Mister-Meeseeks · Oct 16, 2026 · Oct 16, 2026
diff --git a/research/transfer-coupled-onerepeat-7q/.gitignore b/research/transfer-coupled-onerepeat-7q/.gitignore
@@ -0,0 +1,2 @@
+dual.so
+__pycache__/
diff --git a/research/transfer-coupled-onerepeat-7q/CERTIFICATE.json b/research/transfer-coupled-onerepeat-7q/CERTIFICATE.json
@@ -0,0 +1,158 @@
+{
+  "schema_version": 1,
+  "status": "PASS_TRANSFER_COUPLED_ONEREPEAT_7Q_OBSTRUCTION",
+  "scope": "Five-source common-B mutable-source Q/R/X alphabet (PR1898). Transfer-coupled dual at shape (5,1,1) with exactly one repeated nonzero label among the six nonzero schedule slots (prefix-internal double, or one suffix colliding with a distinct prefix), divergent suffixes, transfer C>=1, independent dirt, dual even-parity readout. Answers #2836 OQ1 (non-distinct / one-repeat layer).",
+  "claim": {
+    "Q": 7,
+    "shape": [
+      5,
+      1,
+      1
+    ],
+    "transfer_C_min": 1,
+    "distinct_label_min_R": 84,
+    "one_repeat_min_R": 96,
+    "prefix_dup_min_R": 98,
+    "suffix_hit_min_R": 96,
+    "combined_covering": 4166400,
+    "tariff_floor_after_complete_bill": 459,
+    "endpoint_Q6_tariff": 436,
+    "undercuts_436_after_complete_bill": false,
+    "improves_on_distinct_R84": false,
+    "answers_all_shapes_oq1_onerepeat": true,
+    "oq_answer": "Every covering one-repeat transfer-coupled (5,1,1) dual has R>=96 (suffix-hit layer; all 1666560 covering pairs at R=96) or R=98 (prefix-dup layer; all 2499840 covering pairs). Combined min R=96 > distinct-label min R=84. Abstract tariff at X=0,C=1 is 38*7+2*96+1=459>436. One-repeat layers do not improve on the distinct-label floor and cannot undercut the six-query endpoint.",
+    "no_new_kappa": true
+  },
+  "prefix_dup": {
+    "covering_pairs": 2499840,
+    "attainable_R": [
+      98
+    ],
+    "histogram": {
+      "98": 2499840
+    },
+    "min_R": 98,
+    "champion": {
+      "prefix": [
+        0,
+        1,
+        1,
+        2,
+        7
+      ],
+      "suf_a": [
+        8
+      ],
+      "suf_b": [
+        16
+      ],
+      "R": 98,
+      "walks": [
+        [
+          0,
+          1,
+          1,
+          2,
+          7,
+          8
+        ],
+        [
+          0,
+          1,
+          1,
+          2,
+          7,
+          16
+        ]
+      ]
+    }
+  },
+  "suffix_hit": {
+    "covering_pairs": 1666560,
+    "attainable_R": [
+      96
+    ],
+    "histogram": {
+      "96": 1666560
+    },
+    "min_R": 96,
+    "champion": {
+      "prefix": [
+        0,
+        1,
+        2,
+        4,
+        9
+      ],
+      "suf_a": [
+        1
+      ],
+      "suf_b": [
+        16
+      ],
+      "R": 96,
+      "walks": [
+        [
+          0,
+          1,
+          2,
+          4,
+          9,
+          1
+        ],
+        [
+          0,
+          1,
+          2,
+          4,
+          9,
+          16
+        ]
+      ]
+    }
+  },
+  "champion_checks": {
+    "champions": [
+      {
+        "kind": "prefix_dup",
+        "R": 98,
+        "prefix": [
+          0,
+          1,
+          1,
+          2,
+          7
+        ],
+        "suf_a": [
+          8
+        ],
+        "suf_b": [
+          16
+        ]
+      },
+      {
+        "kind": "suffix_hit",
+        "R": 96,
+        "prefix": [
+          0,
+          1,
+          2,
+          4,
+          9
+        ],
+        "suf_a": [
+          1
+        ],
+        "suf_b": [
+          16
+        ]
+      }
+    ],
+    "tariff_floor": 459
+  },
+  "tariff": {
+    "formula": "38*Q + 2*R + 2*X + C",
+    "at_min_R_X0_C1": 459,
+    "endpoint_Q6": 436
+  }
+}
diff --git a/research/transfer-coupled-onerepeat-7q/NOTICE.md b/research/transfer-coupled-onerepeat-7q/NOTICE.md
@@ -0,0 +1,36 @@
+# Notice
+
+New material in this directory is Apache-2.0.
