KB3k: demote the mu2-constant even-half condition; (4,2) walk shard 29 (outcome=partial)

[navinor]

KB3k: the mu2-constant even-half condition for the 18-point cover question of F2^7

Set-level only, no words enumerated. Outcome = partial. The mu2 "constant per
split" lead of #3108 (agent-83) is pressed. It is not a law of the covering
condition: mu2 varies at a single minimal split, (4,4) in F2^3, and the
variation is explained exactly. For the both-odd target splits (11,7)/(9,9),
mu2 is constant at every scale where covers can be produced, but that is
CHECKED, not forced, and it is not a proof filter. The lead is therefore
demoted. No class is decided; no cover is constructed; no false kill.

Author: Aaron Wang, with help from opencode (substantial AI assistance; no model
named; no independent human review claimed).

Object and chain

In the PR943 prefix-walk model (#943, Ross Richardson) a restored seven-source
singleton word is universal exactly when its prefix set P covers: P + P = F2^7. The 17-point question is closed (#2542, #2634, #2773, #2870); #2911
narrowed the 18-point question to (12,6), (11,7), (10,8), (9,9); #2933
classified the vanishing configurations; #2961 excluded (10,8) and decided the
three tightest classes; #3023 killed (12,6) A-vanish (5,1) and reduced
(4,2)/(3,3) to walkable forms; #3034 demoted the parity shortcut and
localized the m=5 obstruction; #3108 ran a partial (4,2) walk, classified the
even-half self-sum profiles, and recorded the mu2-constancy lead this PR
presses. The reduced covering object (#2911 reduction lemma): for G = F2^m, a
pair (A, B) covers iff A + B = G (cross) and (A+A) u (B+B) = G (even-half).
A P + P = F2^{m+1} cover with parity split (s,t) is exactly a reduced
(A,B) cover of F2^m with |A| = s, |B| = t. The 18-point question is the
reduced cover of F2^6 (64 elements).

Every cited PR author was re-verified at writing with gh pr view NNN --json author,title: #2542/#2634/#2773/#2870/#2911/#2933/#2961/#3023/#3034/#3108 are
Aaron Wang (navinor); #2691/#2973 are Nicholas Stone (NicholasStone); #3005 is
Nick Petalas (petalas); #1185 is Mister-Meeseeks. This PR is strictly SET-level
and enumerates no words (Nicholas Stone's 22 <= f(7) <= 24 word line, #2973 /
#3005, is untouched).

The exact identity (PROVEN)

For any reduced cover (A,B) of G = F2^m, inclusion-exclusion on
(A+A) u (B+B) = G gives mu2 = |(A+A) n (B+B)| = |A+A| + |B+B| - |G|. So
mu2 is constant per split (a,b) if and only if the selfsum-size sum
|A+A| + |B+B| is constant per split. The identity holds at 0 violations over
all 41,341 F2^3 covers and all F2^4 covers checked.

Not forced: the minimal variation split is (4,4) in F2^3 (PROVEN)

Complete enumeration at F2^1, F2^2, F2^3 shows mu2 is constant per split
at F2^1 and F2^2, and first varies at F2^3 at split (4,4). This is the
minimal variation split.

Group	Split	Covers	mu2 values	sum of selfsums
F2^3	(4,4)	3,360	3 and 6	11 and 14
F2^3	(4,k) or (k,4), k=5..8	560 to 3,920	4 and 7	12 and 15
F2^4	(2,8)	15,360	1	17
F2^4	(4,6)	143,360	4	20

Mechanism (exact): mu2 varies with the additive structure of a half. In
F2^3 a 4-set is either a coset of a 2-dim subspace (an affine plane,
|A+A| = 4) or generic (|A+A| = 7). At (4,4) the halves vary independently.

half A	half B	selfsums	sum	mu2	covers
generic	generic	7, 7	14	6	2,688
plane	generic	4, 7	11	3	336
generic	plane	7, 4	11	3	336
plane	plane	n/a	n/a	n/a	0 (never covers)

So mu2 = 6 - 3 * (number of plane halves): one structured (affine-plane) half
lowers |A+A| by 3 and mu2 by 3. The covering condition does not pin this. At
(4,6) in F2^4 the split is constant again (mu2 = 4) only because the two
selfsum sizes are anti-correlated by the cover condition (|A+A| = 4 forces
|B+B| = 16; |A+A| = 7 forces |B+B| = 13; the sum stays 20). So the
constancy that survives is a correlation of a particular split, not a covering
law.

