Research: atomic triple-sum connector does not beat the universal budget (no new kappa)

research/atomic-trio-connector/README.md

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+# Atomic triple-sum connector
+
+**Scoped negative result; no new kappa.** One new complete call
+`TRIO_{j,k,l}: z += B_j x_j + B_k x_k + B_l x_l` does not improve the charged
+universal all-nonzero-row budget on three or five sources. The symbol is
+atomic: it contributes one prefix step of weight three, three source-map
+applications and one root frame. It does not expose the two partial sums.
+
+This refines the connector-alphabet card. The fixed-alphabet premise that
+changes is the ban on combined source queries in
+[the universal connector boundary](../universal-connector-boundary/README.md).
+Weight-2 DUO steps, a second connector, mutable sources and a larger source
+count are not part of this result.
+
+```sh
+python3 -B research/atomic-trio-connector/verify.py
+```
+
+Expected:
+
+```text
+PASS_ATOMIC_TRIO_CONNECTOR three_source_orders=24 misses={3: 8, 5: 8, 6: 8} five_source_words=6048000 full_covers=0 best_coverage=30
+```
+
+Python 3.10+ and the standard library suffice. The registered replay is the
+same command through `scripts/research_package.py`.
+
+## Interface and price
+
+The predecessor interface is unchanged except for the new symbol. Rows `x_j`,
+`y_i` and the connector `z` start arbitrarily dirty. Maps `A_i` and `B_j` are
+exact and need not commute. Sources stay read-only. A schedule may use the old
+calls `B_j: z += B_j x_j` and `A_i: y_i += A_i z`, plus any number of atomic
+triple sums on distinct sources. Paired `A` readouts still see only the
+boundaries between B-side calls. Each nonzero output row therefore still needs
+two boundaries whose prefixes differ by its mask, and every source must appear
+an even number of times for `z` to return to its initial value.
+
+Let `λ_j > 0` be the price of one application of `B_j`, and let `f ≥ 0` be the
+price of one B-side root frame: dispatch, private-scratch restoration, head
+return and entry/exit. A single toggle costs `λ_j + f`. A triple sum costs
+`λ_j + λ_k + λ_l + f`. Charging it as one frame without the three applications,
+or as a free fusion of an arbitrary old word, is a different contract and is
+not used here.
+
+## Why fewer frames cannot cover
+
+A closed B-side word of `k` calls has at most `k` distinct prefixes. Those
+prefixes determine at most `k(k-1)/2` nonzero differences. Covering every
+nonzero mask of `n` sources therefore needs `k(k-1)/2 ≥ 2^n - 1`. Restoration
+makes `k` even once any number `t ≥ 1` of triple sums is included, because the
+application count is `A = k + 2t`.
+
+| Batch | Known repair | Counting | Only shape that could tie that repair | Outcome |
+|---|---|---|---|---|
+| 3 sources, 7 masks | 6 toggles, 6 applications | `k ≥ 6` and `A ≥ 8` | none | strictly more applications |
+| 5 sources, 31 masks | 12 toggles, 12 applications | `k ≥ 10` and, to keep `A ≤ 12`, only `t = 1`, `k = 10`, `A = 12` | 6,048,000 words | none covers all 32 masks |
+
+The three-source repair is the predecessor's four-source word with the idle
+source deleted: `B0 B1 B0 B1 B2 B2`. The five-source repair is the predecessor's
+twelve-toggle word, which can repeat whichever source is cheapest. Against
+either repair, every positive price vector leaves a gap of at least
+`2 min λ_j` on the source side and no frame saving.
+
+The four-call three-source words are not covering candidates: four prefixes
