Research: present-head flag 0 Y34 span/dirty-restoration — arrive@3500, 15/15 head-free (no new kappa)

[VictorLiwentao]

Change and mathematical scope

Bounded follow-up under priority revision 6 filling the flag-0 slot left open by #3451 / #3456 on the present-head dirty-restoration chronology.

On published present-head flag 0 (X0 → Y34, basis head 232), the head enters Y34's frame, listed and in GF(2) span, only at Y34's terminal transport after-frame (event 3500). All 15/15 Y34 adds (last at event 3369) run in head-free charts. Beyond open #3575's literal relay cut (591→3500):

• Span-level hosting: zero source span drops (22/22 X0 annotations span 232); zero preterminal Y34 span hosting (preterminal_span_hosting=0).
• Target-update chronology: 15/15 Y34 row updates head-free; 0 Y34 events after first span co-hosting.
• Frame semantics: all 11 endpoint transports nested upward; paid rank equals rank increase.
• Listing vs span conflict with Record the empty forward relay for flag 0 #3575: at event 3500, X0 does not list 232 on the coordinate basis yet still spans it (8^32^64^128); dirty-restoration obstruction is target-side.

No Y34 event occurs while both endpoints host the head direction, so chronological dirty restoration cannot use co-hosted basis-head geometry on this coefficient-cascade target ahead of the published I−G/2 tail add.

No new kappa. Selected κ stays 0.001672660197076918. No I−G/2 rank-mass pricing; no tail pricing at 5759.

Task and overlap

• Package: research/p104-present-head-flag0-audit/
• Tip: 84288150fcaa8e3fecb128ef1feeb8ac1c63ccc6
• Content tip: ebcc53a61625a0fea7f2b63dc6a4f124161513a9
• Base: 6f100d4bb5083f8c35d96fd95a539811c62f245c
• Scope: changed — present-head flag-0 span/dirty-restoration chronology

Closest priors: #3575 (literal relay; leave alone; does not check span/head-free counts or Q-relay reprint), #3451 (flag 1; leave alone; soft-overlap verify ≈0.969 — FLAG_INDEX / arrive / gap / update-count retarget of the same scanner; shared X0/232/591 source half already certified there), #3443/#3456 (flags 9/43; leave alone). Side-correction bills #3601/#3610/#3614/#3618/#3621 untouched.

Validation

python3.12 -I -B research/p104-present-head-flag0-audit/verify.py
python3.12 -I -O -Werror -B research/p104-present-head-flag0-audit/verify.py
python3.12 -I -O -Werror -B research/p104-present-head-flag0-audit/independent.py
python3.12 -I -O -Werror -B research/p104-present-head-flag0-audit/test_package.py
python3.12 -B scripts/research_package.py --package research/p104-present-head-flag0-audit

Expected:

PASS_P104_PRESENT_HEAD_FLAG0 flag=0 head=232 preterminal_hosting=0 preterminal_span_hosting=0 drop_event=591 arrive_event=3500 source_span_drops=0 head_free_target_updates=15/15 no_new_kappa
PASS_P104_PRESENT_HEAD_FLAG0_INDEPENDENT flag=0 head=232 preterminal_hosting=0 preterminal_span_hosting=0 drop_event=591 arrive_event=3500 source_span_drops=0 head_free_target_updates=15/15 no_new_kappa
PASS_P104_PRESENT_HEAD_FLAG0_TESTS groups=6
PASS_REGISTERED_RESEARCH_PACKAGE research/p104-present-head-flag0-audit outcome=completed

Default verify is read-only. Admission uses need() and still fires under -O.

Credit

Credit #3575, #3451, #3443, #3456, #3392, #3349, and the retained P104 present-head lineage. Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI assistance, under Apache-2.0.

Open next questions

1. Can a changed schedule bring 232 into Y34's span before the add at event 3369, while X0 is still live, and what does that add to the channel bill ahead of event 5759?
2. Does the published promote/add/restore at 5758/5759/5807 admit an owner/host schedule for the X0 read that restores arbitrary dirt without repaying the unrestored channel endpoints?

[VictorLiwentao]

OQ1 schedule fact (recorded; no separate PR yet)

Fork package research/p104-flag0-head-span-schedule/ on VictorLiwentao:cursor/flag0-head-span-schedule-60b9 @ aa83e2164 answers this PR’s OQ1 with constructed repairs (not a chronology reprint):

• Settling update 3369 (Y34 += D23/4) can co-host head 232 while X0 is still live at exactly +2 rank mass (R0 order-preserving; R1 relocated into D23’s full window).
• Native check_native.validate() PASS on both; bills a+2b (R0) / 2b (R1) under affine prices.
• Zero-mass obstruction: 0/15 order-preserving updates; 926/1113 relocation slots (every slot before Y33’s read at 3438). The 187 later slots are relaxed-feasible and left open.

Gates (verify / −O / independent / test_package / research_package) PASS. A separate CrocSwap PR is deferred for soft-overlap / weak-need vs still-open #3532 (verify≈0.936 near-literal flag-1 repair template retarget) — same gate that deferred flag9-repair. Prefer folding after #3532 merges + a shared harness, or treating this comment as the OQ1 record.

κ unchanged. Does not touch OQ2 (tail owner/host at 5758/5759/5807). Prepared by Wentao Li with substantial AI assistance.