[DreamingOfClouds]

No κ claim. This adds research/p104-root-chain-optimality/, a registered research package for the card p104-side-correction.

Claim. For a shortest rank-one root with L = 2g sites, interval masks, y live on the all-field and the host (R2) and singleton hosts (R3), D_root >= f(8L+1) + 2 sum_{m=1..g} [f(8m+1) - f(8m-8)] with f(x) = x ln x. Equality holds for chain(g|g) (the 9|9 root of #6833 at L = 18) and its outer-nested twin only; unbalanced chains are strictly worse. At L = 18 the bound and the chain are both 1418.8458829 per port. Realizability of a family is a sign factorisation sign(i-j) = p_i q_j outside the masks; this gives the size staircase #{fields of size <= t} <= 2t, and convexity of f with increasing f(8m+1) - f(8m-8) gives the bound.

Dropping R2. An exact Gröbner-basis decision over all families within the chain budget at L = 4 and L = 6 finds no gain when every site has a y component; dense y only loses idle discount.

Scope. One loophole is stated, not closed: with one thin query, a site with no y component besides the all-field gives realizable families that save 0.43858 at L = 4, 6 (not computed at L = 18, not built or priced). Doubleton hosts are checked exactly only for L <= 8. Non-coordinate masks, D-dependent bases, non-shortest roots and units of rank above one are outside the statement. No word is constructed and nothing is priced.

How to verify. python3.12 -I -B research/p104-root-chain-optimality/check.py (standard library only, a few seconds) recomputes the bound, the chain and the twin at every even L = 4..18, the unbalanced chains, the staircase over all sign patterns for L <= 16, the exhaustive sign-model minima (697.2194902789 at L = 18, equal to the chain), and the six uncovered-pivot savings; it ends with PASS_P104_ROOT_CHAIN_OPTIMALITY. python3.12 -B scripts/research_package.py --package research/p104-root-chain-optimality reports the package as registered.

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