Research: exact nine-source Johnson-chain connector construction

research/nine-source-johnson-chain-seed923edf/CLAIM.md

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+# Cycle 32 local research claim
+
+**Status:** RESULT SUBMITTED as [PR #2296](https://github.com/CrocSwap/integer-mult-bounds/pull/2296).
+The PR was open and mergeable at head `b4af93f238e69ccf000eb9e90030be4c01cda391`
+against base `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2` at the initial
+submission check; hosted checks were queued. The subsequent handoff metadata
+update advanced the branch; consult the PR for its current head and checks. The
+public Discussion claim remains local-only: the
+authenticated GraphQL overlap refresh hung and was stopped after about two
+minutes, the final pre-submission retry produced no output before timing out
+after 50 seconds at 03:56 UTC on 2026-10-15, and a prior direct REST Discussions
+comment attempt returned 404. No public claim was posted.
+
+- **Track / task / seed:** `recursive-interfaces` / `connector-alphabet-change` /
+  `923edf31ad6e8cfdc58bb11a6c6d6b3d`.
+- **Exact base:** `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`.
+- **Catalogue / selected record:** revision 4; catalogue SHA256
+  `d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7`; selected
+  record SHA256 `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.
+- **Question:** In the established singleton F2 dirty-connector alphabet, can
+  five separate dirty outputs with supports
+  `{01234, 01235, 01256, 01567, 05678}` be computed with nine read-only
+  sources at the complete-call floor? These five 5-subsets form a path in
+  `J(9,5)`, each step replacing one source, and their union is all nine
+  sources.
+- **Changed scope:** The selector card asks to add one connector operation.
+  This investigation adds no operation; it studies one explicit sparse
+  output-support family under the existing alphabet. It is a construction,
+  not a repeat of Cycle 31's eight-source full weight-four two-sweep census.
+- **Candidate screen:** The two-unpaid-coordinate coverage-law question is
+  already claimed in Discussion #1587. Centre-path hosting is claimed in
+  #1516 through 05:00 UTC, with older related nonadjacent-path records #1963
+  and #2058. The joint-query
+  lane has open PRs #2153 and #2268, including the immediately preceding
+  Cycle 30 contribution. The general connector lane has active GF(4) and
+  seven-source claims plus open alphabet/path PRs; this claim narrows it to a
+  different positive n=9, weight-five support chain with no new operation.
+- **Method / domain:** Use the source walk `0 → S1 → S2 → S3 → S4 → S5 → 0`,
+  toggle each symmetric difference, and read each `A_i` once at `q=0` and once
+  at `q=S_i`. Charge all B and A calls. Exhaust all `2^9` source assignments,
+  dirty connector bits and `2^5` dirty-output vectors. Stop after exact replay,
+  the lower-bound proof and the listed failure controls; no search over
+  unrelated support families.
+- **Favorable screen:** Optimistic floor and proposed schedule are both
+  `2*9 + 2*5 = 28` complete calls. Five independent restored row stars cost
+  `5*(2*5+2)=60`. The intended saving is 32 calls in this abstract interface;
+  under stable per-call source prices `c_j`, the weighted saving is exactly
+  `2*sum_j (d_j-1)c_j`, where `d=(5,4,3,2,1,4,3,2,1)`; A-call prices cancel.
+  Position-dependent transitions and concrete supplier placement remain
+  unpriced, so no recursive supplier or kappa is claimed.
+- **Expected evidence:** A standalone verifier, exact source/readout word,
+  arbitrary dirty-state replay, lower-bound argument, negative controls,
+  complete provenance pins, and a focused package PR. Submitted as PR #2296;
+  hosted checks were queued when checked.
+- **Relevant work and distinction:** Open PR #2208 (head
+  `28480ef4fdb4caeab89765ea56595bf1f6f37688`) develops all-word two-root
+  dirty-connector bounds and an exact minimum-read compiler for supplied source
+  walks over as many as 511 outputs. It is adjacent source-walk work, but uses
+  a two-root framed interface and does not present this one-connector, five-
+  mask Johnson chain or its 28-call floor. PR #2271 (head
+  `95fd439596f72575d3869277c2d01aca1ca6ab6a`) is a 12-row bit-register Gray
+  tour through three masks, not this nine-source recursive B/A interface.
