From b154350fbf1e7d49118851ab899a733727483289 Mon Sep 17 00:00:00 2001
From: Mihir Srivastava <mihir2k5@gmail.com>
Date: Thu, 22 Oct 2026 01:05:16 -0700
Subject: [PATCH] Certify all-length receiver-window costs by exact rank
 dominance

---
 .../README.md                                 |   252 +
 .../RESULT.json                               | 10742 ++++++++++++++++
 .../SOURCES.json                              |    60 +
 .../TASK.json                                 |   106 +
 .../VERIFY.json                               |    26 +
 .../verify.py                                 |   329 +
 6 files changed, 11515 insertions(+)
 create mode 100644 research/receiver-window-all-length-floor/README.md
 create mode 100644 research/receiver-window-all-length-floor/RESULT.json
 create mode 100644 research/receiver-window-all-length-floor/SOURCES.json
 create mode 100644 research/receiver-window-all-length-floor/TASK.json
 create mode 100644 research/receiver-window-all-length-floor/VERIFY.json
 create mode 100644 research/receiver-window-all-length-floor/verify.py

diff --git a/research/receiver-window-all-length-floor/README.md b/research/receiver-window-all-length-floor/README.md
new file mode 100644
index 0000000000..9b367f8b39
--- /dev/null
+++ b/research/receiver-window-all-length-floor/README.md
@@ -0,0 +1,252 @@
+# All-length cost floor for the 128 PR4470 receiver windows
+
+For each of the **128 adjacent receiver windows** extracted from the pinned
+PR4470 constructor, the incumbent minimizes the paid rank cost among **all
+finite shear words on the same three persistent fields**, with the same scalar
+endpoint and monotone fixed entrance/exit frames. There is **no length or
+rational-coefficient cap**. The conclusion holds for every price `r^p`,
+`0 < p < 1`, hence every saving `0 < C < 1` with `p = 1-C`.
+
+This is a computer-assisted, scoped cost-floor proof, not a new kappa,
+physical supplier, global optimum, or selected-frontier change. The finite part
+has **1,303 reachable states and 239 window/profile pairs**, checked with exact
+integers. All seven constructor inputs are SHA256-pinned at canonical main
+`f626db5c3a3d781d8f7695038776c621651f75e1`. Every boundary, incumbent and terminal
+rank partition is in [RESULT.json](RESULT.json). The written reduction below
+is essential: a finite calculation alone does not establish an all-length result.
+
+## Delta and coordination
+
+Clear1nsight's [#4546](https://github.com/CrocSwap/integer-mult-bounds/pull/4546)
+at `55b8d34c82fc4df6499c5037e156db6e990b96fe` tested words through four unit
+shears in twelve windows. Ryanson Jonathan's
+[#4567](https://github.com/CrocSwap/integer-mult-bounds/pull/4567) at
+`50fedd98cda709707e4291c542e20ead96bcc786` extends the census to all 128 windows,
+length six and bounded nonunit alphabets; its README explicitly asks for a
+length-independence proof. This package answers the **cost-floor** part with a
+different relaxation: directed potential source dependence and frame dimensions.
+It imports neither predecessor's search/checking code and does not assume their
+finite results. It reuses Cinna Davis's credited #4470 constructor for native data.
+Separate implementation does not mean independent human or formal review.
+
+The `independent-moment` card is refined from its older named suppliers to this
+live handoff. Closing an unbounded search direction in a fixed invoice model
+identifies which premises must change before further construction searches.
+[Discussion #4577](https://github.com/CrocSwap/integer-mult-bounds/discussions/4577)
+is the canonical investigation thread.
+
+**Concurrent work.** The submission refresh discovered
+[#4573](https://github.com/CrocSwap/integer-mult-bounds/discussions/4573) and
+[#4574](https://github.com/CrocSwap/integer-mult-bounds/discussions/4574), posted
+after the initial overlap snapshot, investigating the same all-length question.
+#4573 also proposes a dimension/temporal-contact relaxation; #4574 proposes
+mandatory-contact bounds. This package is an **independent certificate
+implementation and corroborating proof**, not a claim of priority or an
+exclusive new question. Its specific evidence is the complete integer
+capped-rank comparison for every concave price, with geometric and
+premise-removal controls. Coordination updates were posted in both threads.
