promoted from data/fprd-lab/fold-look/bounded-fold-incidence-comparators.md on 2026-07-28 — status: unverified

Two-round targeted comparator map for the bounded-fold incidence run. Withdraws the claim to the 2^k affine-piece bound (Cardinal et al.; lap-number submultiplicativity) and to D<=2^F; confirms the stamp-folding characterisation does not apply to a partial coincidence relation; records two prior necessary conditions on 1D coincidence, one of which became Lemma 16; and closes the crease-count route to fold lower bounds. Gap list included.

Comparator map: bounded-fold incidence and formula resources

Status: targeted primary-source comparison, two rounds, complete for the questions asked. Deliverable 7 of the bounded-fold incidence run. Labels used, exactly: classical result used, direct finite-horizon adaptation, not yet found to be prior, candidate novel result. The word “novel” is not used. Generic fold-count complexity is assumed prior.

Primary sources are held under ~/data/fprd-lab/fold-look/sources/ with SHA-256 recorded there and are never promoted to this repository. Page numbers for Folding and Punching Paper are from an accepted-manuscript extraction and will differ from the published JIP 25 (2017) pagination.

Contents

1. Verdicts

statement	verdict	strongest comparator	label
S1 `Look` = OR over locations of AND over layers	already in the source	An–Demaine–Demaine–Ku, CCCG 2017, §2 p. 2 and Lemma 2 §4 p. 3	`classical result used`
S2 $\varphi_{\mathcal P}$ piecewise $\varepsilon x+c$ on $\le2^k$ intervals	form definitional; count is used folklore	Cardinal et al., Algorithmic Folding Complexity, §2 p. 4 and Lemma 1 p. 4; Eppstein arXiv:2410.07666 §2 pp. 2–3	`classical result used`
$D\le2^F$	same folklore on ply	as S2	`classical result used`
S3 $\sigma(E)$ and $k\ge\lceil\log_4\sigma(E)\rceil$	ingredients prior; assembled measure not found	Ani–Brunner–Demaine arXiv:2202.01293 Lemma 2.2 p. 5 (signs); Cardinal et al. Prop. 1 p. 6 (parity)	ingredients `classical result used`; the measure `not yet found to be prior`
S4 $w_{\mathrm{DNF}}\le D$ , $s_{\mathrm{DNF}}\le S$ , $\ell_{\mathrm{DNF}}\le P_{\mathrm{sites}}$	not found; nearest comparator runs the opposite way	An et al. compile a formula into a folding; Eppstein treats ply as an algorithmic parameter	`not yet found to be prior`
S5 equal continuation complexity, unequal folding demand	not found	Cardinal et al. Thm 1 vs Thm 3 separate within one measure only	`not yet found to be prior`
fold-count lower bound for prescribed point alignment	none exists	`foldpunch` §8 p. 10 states even crease lower bounds are open	`not yet found to be prior`
partial / multi-location alignment as a studied object	not treated, not named	every formulation in custody is single-target	`not yet found to be prior`
shuffle incidence relation realizable?	undecided in the literature	two necessary conditions exist (below)	`not yet found to be prior`
active vs merely present sites	distinction explicit, never charged	An et al. §3 p. 3; Lemma 4 p. 5	`classical result used` / charging it `not yet found to be prior`

What survives as this run’s candidate contribution is narrow: the direction of S4, together with $\sigma(E)$ as a measure. Everything geometric is inherited.

2. Two claims withdrawn on this evidence

Theorem 5 ( $r\le2^k$ ). Withdrawn as a claim. Cardinal et al. open §2 (p. 4) with “Every mountain-valley pattern with $n$ creases requires at least $\lceil\lg(n+1)\rceil$ folds, just to get all the creases folded in some direction” — a statement whose only justification is the $2^k$ count — and Lemma 1 (p. 4) uses the matching structure in the tight case. In general form it is lap-number submultiplicativity from Milnor–Thurston kneading theory, since a 1D all-layers simple fold is a 2-lap piecewise-affine map with slopes $\pm1$ . (That inequality was not located as a pinned sentence in the primary Milnor–Thurston text and is cited as a concept.)