+  PR #2283 (head `21145202e9a27cd9b18ffc7e23383cc83461c4a0`) is Cycle 31's
+  negative result for the full n=8 weight-four shell under two complete
+  sweeps. PR #2229 covers n=7 weights 1--3; PR #2266 changes to GF(4) for two
+  sources. Active neighboring claims #2262 (GF(4)) and #2275 (seven-source
+  paired-call roots) do not use this mask family. Targeted live PR searches at
+  04:00 UTC found no `J(9,5)`, exact five-mask or "nine-source Johnson-chain"
+  match. The complete cached snapshot covered 511 open PRs and 990 Discussions
+  (PR data 03:42 UTC; Discussion data 03:26 UTC, 2026-10-15).
+- **Expiry:** The local claim is released on completion; the recorded planning
+  expiry was 2026-10-15 05:45 UTC.
+
+The selected conditional result is `836330098538459/500000000000000000` at
+this base, reviewed in `docs/research/community-batch1099-review.md`; the
+selected assembly and source manifest are pinned in `SOURCES.json`. The
+selected supplier reproduction is `make frontier1094-verify`; it is not part
+of this connector-level experiment.

research/nine-source-johnson-chain-seed923edf/README.md

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+# Nine-source Johnson-path connector construction
+
+**Exact connector-level construction; no selected-kappa change.** Five
+distinct dirty outputs on nine read-only F2 sources have supports
+`01234, 01235, 01256, 01567, 05678`. A single restored source path computes
+all five outputs in **28 complete calls**, meeting the general floor of two
+calls per active source and two per output. Five independent restored row
+stars cost 60 calls. The construction uses only the existing singleton
+connector alphabet and makes no supplier or multiplication-bound claim.
+
+**Handoff:** [PR #2296](https://github.com/CrocSwap/integer-mult-bounds/pull/2296)
+is open against the exact base `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`.
+At the initial submission check, head `b4af93f238e69ccf000eb9e90030be4c01cda391`
+was mergeable and hosted checks were queued. The subsequent handoff metadata
+update advanced the branch; consult the PR for its current head and checks.
+The public Discussion claim remains unposted because
+the authenticated GraphQL route did not complete and the prior REST comment
+route returned 404; the local claim and PR preserve the coordination record.
+
+Prepared by Cinna Davis with substantial OpenAI Codex assistance in scope
+selection, derivation, implementation and verification. No other contributor
+work is claimed. Apache-2.0.
+
+## Assignment, scope and coordination
+
+- Accepted seed: `923edf31ad6e8cfdc58bb11a6c6d6b3d`; track
+  `recursive-interfaces`; task `connector-alphabet-change`.
+- Excluded draw: `3c0e460e62633c4b16b5923d0f386d45` selected
+  `adversarial-audit` from the prior submitted branch, so it was discarded.
+- Exact base: `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`; catalogue revision 4,
+  catalogue SHA256
+  `d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7`;
+  selected-record SHA256
+  `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.
+- Scope changed from adding a connector operation to keeping the existing
+  alphabet and constructing one sparse support family. The exact local claim
+  is in [CLAIM.md](CLAIM.md). The generalization is limited to support paths
+  that introduce a fresh source at each one-element exchange.
