Research: flag-1 I−G/2 side correction at #3532 charts costs +0 on R1 and +2 on R0 (no new kappa)

[VictorLiwentao]

Change and mathematical scope

Bounded follow-up under priority revision 6, card p104-side-correction. Open #3532 co-hosts flag 1's head at the delivering update via R0/R1 at +2 rank mass and leaves the I − G/2 side correction unpriced. Open #3601 priced flag 43's entry at #3508's cut with co-host discount 0. This package answers the same question for flag 1's entry (7,0) (G[7,0]=−1 ⇒ add +1/2·X0 into Y7) at both of #3532's co-hosted charts.

Seven concrete corrections pass the published check_native.validate() with coefficients set to 0 (three newly lowered recipes, 208 adapters, 280 framed rows, 72 dirty helpers). Real words fail only at Y7's endpoint; only Y7 changes.

Word	Base	Δ events	Δ ADDs	Δ transports	Δ rank vs base	vs published
E (one ADD in FULL after R1's relocated delivering ADD)	R1	+1	+1	0	0	+2
Erow (all 48 row-7 entries there)	R1	+48	+48	0	0	+2
T (Y7: A3→FULL, ADD, FULL→F7)	R0	+2	+1	+1	+2	+4
S (X0: FULL→F7→FULL detour)	R0	+3	+1	+2	+2	+4
P (one private-copy lifetime)	R0	+5	+1	+1	+1	+3
Trow (all 48 entries on T's lift)	R0	+49	+48	+1	+2	+4
Tend (end-cut control)	published	+2	+1	+1	+2	+2

Co-host comparison: at R0 the correction costs +2 vs the end-cut Tend (discount 0, same as flag 43). At R1, E/Erow add 0 rank vs R1, so Tend.rank − E.rank = 2 and the joint co-hosted+corrected word R1+E matches published+Tend at +2 vs published (R1 already lifts Y7 into the full chart where X0 sits).

Obstruction. X7's own head 387 is outside F7 (G[7,7]=2 even) and reaches X0 at event 1074 along X7 –616– G11 –679– X20 –1070– G18 –1074– X0. A least-frame sweep with the delivering ADD dropped blocks all 18 insertion classes, covering every one of the 1,116 delivering-ADD placements. Per-register mass is even ⇒ strict minimum exactly 2.

Joint theorem. Words that both co-host the head and apply the correction: with relocation the minimum is exactly 2 (R1+E); order-preserving strict words need at least 4 non-private rank (R0+T and R0+S attain 4). Only D17's freedom rescues co-hosting alone, and with D17 free all 19 correction classes stay blocked. The private route costs exactly 1 (5/5 early deliveries blocked).

Scoped bill. No new kappa. Selected κ stays 0.001672660197076918.

Task and overlap

• Package: research/p104-flag1-side-correction-bill/
• Tip: 4352496b3aa56ad17446ddbd1d78931f3d392996
• Content tip: f5c09d4688c848f02d546a08191b047cd7b3687f
• Base: 20ecf2891d80b5acf18d980b2164fb44ea7faa90 (contains merged Research: flag-43 head-step repair co-hosts at update 3364 with no added channel mass (no new kappa) #3508 and Research: present-head flag 1 reaches target span only after its last update (no new kappa) #3451; requested pin 6f100d4bb is an ancestor)
• Scope: refined — flag-1 entry only, at Research: flag-1 head-step repair co-hosts at update 2797 for exactly two added rank units (no new kappa) #3532's R0 and R1 charts
• Research: flag-1 head-step repair co-hosts at update 2797 for exactly two added rank units (no new kappa) #3532 inputs vendored byte-identically under inputs/pr3532/ from head 8fecbf7df7b48c9004bd10a0da6a81f33d455737

Closest priors: #3532 (R0/R1 +2 co-host floor; leave-alone — this package does not re-certify R0/R1 optimality or the late-slot OQ), #3601 (flag-43 format prior; discount 0), #3451 (flag-1 chronology, cited only), #3508 (method / class Z), research/p104-side-correction-rank-obstruction (end-cut +2; Tend is a control only). Soft-overlap with #3601 is template-level; scientific novelty is R1's free-ride entry (added_vs_base.rank_mass=0), the joint 2-vs-4 theorem with single-register census, the X7/G11/X20/G18 forcing chain, and coverage of every delivering-ADD placement. Not a supplier. Does not reopen deferred flag-9 repair / pinned-late / peeled-leaf / rho=0 / V45 / group-B reprints, or the merged readout series through #2883.

Validation

python3.12 -I -B research/p104-flag1-side-correction-bill/verify.py
python3.12 -I -O -Werror -B research/p104-flag1-side-correction-bill/verify.py
python3.12 -I -O -Werror -B research/p104-flag1-side-correction-bill/independent.py
python3.12 -I -O -Werror -B research/p104-flag1-side-correction-bill/test_package.py
python3.12 -B scripts/research_package.py --package research/p104-flag1-side-correction-bill

Expected:

PASS_P104_FLAG1_SIDE_CORRECTION_BILL flag=1 entry=(7,0) gram=-1 coefficient=+1/2 zero_mass_placements=0/18(delivery_anywhere_in_1116_slots) forcing=X7(387)-616-G11-679-X20-1070-G18-1074-X0 strict_min_added_rank=2 R0:T=+1transport+1add+2rank S=+2transport+1add+2rank P=+1rank+1private_lifetime Trow=+48add+2rank R1:E=+0transport+1add+0rank Erow=+48add+0rank row7=identity cohost_discount_R0=0 cohost_discount_R1=2 joint_min_relocated=2 joint_min_order_preserving_strict=4 native_validate=7/7 no_new_kappa
PASS_P104_FLAG1_SIDE_CORRECTION_BILL_INDEPENDENT entry=(7,0) coefficient=+1/2 strict_classes_obstructed=18/18(delivery_dropped)+19/19(published_order) early_private_obstructed=5/5 forcing_head=387(X7) R0:T,S,Trow=+2rank P=+1rank R1:E,Erow=+0rank Tend=+2rank joint_order_preserving_single_rescue=none rows_changed=Y7_only row7=identity new_recipes=3 no_new_kappa
PASS_P104_FLAG1_SIDE_CORRECTION_BILL_TESTS cases=18 clean_read_only=1 scanners=2
PASS_REGISTERED_RESEARCH_PACKAGE research/p104-flag1-side-correction-bill outcome=completed

Default verify is read-only. Admission uses need() and still fires under -O. Duplicate JSON keys are rejected. Corruption controls include certificate bill/obstruction/census/discount flips, vendored #3532 pin flips, wrong common chart, unrestored source, and unclosed private lifetime.

Credit

Credit #3532, #3601, #3451, #3508, the side-correction-rank-obstruction authors, Nick Petalas / #2981, and Chafik Boukhalfa. Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI assistance, under Apache-2.0.

Open next questions

1. Is there a co-hosted and corrected flag-1 word with joint added mass 1 (zero-mass late relocation plus a private-copy correction)? Blocked by Research: flag-1 head-step repair co-hosts at update 2797 for exactly two added rank units (no new kappa) #3532's late-slot OQ1.
2. Does every present-head target row need +2 for its strict side-correction entry?
3. Under the repo's actual per-rank weights, is R1+E (H8 − H6) cheaper than Tend (H1 − H6 + H7)?