+
+## Model and theorem
+
+Label the persistent fields `a,b,x` by `0,1,2`. Their arbitrary initial scalar
+values are independent formal variables. The required scalar endpoint is
+
+```
+a <- a + ca*x,   b <- b + cb*x,   x <- x,
+```
+
+where the source windows have `ca,cb` in `{+1,-1}`. A replacement is any finite
+word `y_i <- y_i + c*y_j`, `i != j`, with rational coefficients and that exact
+endpoint. Both operands of every gate have one common binary frame in `F2^9`.
+Each field's frames grow by inclusion from fixed `S_i` to fixed `E_i`. Every
+positive dimension increment, including the exit gap, is charged by its rank.
+No extra/owned field, frame decrease, free transport or changed boundary is
+allowed. Scalar-height charges are omitted in the replacement's favor; adding
+nonnegative charges cannot defeat the lower bound. The common moment
+normalization does not affect this local comparison.
+
+**Theorem.** In each extracted window, every replacement in this model costs
+at least the incumbent under `r^p` for every `0<p<1`. The incumbent has the
+required endpoint and a legal monotone path, so the minimum equals its cost.
+Windows are considered individually: no overlapping gains may be summed.
+
+### 1. A finite relaxation contains every legal word
+
+The state is `(d_0,d_1,d_2,M)`, with current dimensions and a three-bit mask.
+Initially `d_i=dim S_i` and `M=100` (binary). At a gate with destination `i`
+and source `j`, update
+
+```
+M' = M union {i if j is in M}.
+```
+
+Inductively, any field with a nonzero exact coefficient of initial `x` belongs
+to `M`: a shear cannot create it unless it was present in its destination or
+source. Cancellation can only make the mask overestimate dependence. The
+required endpoint has nonzero `x` coefficients in both receivers, so valid
+termination requires `M=111`. Ignoring the exact coefficients and the other
+initial variables enlarges the candidate set. There is no coefficient bound.
+
+The common frame `F` at a contact necessarily satisfies
+
+```
+S_i + S_j <= F <= E_i intersection E_j.
+```
+
+Exclude a pair if either start is outside the other exit. Otherwise enumerate
+every integer common dimension
+
+```
+max(d_i, d_j, dim(S_i+S_j)) <= h <= dim(E_i intersection E_j).
+```
+
+Set both dimensions to `h`, update `M`, and append positive ranks `h-d_i` and
+`h-d_j`. Actual intermediate subspaces and tighter chronological bounds are
+forgotten. Every legal common frame maps to such a transition; the converse
+is not asserted.
+
+At any state with `M=111`, termination appends every positive rank
+`dim E_i-d_i`. If an implementation splits a climb, aggregate increments
+between consecutive contacts on that field, or between its last contact and
+exit. For concave `f` with `f(0)=0`, concavity gives
+`f(u+v) <= f(u)+f(v)`. Thus aggregation can only reduce cost. Every actual
+word maps to a relaxed path with cost no greater than its actual bill.
+
+### 2. The enumeration has no hidden length cutoff
+
+Every dimension and dependency bit is nondecreasing. An edge that leaves the
+state unchanged adds no positive rank. Removing it preserves all future
+transitions and terminal options **in this relaxation**; this is not a claim
+that its shear can be deleted from the actual scalar word.
+
+Every other edge strictly increases
+`Phi=d_0+d_1+d_2+popcount(M)`. The reduced graph is therefore a DAG, with at
+most `sum_i(dim E_i-dim S_i)+2 <= 29` nontrivial edges on a path and at most
+`4*product_i(dim E_i-dim S_i+1)` states. The `x` bit always stays set.
+
+`relaxed_profiles` explores every allowed edge and unions its appended ranks
+with every suffix partition, also including every permitted terminal option.
+Induction on remaining `Phi` proves it enumerates all relaxed terminal
+partitions. Removed self-loops add no ranks. Every terminal partition has
+telescoping mass `sum_i(dim E_i-dim S_i)`, which the code checks explicitly.
+
+### 3. Integer dominance proves every concave-price comparison
+
+For a partition `v`, set `B_k(v)=sum_{r in v} min(r,k)` for `k=1,...,9`.
+For every terminal `v` and incumbent `u`, the replay checks
+
+```
+sum(u) = sum(v),     B_k(u) <= B_k(v) for k=1,...,9.
+```
+
+These inequalities are a discrete majorization certificate. For completeness,
+put `delta_k=f(k)-f(k-1)`. Concavity makes these deltas nonincreasing. For
+integer `0<=r<=9`, telescoping yields
+
+```
+f(r) = delta_9*r
+       + sum_{k=1}^8 (delta_k-delta_{k+1})*min(r,k).