+- Overlap: the complete cached snapshot held 511 open PRs and 990 Discussions
+  (no warnings). The exact local overlap-helper searches for the nine-source
+  weight-five path returned no direct matches. PR data was fetched at 03:42 UTC
+  and Discussion data at 03:26 UTC on 2026-10-15. A newer authenticated
+  GraphQL refresh hung and was stopped after about two minutes. The final
+  pre-submission refresh produced no output and
+  timed out after 50 seconds at 03:56 UTC, so the full cached snapshot was not
+  refreshed. Targeted live PR searches at 04:00 UTC found no `J(9,5)`, exact
+  five-mask, or "nine-source Johnson-chain" match. A broader search surfaced
+  open PR #2208 at head
+  `28480ef4fdb4caeab89765ea56595bf1f6f37688`, which studies all-word two-root
+  dirty-connector bounds and a minimum-read source-walk compiler for up to 511
+  outputs. That work is adjacent, but does not give this single-connector
+  five-mask path or its 28-call floor. The nearest small source-path result,
+  PR #2271 at
+  `95fd439596f72575d3869277c2d01aca1ca6ab6a`, is a 12-row local bit-register
+  tour through three different masks. Cycle 31 PR #2283 at
+  `21145202e9a27cd9b18ffc7e23383cc83461c4a0` is a negative result for the
+  entire n=8 weight-four shell under two complete sweeps. PR #2229 studies n=7
+  weights 1--3; PR #2266 changes the alphabet to GF(4) for two sources.
+  Active nearby claims #2262 (GF(4)) and #2275 (seven-source paired-call root
+  counts) use different objects and support families. PR #2208 changes the
+  interface to two roots and studies many outputs/readout tariffs; this result
+  uses one F2 connector and five specified outputs, with no frame transitions.
+- Discussion claim status: local-only because the authenticated GraphQL
+  request hung, the final pre-submission refresh timed out without output, and
+  a prior direct REST Discussions comment attempt returned 404. Claim expiry
+  recorded in [CLAIM.md](CLAIM.md) is 2026-10-15 05:45 UTC.
+
+## Model and fully charged cost screen
+
+Use the pinned singleton F2 interface: `B_j: z += B_j x_j` for read-only
+sources `j=0,...,8`, and `A_i: y_i += A_i z` for five separate outputs. At
+entry, all source rows, targets and connector `z` are arbitrary and dirty.
+Each complete call restores its private scratch and tape heads. The interface
+and predecessor are pinned in [SOURCES.json](SOURCES.json), especially the
+[universal connector statement](../universal-connector-boundary/README.md).
+
+Before implementation, the target was 28 complete calls: every one of the
+nine active source maps must be toggled an even positive number of times to
+restore dirty `z`, so it costs at least 18 B calls. Every nonzero output map
+must be read an even positive number of times to cancel arbitrary dirty `z`,
+so five outputs cost at least ten A calls. The independent restored-row
+baseline is 60 calls: for each weight-five row, read its `A_i` at `q=0`,
+toggle five sources into `z`, read `A_i` at `q=S_i`, and toggle those five
+sources back, for `5+2+5=12` calls per row. The path below reaches 28 exactly,
+saving 32 complete calls (53 1/3%) in the unit-call model.
+
+For the lower bound, specialize every operator to scalar identity maps and
+vary one source and the dirty connector independently. Restoring `z` forces
+each active `B_j` count even, while producing every nonzero output and
+cancelling arbitrary `z` forces each `A_i` count even and positive. This
+gives `2*9 + 2*5` for any correct schedule in the stated alphabet.
+
+The same comparison holds with unequal **per-call** prices. Let `c_j > 0` be
+the complete price of `B_j`, `a_i > 0` the complete price of `A_i`, and
+`d_j` the number of outputs containing source `j`. Here
+`d=(5,4,3,2,1,4,3,2,1)`. The shared path costs
+`2 sum_j c_j + 2 sum_i a_i`; independent row stars cost
+`2 sum_j d_j c_j + 2 sum_i a_i`. Thus the exact saving is
+`2 sum_j (d_j-1)c_j > 0`. Readout prices cancel; the saving comes from
+source maps that occur in multiple rows. This weighted invoice charges each
+complete B/A call, including its recursive child bill when such a bill is
+assigned. It assumes a call's price is stable across positions in the word.
+
+The path adds no clean rows, copies, helper operations or separate frames, and
+returns `z` to its dirty entry value. The abstract connector model has no
+position-dependent frame-transition charge. A concrete supplier would still
+need its actual child prices, any schedule-dependent transition or host cost,
+precision and tape bill, and a placement proof. This local cost reduction is
+not a kappa claim.
+
+## Construction
+
+Let the five supports be
+
+```text
+S1 = {0,1,2,3,4}
+S2 = {0,1,2,3,5}
+S3 = {0,1,2,5,6}
+S4 = {0,1,5,6,7}
+S5 = {0,5,6,7,8}.
