Research: census of staged entries on PR3734's other helpers; none found that pays (no new κ)

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This asks whether the staged-entry mechanism of #3605 (helpers 644/660) and #3734 (helper 544) lowers the hosted width of any other helper of #3734's word, from 9 to 8 or from 8 to 7, and whether it pays. It is stacked on #3734's head fa041037a3374eaa08feeca0b26a5d6893194147 and adds only research/staged-entry-census-3734/. No new κ.

Result

• The identities exist. Exact prefix identities (P·v = v for Reduce helpers 644/660 from width 8 to 7; conditional κ > 0.00173 #3605/Stage helper544 after a returning producer excursion; improve conditional kappa #3734, or the general form with releases a = M_c v − e_h into any register) exist for 25 width-8 and all 38 width-9 candidates whose blocking contacts are reads before the producer.
• With the other frames fixed, no construction exists. 9,862 (helper, raise, cut) cases per provider were tested, with releases allowed into any frame-compatible register, targets included. The same code rediscovers the three published cases on their parent words: 544 on Consume retimed staged releases with smaller T360 stock #3682's word, 644 and 660 on research(rank7-coupled-586-588-carrier-fold): lower helpers 586 & 588 via coupled B+restore carrier folds (κ = 0.0017297760894) #3133's word, with the same donors and coefficients.
• With frames re-solved, only helper 472 reaches width 7. Feasibility is decided by an exact minimal-assignment test: it is pointwise minimal among assignments keeping A, the entrances and the terminals, so its failure is a proof. 472 is staged by 624 −= 472 and 628 −= 472/2 before its producer B538, with entrance span(429,51). It passes Stage helper544 after a returning producer excursion; improve conditional kappa #3734's frame bill, hygiene and full arbitrary-dirty semantics on both providers.
  • Its frame cascade costs {1:+15,2:−9,3:−2,4:+2} per call.
  • An exact dual certificate adapted from Research: certify the staged544 packing ceiling and reach it on PR3734's T360 clock (no new κ; same value as #3802) #3815 shows that every owner fill of this inventory and bill, for any clock and any replica count, has root < 0.0017305031. The parent ceiling 0.001730578437266246818740 (attained by Improve conditional kappa beyond T488 with a T385 combined-clock allocation #3802/Research: certify the staged544 packing ceiling and reach it on PR3734's T360 clock (no new κ; same value as #3802) #3815) is 7.5e-8 higher.
  • In Research note: staged544 width-unit prices, the exact staging break-even in the fixed-T360 ideal (no new κ) #3862's terms, 472's signature nets −0.00154 per width unit; the 544 signature nets +0.00326.
• The width-9 helpers fail as well. Leaving the zero-frame compensation pushes a target out of its port hyperplane or over its cut level, for every placement tried.

Why it fails. For 544, 644 and 660, the donors' next frames already contained the raised entrance. For every other candidate, the producer's frame and the recipients' next frames meet only in the old entrance. The new direction therefore propagates and splits partner climbs. For 472, {624, 628} is the only donor set of size three or less.

Verification

python3 scripts/research_package.py --package research/staged-entry-census-3734 --check-only
python3.12 -B scripts/research_package.py --package research/staged-entry-census-3734
python3 -B scripts/check_research_commit.py --package research/staged-entry-census-3734 --check-source-history

All three pass locally (CPython 3.12.14 on macOS arm64; the replay takes about 11 s, standard library only).

verify.py does the following:

• rebuilds the 472 word from Stage helper544 after a returning producer excursion; improve conditional kappa #3734's words and compares it with the frozen words;
• runs the audits above, and rejects a flipped release and releases placed after the producer;
• shows that the pure retiming of 472's early read is scalar-correct but frame-infeasible;
• rebuilds 544, 644 and 660 on their parent words (Reduce helpers 644/660 from width 8 to 7; conditional κ > 0.00173 #3605's parent is read from Stage helper544 after a returning producer excursion; improve conditional kappa #3734's archive, with its digest checked);
• checks the exact certificate: tight to 1e-10, with the parent bracket and a perturbed dual as controls;
• authenticates the recorded census and recomputes its summaries.

The census itself (numpy 2.5.4, scipy 1.18.1) is recorded in evidence/ with its commands. The optional replay_subset.py reruns a subset in about 2.5 min and reproduces the recorded rows exactly. It is not registered, because float decisions could differ across BLAS builds.

Scope and limits

• Sampled cuts. 24 per helper plus the producer cut with fixed frames; 5 or 6 per helper with re-solved frames, with up to 4 placements per release recipient. Infeasibility is proved per placement.
• What stays fixed. Other entrances and terminals, the prelude and the native maps. Joint staged entries and clock changes are not searched.
• 472's frames are not proved optimal. The certificate holds for the constructed frames. Two local optimisers (B; A+B) and a 4,200-slot retiming search find nothing better, but an exact frame solve of the 472 word was not run, and a cheaper assignment is not excluded.
• Raise conditional staged544 κ by filling idle owner slots #3945 (4→5 widening). With the same cascade, 472 also loses about 7.5e-8 on the ideal ceiling. This is a float estimate from Raise conditional staged544 κ by filling idle owner slots #3945's published profile; its words were not used.
• Inherited. All of Stage helper544 after a returning producer excursion; improve conditional kappa #3734's conditional hypotheses. No selected-frontier change. If Stage helper544 after a returning producer excursion; improve conditional kappa #3734 changes head, this needs a rebase and a rerun.

Open next questions

1. Joint staged entries of two helpers whose donors overlap, as Reduce helpers 644/660 from width 8 to 7; conditional κ > 0.00173 #3605 did for 644/660.
2. An exact frame solve of the 472 word on its nine cascade registers; absorbing the cascade by changing partner terminals; or 472 on Raise conditional staged544 κ by filling idle owner slots #3945's word.
3. Every cut rather than a sample: a matter of compute.

The details are in TASK.json, README.md and FINDING.md.

Credit

The staged-entry mechanism is due to Ross Richardson (#3605) and dataisfire (#3734). The frame bill and semantics are Sagar Thakrar's, from #3734's retained audit_core.py and audit_verify.py; rational_price.py is dataisfire's. The exact dual is adapted from #3815, and the width-unit prices are from #3862 (thakrarsagar) and #3890 (ToikY). Related but distinct work: #3933, #3886/#3941/#3942 and #3945.

Prepared by Erwan Secousses with substantial assistance from Claude (Anthropic, Claude Code). There is no independent human or formal review. Apache-2.0.

🤖 Generated with Claude Code