Research: transfer-coupled (5,1,1) 7q readout obstruction (no new kappa)

research/transfer-coupled-7q-obstruction/CERTIFICATE.json

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+{
+  "schema_version": 1,
+  "status": "PASS_TRANSFER_COUPLED_7Q_OBSTRUCTION",
+  "scope": "Five-source common-B mutable-source Q/R/X alphabet (PR1898). Transfer-coupled dual at Q=7 with shape (p,a,b)=(5,1,1): shared prefix length 5, priced transfer C>=1, divergent length-1 tips, independent dirt, dual even-parity readout, fair shared-prefix XOR billing. Covering min R=84; min fair X at R=84 is 9; tariff floor 452>436. Answers transfer-coupled OQ for the maximal-sharing fork.",
+  "claim": {
+    "Q": 7,
+    "shape": [
+      5,
+      1,
+      1
+    ],
+    "covering_pairs": 8749440,
+    "min_R": 84,
+    "min_fair_X_at_R84": 9,
+    "tariff_floor": 452,
+    "endpoint_Q6_tariff": 436,
+    "undercuts_436_after_complete_bill": false,
+    "transfer_C_min": 1,
+    "answers_transfer_coupled_511": true,
+    "oq_answer": "Every covering distinct-label transfer-coupled schedule of shape (5,1,1) at total Q=7 (shared restored prefix of length 5, two divergent nonzero tips, priced transfer C>=1, independent dirt, dual even-parity readout, fair shared-prefix XOR billing) has R>=84. Among the 1458240 covering pairs with R=84, every fair-X bill is >=9 (witness X=9). Abstract tariff floor is therefore 38*7+2*84+2*9=452>436. C=0 is rejected. Other (p,a,b) shapes are out of full census here; spot samples show observed min R>=86, which already cannot undercut because R+X>=86>85.",
+    "no_new_kappa": true
+  },
+  "census_5_1_1": {
+    "kind": "transfer_coupled_5_1_1",
+    "Q": 7,
+    "shape": [
+      5,
+      1,
+      1
+    ],
+    "covering_pairs": 8749440,
+    "n_R84": 1458240,
+    "attainable_R": [
+      84,
+      88,
+      90
+    ],
+    "histogram": {
+      "84": 1458240,
+      "88": 2083200,
+      "90": 5208000
+    },
+    "min_R": 84,
+    "min_fair_X_at_R84": 9,
+    "champion": {
+      "prefix": [
+        0,
+        1,
+        3,
+        6,
+        12
+      ],
+      "tips": [
+        8,
+        16
+      ],
+      "R": 84,
+      "fair_X": 9,
+      "tariff": 452,
+      "transfer_C": 1
+    },
+    "tariff_floor": 452,
+    "endpoint_Q6": 436
+  },
+  "champion": {
+    "prefix": [
+      0,
+      1,
+      3,
+      6,
+      12
+    ],
+    "tips": [
+      8,
+      16
+    ],
+    "walks": [
+      [
+        0,
+        1,
+        3,
+        6,
+        12,
+        8
+      ],
+      [
+        0,
+        1,
+        3,
+        6,
+        12,
+        16
+      ]
+    ],
+    "R": 84,
+    "fair_X": 9,
+    "transfer_C": 1,
+    "tariff": 452
+  },
+  "transfer_C_controls": {
+    "C0_rejected_by_need": true,
+    "C1_fair_X": 9,
+    "transfer_C_min": 1
+  },
+  "other_shape_samples": {
+    "threshold_R_plus_X": 85,
+    "shape_4_1_2": {
+      "covering_samples": 3205,
+      "min_R_observed": 86,
+      "best": {
+        "P": [
+          7,
+          21,
+          30
+        ],
+        "A": [
+          19
+        ],
+        "B": [
+          16,
+          17
+        ],
+        "R": 86
+      },
+      "cannot_undercut_if_R_ge_86": true
+    },
+    "shape_3_2_2": {
+      "covering_samples": 3185,
+      "min_R_observed": 90,
+      "best": {
+        "P": [
+          14,
+          19
+        ],
+        "A": [
+          23,
+          31
+        ],
+        "B": [
+          18,
+          24
+        ],
+        "R": 90
+      },
+      "cannot_undercut_if_R_ge_86": true
+    },
+    "shape_3_1_3": {
+      "covering_samples": 3186,
+      "min_R_observed": 88,
+      "best": {
+        "P": [
+          18,
+          22
+        ],
+        "A": [
+          11
+        ],
+        "B": [
+          8,
+          9,
+          24
+        ],
+        "R": 88
+      },
+      "cannot_undercut_if_R_ge_86": true
+    },
+    "note": "Samples only; not a full other-shape census. Observed minima \u226586 already forbid undercutting at Q=7 because R+X\u226586>85."
+  },
+  "unit_tariff": {
+    "b": 38,
+    "a": 2,
+    "x": 2
+  },
+  "tariff_check": {
+    "formula": "38*Q + 2*R + 2*X  (C folded into fair X)",
+    "floor_R84_X9": 452,
+    "endpoint_Q6": 436,
+    "undercut_requires_R_plus_X_lt": 85
+  }
+}

research/transfer-coupled-7q-obstruction/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.
+- 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 #2687 — leave-alone; 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`; not edited.
+- Pending / open partitioned independent-walks package
+  `research/independent-walks-7q-obstruction` — transfer-coupled OQ answered
+  here for shape `(5,1,1)` without editing that package.
+- Live pin: `research/mutable-source-shear-cost/CERTIFICATE.json`.
+
+Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI
+assistance.

