[DreamingOfClouds]

The supplier tree of the #6726 caller is replaced by one root with a full site for every native call and a chain of whole idles, on #6824's unchanged native word. The complete 18-call construction supports conditional κ=0.00337180, up 6.36e-5 (+1.92%) from #6824's 0.00330823. The raw crossing is in (0.0033718004754, 0.0033718004755), above the tested saving 0.00337180035.

Root: fields A, f_0..f_8, h_0..h_8 and one empty field (n=20).

• L site i: x=A+f_i+2Σ_{j<i}f_j, y=A−f_i, support {A,f_0..f_i}. R site m mirrors it: x=−A+h_m+2Σ_{l>m}h_l, y=A*+h_m*, support {A,h_l: l≥m}. Three thin queries complete Π(I−2R_k)=−I.
• Each inner field has one idle line φ=f_{j−1}, γ=f*_{j−1}−2A* (mirrored), killed by y_i and x_i on all later sites of its group. It commutes across them, so its stage factors merge into one whole idle of rank 8·(sites covered).

Per port: T={1:1, 9:2, 17:2, ..., 73:2, 145:1}, P={1:3, 8s:2 for s=1..8}, no binary nodes. Roles 34→20, thin queries 15→3, largest paid rank 40→64, productive fields 13,688→12,232, paid requests 66,888→64,184, paid mass 178,044→166,812, surplus 760 unchanged. Caller scalars are integers.

PROOF.md gives the arbitrary-input identity, with hand proofs of the root endpoint (sign Gram lemma), the stage-factor move and the chain-merge lemma. Gate: the same family at (4,4,1|1,4,4) has #6726's invoice minus one cancelling T32/P32 pair, and both interval engines place its crossing in #6824's interval (0.00330823456614, 0.00330823456615).

Base: 7c49fe3.
#6824 native: 2a7ca44, payload SHA256 d3ccfce7bdbca3b79771844e6c9fee48f1a8066444a19218995648df96982821 (rebuilt from #6702 through #6805). #6726 caller, retained as baseline: 435bbe9. #6763 audit: c3e0500.
New caller event SHA256: 03ecddfd75d0dbe325fcc489a50ae7657364482f0d887d81fa131ff6bf41fe70.
All required bytes and Git/SHA256 pins are retained.

Reproduce (Python 3.12, stdlib):

python3.12 -B scripts/research_package.py --package research/p104-chain-nested-root

Expected: PASS_CHAIN_NESTED_ROOT and PASS_REGISTERED_RESEARCH_PACKAGE.

Executed: registered normal and optimized replays (Python 3.12) returned PASS_CHAIN_NESTED_ROOT and PASS_REGISTERED_RESEARCH_PACKAGE, as did standalone normal and -O replays; the receipt is reproduced exactly. Checked: root algebra and chain-merge hypotheses, 400 forward and reverse caller coefficients, a separate symbolic root audit, the #6824 native word (801x801 endpoint coefficients, 2,940 maps with phases), 7,488 caller contacts, adapters, allocation, independent invoice, both interval engines, compiler and 54+4 outer inequalities. Negative controls reject deleted LOCAL, QUERY, IDLE_ONE, LAMBDA and ADD, a shrunk idle, wrong phases, aliased inputs and omitted adapters. Recomputed bounds: ordinary 488,177,792 (was 535,415,008), compiler gap 1/20,000,000,000 (was 1/1,750,000,000), contacts 7,488 (same count). Details in VALIDATION.md.

Caveats: all five assumptions remain: native_all_columns, owned_erasure, compiler, precision_tape, analytic_recovery. Selected κ=0.003125804 is unchanged. External mathematical review remains open; no formal proof-kernel or unconditional theorem claim. Whole idles at ranks 48–64 extend the rule #6726 used at rank 40, and this PR gives no new compiler proof for them. The merged lines also cross sites with different ordinary bases, unlike #6716's shared-basis coordinates: ADAPTERS.md and PROOF.md section 6 state the reading for reviewers to check. κ stays below the BIT cap .0034, so #6805's pin stays.

Credit: thu-nmrc (#6716, common idle fusion), zxv (#6726), @CubeJerry (#6805), Ali Romman (independent-helper basis, exact caller replay, native input lifting #6349, #6353, #6372), #6763, Jacob Alan Sussman (compiler), DreamingOfClouds (#6702, #6375, #6824), and retained native/interval authors. Original notices preserved; Apache-2.0.

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