Research: one-repeat partitioned (2,6) 8q priced obstruction (no new kappa)

research/independent-walks-onerepeat-8q/CERTIFICATE.json

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+{
+  "schema_version": 1,
+  "status": "PASS_INDEPENDENT_WALKS_ONEREPEAT_8Q_OBSTRUCTION",
+  "scope": "Five-source common-B mutable-source Q/R/X alphabet (PR1898) at (b,a,x)=(38,2,2). Freestanding one-repeat layer inside partitioned independent dual walks at total Q=8 on the (2,6) Hamilton champion shape: every covering one-repeat length-6 long walk paired with short (0,\u03c3) has exact R=112; tariff floor at X=0 is 528>436. Answers independent-walks-8q OQ1 without editing leave-alones through #2846 or pending iw8q.",
+  "claim": {
+    "Q": 8,
+    "partition": [
+      2,
+      6
+    ],
+    "one_repeat_covering": 59996160,
+    "one_repeat_min_R": 112,
+    "valid_one_repeat_walks": 4530960,
+    "rank4_one_repeat_walks": 3749760,
+    "valid_rank_lt4_walks": 781200,
+    "covering_rank_lt4": 0,
+    "distinct_partition_26_min_R": 92,
+    "tariff_floor_at_X0": 528,
+    "endpoint_Q6_tariff": 436,
+    "undercuts_436_after_complete_bill": false,
+    "beats_distinct_92": false,
+    "answers_independent_walks_8q_oq1_onerepeat": true,
+    "oq_answer": "Every covering one-repeat partitioned dual on shape (2,6) (short walk (0,\u03c3); long walk of length 6 whose five nonzero slots contain exactly one repeated nonzero label; independent dirt; dual even-parity readout) has R=112 exactly (covering 59996160; spectrum {112}). Only rank-4 longs cover, always with \u03a3e=40 and \u03c3 outside the long span (16 choices), so R=2\u00b740+32=112 by the rank-4-outside identity; all 781200 rank<4 valid longs are admitted with covering_rank_lt4=0. Distinct-label (2,6) floor R=92 is not improved. At X=0 the abstract tariff is already 528>436 under (b,a,x)=(38,2,2). Leave-alones through #2846 and pending independent-walks-8q are not reopened.",
+    "no_new_kappa": true
+  },
+  "one_repeat_2_plus_6": {
+    "kind": "one_repeat_partition_2_plus_6",
+    "Q": 8,
+    "valid_one_repeat_walks": 4530960,
+    "rank4_one_repeat_walks": 3749760,
+    "valid_rank_lt4_walks": 781200,
+    "covering_rank_lt4": 0,
+    "covering_pairs": 59996160,
+    "attainable_R": [
+      112
+    ],
+    "histogram": {
+      "112": 59996160
+    },
+    "min_R": 112,
+    "sum_even_rank4": 40,
+    "sigmas_outside_rank4": 16,
+    "first_witness": {
+      "112": [
+        [
+          0,
+          16
+        ],
+        [
+          0,
+          4,
+          1,
+          8,
+          2,
+          1
+        ]
+      ]
+    }
+  },
+  "champion": {
+    "champion": {
+      "walks": [
+        [
+          0,
+          16
+        ],
+        [
+          0,
+          4,
+          1,
+          8,
+          2,
+          1
+        ]
+      ],
+      "R": 112,
+      "path_X": 8,
+      "tariff": 544,
+      "structural_rank4_outside": {
+        "applies": true,
+        "rank": 4,
+        "sigma_in_span": false,
+        "R": 112,
+        "sum_even": 40
+      },
+      "one_repeat_long": true
+    },
+    "tariff_floor_at_X0": 528,
+    "endpoint_Q6": 436,
+    "distinct_partition_26_min_R": 92,
+    "one_repeat_min_R": 112
+  },
+  "unit_tariff": {
+    "b": 38,
+    "a": 2,
+    "x": 2
+  },
+  "tariff_check": {
+    "formula": "38*Q + 2*R + 2*X",
+    "at_R112_X0": 528,
+    "champion": 544,
+    "endpoint_Q6": 436,
+    "distinct_26_floor_at_R92_X0": 488
+  }
+}

research/independent-walks-onerepeat-8q/NOTICE.md

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+# 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 `(b,a,x)=(38,2,2)`.
+- 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 at Q=8; not edited.
+- Open #2779 `research/independent-walks-7q-obstruction` — partitioned dual at
+  Q=7; not edited.
+- Open #2846 `research/transfer-coupled-onerepeat-7q` — transfer-coupled
+  one-repeat at Q=7; leave-alone; not edited.
+- Open #2840 / #2836 / #2817 / #2764 / #2761 / #2715 / #2687 / #2686 / #2643 /
+  #2593 / #2487 — leave-alone through #2846; not edited.
+- Pending `cursor/independent-walks-8q-60b9@379aaf8f`
+  `research/independent-walks-8q-obstruction` — distinct partitioned dual at
+  Q=8; OQ1 (one-repeat on (2,6)) answered here; not edited or competed.
+- Live pin: `research/mutable-source-shear-cost/CERTIFICATE.json`.
+
+Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI
+assistance.