+```
+
+The mass term cancels; every remaining coefficient is nonnegative. Hence
+`sum f(u)<=sum f(v)` for every concave `f` on these ranks with `f(0)=0`, in
+particular `r^p`, `0<p<1`. No floating point, logarithm, root enclosure or
+rounding is used. Combining the three steps proves the theorem.
+
+## Source binding and evidence
+
+The pinned `native_joint.construct()` runs in an isolated child Python with a
+fresh private bytecode cache. No unmerged Git object or predecessor verifier
+is needed. The new extractor scans the forward program with climb events
+omitted, retaining each persistent field's last frame. It selects consecutive
+ADDs in `B` or `cleanup_tail` with a common source, distinct receivers and unit
+coefficients. Each exit is its field's next scalar/warp entrance or final frame.
+The two original contacts and every exit gap determine the incumbent; their
+inclusions are checked exactly.
+
+| Evidence | Result |
+|---|---:|
+| Eligible windows | 128: 123 B, 5 cleanup_tail |
+| Reachable states, summed over windows | 1,303 |
+| Terminal window/profile pairs | 239 |
+| Incumbent admitted and dominating every profile | All 128 windows |
+| Profile-count distribution | 93x1, 9x2, 6x3, 2x4, 13x5, 4x7, 1x9 |
+| Exact unit-shear endpoint words through length 4 in a control fixture | 84 |
+| Legal `F2^2` common-frame paths for those words inside the relaxation | 2,868 |
+
+The small geometric fixture tests the abstraction on concrete paths; it does
+not prove its universal soundness. Three weakened-model controls expose strict
+false savings for every `0<p<1`:
+
+| Removed premise | Window | Incumbent | Falsely admitted |
+|---|---:|---|---|
+| Directed source-dependence requirement | 10557 | `(1,1,1)` | `(2,1)` |
+| Exit-intersection dimension cap | 11078 | `(1,1,1,1)` | `(2,2)` |
+| Terminal exit charges | 10554 | `(2)` | empty |
+
+The normal model excludes each false gain. Dimension mass or uncharged cleanup
+alone therefore cannot establish the floor.
+
+## Reproduce
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/receiver-window-all-length-floor
+python3.12 -I -O -B research/receiver-window-all-length-floor/verify.py
+```
+
+Both recompute and compare the complete `RESULT.json` receipt, without
+rewriting it. `--write` explicitly authors the receipt. Windows/CPython 3.12.0
+takes about three seconds. All acceptance checks remain active under `-O`.
+The PR records publication and export checks. Full supplier/native/physical
+and selected-frontier replays are outside scope. The historical baseline was
+attempted in the sparse checkout but stopped at a missing selected-package
+file; no baseline success is claimed.
+
+## Overlap and limitations
+
+Initial overlap covered 355 open PRs and 3,470 Discussion records with complete
+remote pagination around 2026-10-22 07:48–07:50 UTC. Local search was sparse,
+7/5,860 note paths; relevant main inputs and live #4546/#4567 material were read
+directly. This is not a novelty guarantee. The selected result remains
+`23/10000`. The central ledger labels #4470 partially reviewed; this result
+does not upgrade its admission status.
+
+At the submission refresh main remained `f626db5c3a3d781d8f7695038776c621651f75e1`.
+#4567 had advanced to `5ac594d1bc1e6d91f20ef8157f38bb9b29768bfd`, adding 88
+nonadjacent pairs and a whole-word single-contact retiming check. Those newer
+scopes are outside this package; none of its runtime inputs come from #4567.
+
+The theorem covers pinned **forward adjacent windows with fixed boundaries**.
+It does not assert equality of #4546's whole relaxed-profile set at every
+length, exact scalar realizability of every state path, or geometric
+attainability of every partition. These stronger statements are unnecessary
+for the lower bound. Extra/owned fields, nonmonotone frames, new operations,
+nonadjacent or jointly overlapping windows and changed producer/consumer
+contracts are outside scope. Native correctness, physical lowering, precision,
+stock, hosting, reverse admission and all-size multiplication remain separate.
+
+## Open next questions
+
+1. **Change a boundary premise before searching again.** Extra fields or
+   nonmonotone transports could escape the DAG. First name one actual window
+   and specify all entrance, exit and restoration costs of the changed model,
+   then derive its optimistic graph. The missing complete cost contract is
+   the blocker; unknown costs cannot be zero.