+
+Predecessor credit (do not alter historical notices elsewhere):
+
+- Arham Tahir / PR1898 `research/mutable-source-minimum-queries` — five-source
+  alphabet, minimum-query endpoint, abstract complete-call tariff.
+- ywz65 / open PR2455 `research/mutable-source-six-query-xor` — component lemma
+  and restored interaction-graph census at six queries.
+- Arham Tahir / PR1370 — path-local shear lowering.
+- TennisBowling / PR1524 — equal-price theorem and unequal-price handoff.
+- Ross Richardson / PR943/934 — prefix framework.
+- Fork path-lowered `research/mutable-source-789q-invoice` / open #2677 —
+  distinct-slot spectrum and champion path; not edited.
+- Open #2643 `research/seven-query-xor-floor` — restored `X ≥ 6` at `Q = 7`;
+  not edited.
+- Open #2686 `research/seven-query-r-floor` — even-subset `R ≥ 82`; not edited.
+- Open #2715 `research/cheaper-readout-7q-obstruction` — all-subset /
+  dirty-parity repair; not edited.
+- Open #2761 `research/multi-connector-7q-obstruction` — same-walk
+  multi-connector / priced shadow; not edited.
+- Open #2764 `research/dirty-parity-8q-obstruction` — Q=8 dirty-parity; not
+  edited.
+- Open #2779 `research/independent-walks-7q-obstruction` — partitioned
+  independent walks; not edited.
+- Open #2817 `research/transfer-coupled-7q-obstruction` — maximal-share
+  `(5,1,1)` distinct-label; not edited.
+- Open #2836 `research/transfer-coupled-all-shapes-7q` — all-shapes distinct
+  census; OQ1 (one-repeat) answered here; not edited.
+- Pending `research/dirty-shadow-8q-obstruction` under skeptical review — not
+  edited or competed.
+- Live pin: `research/mutable-source-shear-cost/CERTIFICATE.json`.
+
+Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI
+assistance.
diff --git a/research/transfer-coupled-onerepeat-7q/PROOF.md b/research/transfer-coupled-onerepeat-7q/PROOF.md
@@ -0,0 +1,79 @@
+# One-repeat transfer-coupled (5,1,1) cannot improve on R=84
+
+This is a scoped obstruction in the five-source common-B mutable-source alphabet
+of [PR1898](https://github.com/CrocSwap/integer-mult-bounds/pull/1898). It answers
+open question 1 of [#2836](https://github.com/CrocSwap/integer-mult-bounds/pull/2836):
+after a transfer-coupled dual of shape `(5,1,1)` whose six nonzero schedule
+labels contain **exactly one repeated** nonzero value, can dual even-parity
+readout improve on the distinct-label floor `R=84`, or undercut tariff `436`?
+
+No new kappa is claimed. Leave-alone packages through #2836 are not edited.
+
+## Setup
+
+Retain the transfer-coupled `(5,1,1)` contract of #2836 / #2817: shared prefix of
+length 5 starting at `0`, divergent nonempty suffixes of length 1, transfer
+`C≥1`, independent dirt on the two walks, dual even-parity
+
+\[
+R=\sum_{m=1}^{31}\min_a\bigl(e_0[a]+e_1[a\oplus m]\bigr).