Theorem 12’s word “requires”. Weakened to a disjunction. See §3.

3. The stamp-folding question, and why it does not apply

Round 1 found the classical characterization of foldable permutations — Koehler 1968 via Legendre 2014, Lemma 2.1 p. 276 — implemented the criterion, validated it against OEIS A000136 for $n\le8$ (1, 2, 6, 16, 50, 144, 462, 1392), and found that reversal, identity and cyclic shift are foldable while the even–odd shuffle is not, for every $n\ge4$ through $n=16$ .

Round 2 withdrew the tension this appeared to create, and the withdrawal is correct. Legendre’s proof (pp. 276–277) locates the obstruction in single-pile geometry: perforation arcs on one side of a single pile would have to cross. That arc diagram exists only because all $n$ cells occupy one location under one total layer order. The relation $E_\pi$ requires only that specified pairs coincide, at up to $n/2$ distinct locations, with the rest of the strip free to remain spread out. No single pile, no arc assignment, no crossing obstruction.

So Koehler unrealizability of the shuffle is a statement about a strictly stronger object and does not bear on $E_\pi$ . The Q4 hit stands as prior art for “which permutations are realizable by folding into one pile”; its relevance to Theorem 12 is none.

4. Two prior necessary conditions on 1D coincidence

Both were located by the comparison and are usable instruments, not just context.

Parity. Cardinal et al., Proposition 1, p. 6: “Two creases $i$ and $j$ can be piled at the same point if and only if $i-j$ is even.” Stated for creases at integer positions.

Orientation / IVT. orthofoldcut Lemma 2.2, p. 5 is stated for the single-target problem, but the core step of its proof does not use the single-target hypothesis and therefore survives the weakening: a coincident pair at which the location map has the same orientation forces a third coincident point strictly between them. The lemma’s alternate conclusion does not transfer, because it depends on exclusivity; the IVT step does.

This second condition was converted into Lemma 16 (orientation forcing) in docs/probes/k-fold-incidence-and-permutations.md: combined with the normal form, it shows every required width-two pair must be opposite-orientation, hence must lie on a reflection-type line. That is why reversal is the one-fold case, and it is a direct gain from the comparison rather than from the lab’s own work.

5. The crease/fold gap — a route that is closed

Everything in the alignment literature that reads like a lower bound is a crease count, and crease counts do not transfer to fold counts. Cardinal et al., Theorem 1, p. 4: “The folding complexity of a uniform string and of a pleat of length $n$ in the all-layers fold, all-unfold model is at most $\tfrac32\lg^2 n$ ” — a pleat has $n$ creases. So any argument of the form “this alignment forces $n$ creases, therefore many folds” is refuted.

Corollary 7 was checked against this and routes through doubling, not through creases, so it is safe. The related consequence — that excluding Unfold is load-bearing for Theorem 5 — is recorded in the piecewise-isometry memo.

6. Gaps, stated rather than papered over

“Optimal Simple Fold-and-Cut of a Polygonal Line”, doi 10.1007/978-981-96-6561-7_26 — not read; Springer redirected to an authentication endpoint. Authorship and venue unconfirmed (search suggested Uehara / 8OSME 2024; the DOI prefix suggests a 2025 volume). It is the only fold-count optimization result located, so if anything in the literature collides with Corollary 7 it is most likely there or in its references. Highest-value unresolved item.
No forward-citation sweep of Folding and Punching Paper was attempted (rate-limited). A later paper could have added lower bounds or a partial alignment variant, so the “not found” verdicts on fold-count lower bounds and on partial alignment are provisional on that sweep.
Milnor–Thurston lap-number inequality cited as a concept, not a located sentence.
Page numbers for Folding and Punching Paper are preprint pagination.

Verdicts labelled not yet found to be prior are statements about what this comparison found. They are not novelty claims and must not be upgraded without closing gaps 1 and 2.