+2. **Independently review the all-length reduction.** Check the dependency
+   induction, state-edge deletion and concave aggregation against the exact
+   caller, then replay the integer certificate. Finite tests alone cannot
+   establish that reduction. No independent human or formal review is claimed.
+   No further length/coefficient census is justified inside the proved model.
+
+## Credit and license
+
+Credit Chris R / Clear1nsight (#4546's model and finite ceilings), Ryanson
+Jonathan (#4567's census and explicit all-length question), Cinna Davis
+(#4470's constructor), zxv (productive-helper and receiver-sharing work),
+Jacob Alan Sussman, and the native/Clifford/geometry/outer contributors credited
+in the source packages. Prepared for mihir-s-05 with substantial OpenAI Codex
+assistance. Apache-2.0. Predecessor packages and frozen evidence are unchanged.
+The concurrent mihir-s-05/Codex investigations #4573 and #4574 are acknowledged
+above; no independent human review is implied by their separate sessions.
diff --git a/research/receiver-window-all-length-floor/RESULT.json b/research/receiver-window-all-length-floor/RESULT.json
new file mode 100644
index 0000000000..3fec4b9b51
--- /dev/null
+++ b/research/receiver-window-all-length-floor/RESULT.json
@@ -0,0 +1,10742 @@
+{
+  "failure_controls": {
+    "drop_directed_dependency": {
+      "falsely_admitted": [
+        2,
+        1
+      ],
+      "incumbent": [
+        1,
+        1,
+        1
+      ],
+      "index": 10557
+    },
+    "drop_exit_intersection": {
+      "falsely_admitted": [
+        2,
+        2
+      ],
+      "incumbent": [
+        1,
+        1,
+        1,
+        1
+      ],
+      "index": 11078
+    },
+    "drop_terminal_charge": {
+      "falsely_admitted": [],
+      "incumbent": [
+        2
+      ],
+      "index": 10554
+    }
+  },
+  "geometric_control": {
+    "concave_basis_checks": 90,
+    "endpoint_words": 84,
+    "geometric_paths": 2868
+  },
+  "new_kappa": false,
+  "profile_count_histogram": {
+    "1": 93,
+    "2": 9,
+    "3": 6,
+    "4": 2,
+    "5": 13,
+    "7": 4,
+    "9": 1
+  },
+  "reachable_states": 1303,
+  "schema": "receiver-window-all-length-floor/v1",
+  "source_base": "f626db5c3a3d781d8f7695038776c621651f75e1",
+  "source_files_checked": 7,
+  "status": "PASS_ALL_LENGTH_RECEIVER_FLOOR",
+  "window_count": 128,
+  "window_profile_pairs": 239,
+  "windows": [
+    {
+      "coefficients": [
+        -1,
+        1
+      ],
+      "contact_bases": [
+        [],
+        []
+      ],
+      "exit_bases": [
+        [],
+        [
+          224,
+          30
+        ],
+        []
+      ],
+      "incumbent": [
+        2
+      ],
+      "index": 10554,
+      "phase": "B",
+      "reachable_states": 4,
+      "roles": [
+        162,
+        172,
+        513
+      ],
+      "start_bases": [
+        [],
+        [],
+        []
+      ],
+      "terminal_profiles": [
+        [
+          2
+        ]
+      ]
+    },
+    {
+      "coefficients": [
+        1,
+        1
+      ],
+      "contact_bases": [
+        [],
+        []
+      ],
+      "exit_bases": [
+        [
+          224,
+          30
+        ],
+        [],
+        []
+      ],
+      "incumbent": [
+        2
+      ],
+      "index": 10555,
+      "phase": "B",
+      "reachable_states": 4,
+      "roles": [
+        172,
+        196,
+        513
+      ],
+      "start_bases": [
+        [],
+        [],
+        []
+      ],
+      "terminal_profiles": [
+        [
+          2
+        ]
+      ]
+    },
+    {
+      "coefficients": [
+        1,
+        -1
+      ],
+      "contact_bases": [
+        [
+          30
+        ],
+        [
+          30
+        ]
+      ],
+      "exit_bases": [
+        [
+          416,
+          30
+        ],
+        [
+          30
+        ],
+        [
+          30
+        ]
+      ],
+      "incumbent": [
+        1,
+        1,
+        1
+      ],
+      "index": 10557,
+      "phase": "B",
+      "reachable_states": 9,
+      "roles": [
+        140,
+        144,
+        502
+      ],
+      "start_bases": [
+        [],
+        [],
+        [
+          30
+        ]
+      ],
+      "terminal_profiles": [
+        [
+          1,
+          1,
+          1
+        ]
+      ]
+    },
+    {
+      "coefficients": [
+        -1,
+        1
+      ],
+      "contact_bases": [
+        [
+          30
+        ],
