[petalas]

Summary

The rank-cap continuation of the selected complex local word's packing ceiling prices the local
paid mass with integer child counts but leaves the idle remainder in a fractional block
(1646/51 rank-51 children). Its integrality-loss column therefore measures only half of the
rounding, and its ceilings are not attainable by any real packing.

This package computes the attained optimum over joint integer packings for every cap
s = 1..51 the outer halving guard allows: local paid mass 5586 in integer children of rank
<= s, idle remainder 1646 in integer children of rank <= 51, both by dynamic programming over
the mass axis. Every cap is certified in both directions: an explicit joint integer packing whose
certified moment is below one, and a certified lower bound of every joint packing above one.

Quantity	Value
Cap at which the joint integer class first clears a = 1/99	44 (cap 43 fails at 0.00993809)
Joint ceiling at cap 44	a = 0.01031470, κ ≤ 0.01020939, ×6.10 selected κ
Joint ceiling at the guard cap 51	a = 0.01349051, κ ≤ 0.01331094, ×7.96 selected κ
Worst integrality loss on the curve	1.35033 % (paid rounding 0.66054 % + idle rounding 0.68979 %)

So the rank-shape consequence survives integrality — the target is cleared at cap 44 on an
attainable object — but the guard-cap figure is 0.99 % lower than the fractional-idle reading and
the worst-case integrality loss is about 2.04× the published figure that counts paid-mass rounding
only.

Controls include: the class relaxation equals the pinned parent ceiling() on the reshaped
profile at caps 8, 16 and 51 (the parent's rank-8 and rank-16 lever rows); the cited live column is
reproduced independently at its own grid at caps 44 and 51; the paid dynamic-program optimum equals
the largest-first greedy cost and histogram at all 51 caps, extending a recorded 11-cap check;
both histograms conserve mass and respect their caps at every cap; brackets are witnessed on both
sides; 25 sampled joint packings per cap never reach the rejected saving; and the 1e-8 bracket
contains the 1e-12 bracket at three caps.

Scoped bound inside the parent ceiling model, with one fixed paid local mass, one fixed idle
remainder, m = 73, 896 families over nine stages, centre charge 64, the coverage law and the
halving guard. No new κ, no supplier, no change to the selected frontier.

Related open work

Three open peer packages price the same selected complex local word in the same parent ceiling
model; none is modified here and each is cited for comparison only.

• feat(research): exact integer idle-histogram ceiling for the selected complex word #3729 research/complex-idle-integrality-ceiling — incumbent paid profile, integral idle
  counts per call, optimum {51: 32, 14: 1}, ceiling a < 0.001688603446527. Agreement: this
  package uses exactly that idle packing at every cap and the same per-call premise. Differs:
  feat(research): exact integer idle-histogram ceiling for the selected complex word #3729 re-optimises the idle remainder of the incumbent word; the cap-s rows here reshape the
  paid mass into rank-s children, so the cap-8 row 0.00273511 is the integer-idle version of
  the parent's rank-8 lever row, not feat(research): exact integer idle-histogram ceiling for the selected complex word #3729's incumbent ceiling.
• Certify the idle rank-cap sensitivity of the selected packing endpoint (no new kappa) #3805 research/idle-rank-cap-sensitivity — sweeps the idle rank cap and separates
  aggregate pooling (4647×51 + 27 at cap 51, +0.870319 %) from a per-call-forced optimum
  (32×51 + 14 per call, +0.784307 %). This package uses the per-call premise throughout
  (each call packs its own integral 1646), so its rows belong with Certify the idle rank-cap sensitivity of the selected packing endpoint (no new kappa) #3805's per-call row and with
  feat(research): exact integer idle-histogram ceiling for the selected complex word #3729 and are not comparable with Certify the idle rank-cap sensitivity of the selected packing endpoint (no new kappa) #3805's aggregate rows. The axes are complementary: Certify the idle rank-cap sensitivity of the selected packing endpoint (no new kappa) #3805
  varies the idle cap at a fixed paid word, this package varies the largest paid child rank with
  the idle cap held at the guard.
• Research: price the paid-mass elimination budget of the selected complex word (screening budget; no new kappa) #3799 research/complex-paid-mass-budget — free removal of paid mass at fixed rank shape.
  Its full-elimination endpoint 0.013675160009708 is the same object as this package's cap-51
  class relaxation, and that agreement is registered as a control (below 1e-9); its φ = 0 row
  is the parent's inherited ceiling 0.001690054109089. Differs: its 1/99 threshold
  φ ≥ 92.7698485 % is a mass-removal budget, whereas cap 44 here clears 1/99 with no mass
  removed, by delivering the same paid mass in rank-≤44 children. Same model, same target,
  different lever; neither statement bounds the other.

Test plan

• python3.12 -B research/complex-joint-integer-cap-ceiling/verify.py (~2.5 min, PASS_COMPLEX_JOINT_INTEGER_CAP_CEILING)
• python3.12 -I -B research/complex-joint-integer-cap-ceiling/verify.py (registered VERIFY.json command, same PASS)
• python3.12 -B scripts/research_package.py --package research/complex-joint-integer-cap-ceiling → PASS_REGISTERED_RESEARCH_PACKAGE
• python3.12 -B scripts/check_research_commit.py --package research/complex-joint-integer-cap-ceiling --check-source-history → PASS_RESEARCH_COMMIT_BYTES
• Maintainer replay from a clean checkout at the parent commit

Prepared by Nick Petalas with AI assistance. Apache-2.0.