[JoyboyBrian]

The actual balanced CRT caller reconstructs a left integer from current transformed child coordinates. Existing encoded triangular offsets use original lower coordinates, and the fixed key port consumes supplied ROM data. This adds the missing encoded current-child numerical producer and completes its right-node offset/padded-coordinate consumer.

Concrete source witness: moduli (5,7) and current coordinates (1,5) reconstruct 11; right modulus 13 gets offset 7. Treating those coordinates as original mixed-radix digits yields 26 and wrong offset 0, despite both being individually valid.

The new Arithmetic explicitly admits any modulus 1<M<2^(P-1) in a common P-bit arena and implements modular operations from retained unsigned bit-tape primitives. It does not call or bypass the frozen Ring constructor, whose stricter significant-bit guard remains unchanged and tested. No runtime integer decoding, word-backing read, bit_length or times_small(decoded modulus) is used.

The written proof covers arbitrary finite nonempty left/right groups with pairwise coprime moduli and complete product below 2^(P-1). It proves locally scaled reconstruction, all successive right prefixes, invalid-left disabling, preservation of padded right coordinates, borrowed byte/head preservation and complete owned cleanup. At most 32 numerical slots are live in a fixed 64-slot arena; a conservative 2^20*(P+2)^4 bound counts source-tape/control operations. This is not unconditionally substituted for a complete literal binary E(p,ell) implementation.

Validation (CPython 3.12, standard library):

python3.12 -B scripts/research_package.py --package research/encoded-current-crt --check-only
python3.12 -B scripts/research_package.py --package research/encoded-current-crt
python3 -B scripts/check_research_commit.py --package research/encoded-current-crt --check-source-history

• Ordinary and optimized registered replay pass: 1239 complete modular operand pairs, 119 unit/nonunit inverses, all 64 eight-valued current left-coordinate pairs, invalid padding and moduli exceeding the finite small-multiplier vocabulary.
• Varied 1+1/1+2/2+2/3+3 groups at P=4,8,12,20,32 use the same producer. Eight dirty borrowed words at noninitial heads and the arena template remain unchanged; observed peak is at most 26 slots.
• Four original balanced-root cases and four complete original mixed-width Callers check every numerical offset/level endpoint against unchanged source Beneš rows at declared widths (4,4,4,5). Each module receives an encoded source-derived fixture; this is not an uninterrupted physical whole-tree execution.
• Three actual semantic mutants misinterpret current coordinates, omit a preceding right modulus or reduce padded values. Eleven rejection cases check noncoprime inputs, capacity/domain limits, empty groups, short framing and late failure after output production; cleanup is inspected before fixture disposal.
• Source-history/committed bytes, official export integrity, read-only Git-free ordinary/optimized replay, live-source corruption and type-changing frozen-receipt controls pass. These are reproduction and source-binding evidence, not human or proof-kernel acceptance.

Canonical coordination: #4709 (comment) . Card uniform-transfer-audit, refined to the actual current-child numeric boundary. Scientific base f259dabc7d70e0db8d6d36bfda888f008bbfecc1, priority revision 7. Initial overlap covered all 6710 notes, 320 open PR bodies and 4130 Discussion records with complete inventory/pagination; arbitrary diffs and unpublished work remain limits.

Family decision: narrow to this complete local numerical module, then stop. Initialized arena/parameter/descriptor production and transport, literal binary control compilation, complete ROM/Beneš construction, raw macro, sparse repair, ordinary interchanges, full E and native/outer transfer remain external. No supplier, selected frontier, complete multiplication theorem, speedup or new kappa is claimed. README/TASK preserve bounded next questions.

Credit Cinna Davis/#4100; Hiroki KUMAZAKI/#4169/#4174 and the mixed-width caller; Tekalign's already encoded original-coordinate triangular offset producer; ToikY/#4192, unpaidmoon/#4377; Jose Manuel and the retained physical Word/caller, Douglas Colkitt/John-Michael Smith and NOTICE lineage. Frozen sources are unchanged. Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0. Independent human and formal proof-kernel acceptance are unclaimed.

Submission refresh: canonical main 4c0a359d88f22f05189d51f48387750b801a45d4; governing guidance, selected record and this contribution's source inputs remain unchanged from the scientific base. Fresh overlap covers 309 open PR bodies, 4,156 Discussion records and all 6,760 local notes with complete reported pagination/inventory. New public deltas were checked; this remains advisory text coverage, excluding arbitrary diffs and unpublished work.