[unpaidmoon]

Change and mathematical scope

Scoped tariff lemma. No new κ. The selected conditional exponent stays 605457647/200000000000. #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 #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. Priority revision 7. Task card uniform-transfer-audit. Scope refined.

Nearest overlap: research/pr4100-guard-coloring (Ryanson Jonathan); Hiroki KUMAZAKI's #4174 layout; unpaidmoon #4321. The literal many-target tape macro asked by research/crt-mask-repair-bridge stays unpinned.

Validation

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 and PASS_REGISTERED_RESEARCH_PACKAGE research/crt-occupied-color-macros outcome=completed.

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 #4100. The batch cap is Hiroki KUMAZAKI's #4174 layout, as vendored by Ryanson Jonathan. #4321 (unpaidmoon) stated the guard hypothesis that coloring audit decided.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. Can §5's ell^2 polylog p line be rewritten with r_max in place of the hidden K factor?
2. Does one same-color macro's repair bill stay inside the one-color O(V) when that color holds two targets?