research/independent-walks-onerepeat-8q/PROOF.md

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+# One-repeat partitioned (2,6) cannot improve on R=92 at Q=8
+
+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) at unit
+tariff `(b,a,x)=(38,2,2)`. It is a freestanding `#840`-style priced alphabet at
+eight queries: the **one-repeat** companion of the partitioned independent dual
+on shape `(2,6)`. It answers open question 1 of pending
+`cursor/independent-walks-8q-60b9`: whether allowing one repeated nonzero label
+in the length-6 walk improves on distinct-label `R=92`.
+
+No new kappa is claimed. Leave-alone packages through #2846 are not edited.
+Pending independent-walks-8q is not edited or competed.
+
+## Setup
+
+Retain queries `Q_j`, readouts `R_m: y_m += A_m z`, and paid source shears
+`X_(s,t)`. A **restored walk** of length `n≥2` is an open slot list starting at
+`0` whose closed completion has only nonzero edges.
+
+A **one-repeat (2,6) dual** is a pair
+
+- short walk `(0,σ)` with `σ∈{1,…,31}`,
+- long walk `(0,ℓ_1,…,ℓ_5)` whose five nonzero slots contain **exactly one**
+  repeated nonzero label (multiset shape `{d:2,o₁,o₂,o₃}`),
+
+with independent dirt and dual even-parity cost
+
+```text
+R = Σ_{m=1}^{31} min( e_long[m],  2 + e_long[m ⊕ σ] ).
+```
+
+The abstract complete-call tariff at `(b,a,x)=(38,2,2)` is `38·Q + 2·R + 2·X`.
+
+## Length-1 exclusion
+
+A length-1 list `(0,)` closes to `(0,0)`. That edge has XOR `0`, so it is not a
+restored walk. The short component of `(2,6)` therefore cannot itself host a
+one-repeat layer.
+
+## Structural identity
+
+**Lemma (Σe = 40 on rank-4 one-repeat longs).** Every valid restored length-6
+walk whose nonzero multiset has shape `{d:2,o₁,o₂,o₃}` and whose four distinct
+nonzero labels span a rank-4 subspace has
+
+```text
+Σ_{v ∈ V \ {0}} e_long[v] = 40,
+```
+
+where `V` is the long span. Machine-checked on all `3749760` such walks.
+
+**Lemma (rank-4 outside).** When the long span has rank `4` and `σ ∉ V`,
+
+```text
+R = 2 · Σ_{v ∈ V \ {0}} e_long[v] + 32 = 2·40 + 32 = 112.
+```
+
+**Lemma (inside span never covers).** If `σ ∈ V`, the dual cost returns `None`
+on every one-repeat rank-4 long (machine-checked: zero inside-span covers).
+
+**Lemma (rank-3 never covers).** If the four distinct nonzero labels have rank
+`<4`, no `σ` yields a covering dual. Admitted in `model.py` /
+`independent.py` census: every valid rank`<4` long is dual-scanned for all
+`σ`, with `valid_rank_lt4_walks=781200` and `covering_rank_lt4=0`.
+
+## Census
+
+Valid one-repeat length-6 walks: `4530960`, of which rank-4: `3749760`.
+Outside-span short labels per rank-4 long: `31 − 15 = 16`. Covering pairs:
+
+```text
+3749760 · 16 = 59996160.
+```
+
+Attainable spectrum: `{112}` with histogram `{112: 59996160}`. Minimum `R=112`.
+
+Champion path `((0,16),(0,4,1,8,2,1))` has path `X=8`, tariff `544`. The
+`X=0` figure `528` is only the abstract lower bound from `R=112`.
+
+## Tariff consequence
+
+Every covering one-repeat `(2,6)` dual has `R=112`. Even at `X=0`,
+
+```text
+38·8 + 2·112 + 2·0 = 528 > 436.
+```
+
+Positive path-`X` only increases the bill. Distinct-label `(2,6)` floor `R=92`
+(pending iw8q) is strictly better than one-repeat `R=112`, so one-repeat does
+not reopen an undercut of the six-query endpoint under the PR1898 abstract
+tariff.
+
+## Scope
+
+This is a priced-alphabet obstruction, not a multiplication theorem and not a
+new kappa. Leave-alone packages through #2846 and pending
+`cursor/independent-walks-8q-60b9@379aaf8f` are not edited. Deferred
+soft-overlaps (transfer-coupled-dual/shapes; shared-xor; host screens) are not
+opened.
+
+Prepared by Wentao Li with substantial AI assistance.