+give at most six nonzero differences. They do meet that ceiling. All 24 orders
+of `{e0+e1+e2, e0, e1, e2}` have four distinct prefixes and miss exactly one
+weight-2 mask, eight times each for masks 3, 5 and 6. The word `TRIO, B0, B1, B2`
+has prefixes `0, 7, 6, 4` and misses 5. The missing mask is the sum of the four
+prefixes in every order.
+
+## Five-source census
+
+Appearances totaling 12, each even and at least 2, must be `4,2,2,2,2`. With
+one triple sum, the source of multiplicity 4 may lie inside or outside the
+triple. There are 30 inside-heavy patterns and 20 outside-heavy patterns.
+
+```text
+30 * 10! / (3! 2! 2!) + 20 * 10! / (4! 2!) = 6048000
+```
+
+Two generators enumerate every word: multiset permutations with the triple sum
+inserted in each slot, and direct placement by index combinations. They agree
+on the coverage histogram
+
+| Masks covered | 16 | 19 | 21 | 22 | 23 | 24 | 25 | 26 | 28 | 29 | 30 |
+|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
+| Words | 140400 | 878400 | 1162800 | 208800 | 993600 | 1022400 | 399600 | 658800 | 396000 | 57600 | 129600 |
+
+The largest coverage is 30 of 32. No word covers 32. A listed control,
+`TRIO_{0,1,2} B0 B0 B0 B1 B2 B3 B3 B4 B4`, is restored and misses at least one
+mask; no paired readout emits that mask from the all-ones input.
+
+The predecessor controls are also checked, not re-proved: the ten-toggle word
+misses only mask 21, and the twelve-toggle word covers every mask. Identity
+schedules for the three-source repair and the twelve-toggle repair are replayed
+on basis and mixed inputs, with a dirty connector, forward and backward. The
+three-source repair is additionally replayed at length 2 with three
+noncommuting source maps. Dropping the mask-7 row is a failure control.
+
+## Open next questions
+
+Could a different single instruction, whose step is not an atomic weight-3 sum
+and not a weight-2 DUO, cover all 32 five-source masks in ten B-side calls
+while paying at most twelve source applications? That is the only remaining
+counting window in which a new B-side step could tie the twelve-toggle leaf
+bill and still reduce the frame count. The first step is to fix the step subset
+and the even-appearance equation before searching. Weight-2 DUO is discussion
+#1403. Two-connector transfers are #1412, #1486 and PR #1438; #1486 has since
+reported that four paid transfers do not cover these masks, and this package
+does not continue that search.
+An interruptible triple that exposes its partial sums is the old toggle
+alphabet inside one frame, which this price model does not treat as free.
+
+## Credit and limits
+
+The predecessor is Ross Richardson's universal connector boundary, with the
+PR934 binary-query contract. This package adds the atomic triple-sum price and
+the three-source and five-source exclusions above. It does not claim priority
+for that boundary, for Navinor's DUO proposal, or for the two-connector
+transfer results.
+
+No selected supplier, certificate, review ledger or kappa changes. Common-label
+realization, frames, hosting, precision, tape and the analytic hypotheses of the
+selected construction remain open. The census is a finite exclusion of one
+charged symbol on two batch sizes, not a proof-assistant theorem and not a
+lower bound for every other connector alphabet.
+
+Prepared by Riquelme Soares (@hevlyo) with substantial Cursor agent assistance.
+New work is Apache-2.0. No independent human peer review is asserted.