The both-odd regime of (11,7)/(9,9) (CHECKED, not forced)

The target splits (11,7) and (9,9) are both-odd. At every both-odd split
where covers can be produced, mu2 is constant: F2^3 both-odd splits
(complete), and F2^5 (7,7) (310 covers, all mu2 = 12 with both selfsums
pinned at 22). But this is not forced. The (4,4) F2^3 counterexample shows
the covering condition does not pin mu2 in general, and both-odd halves at
F2^6 (|A| = 11, 9) can still have varying |A+A| (a subspace plus points).
Direct F2^6 (9,9)/(11,7) covers were not produced in a bounded hill-climbing
search (0 in 300 restarts each), so mu2 variation at the exact target splits is
neither observed nor ruled out.

Verdict: mu2-constancy is scale- and split-dependent, false for both-even
splits, empirical for both-odd splits, unforced everywhere. The planned lift
("carry mu2 = const to #2911's 501/196 rows as a filter, then lift to m=6") is
not justified and is demoted. This is the same shape as the demoted
scale-dependent self-sum lift of #3034. (PR 2911's 501/196 rows are
pair-representation row configurations, not concrete (A,B) covers, so mu2
cannot be evaluated on them without the still-open row-to-cover map.)

The (4,2) walk (continuation)

The (12,6) A-vanish (4,2) decision walk is continued as a budget-honest
partial, never an exclusion. #3108's sharded walk is extended by shard 29 with a
per-shard receipt (explicit complete/budget_hit flags, budget checked after
every U). This session's shard ran from empty receipts and was re-run from empty
receipts with identical counters to prove recomputation. The inherited 29 shards
plus shard 29 give 30 of 2048 shards complete, 2,490 U slices, 257,845,500
normalized pairs, w_candidates = 0 and covering_configs = 0 on every walked
pair. The complete walk is far outside one session (#3108 sized it at ~608
core-hours), so this is a narrowed remainder.

Anchors reproduced before any new number

Anchor	Expected	Status
PR 2870 OBSTRUCTION	parseval=696 allowed=504 deficit=192	recomputed live
PR 2933 CLASSIFY	tight=2 cases=6720	recomputed live
PR 2961 FOURIER (7,1)	need=175 lower=239 contradiction=True	recomputed live
PR 2961 CLASS82	checked=28,815,360 covering=0	read back
PR 2961 CLASS60	d_checked=169,911	read back
PR 3023 AV51	uv_pairs=3,049,492,320 covering=0	read back
PR 3108 SELFSUM	(5,5)=3,136 and (7,7)=64 profile (8,8,8)	read back
PR 3108 CENSUS33	(3,3) covers=296,352	read back
mu2 identity	0 violations over 41,341 F2^3 covers	recomputed live

Reproduce

Standard library only, CPython 3.12. From the repository root:

python3 -I -B research/f7-mu2-evenhalf-kb3k-88a1/verify.py quick
python3 -I -B research/f7-mu2-evenhalf-kb3k-88a1/mu2_search.py m3
python3 -I -B research/f7-mu2-evenhalf-kb3k-88a1/mu2_mechanism.py

Markers: PASS_KB3K_QUICK, MU2_M3_DONE, MU2_MINIMALITY_DONE. Heavier
receipts re-run with mu2_search.py m4, mu2_search.py m5odd,
mu2_search.py m6, mu2_odd_search.py. All arithmetic is exact Python integers.
Vendored predecessor engines are byte-exact with SHA-256 pins in SOURCES.json.

Scope and limitations

Set-level only inside PR943's abstract interface. The verdict is that
mu2-constancy is not a covering law and is demoted as a proof filter; it is
not a proof that (11,7)/(9,9) are excluded, and no cover is constructed or
excluded here. (4,4) variation is PROVEN (complete F2^3); both-odd
constancy is CHECKED at F2^3 (complete) and F2^5 (7,7) (310 covers) only;
F2^6 (9,9)/(11,7) mu2 is untested (no covers produced). Not checked:
anything word-level (#2973/#3005 hold 22 <= f(7) <= 24), the row-to-cover map,
the (12,6) D-vanish classes, the surviving (10,8) classes, the 19-point
question, and any kappa, supplier or selected-frontier implication. Frontiers
unchanged.

Credit

PR 2542/2634/2773/2870/2911/2933/2961/3023/3034/3108 (Aaron Wang): the 17-point
exclusion chain, the pair-representation rows and Fourier/Parseval identity, the
(A,D) reduction and vanishing-character restriction, the class floors and
tight-class decisions, the (5,1) decision and walk engine, the KB3i remainder,
and the KB3j mu2 lead and (4,2) walk this PR presses and continues. PR 943
(Ross Richardson): the prefix-walk model. PR 1185 (Mister-Meeseeks): the pinned
f(5)=12, f(6)=16 record. PR 2691/2973 (Nicholas Stone) and PR 3005 (Nick
Petalas): the word-level census line (recorded state; the set-level chain is
unaffected). Vendored engines are byte-exact with SHA-256 pins and credited in
NOTICE.md/SOURCES.json.