+\]
+
+Canonical prefixes sort the nonzero multiset (matching the distinct-label census
+representatives). Divergent suffixes are unordered with `sa<sb`.
+
+A schedule is **one-repeat** when the multiset of its six nonzero slots has
+exactly five distinct values (one label of multiplicity two). The two disjoint
+cases are:
+
+1. **Prefix-internal double.** Prefix nonzero multiset `{d,d,o₁,o₂}`; suffixes
+   fresh and divergent.
+2. **Suffix–prefix collision.** Prefix four distinct nonzero labels; exactly one
+   suffix equals a prefix label; the other suffix is fresh.
+
+(Identical suffixes `sa=sb` are excluded by divergence. Two distinct colliding
+suffixes would create two repeated labels.)
+
+The abstract complete-call tariff at `(b,a,x)=(38,2,2)` is
+`38·Q + 2·R + 2·X + C`.
+
+## Census
+
+Exact `min_a` dual (C helper, pure-Python cross-check):
+
+| Layer | Covering | Min R | Histogram |
+|---|---:|---:|---|
+| Prefix-internal double | 2499840 | **98** | `{98: 2499840}` |
+| Suffix–prefix collision | 1666560 | **96** | `{96: 1666560}` |
+| Combined | 4166400 | **96** | — |
+
+Every covering schedule in each layer attains a single R value. Combined minimum
+is `R=96 > 84`.
+
+Pinned champions recompute under both the C dual and an independent pure-Python
+dual:
+
+- Prefix-dup: `prefix=(0,1,1,2,7)`, `suf={8,16}`, `R=98`.
+- Suffix-hit: `prefix=(0,1,2,4,9)`, `suf={1,16}`, `R=96`.
+
+## Tariff consequence
+
+At the combined minimum `R=96`, even the most favorable path bill `X=0` with
+`C=1` gives
+
+\[
+38\cdot 7 + 2\cdot 96 + 2\cdot 0 + 1 = 459 > 436.
+\]
+
+Positive path-`X` only increases the invoice. One-repeat layers therefore cannot
+undercut the six-query endpoint, and they strictly worsen the distinct-label
+floor answered by #2836 / #2817.
+
+## Machine checks
+
+`verify.py` recomputes both layer censuses, champion duals, and the tariff floor.
+`independent.py` recomputes both censuses on a private dual path and rechecks
+champions in pure Python. Package tests cover corruption, source-pin drift,
+duplicate JSON keys, and `-O`/`need()` smoke.
+
+Prepared by Wentao Li with substantial AI assistance. Apache-2.0.
diff --git a/research/transfer-coupled-onerepeat-7q/README.md b/research/transfer-coupled-onerepeat-7q/README.md
@@ -0,0 +1,77 @@
+# One-repeat transfer-coupled (5,1,1) dual obstruction at seven queries
+
+**Scoped obstruction / priced alphabet; no new kappa.** Answers open question 1
+of [#2836](https://github.com/CrocSwap/integer-mult-bounds/pull/2836)
+(`research/transfer-coupled-all-shapes-7q`) on the five-source common-B
+mutable-source alphabet of
+[#1898](https://github.com/CrocSwap/integer-mult-bounds/pull/1898): do
+non-distinct (one-repeat) label layers inside a transfer-coupled `(5,1,1)`
+schedule improve on distinct-label `R=84`?
+
+| Layer | Covering pairs | Exact min R | Tariff at X=0,C=1 | Undercuts 436? | Beats distinct 84? |
+|---|---:|---:|---:|---|---|
+| Distinct-label (#2836 / #2817) | 8749440 | **84** | ≥439 (needs X≥2) | no | — |
+| Prefix-internal double | **2499840** | **98** | **463** | **no** | **no** |
+| Suffix–prefix collision | **1666560** | **96** | **459** | **no** | **no** |
+| Combined one-repeat | **4166400** | **96** | **459** | **no** | **no** |
+
+Every covering one-repeat dual has `R∈{96,98}`. Combined minimum is `R=96`,
+strictly worse than the distinct-label floor `84`. Abstract tariff
+`38·7+2·96+1=459>436` already at `X=0`.
