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Proposition and corollary

FPRD-T67

Material–precision tradeoff for Fold-and-Look

Exact statement

Under FPRD-D29, Psites≤2F(Λfp/δ+1)P_{\mathrm{sites}}\le 2^F(\Lambda_{\mathrm{fp}}/\delta+1). Consequently a presentation of parity on mm bits satisfies Λfp/δ+1≥m2m−1−F\Lambda_{\mathrm{fp}}/\delta+1\ge m2^{m-1-F}, which is exponential in mm whenever the fold budget is, for example, F=O(log⁡m)F=O(\log m).

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

Context

Material length and feature precision are not independent resources. Packing many potential sites onto a short strip necessarily requires fine spacing, and the number of affine pieces created by the folds controls how often that packing can repeat.

Definitions

  • Λfp\Lambda_{\mathrm{fp}} is the length of the final footprint and δ\delta is the minimum separation between feature sites within one affine piece.

Hypotheses and scope

  • All-layers folds, so the address map has at most (2^F) affine pieces.
  • Feature sites in one affine piece are separated by at least (delta>0).
  • The parity corollary uses the exact DNF literal count (m2^{m-1}) and FPRD-T64.

Proof or evidence

Each affine piece maps isometrically into the final footprint and therefore contains at most Λfp/δ+1\Lambda_{\mathrm{fp}}/\delta+1 separated sites. Multiplying by at most 2F2^F pieces gives the first inequality. Parity has m2m−1m2^{m-1} essential literal occurrences, so FPRD-T64 supplies the matching lower bound on PsitesP_{\mathrm{sites}}; rearrangement gives the displayed tradeoff.

Fold, material, and precision coupling →

Verification notes

This result arose from an adversarial audit of an earlier resource table that treated material and precision as separate coordinates. The argument and parity consequence were checked again in the later Fold-and-Look re-audit. The exponential conclusion is explicitly conditional on a small fold budget rather than claimed for arbitrary F.

Limitations

  • The inequality constrains a ratio; it does not prove that material length alone must be exponential.
  • The final Look operation's fan-in and the work needed to encode the input remain uncharged.
  • No external or specialist review is documented.

Open work

Extend the resource table to charge encoder work and the fan-in of the final observation.

Notes

Closes a gap this lab had left open: the resource table reported material and precision as independent coordinates and stated that material lower bounds do not follow. Both were true and both concealed that the trade is constrained. The parity consequence does not depend on the S coordinate, whose lower bound is separately reinstated under D29. Two resources remain uncharged and are recorded as such: Look's unbounded fan-in and encoder work.