Heterogeneous 3+7 order-four boundary across two distinct source banks

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Scope

recursive-interfaces, card persistent-boundary (scope key 026be4b6e61971d51838), chooser OS nonce ba18ac446df7d55ddc50d91395fd9a57, priority revision 3. Base e112d71c8d7bb32435debcc54df932593dd8223a, branch research/agent-38-persistent-boundary-ba18ac44. Package: research/z4-hetero-two-bank-seedba18/.

Card selection note (pair-collision): the first-choice card centre-path-sharing matches already-claimed recent objects (centre-sharing claims in #1166 at 12:58 and 13:03 UTC on 2026-10-14), so the second-choice card persistent-boundary was taken. This is recorded in TASK.json.

Object. The one uncovered cell of the pinned PR 1391 order-four typed boundary interface: two consecutive root echoes with heterogeneous reader schedules across two distinct source banks. Type A has three children reading source octets [0,1,2] of bank X; type B has seven children reading [2,1,0,2,1,0,2] of bank Y; one arbitrary dirty output octet and one 24-coordinate arbitrary dirty workspace must both be restored exactly. Predecessors cover 5|5 identical calls on two banks with a midpoint handoff (research/z4-distinct-source-seed55) and heterogeneous 3|7 calls on one shared source (research/z4-heterogeneous-pair-seed3x7, #1768); neither covers this cell, where the charged 48-ADD source handoff sits at an asymmetric 3|7 boundary. Claim: #1166 (comment)

Result

One joint ten-child phase-indexed typed echo with the handoff W += P^t(Y-X) after child 2 replaces the two separate calls. Exact maps pass on all 80 basis columns in both directions.

Execution	ADD	NEG	Primitives	Reads	Prefix row-L1
A then B, separate	1,552	144	1,696	3,248	18
Joint, asymmetric 3-to-7 handoff	1,408	144	1,552	2,960	21

Saving: 144 primitives and 288 reads (8.5%), all costs charged (both forward and literal-inverse child bodies, per-call 48-ADD copy/restore wrappers, the 48-ADD handoff, and both traversals of every phase boundary).

Boundary law (exact for this alphabet). The workspace boundary charge has period 4 under the exact recurrence F_0(t)=A(t), F_(j+1)(s)=min_t F_j(t)+A((s-t-1) mod 4), terminal phase 0, with A = [(0,0),(54,6),(36,12),(54,6)]: g(k) = g(k mod 4), cross-checked against exhaustive coverage at 64 + 1,024 + 16,384 + 1,048,576 phase paths (k = 3, 5, 7, 10) and for k = 1..16. Consequently separate(k1,k2) - joint(k1,k2) = 2(g(k1)+g(k2)-g(k1+k2)), which at total 10 is 144 primitives exactly when both child counts are odd and 0 when both are even. Both sides are confirmed by emitted words: the heterogeneous 3|7 pair realizes the full 144 (invoice identical to the 5|5 case), and a 2|8 pair with the same banks and child family passes both maps with zero saving.

Evidence and reproduction

python3 -I -B research/z4-hetero-two-bank-seedba18/verify.py
python3 -B scripts/research_package.py --package research/z4-hetero-two-bank-seedba18 --check-only

Observed (CPython 3.12, Windows):

PASS_Z4_HETERO_TWO_BANK_3_7 g3=60 g7=60 g10=48 separate=1696 joint=1552 saved=144 reads_saved=288 law=odd_splits_only even28_saved=0 paths=64+1024+16384+1048576 basis=80 controls=6
PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)

verify.py re-checks the pinned PR 1391/PR 1410 input blobs and the selected-record hash at base, the 12 typed-child identities Q^(-t)R_b P^t = R_b and gate counts, recurrence-versus-exhaustive agreement, the exact 5|5 predecessor figures (positive control), the 3|7 and 2|8 exact maps and invoices, and six negative controls (midpoint-instead-of-3|7 switch, wrong handoff phase, mutated A reader, omitted B call, missing terminal Y restoration, missing literal inverse), each rejected by the exact map. RESULT.json must match the recomputation byte-for-byte on a second run (it does).

Limitations and remaining obligations

• Exact only for the declared order-four phase-indexed child alphabet and these fixed reader schedules; not a lower bound for arbitrary ADD/NEG words.
• Output-row reindexing is free in this word model (it is the typed ADD target). Address calculation, physical row routing, source hosting, tape travel, fixed-precision admission, recursive assembly into a selected supplier and analytic transfer are unproved.
• The joint word raises the prefix row-L1 peak from 18 to 21; no precision improvement is claimed.
• No new κ, selected moment, supplier or frontier change. The typed interface remains outside the selected supplier chain.
• The registered package replay (scripts/research_package.py --package research/z4-hetero-two-bank-seedba18 without --check-only) was not launched here; the same verify.py was run twice under CPython 3.12 with identical output.

Credit

Aaron Wang, with help from opencode (substantial AI assistance; no independent human review claimed). Predecessor credit: Cinna Davis (1337isnot1337) for the order-four child family (PR 1391), the same-source persistent boundary (PR 1410), research/z4-persistent-pair-seed57, research/z4-paired-frame-restoration-seed56 and research/z4-distinct-source-seed55 (typed two-bank 5|5); 1337isnot1337 for research/z4-heterogeneous-pair-seed3x7 (#1768). The selected-result context is Rohan Arun's #1094 on Jeremy Kothe's PR 1087 construction. All predecessor inputs are pinned byte-for-byte in inputs/ and SOURCES.json.

Workflow note: make is not installed on this host, so the selected record's make frontier1094-verify cold replay was not run; scripts/research_baseline.py printed PASS_RESEARCH_BASELINE_BINDINGS_ONLY for orientation.