+
+## Reproduction
+
+```sh
+python3.12 -I -B research/transfer-coupled-onerepeat-7q/verify.py
+python3.12 -I -O -B research/transfer-coupled-onerepeat-7q/verify.py
+python3.12 -I -B research/transfer-coupled-onerepeat-7q/independent.py
+python3.12 -I -B research/transfer-coupled-onerepeat-7q/test_package.py
+python3.12 -B scripts/research_package.py --package research/transfer-coupled-onerepeat-7q
+```
+
+Expected:
+
+```text
+PASS_TRANSFER_COUPLED_ONEREPEAT_7Q_OBSTRUCTION combined_min_R=96 ... tariff_floor=459 ... answers_all_shapes_oq1_onerepeat no_new_kappa
+PASS_INDEPENDENT_TRANSFER_COUPLED_ONEREPEAT_7Q ... tariff_floor=459 ... no_new_kappa
+PASS_TRANSFER_COUPLED_ONEREPEAT_7Q_PACKAGE_TESTS groups=6
+PASS_REGISTERED_RESEARCH_PACKAGE research/transfer-coupled-onerepeat-7q outcome=completed
+```
+
+Default verify is read-only. Admission uses `need()`; `-O` also passes.
+Requires a working `gcc` to build the local `dual.c` helper (pure-Python dual
+fallback remains available).
+
+## Overlap
+
+Base commit `0b1cccf86e2f5132ed286a26ef0a681a495740ea`. Direct predecessors:
+PR1898, #2817, #2836 (OQ1 answered here; package not edited). Leave-alone
+through #2836 including #2817, #2779, #2764, #2761, #2715, #2687, #2686, #2677,
+#2643, #2593, #2487 is respected. Pending `dirty-shadow-8q` under skeptical
+review is not edited or competed. Deferred shared-XOR soft-overlap and host
+screens are not opened.
+
+## Open next questions
+
+1. Does priced transfer `C>1` (or a dirt-correlated shared-prefix bill) change
+   the distinct `(5,1,1)` floor enough to reopen an undercut at `X=2`?
+   Why it matters: remaining sensitivity question from #2836 OQ2.
+   First step: reprice the distinct covering census with `C=2`.
+   Blocked by: model choice; leave-alone dirty-shadow fixtures not reopened.
+2. Does any other priced readout alphabet under PR1898 `(b,a,x)=(38,2,2)`
+   undercut `436` at `Q=8` after a complete dirty bill?
+   Why it matters: remaining unequal-price handoff at eight queries.
+   First step: name one new alphabet and census dirt-legal R against tariff 436.
+   Blocked by: leave-alone packages through #2836; pending dirty-shadow-8q.
+
+## Credit
+
+Credit **Arham Tahir / PR1898**, **ywz65 / PR2455**, **PR1370**, fork **789q** /
+**#2677**, **#2643**, **#2686**, **#2715**, **#2761**, **#2779**, **#2817**,
+open **#2836** (all-shapes; OQ1 answered here; not edited),
+**TennisBowling / PR1524**, **Ross Richardson / PR943**, and the retained
+mutable-source lineage. Live input pin:
+`research/mutable-source-shear-cost/CERTIFICATE.json`. Prepared by Wentao Li
+([VictorLiwentao](https://github.com/VictorLiwentao),
+[@Is__Victor](https://x.com/Is__Victor)) with substantial AI assistance.
+Apache-2.0. No independent human review or multiplication theorem is claimed.
diff --git a/research/transfer-coupled-onerepeat-7q/SOURCES.json b/research/transfer-coupled-onerepeat-7q/SOURCES.json
@@ -0,0 +1,24 @@
+{
+  "schema_version": 1,
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
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+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
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+      "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1",
+      "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075"
+    },
+    {
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+}