research/independent-walks-onerepeat-8q/README.md

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+# One-repeat partitioned independent-walk obstruction at eight queries
+
+**Scoped obstruction / priced alphabet; no new kappa.** Freestanding
+`#840`-style priced obstruction under
+[#1898](https://github.com/CrocSwap/integer-mult-bounds/pull/1898) at
+`(b,a,x)=(38,2,2)`. Names the one-repeat companion of the partitioned
+`(2,6)` Hamilton layer at `Q=8` outside leave-alones through #2846 and
+pending `cursor/independent-walks-8q-60b9`. Answers open question 1 of that
+pending package: do non-distinct (one-repeat) label layers inside a
+partitioned `Q=8` walk improve on distinct-label `R=92` at `(2,6)`?
+
+| Dual schedule | Covering pairs | Exact min R | Tariff floor (X≥0) | Undercuts 436? | Beats distinct 92? |
+|---|---:|---:|---:|---|---|
+| Distinct-label `(2,6)` (iw8q) | 3916416 | **92** | ≥488 | no | — |
+| One-repeat long `(2,6)` (this) | **59996160** | **112** | **≥528** | **no** | **no** |
+
+Every covering one-repeat dual has exact `R=112`. Only rank-4 longs cover,
+always with `Σe=40` and `σ` outside the long span (16 choices), so
+`R=2·40+32=112`. Abstract tariff `38·8 + 2·112 + 2·0 = 528 > 436`. Distinct-
+label floor `92` is not improved. Transfer-coupled / onerepeat / dirty-shadow
+/ iw8q fixtures are not edited.
+
+## Reproduction
+
+```sh
+python3.12 -I -B research/independent-walks-onerepeat-8q/verify.py
+python3.12 -I -O -B research/independent-walks-onerepeat-8q/verify.py
+python3.12 -I -B research/independent-walks-onerepeat-8q/independent.py
+python3.12 -I -B research/independent-walks-onerepeat-8q/test_package.py
+python3.12 -B scripts/research_package.py --package research/independent-walks-onerepeat-8q
+```
+
+Expected:
+
+```text
+PASS_INDEPENDENT_WALKS_ONEREPEAT_8Q_OBSTRUCTION one_repeat_min_R=112 ... tariff_floor=528 ... answers_independent_walks_8q_oq1_onerepeat no_new_kappa
+PASS_INDEPENDENT_WALKS_ONEREPEAT_8Q covering=59996160 ... one_repeat_min_R=112 tariff_floor=528 ... no_new_kappa
+PASS_INDEPENDENT_WALKS_ONEREPEAT_8Q_PACKAGE_TESTS groups=6
+PASS_REGISTERED_RESEARCH_PACKAGE research/independent-walks-onerepeat-8q outcome=completed
+```
+
+Default verify is read-only. Admission uses `need()`; `-O` also passes.
+
+## Overlap
+
+Base commit `0b1cccf86e2f5132ed286a26ef0a681a495740ea`. Direct predecessors:
+PR1898, pending iw8q (distinct partitioned dual at Q=8; OQ1 answered here),
+#2779 (partitioned walks at Q=7), #2846 (transfer-coupled onerepeat at Q=7).
+Leave-alone through #2846 including #2840, #2836, #2817, #2779, #2764, #2761,
+#2715, #2687, #2686, #2677, #2643, #2593, #2487 is respected; those packages
+are not edited. Pending `cursor/independent-walks-8q-60b9@379aaf8f` is not
+edited or competed. Deferred soft-overlaps (transfer-coupled-dual/shapes;
+shared-xor; host screens) are not opened.
+
+## Open next questions
+
+1. Do one-repeat layers on partitions `(3,5)` or `(4,4)` improve on their
+   distinct-label floors (`98` / `100`)?
+   Why it matters: closes the remaining one-repeat companions at eight queries.
+   First step: allow one repeated nonzero label in either walk of `(3,5)`.
+   Blocked by: larger search; expected to worsen like `(2,6)`.
+2. Does any third priced readout alphabet under PR1898 `(b,a,x)=(38,2,2)`
+   (beyond same-walk / dirty-shadow / partitioned dual / one-repeat) undercut
+   `436` at `Q=8`?
+   Why it matters: remaining unequal-price handoff after this freestanding
+   alphabet.
+   First step: name one new alphabet outside leave-alones and census dirt-legal R.
+   Blocked by: leave-alone packages through #2846; pending iw8q.
+
+## Credit
+
+Credit **Arham Tahir / PR1898**, **ywz65 / PR2455**, **PR1370**, fork **789q** /
+**#2677**, **#2643**, **#2686**, **#2715**, **#2761**, **#2779**, **#2846**,
+pending **independent-walks-8q** (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.

research/independent-walks-onerepeat-8q/SOURCES.json

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