research/transfer-coupled-7q-obstruction/PROOF.md

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+# Transfer-coupled (5,1,1) duals cannot undercut tariff 436 at Q=7
+
+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
+the **transfer-coupled** open question left by partitioned independent walks:
+can a shared prefix, priced transfer `C≥1`, and divergent suffixes undercut the
+six-query endpoint tariff `436` at total `Q = 7` after a complete dirty bill?
+
+The claim is restricted to shape `(p,a,b) = (5,1,1)` (maximal sharing / longest
+walks). No new kappa is claimed. Leave-alone packages through #2764 / #2761 are
+not edited. Pending independent-walks / dirty-parity / multi-connector fixtures
+are not competed with on their own claims.
+
+## Setup
+
+Retain queries `Q_j`, readouts `R_m: y_m += A_m z`, and paid source shears
+`X_(s,t)`. A **transfer-coupled schedule** of shape `(p,a,b)` with
+`p+a+b = 7` consists of:
+
+1. a shared restored **prefix** of length `p` starting at `0`;
+2. a priced **transfer** of cost `C ≥ 1` forking the connector state;
+3. two divergent **suffixes** of lengths `a` and `b`.
+
+The two walks are `prefix∥suffix_a` and `prefix∥suffix_b`. Shared prefix queries
+are counted once, so the query bill is `Q = p+a+b`. Each walk carries independent
+dirt, so dual even-parity readout costs
+
+```text
+R = Σ_{m=1}^{31} min_a (e0[a] + e1[a ⊕ m]).
+```
+
+**Fair shared-prefix XOR billing** charges prefix edges once, adds the transfer
+`C`, then charges each tip chain `(prefix_end → tip → 0)` separately. The
+abstract complete-call tariff at `(b,a,x)=(38,2,2)` is `38·Q + 2·R + 2·X` with
+`C` folded into the fair-`X` bill.
+
+## Why `(5,1,1)`
+
+Up to swap, shapes with `p≥1` and `a,b≥1` are the fifteen triples listed in the
+independent-walks OQ. Shape `(5,1,1)` maximises the shared prefix and produces
+two length-6 walks at `Q=7` — the most threatening dual-row configuration after
+partitioned walks collapsed at `R=112`. Spot samples of `(4,1,2)`, `(3,2,2)` and
+`(3,1,3)` observe covering minima `R≥86`; already `R+X≥86>85` forbids undercutting
+`436` at `Q=7`, but those shapes are not fully censused here.
+
+## Undercut threshold
+
+At `Q=7`,
+
+```text
+38·7 + 2(R+X) < 436  ⟺  R+X < 85.
+```
+
+Any covering schedule with `R≥86` is therefore blocked even at `X=0`. The only
+remaining threat is `R≤84` with very small fair `X`.
+
+## Census of distinct-label `(5,1,1)`
+
+Prefix nonzero labels form a 4-set `P ⊂ {1,…,31}`; tips `a≠b` lie outside `P`.
+Even-subset tables depend only on the slot **set**, so the dual `R` census is
+over unordered tip pairs. Covering count:
+
+```text
+covering = 8 749 440,
+attainable R = {84, 88, 90},
+histogram: 84 ↦ 1 458 240,  88 ↦ 2 083 200,  90 ↦ 5 208 000.
+```
+
+In particular **min `R = 84`**. (Machine-checked dual `min_a`, not a copied
+constant; independent checker reproduces the same histogram.)
+
+## Fair-`X` floor on the `R=84` layer
+
+For each `R=84` system, the fair-`X` cost decomposes by path end-point:
+
+```text
+X = min_pend ( path_0→P→pend + C + edge(pend,a) + edge(pend,b)
+               + edge(a,0) + edge(b,0) ),
+```
+
+with `C=1`. Enumerating all `1 458 240` systems yields
+
+```text
+min fair X |_{R=84} = 9,
+```
+
+attained at prefix `(0,1,3,6,12)` with tips `(8,16)` (and the tip swap). No
+`R=84` schedule has fair `X≤8`. Consequently every covering `(5,1,1)` bill
+satisfies `R+X ≥ 84+9 = 93 > 85`.
+
+## Tariff consequence
+
+```text
+38·7 + 2·84 + 2·9 = 452 > 436.
+```
+
+Positive deviations of `R` or `X` only increase the bill. Transfer-coupled
+`(5,1,1)` duals therefore cannot undercut the six-query endpoint under the
+PR1898 abstract tariff after charging walks, transfer, and dirt.
+
+`C=0` is rejected by the priced-transfer interface (`need(C≥1)`).
+
+## Scope limits
+
+- Fixes transfer-coupled shape `(5,1,1)` with distinct nonzero labels.
+  Other `(p,a,b)` triples are out of full census (spot samples only).
+- Same-walk multi-connector / priced shadow remain #2761; dirty-parity at
+  `Q=8` remains #2764; partitioned independent walks remain their own package.
+- Shared non-path XOR schedules remain deferred (soft-overlap with #2643).
+- Host screens are deferred.
+- Leave-alone through #2764 / #2761 (including #2593, #2643, #2677, #2686,
+  #2687, #2715, #2761, #2764) is not edited.
+- No new kappa, physical supplier, or multiplication theorem.
+
+## Relation to predecessors
+
+| Prior | Already proves | This package adds |
+|---|---|---|
+| PR1898 | Min `Q=6`, endpoint tariff 436 | — |
+| #2643 | `X≥6` at single-walk `Q=7` | — |
+| #2686 | Even-subset `R≥82` on one walk | — |
+| #2715 | All-subset collapses under dirty-parity | — |
+| #2761 | Same-walk dual / dirty shadow stay at `R≥82` | — |
+| Independent-walks | Partitioned duals have exact `R=112` | transfer-coupled OQ |
+| **This** | — | **`(5,1,1)` transfer-coupled duals have min `R=84`, min fair `X=9` at that layer; tariff floor `452>436`** |