[DreamingOfClouds]

No new κ. Four scoped statements about the chain-nested root of #6833, each with its hypotheses, proved or checked exactly on the literal words.

1. Idle cover: for the root of shape (g|g) (site i has |M_i| - 2 = i idle directions), every cover by whole idle jobs that lie on one side, with a native active part of rank at most two and price (8m)^t for 0 < t ≤ 1, costs at least 2 Σ_{s=1}^{g-1} (8s)^t; for 0 < t < 1 equality holds exactly for the chain's job lengths g-1, ..., 1. Proof: rank subadditivity, a Gale-Ryser majorization and Karamata's inequality. Targets are fixed by the masks, so any other cover has a strictly larger paid side. Jobs on both sides, native stages of rank three, or a convex price lower the minimum (controls). No idle line is transported from a left to a right site through the three thin queries (exact over Q, g = 3, 4, 9).
2. Top targets: for a shortest root (n fields, n + 1 square-zero queries, constant bases, one variable per site, J = -I) the product identity makes a conjugated Gram matrix rank one, which forces a field in every full mask (target rank 145 at 18 stages) and, through a three-site sign argument, a prefix field and a suffix field of total weight at least 18. The three largest target ranks sum to at least 291; {145, 73, 73} of Conditional κ = 0.00330720 (+3.000e-6, +0.09079% over #6716): two paid idle fragments unlock independent outputs #6726 and Conditional κ = 0.00337180 (+6.36e-5, +1.92% over #6824): replace the supplier tree by a chain-nested root #6833 attains it. The sum is forced, the split 73 + 73 is not.
3. Retirement: removing the two pivots of Conditional κ = 0.00330720 (+3.000e-6, +0.09079% over #6716): two paid idle fragments unlock independent outputs #6726's published retirements from its masks gives exactly the masks of the chain gate (4,4,1|1,4,4) up to relabelling, and the outer histograms differ by one cancelling pair of rank-32 jobs; the literal chain (9|9) word has no retirement candidate under the criterion of Prove the pure dual-pair retirement criterion and its unit-divisor count (no κ) #6827 (its census finds the three known cuts of the unretired Conditional κ = 0.00330720 (+3.000e-6, +0.09079% over #6716): two paid idle fragments unlock independent outputs #6726 caller). That every retirement of a shortest root is a mask reduction is not proved.
4. Shape: at the 18 stages of the retained native word, (9|9) is the unique best chain shape on the complex histogram (crossing 0.00337180047549672) and on the binary histogram (crossing in (0.0038525393934, 0.0038525393935)), over all 2^17 compositions per side with the best binary tree per child; the runner-up (8|10) loses about 1.5e-6 (complex) and 1.9e-6 (binary) in κ at first order. A scaled-native model for 10 to 40 stages also peaks at 18.

Exact checks: literal idle jobs of #6833 against the demand of all 18 sites; exhaustive minimum covers for g = 2..9 and 4,000 random covers; root identity at random D, rank-one Gram on all triples and the residual identity on the chain and #6726 roots; ledger against the literal events of four built words and CALLER.json; fourteen negative controls.

Base: 7c49fe3. Inputs are unmodified copies pinned in retained/SOURCE-PINS.json to #6833 at ad24ad9, #6726 at 435bbe9 and #6827 at f0fbfd6.

Reproduce (CPython 3.12, standard library):

python3 -B scripts/research_package.py --package research/p104-chain-idle-cover-optimal

Expected: PASS_CHAIN_IDLE_COVER_OPTIMAL (normal and -O) and PASS_REGISTERED_RESEARCH_PACKAGE.

Scope: one-sided idle jobs with active rank two; J = -I and exactly n + 1 queries for statement 2; finite statements for 3; the additive ledger of the chain-nested family for 4, with only N = 18 literal. Cross-side jobs conjugated through other sites, other mask patterns and non-chain roots are not covered. Pricing, native phases and the five framework assumptions of the cited words are not re-derived and no κ changes.

Overlap: #6832 gives the N ≥ r + 1 count used in the setting; #6774 excludes the four-site coupled root with a different method and scope; #6834 transfers the chain word to the binary native word and its crossing is reproduced here. No open or merged pull request states the idle-cover bound, the top-target sum or the exhaustive shape search.

Credit #6833 (DreamingOfClouds) for the chain-nested root and caller, zxv (#6726), thu-nmrc (#6716), CubeJerry (#6805), Ali Romman and the caller/verifier lineage. Apache-2.0.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.

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