+        [
+          30
+        ]
+      ],
+      "exit_bases": [
+        [
+          30
+        ],
+        [
+          30
+        ],
+        [
+          30
+        ]
+      ],
+      "incumbent": [
+        1,
+        1
+      ],
+      "index": 10558,
+      "phase": "B",
+      "reachable_states": 9,
+      "roles": [
+        144,
+        187,
+        502
+      ],
+      "start_bases": [
+        [],
+        [],
+        [
+          30
+        ]
+      ],
+      "terminal_profiles": [
+        [
+          1,
+          1
+        ]
+      ]
+    },
+    {
+      "coefficients": [
+        1,
+        1
+      ],
+      "contact_bases": [
+        [
+          51
+        ],
+        [
+          51
+        ]
+      ],
+      "exit_bases": [
+        [
+          51
+        ],
+        [
+          388,
+          51
+        ],
+        [
+          51
+        ]
+      ],
+      "incumbent": [
+        1,
+        1,
+        1
+      ],
+      "index": 10560,
+      "phase": "B",
+      "reachable_states": 9,
+      "roles": [
+        168,
+        150,
+        548
+      ],
+      "start_bases": [
+        [],
+        [],
+        [
+          51
+        ]
+      ],
+      "terminal_profiles": [
+        [
+          1,
+          1,
+          1
+        ]
+      ]
+    },
+    {
+      "coefficients": [
+        1,
+        -1
+      ],
+      "contact_bases": [
+        [
+          51
+        ],
+        [
+          51
+        ]
+      ],
+      "exit_bases": [
+        [
+          388,
+          51
+        ],
+        [
+          51
+        ],
+        [
+          51
+        ]
+      ],
+      "incumbent": [
+        1,
+        1,
+        1
+      ],
+      "index": 10561,
+      "phase": "B",
+      "reachable_states": 9,
+      "roles": [
+        150,
+        115,
+        548
+      ],
+      "start_bases": [
+        [],
+        [],
+        [
+          51
+        ]
+      ],
+      "terminal_profiles": [
+        [
+          1,
+          1,
+          1
+        ]
+      ]
+    },
+    {
+      "coefficients": [
+        1,
+        -1
+      ],
+      "contact_bases": [
+        [
+          51
+        ],
+        [
+          51
+        ]
+      ],
+      "exit_bases": [
+        [
+          196,
+          51
+        ],
+        [
+          51
+        ],
+        [
+          51
+        ]
+      ],
+      "incumbent": [
+        1,
+        1,
+        1
+      ],
+      "index": 10563,
+      "phase": "B",
+      "reachable_states": 9,
+      "roles": [
+        131,
+        117,
+        560
+      ],
+      "start_bases": [
+        [],
+        [],
+        [
+          51
+        ]
+      ],
+      "terminal_profiles": [
+        [
+          1,
+          1,
+          1
+        ]
+      ]
+    },
+    {
+      "coefficients": [
+        -1,
+        -1
+      ],
+      "contact_bases": [
+        [
+          51
+        ],
+        [
+          51
+        ]
+      ],
+      "exit_bases": [
+        [
+          51
+        ],
+        [
+          51
+        ],
+        [
+          51
+        ]
+      ],
+      "incumbent": [
+        1
+      ],
+      "index": 10564,
+      "phase": "B",
+      "reachable_states": 6,
+      "roles": [
+        117,
+        584,
+        560
+      ],
+      "start_bases": [
+        [],
+        [
+          51
+        ],
+        [
+          51
+        ]
+      ],
+      "terminal_profiles": [
+        [
+          1
+        ]
+      ]
+    },
+    {
+      "coefficients": [
+        -1,
+        -1
+      ],
+      "contact_bases": [
+        [],
+        []
+      ],
+      "exit_bases": [
+        [
+          30
+        ],
+        [
+          491,
+          30
+        ],
+        [
+          459,
+          38,
+          30
+        ]
+      ],
+      "incumbent": [
+        3,
+        2,
+        1
+      ],
+      "index": 10575,
+      "phase": "B",
+      "reachable_states": 20,
+      "roles": [
+        205,
+        170,
+        452
+      ],
+      "start_bases": [
+        [],
+        [],
+        []
+      ],
+      "terminal_profiles": [
+        [
+          2,
+          1,
+          1,
+          1,
+          1
+        ],
+        [
+          2,
+          2,
+          1,
+          1
+        ],
+        [
+          3,
+          1,
+          1,
+          1
+        ],
+        [
+          3,
+          2,
+          1
