Research: flag-1 head-step repair co-hosts at update 2797 for exactly two added rank units (no new kappa)

[VictorLiwentao]

Change and scope

Bounded follow-up under priority revision 6, card p104-side-correction. #3451 shows that on present-head flag 1 (X0 → Y7, head 232) no published Y7 update co-hosts the basis head. #3508 gave a zero-added-mass repair for flag 43. This package shows flag 1 is different: the delivering update can be repaired, but every checked repair costs at least +2 rank mass.

Event 2797 (Y7 += D17/4) is the only Y7 update that carries X0 (coefficient -1/2, the final value). Two concrete schedules make that ADD co-host the head while X0 is still live; both pass the published check_native.validate() unchanged.

	Events	ADDs	Transports	Rank mass	Profile delta	Surcharge on
R0 (order-preserving)	+1	0	+1	+2	-H1 + 2H2 + H5 - H6	D17
R1 (relocated into D17's full window)	0	0	0	+2	H8 - H6	Y7

Under an affine price the extra cost is positive (a+2b / 2b). Under convex H=r², R0's profile delta is -4 (rank mass still +2).

Obstruction. The #3508 head-closure rule blocks all six Y7 updates (first blocker for 2797 is Y8 at 2799). An exact least-frame sweep shows zero added non-private mass infeasible for all 6 order-preserving updates and for 868 of 1,115 relocation slots before event 3366. Excess over the floor is always even, so +2 is the floor in those classes; R0 is mass-optimal among order-preserving repairs. 247 later relocation slots remain open (scoped floor, not a global relocation floor).

Scoped repair bill. No new kappa. Selected κ stays 0.001672660197076918. Does not reprint #3451 drop/arrive chronology.

Result

Package: research/p104-flag1-head-step-repair/.
Tip: 8fecbf7df7b48c9004bd10a0da6a81f33d455737.
Content tip: 053bff9e760e05354687cc669461dd6d8effed18.
Base: 6f100d4bb5083f8c35d96fd95a539811c62f245c.

Validation

python3.12 -I -B research/p104-flag1-head-step-repair/verify.py
python3.12 -I -O -Werror -B research/p104-flag1-head-step-repair/verify.py
python3.12 -I -O -Werror -B research/p104-flag1-head-step-repair/independent.py
python3.12 -I -O -Werror -B research/p104-flag1-head-step-repair/test_package.py
python3.12 -B scripts/research_package.py --package research/p104-flag1-head-step-repair

Expected:

PASS_P104_FLAG1_HEAD_STEP_REPAIR flag=1 head=232 delivering_update=2797 R0=order_preserving added_events=1 added_adds=0 added_transports=1 added_rank_mass=2 profile_delta=-H1+2H2+H5-H6 R1=relocated added_events=0 added_transports=0 added_rank_mass=2 profile_delta=-H6+H8 native_validate=PASS zero_mass_order_preserving=0/6 zero_mass_relocation_obstructed=868/1115 no_new_kappa
PASS_P104_FLAG1_HEAD_STEP_REPAIR_INDEPENDENT … rows_equal=280/280 … zero_mass_relocation_obstructed=868/1115 no_new_kappa
PASS_P104_FLAG1_HEAD_STEP_REPAIR_TESTS cases=17 clean_read_only=1 scanners=2
PASS_REGISTERED_RESEARCH_PACKAGE research/p104-flag1-head-step-repair outcome=completed

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

Overlap and credit

Closest priors: #3451 (flag 1 obstruction; this answers its repair OQ), #3508 (flag 43 zero-mass method prior; flag 1 has a different floor). Soft-overlap: verify/independent reuse helper/splice structure from #3508; scientific novelty is the +2 floor and 0/6 zero-mass obstruction. Soft-overlap checked against open #3363 (only the single -1/2 delivery at 2797 is used). Does not reopen #3456/#3443/#3441/#3436/#3421/#3411/#3392, deferred peeled-leaf/rho=0/V45, or the merged readout series through #2883.

Credit #3451, #3508, Nick Petalas / #2981, Chafik Boukhalfa (witness/checker/compiler), and #3363. Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI assistance, under Apache-2.0.

[VictorLiwentao]

Follow-up on OQ1 (late relocation slots), scoped to the C-pinned class — not a competing PR.

A fork package research/p104-flag1-pinned-late-relocation/ (cursor/flag1-pinned-late-relocation-44d0 @ d4d3d634e) ran the C-pinned least-frame sweep (private register C keeps its published events — the class containing this PR’s R0/R1):

Scan of 1,115 slots	Obstructed	Late (after 3366)
C relaxed, head forced (this PR)	868	0 / 247
C pinned, head forced	1,115	247 / 247
C pinned, head-free	458	247 / 247 (all after 3183)

So under C-pinned, the +2 floor covers the full relocation window. Late blocker is final(Y7)=ker(61); after 3366, D17’s pinned least frame exits that hyperplane (head not required). Exact private core K={2480,2482,2504,2511} (D8/D10/D20/D23 += C): pinning any one closes all 247 late slots; releasing only those four opens them.

Does not close this PR’s OQ1 in the C-relaxed model (zero-mass existence after 3366 with free C charts remains open). Independent skeptical review DEFER’d a standalone CrocSwap PR as soft-overlap / absorb-here; recording the fact on this thread instead.