research/atomic-trio-connector/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "c05fafaa25ba95b2d5af86b1800427198e3d20eb",
+      "sha256": "89b5928bc92c81e2647b6d702a7ad85a6795b8f44e2eb267b2ff3145d8f2b2ce"
+    },
+    {
+      "path": "research/universal-connector-boundary/README.md",
+      "usage": "input",
+      "git_blob": "d277f7c5830b1f2bc42e1ff41ab86e7d7564929f",
+      "sha256": "23415f7829ab8d80a783142f10f7d31abbce84579362769984680af3df954476"
+    }
+  ]
+}

research/atomic-trio-connector/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "connector-alphabet-change",
+  "priority_revision": 2,
+  "base_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6",
+  "original_card": {
+    "id": "connector-alphabet-change",
+    "question": "Does one explicit extension of the connector alphabet offer a favorable completed budget?",
+    "starting_points": [
+      "research/universal-connector-boundary/README.md",
+      "certificates/selected-result.json"
+    ],
+    "methods": [
+      "Construct and charge a legal extra operation on a minimal instance.",
+      "Prove a target-specific bound after adding one specified operation family."
+    ],
+    "acceptance_gate": "State which fixed-alphabet premise changes and charge its implementation; do not declare it free.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/universal-connector-boundary/verify.py"
+      ],
+      "expected": "PASS_UNIVERSAL_CONNECTOR_BOUNDARY",
+      "inspect": "Reproduce the 945-pairing control, then read Exact interface to list the allowed calls.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Add exactly one legally priced operation and test a tiny dirty-input instance before enumerating larger cases.",
+    "bottleneck_link": "A new legal connector operation could change recursive interface cost; a larger census in the same alphabet is a separate side result.",
+    "allowed_variations": [
+      "Choose one additional operation and a small source count for its first counterexample."
+    ],
+    "scope_change_triggers": [
+      "Keeping the alphabet and only increasing the source count; record this as a changed assignment."
+    ],
+    "overlap_terms": [
+      "universal connector",
+      "source-toggle",
+      "connector alphabet"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Does one atomic triple-sum call, priced as three source-map applications plus one root frame, lower the completed universal all-nonzero-row budget on three sources or on five sources?",
+  "scope_change_reason": "The card allows one additional operation on a small source count. The operation is fixed as the atomic weight-3 sum TRIO_{j,k,l}, and the batches are the universal 7-mask and 31-mask targets. The old alphabet, weight-2 DUO steps, a second connector, mutable sources and a larger source count are excluded.",
+  "bottleneck_link": "A cheaper legal connector operation would lower recursive interface cost. This symbol does not: on the universal batches its fully charged source-side bill stays at least two cheapest applications above the known repairs, so it does not change the paid recursive profile.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the three-source counting bound and the five-source ten-call census are checked. Do not search longer TRIO words, six sources, internal partial-sum readouts, DUO steps or a second connector.",
+  "outcome": "negative-result",
+  "evidence": "verify.py: all 24 three-source four-call orders miss one weight-2 mask; both generators agree that 0 of 6048000 five-source ten-call words cover 32 masks (best 30); identity and length-2 matrix replays of the known repairs pass; dropping mask 7 fails closed.",
+  "author": "Riquelme Soares (@hevlyo)",
+  "ai_assistance": "Substantial Cursor agent assistance in the obstruction argument, the dual census, the verifier and the write-up. No proof assistant was used.",
+  "open_questions": [
+    {
+      "question": "Is there a single new B-side instruction, other than an atomic weight-3 sum or a weight-2 DUO, that covers all 32 five-source masks in ten calls while paying at most twelve source applications?",
+      "why_it_matters": "That is the only counting window in which a new B-side step could tie the twelve-toggle leaf bill and still reduce the frame count.",
+      "first_step": "Fix the step subset and the even-appearance equation, and reject the symbol if its application count already exceeds twelve before searching words.",
+      "blocked_by": "Discussion #1403 claims weight-2 DUO. Discussions #1412 and #1486 plus PR #1438 study two-connector transfers; #1486 reports that four paid transfers do not cover these masks, and this package does not continue that search. An interruptible triple is the old toggle alphabet inside one frame, which this price model does not treat as free."
+    }
+  ]
+}

research/atomic-trio-connector/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/universal-connector-boundary/README.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 300,
+      "checks": "Three-source weight-2 obstruction, five-source candidate count, dual census of all 6048000 ten-call words, known ten- and twelve-toggle controls, dirty identity and length-2 matrix replays, and the positive leaf-price gap."
+    }
+  ]
+}