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Computational finding · failed scarcity proxy

FPRD-FAIL-COLLATZ-LF4

Zero nonforced mass is not trace scarcity

Exact statement

At layer (K,S)=(27,17)(K,S)=(27,17) and depth h=14h=14, exact reconstruction gives 2,3212{,}321 live prefixes, all with capped multiplicity one, so P14=2,321P_{14}=2{,}321 and F14=0F_{14}=0; forced continuation therefore does not by itself mean that few traces survive.

StatusIndependently reproduced with an exact-integer checker
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The branching-stripped statistic counts excess continuation multiplicity, not the number of surviving deterministic threads.

Definitions

  • Fh=∑∣u∣=h(mu−1)+F_h=\sum_{|u|=h}(m_u-1)_+.

Hypotheses and scope

  • Finite computation at the single recorded layer (27,17) and depth 14.

Proof or evidence

A new exact-integer checker reconstructs each depth-14 prefix from its guard, applies the maximal-completion budget, removes full-guard representatives 0,1,20,1,2, caps guard slots by Wu=(1317−∣u∣1)W_u=\binom{13}{17-|u|_1}, and obtains 2,321 live prefixes, 2,321 capacity-one prefixes, P14=2,321P_{14}=2{,}321, and F14=0F_{14}=0.

Exact LF4 reconstruction and evidence boundary →

Verification notes

The result was independently reconstructed in JavaScript rather than copied from the archived Python helper. The apparent discrepancy with that helper was resolved: its generic row reports raw guard slots without the greater-than-two admissibility adjustment or the fixed-weight word-supply cap used by F_h. The production build now reruns the exact row.

Limitations

  • This is a finite observation at one layer and depth, not an asymptotic theorem.
  • Reproduction validates the reported row but does not constitute external mathematical review.

Open work

Preserve the executable regression and avoid treating this finite row as an asymptotic contraction result.