[DreamingOfClouds]

BIT dependency bound only; no overall κ. The chain-nested root caller of the complex companion #6833 ( n=20, 3 thin queries, 18 local calls, 16 merged idles, no supplier tree) is integral, so it transfers to the retained BIT native word (SHA-256 51203e7fc75ba10bd3ac3f8e675759f47e13d262bea66b6600d1e515095cad78) by the same procedure as #6726: specialise q²=1 so the ordinary normalisations become one, reduce mod two (odd ADD becomes XOR, even ADD disappears, SCALE ±1 becomes the identity).

Result: a 670-event binary caller with 20 independent inputs and 18 native stages; complete invoice of 10,086 productive fields and 53,787 paid jobs, rank-mass surplus 753; raw crossing in (0.0038525393934, 0.0038525393935), against 0.0037667234776 for #6726. Finite admission at tested saving 0.0038525 admits ordinary BIT saving 0.00385248 (previous 0.00376669). The ordinary wrapper is two strict steps 0.00374998 → 0.00385 → 0.00385248 with atom 0.004.

The basis matters: the sites use x = A + f_i + 2Σf_j, whose coefficient 2 disappears mod 2, but the basis matrices have determinant ±1, so the identity holds over Z first. A construction with power-of-two pivots yields factors 1/2 and 2 and is rejected; this one uses ±1 pivots only (checked on all 1237 complex events).

Exact checks (400 coefficients each): forward over F2 with stage monomials; forward over Z[q,D]/(q²−1); the retained independent reverse audit; the complex caller replayed over Z[i] with its normalisations. Crossing by the retained interval engine and an independent Decimal engine. Controls reject a missing input or output basis, LOCAL, QUERY, merged idle, normalisation, even ADD, a stray XOR, a shrunk idle, the non-integral basis and a saving above the crossing.

Base: 7c49fe3. Retained inputs are unmodified copies of #6726 at 435bbe9, pinned in retained/SOURCE-PINS.json.

Reproduce (CPython 3.12, standard library):

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

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

Scope: the same five framework assumptions as #6726 (native_all_columns, owned_erasure, compiler, precision_tape, analytic_recovery); the native theorem is unchanged. A complex construction must keep its own κ below this saving; none is claimed here. Using the saving in a package ledger needs the two-step wrapper envelope and atom 0.004.

Credit zxv (#6726: integral transfer, BIT native word, finite admission) and thu-nmrc (#6716). Apache-2.0.

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

🤖 Generated with Claude Code