FPRD LabShare an approach

← Claims

Representation no-go theorem

FPRD-T123

Positive atomic coordinate scores cannot cover the half-line bijectively

Exact statement

A positive additive score on a Boolean support subgroup containing every singleton, with finite score on every allowed support, cannot be a bijection onto the full nonnegative half-line.

StatusProved and internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

A tempting construction labels Boolean coordinates with positive weights and reads a support by summing them. The obstruction shows why no such globally bijective atomic presentation can realize the whole ray.

Hypotheses and scope

  • The Boolean subgroup contains every singleton support.
  • All coordinate weights are positive and every allowed support has finite additive score.
  • The score is assumed globally bijective onto [0,infinity).

Proof or evidence

Singleton toggles force parity blocks, factor-two weight separation, and then a complete two-sided dyadic chain. The unique supports for u/3 and 2u/3 have alternating tails whose symmetric difference collides at score u with the singleton of weight u.

Atomic-score obstruction →

Verification notes

The parity formula, dyadic-chain forcing, dual-tail collision, infinite-support scope, and reparameterization boundary were independently reconstructed. Kakeya is comparator prior art, not the proof.

Limitations

  • Signed scores, interaction terms, non-atomic encodings, and non-coordinate actions are outside the theorem.
  • No novelty claim follows from the incomplete source search.

Open work

Investigate interaction-dependent, signed, and non-coordinate presentations outside the theorem's hypotheses.

Notes

Singleton toggling forces block parity, factor-two separation, the full two-sided dyadic weight chain, and a dual-tail collision at the same score. The theorem is scoped to positive coordinate-additive scores with all singleton supports and finite score for every allowed support; signed scores, interaction terms, non-atomic presentations, and global non-coordinate actions are outside it. Kakeya is comparator prior art for positive subsums, not a proof of this full-ray obstruction. No novelty claim is made.