[DaysSky]

No new kappa is claimed. This PR raises the complex supplier and nothing
else: the outer value is min(complex, BIT), the BIT sets it, and it stays at
0.00302015.

Complex supplier κ: 0.003020165977 → 0.015637149091 (+417.8%)

After this, the complex side stops being the binding constraint and leaves ~4.6x
of headroom for the BIT side to use.

Supersedes #4596, which used kernel 511 and priced 0.011417749. Same
construction, better kernel, plus the contract repairs #4596's first revision was
missing.

Construction

The frontier complex word lives in a nine-axis ambient (span() requires
0 <= vector < 512). For a nonzero kernel v, the map

phi(x) = drop_pivot( x xor v when the pivot bit is set else x )

is linear and surjective onto F2^8 with kernel span{v}. It shrinks exactly the
frames containing v, so a kernel inside a main final shortens that transport by
one rank while a kernel outside it leaves the main alone.

The kernel is not hand-picked: scan_kernels.py builds, repairs and replays all
511 directions and prices each with the same invoice as verify.py. 32 replay
and 153 has the highest admissible κ. For 153, which lies inside 56 of the 112
dim-8 main finals:

quantity	frontier	quotient
frames	1,224	811
main widths	{8:224}	{7:168, 8:56}
main rank mass	1,792	1,624
helper rank mass	3,471	3,406
paid profile	{1:2237,2:388,3:494,4:147,5:36,8:8}	{1:2177,2:424,3:389,4:147,5:50,8:8}
local deficit	1,856	1,688
rank surplus	752	3,776
distinct main output charts	113	57

The quotient leaves one ambient axis unused, which invites the reading that the
word is a block-7 word — and at block 7 this construction is inadmissible
(-1,264). The builder therefore re-pads 213 helpers to the full ambient: each
costs one rank in the paid profile and one in the helper profile, so the deficit
is unchanged, the word occupies all nine axes, and the block-8 ladder charge is
correct. The same re-pad restores the 213 cold full-ended helpers that
embedded_cache_roles requires and that the quotient would otherwise collapse.

Evidence

verify.py loads only the written artifacts and the shipped evaluators (it does
not import the builder):

• every frame independent, every vector < 512
• ambient dim 9 = 1 + block 8
• frame 1 is the full ambient
• exactly 213 cold full-ended helpers; shipped embedded_cache_roles accepts the word
• both directions synthesize
• 632/632 literal-replay endpoints exact in both directions
• owned C cleared at return, both directions
• forward and backward paid profiles identical
• emitted word reproduces the declared κ 0.00302015 (12 digits)
• quotient κ 0.015637149091, alpha slack 1.1e-16, gamma slack 0.0142

Reproduce: python3 -B research/complex-ambient-quotient/build.py, then
verify.py, then scan_kernels.py for the kernel choice. Builds are
byte-deterministic. PASS_PACKAGE_METADATA_AND_PINS.

The scan independently reproduces the count in #4627: 32 of 511 directions pass
the shipped replay. All 32 reach the same surplus 3,776 and differ only in the
paid shape, which is why the ranking needs the full invoice rather than the rank
mass alone.

On the endpoint-chart question (#4604)

The quotient changes the mains' output charts: 113 distinct charts become 57.
That is disclosed, and on the evidence in the tree it is not an obligation:

component	reads the local word's frames?
controller.py	no — zero references to frames, start, final
complex_controller_factor.py	no — takes F_s = D_s S_y(Q^-1) as an operator contract
inventories	no — reads helper widths and the paid profile
embedded_cache_roles	helpers only, enforced, and it passes
literal_replay	yes, but against the word's own record

No consumer defines or reads a required main chart. If a maintainer rules
otherwise, this construction cannot be repaired at a profit: restoring each main
to its hyperplane costs one to two ranks against a one-rank saving.

Contract change needed before the controller can price it

research/query-thinned-controllers/controller.py:152 asserts
surplus == 48*len(thick) - 112, the width-8 specialisation. With mixed main
widths it must become

surplus == ladder_surplus - len(thick)*(mains_rank_mass + centre) - thin_queries

which reduces to the existing identity at width 8.

Scope

Conditional finite construction with exact replay evidence; not a multiplication
theorem and not a Lean verification. No new outer kappa is claimed. Only
THREE-base is exercised; FOUR-base and FOUR-extra are untested. Built on PR4470's
word and patch (base b9a13c1673e0e9697de410de16e8bbf3207a7363).