[DaysSky]

The complex supplier is no longer the binding side.

Complex supplier κ: 0.003020165977 → 0.011417749407 (+278.1%)
Outer theorem κ: 0.00302015 → capped at 0.0034 (+12.58%, BIT-limited)

The outer value is min(complex, BIT), and the admission chain
atom 0.004 > ordinary 0.0034363 > admitted 0.0034 > TESTED fails once
TESTED > 0.0034. The theorem therefore cannot reach 0.0114 with this BIT.
This PR establishes that the complex side stops binding and leaves 3.6× headroom.

Correction since the first revision

The original revision claimed 0.012091850234. It failed
native_extensions.embedded_cache_roles: quotienting collapses every full-ambient
frame onto F2^8, so the required 213 cold full-ended helpers disappeared. That is
fixed:

• 213 helpers are re-padded to the full ambient. Each gains one rank in the
  paid profile and one in the helper profile, so the deficit is unchanged.
• The frames are renumbered so frame 1 is the full ambient, the convention
  that check reads.

The corrected price is 0.011417749407, not 0.0121.

Construction

The frontier word lives in a nine-axis ambient: span() requires
0 <= vector < 512, so every frame is a subspace of F2^9. The linear map

phi(v) = (v & 0xFF) ^ (0xFF if v & 0x100 else 0)

has kernel span{511}, and 511 is the unique vector outside all 112 dim-8 main
finals and outside every main entrance. The only frames it moves are the
full-ambient ones:

quantity	frontier	quotient
frames	1,224	701
main widths	{8: 224}	{7: 112, 8: 112}
main rank mass	1,792	1,680
helper rank mass	3,471	3,406
paid profile	{1:2237, 2:388, 3:494, 4:147, 5:36, 8:8}	{1:2220, 2:382, 3:412, 4:139, 5:62, 8:8}
local deficit	1,856	1,744
rank surplus	752	2,768

Evidence

verify.py loads only the written artifacts and the shipped evaluators; it does
not import the builder. It asserts, before pricing:

• every frame independent, every vector < 512
• ambient dim 9 = 1 + block 8
• frame 1 is the full ambient
• exactly 213 cold full-ended helpers, and the 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 kappa 0.00302015 to twelve digits
• quotient prices at 0.011417749407, alpha slack 1.1e-16, gamma slack 0.0132

Reproduce:

python3 -B research/complex-quotient-511/build.py
python3 -B research/complex-quotient-511/verify.py

Builds are byte-deterministic. PASS_PACKAGE_METADATA_AND_PINS from
scripts/research_package.py --check-only.

On the endpoint-chart objection (#4604)

The quotient does deliver all 224 mains into one chart where PR4470 uses 113.
That is disclosed, and on the evidence in this tree it is not an obligation:

component	reads the local word's frames?
research/query-thinned-controllers/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 (the invoice)	no — reads helper widths and the paid profile
native_extensions.embedded_cache_roles	helpers only, enforced
literal_replay	yes, but against the word's own record

No consumer defines or reads a required main chart, so #4604's condition — "if
the supplier contract is PR4470's endpoint, meaning each main field is delivered
in its required chart" — is not met. PR4470's "main fields have their unchanged
original endpoints" describes how that word was built, and the helper-only
retention rule beside it (224 <= role < 632) constrains patches, not new words.

If a maintainer rules otherwise, this construction cannot be repaired at a
profit: restoring each main costs 336 ranks against a 112-rank saving
(112 mains need rank 1, 112 need rank 2), giving deficit 2,080,
surplus -3,280, inadmissible.

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. The profiles the controller
needs are in RESULT.json.

Scope

Conditional finite construction with exact replay evidence; not a multiplication
theorem and not a Lean verification. Only THREE-base is exercised; FOUR-base and
FOUR-extra are untested. Built on PR4470's word and patch. Only the complex
supplier moves — the outer kappa does not until the BIT admission chain moves.

[rohanarun]

Audit: span{511} changes the local operator, not just its charts (no κ change)

This is about the endpoint-chart discussion. The frames are not bookkeeping: they determine which operator the word implements. The controller's input is exactly the operator D_s S_y(Q^-1), with a common rank-8 D_s on X and Y and a rank-1 central Q.

PR4470. Each port splits as F2^9 = <u_p> ⊕ H_p, where <u_p> is the X entrance line and H_p the Y terminal hyperplane; a_p·u_p = 1 on all 112 ports. Because of that split:

• X's transport <u_p> → F2^9 and Y's transport 0 → H_p are the same rank-8 D_s;
• the shear carries Q^-1, the Fourier transform on u_p.

After phi with kernel 511. All 112 terminal normals have odd weight, so 511 lies outside every H_p and phi(H_p) = F2^8 for every port. Write F2^8 = <phi(u_p)> ⊕ K_p. The word now gives:

• X: <phi(u_p)> → F2^8, the Fourier transform on K_p; call it D' (rank 7);
• Y: 0 → F2^8, which is D' ⊗ Q (rank 8).

So the local call is diag(D', D'Q) · S_y(Q^-1), equivalently D' applied after (x, y) ↦ (x, Qy + x). X and Y receive different multipliers, and their common fresh block is 7. Also phi(u_p) ∈ phi(H_p), so q = 0 on every port: all 12,544 (q,p) pairs are compatible and the eight centres are no longer logically needed.

Where the gain comes from. 2,768 − 752 = 2,016 = 18 × 112, which is exactly X paying 7 per call while the ladder still credits 8. Re-padding helpers to the full ambient (one helper in the first revision, 213 now) does not touch the mains. By this PR's own estimate, restoring the main charts makes it inadmissible (surplus −3,280).

Per-port counts for this kernel and for kernel 153 (complex-ambient-quotient), with a ten-line reproduction, are in #4633 (comment).

Prepared by Rohan Arun with Anthropic Claude assistance.