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Theorem

FPRD-T63

Fold-and-Look incidence normal form

Exact statement

Under the hypotheses of FPRD-D29, the final observation is Look⁡(w)=⋁y∈Yeff ⋀p∈Φ(y)ℓp(w)\operatorname{Look}(w)=\bigvee_{y\in Y_{\mathrm{eff}}}\ \bigwedge_{p\in\Phi(y)}\ell_p(w), where YeffY_{\mathrm{eff}} contains the interior final locations with a nonempty fibre consisting entirely of feature sites.

StatusRepaired after counterexample and rechecked within FPRD Lab
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The theorem converts the geometry of a fixed folded strip into a finite disjunctive normal form: one conjunction for each final location that can become a genuine through-hole.

Definitions

  • Φ(y)\Phi(y) is the set of material points stacked above final coordinate yy.
  • A location belongs to YeffY_{\mathrm{eff}} only when it lies inside the final footprint, Φ(y)≠∅\Phi(y)\ne\varnothing, and every point of the fibre is a feature site.

Hypotheses and scope

  • All five standing conditions of FPRD-D29, including uniform overlay and positive punch radius.
  • Each feature site is controlled by one literal or a Boolean constant.

Proof or evidence

Punches map isometrically to disjoint or identical intervals. Uniform overlay fixes which affine pieces cover each such interval; every inverse point then tracks the image point isometrically. A punch image is see-through exactly when every covering material point is an open feature site, giving the displayed conjunction, and the final observation takes the disjunction over all effective locations.

Incidence normal form and proof →

Verification notes

An earlier version omitted uniform overlay and was refuted by a one-fold, one-site counterexample. The repaired theorem was rechecked analytically and by an independent internal finite harness. The audit also found and repaired the zero-radius degeneracy and a misleading claim that the fibre itself remains constant across a punch image.

Limitations

  • Without uniform overlay the displayed fibre formula is false, although a weaker full-piece normal form survives.
  • The theorem is an extraction of the OR-over-stacks and AND-over-layers mechanism already present in the cited fold-and-punch construction; no novelty claim is made.
  • No external or specialist review is documented.

Open work

Compare the fibre formulation directly with the source construction in An–Demaine–Demaine–Ku and add an exact bibliographic locator.

Notes

An extraction of the OR-over-stacks/AND-over-layers semantics already in An, Demaine, Demaine and Ku, Lemma 2; not claimed as new. The first version omitted (UF) and was REFUTED by independent audit with an explicit one-fold one-site counterexample. The repair restores the uniformity that the source construction has and this lab had dropped. A subsequent independent re-audit confirmed the repaired theorem for R>0. The corrected proof uses a constant set of covering pieces over each open punch image, with their inverse points tracking the image point by isometry; it does not assert the false equality Phi(z)=Phi(y).