[unpaidmoon]

Change and mathematical scope

Scoped tariff lemma. No new κ. The selected conditional exponent stays 605457647/200000000000. CrocSwap#4100 is not accepted.

CRT.md §4 colors active targets by position modulo K and charges a switch layer for O(K) colors. The macro that paragraph prices is one occupied color. For every batch cut by the pinned CrocSwap#4174 layout,

c ≤ r ≤ r_max.

r is the number of active targets, c is the number of distinct residues modulo K, and r_max is the layout's batch cap. Empty residues are not macros. Two targets exactly K apart are one macro.

At d = 65536, ell = 24, and alternating widths 24/23, with the constructor call p = 6L used by research/pr4100-guard-coloring, the parameters recompute as L = 1540096, K = 196608, r_max = 5, and H = 15360. The §4 color factor may be 5 instead of 196608. The ratio is exactly 196608/5. The all-short width 23 row recomputes as r_max = 4 and ratio 49152.

The singleton-color hypothesis in the predecessor's open question is sufficient but not necessary: the batch cap already forces c ≤ r_max when a color collides.

This is not a tape program, not a census replay, and not a change to the §5 aggregate O(V p^(1-a) ell^2 polylog p), which still hides K.

Task and overlap

Base 32890bc5de034ab8fc93ac88ead7279a6a416d04 (upstream CrocSwap/integer-mult-bounds main). Priority revision 7. Task card uniform-transfer-audit. Scope status is refined: the card asks whether one quantified step of CrocSwap#4100 holds at its caller. This package prices the occupied-color factor in the CRT.md §4 layer sentence.

Session r44, item 12 of the ordered eligible list in research/crt-occupied-color-macros/README.md. Open handoffs from the 19 October index come first, then still-open successors of the priority table in the order recorded by research/pr4100-ledger-boundary-exponent. Answered rows, open-PR coverage, this account's deliveries, and fork cursor/* topics were dropped before the cut.

Nearest overlap:

• research/pr4100-guard-coloring (Ryanson Jonathan) proves the (K−1)-bit guard lemma and leaves "price macros per occupied color, not per residue" open. This package answers that price. It does not rerun the 104,864-plan census.
• Hiroki KUMAZAKI's Audit CRT reservoir placement and split-axis coordinate descriptors CrocSwap/integer-mult-bounds#4174 layout supplies r_max = L // (100 G), vendored in that coloring package.
• unpaidmoon Audit the compact-switch coefficient and guard contract. CrocSwap/integer-mult-bounds#4321 stated the disjoint-guard hypothesis the coloring audit decided. This note does not reopen that contract.
• The literal many-target tape macro asked by research/crt-mask-repair-bridge stays unpinned. Occupied-color pricing is the unblocked tariff half of that question.

Validation

Python 3.12, standard library only, from the repository root:

python3 -B research/crt-occupied-color-macros/verify.py
python3 -B scripts/research_package.py --package research/crt-occupied-color-macros

Observed:

PASS_CRT_OCCUPIED_COLOR_MACROS K=196608 r_max=5 ratio=196608/5 uniform23_r_max=4 ratio=49152 same_color_pair_macros=1 no_new_kappa=true
PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
PASS_REGISTERED_RESEARCH_PACKAGE research/crt-occupied-color-macros outcome=completed

The checker hashes CRT.md (02c3a1e38cbee25aefe2640cc72676da62ba10eb7519fbcb9adb9d48b74eff2b) and the layout allocator (b764195fc6291e3c8fa8297b1f4ceaefc119d5664d8ff348b49eac8a2e79c688), checks the §4 sentences, exhausts the 2^12 subsets of a 12-point universe at K = 6, and recomputes both production rows.

Not run: the 104,864-plan census, Lean, Docker, and the selected cold replay.

Credit

The coloring, the O(K) layer sentence, and the one-color cost are Cinna Davis's CrocSwap#4100. The batch cap is Hiroki KUMAZAKI's CrocSwap#4174 layout, as vendored by Ryanson Jonathan. CrocSwap#4321 (unpaidmoon) stated the guard hypothesis that coloring audit decided.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human review and no formal-kernel acceptance are claimed.

Open next questions

1. Can §5's ell^2 polylog p line be rewritten with r_max in place of the hidden K factor, and does any later numerical budget use the looser factor? The first step is to expand O(log d) levels times O(ell) layers times c ≤ r_max colors. The blocker is that §5 does not display K as a separate multiplier.
2. Does one same-color macro's repair bill stay inside the one-color O(V) when that color holds two targets? The first step is the bad-density line at r = 2 and this K. The literal many-target tape macro is still unpinned, so this question does not ask for that program.

Workflow feedback (optional)

The ordered-list cut is recorded in the package README. No separate session report.