FPRD LabShare an approach

← Claims

Theorem and exact separation

FPRD-T68

Causal width and the late-selector separation

Exact statement

For the explicit even-palindrome survivor recurrence, virtual axes are stationary and query events form a two-dimensional dominance order, so offline persistent-track demand equals poset width. Width two can force three tracks online. With (r) late selector bits, no earlier demand, and targets (q(z)=(r+1)operatorname{value}_2(z)), every completed path has width one while exact pre-query demand is (2^r).

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

Context

A completed computation may need only one causal chain even though the machine must provision tracks for every selector value before learning which branch will be demanded.

Definitions

  • Events satisfy (t,p)⪯(t′,p′)(t,p)\preceq(t',p') when p′≤pp'\le p and p−p′≤t′−tp-p'\le t'-t.
  • The virtual axis of survivor (r) at time (T) is ((T+r-1)/2).

Hypotheses and scope

  • The survivor set is the explicitly displayed even-palindrome suffix recurrence.
  • The selector has r bits, appears after pre-query placement, and targets are spaced r+1 apart.

Proof or evidence

Dilworth gives the offline equality after an affine change to dominance order. The online lower bound uses a four-event adaptive adversary. Target spacing makes every r-step backward cone contain at most one selected address.

Causal width theorem →

Verification notes

A dependency-free implementation checked 65,534 recurrence extensions, 64,770 stationary lineage steps, 502 finite causal posets, both adaptive adversary completions, and 20,490 selector positions.

Limitations

  • The r+1 spacing is load-bearing; spacing r already lets one track see two adjacent targets.
  • Pathwise width is an offline measure and does not include query-oblivious preplacement.

Open work

Determine which selector encodings retain an exponential gap when target spacing or one-way movement is relaxed.

Notes

Produced by the review lane, not by this lab; captured with attribution and independently verified here by fresh implementation. The exponential is produced by pre-query placement plus the no-random-access axiom. Random-access placement directly at the selected leaf collapses the demand to one; an ordinary birth constrained by the same travel cone does not provide that operation. The exact theorem above keeps the original pre-placed-track model and makes no claim about every possible birth rule. The offline theorem tacitly grants free initial positioning, which is exactly what the causal theorem charges. Two received statements arrived garbled and were reconstructed; the verdicts do not transfer if the intended definitions differ.