gives the complete syntax, semantics, resource coordinates, and geometric assumptions. A later re-audit found that allowing zero-radius punches makes the intended through-hole semantics fail, so positive radius is now part of the definition.

Definition and standing hypotheses →

Verification notes

The current statement was compared with the corrected theorem memo and the independent internal re-audit. The re-audit also corrected the meaning of uniform overlay: covering pieces remain constant, not the material points in the moving fibre.

Limitations

  • This model does not describe paper folding in general.
  • The feature sites are prepared before the fold program; no new punch or cut is made during computation.
  • The present results use all-layers folds and one final Boolean observation.

Open work

Add one worked example and one example excluded by each geometric hypothesis.

Notes

The doctrine is a prepared-substrate specialization of the fold-and-punch machine of An, Demaine, Demaine and Ku (CCCG 2017), including its 1.5D narrow-strip variant. Punch is native to that model; what is non-native is that the punches arrive before the program rather than being produced by it. R>0 was added after an independent re-audit found the theorem false at R=0 with all other hypotheses holding vacuously.