+```
+
+They form a path in `J(9,5)`: each next support removes one current source
+and introduces a source not used before. Their union is all nine sources, and
+only source 0 is common to all five outputs; the route does not rely on a
+large shared core.
+The source word follows the parity states
+
+```text
+0 -> S1 -> S2 -> S3 -> S4 -> S5 -> 0
+```
+
+by toggling each symmetric difference:
+
+```text
+B0 B1 B2 B3 B4 | B4 B5 | B3 B6 | B2 B7 | B1 B8 | B0 B5 B6 B7 B8
+```
+
+At the initial `q=0` prefix, call each `A_i` once. At the later prefix
+`q=S_i`, call that same `A_i` once more. The resulting full schedule uses 18
+source calls and ten output readouts. Each `B_j` appears exactly twice, so
+arbitrary connector contents are restored.
+
+For each output, the two `A_i` calls contribute
+
+```text
+A_i z_initial + A_i(z_initial + sum_{j in S_i} B_j x_j)
+= sum_{j in S_i} A_i B_j x_j,
+```
+
+because the interface is over F2 and `A_i` is linear. Thus arbitrary initial
+target and connector contents cancel in the update, while every source input
+remains unchanged. The scalar verifier exhausts all source values and dirty
+bits; this algebraic identity establishes the same schedule for arbitrary
+compatible linear maps.
+
+More generally, take `p` distinct supports `S_1,...,S_p` of size `r`, where
+each transition replaces one current source by a fresh source and
+`n=r+p-1`. The same closed path `0,S_1,...,S_p,0` uses
+`r+2(p-1)+r=2n` source calls; each source appears twice. Two A readouts per
+output give `2n+2p`, the complete-call lower floor. Separate row stars cost
+`p(2r+2)`. In the recorded instance `n=9`, `r=p=5`, these are 28 and 60.
+
+## Verification
+
+Run from repository root with Python 3.10 or later and the standard library:
+
+```sh
+python3 -B research/nine-source-johnson-chain-seed923edf/verify.py
+python3 scripts/research_package.py --package research/nine-source-johnson-chain-seed923edf --check-only
+```
+
+The exact verifier checks the five supports, all nine source labels, every
+Johnson-path transition, the 18-toggle word, the ten readouts and the 28-call
+floor. For each of 512 source assignments, both dirty connector bits and all
+32 dirty-output vectors (32,768 cases), it replays both the shared path and an
+independent 60-call row-star schedule. Negative controls expose single-readout
+connector leakage, a wrong output boundary, and a missing restoration call.
+The verifier also checks the weighted formula at the nonuniform prices
+`c_B=(1,...,9)` and `a=(1,...,5)`, where the saving is 120 price units.
+
+Expected output starts with:
+
+```text
+PASS_NINE_SOURCE_WEIGHT_FIVE_JOHNSON_CHAIN
+supports=01f,02f,067,0e3,1e1
+common_sources=0
+source_calls=18
+readout_calls=10
+total_complete_calls=28
+independent_row_stars=60
+saved_calls=32
+weighted_price_control_saving=120
+dirty_state_cases=32768
+```
+
+The registered package replay uses Python 3.12. This host has Python 3.14.7
+and no `python3.12`; the direct verifier and package metadata/pin check were
+run locally. Hosted replay remains an obligation.
+
+## Selected context
+
+The selected conditional result remains
+`836330098538459/500000000000000000`, reviewed in
+`docs/research/community-batch1099-review.md`. Its source manifest,
+`ASSEMBLY.json`, reproduction `make frontier1094-verify`, Makefile rule and
+actual verifier are pinned in [SOURCES.json](SOURCES.json). That reproduction
+was resolved but not rerun: this contribution changes no selected source,
+certificate, supplier or frontier record.
+
+## Open next questions
+
+The next useful question is whether nonuniform B/A call prices preserve a net
+saving over separate row stars for this fresh-source Johnson-path family. That
+requires assigning actual recursive child prices before claiming any effect
+on the selected bound.