+        ]
+      ]
+    },
+    {
+      "coefficients": [
+        -1,
+        -1
+      ],
+      "contact_bases": [
+        [
+          30
+        ],
+        [
+          30
+        ]
+      ],
+      "exit_bases": [
+        [
+          30
+        ],
+        [
+          30
+        ],
+        [
+          448,
+          21,
+          11
+        ]
+      ],
+      "incumbent": [
+        2,
+        1,
+        1
+      ],
+      "index": 10577,
+      "phase": "B",
+      "reachable_states": 9,
+      "roles": [
+        118,
+        194,
+        453
+      ],
+      "start_bases": [
+        [],
+        [],
+        [
+          30
+        ]
+      ],
+      "terminal_profiles": [
+        [
+          2,
+          1,
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diff --git a/research/receiver-window-all-length-floor/SOURCES.json b/research/receiver-window-all-length-floor/SOURCES.json
new file mode 100644
index 0000000000..70a88c8c48
--- /dev/null
+++ b/research/receiver-window-all-length-floor/SOURCES.json
@@ -0,0 +1,60 @@
+{
+  "schema_version": 1,
+  "base_commit": "f626db5c3a3d781d8f7695038776c621651f75e1",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ed33caef5c5827693e9572d252a66f99cbb73d40",
+      "sha256": "5732707cc1e460b090b53f2463162749ad860de878a18f7408938671d295a9d9"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65",
+      "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96"
+    },
+    {
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+      "usage": "input",
+      "git_blob": "45744fedd0e28967aa72586048b8bc0c72f419a9",
+      "sha256": "cde73b8ff5b3c0d06e69e2293b991b24fd14a16dd4ccab7ff6b3439074a94624"
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+      "sha256": "9b55a2d3c3972148fae9d112e745d9244968166bd89bd900c938f429fc76ab29"
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+    },
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+      "path": "research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-CANONICAL.json.gz",
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diff --git a/research/receiver-window-all-length-floor/TASK.json b/research/receiver-window-all-length-floor/TASK.json
new file mode 100644
index 0000000000..f13d3c11f0
--- /dev/null
+++ b/research/receiver-window-all-length-floor/TASK.json
@@ -0,0 +1,106 @@
+{
+  "schema_version": 1,
+  "task_id": "independent-moment",
+  "priority_revision": 7,
+  "base_commit": "f626db5c3a3d781d8f7695038776c621651f75e1",
+  "original_card": {
+    "id": "independent-moment",
+    "question": "Does a new paid-profile or mixture-ceiling claim have independent source-bound coverage of its actual changed ledger?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/staged544-mixed-fourth-entry/README.md",
+      "research/staged-fourth-mixture-ceiling/README.md",
+      "research/r437-two-window-carrier/README.md"
+    ],
+    "methods": [
+      "Independently reconstruct one changed paid ledger or pooled dual with exact source and model binding.",
+      "Check a genuinely uncovered moment implication with a different enclosure method and semantic controls."
+    ],
+    "acceptance_gate": "Demonstrate independence, source binding, positive cases and semantic failure controls. Name the existing checks and the missing coverage. More digits or another equivalent evaluation of an unchanged root is not by itself new verification coverage.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Choose one uncovered implication in #4088 literal mixed fill, #4091 pooled dual/model binding or #4067 changed frame/host bill. #4091 pins an earlier #4088 head: validate word, capacity and stock equivalence before applying that ceiling to the later fill.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Independently reconstruct one changed ledger or pooled certificate with a positive case and a semantic failure control. Compare exact sources and name what existing checks miss. Keep this separate from supplier-boundary-audit (ordinary/outer dispatch) and uniform-transfer-audit (all-size proof). Stop after the chosen implication; more digits of an unchanged root are not additional mechanism coverage.",
+    "bottleneck_link": "Independent pricing prevents unsupported frontier claims and can reveal genuine accounting slack.",
+    "allowed_variations": [
+      "Reconstruct either a source ledger or a numeric enclosure with a different method."
+    ],
+    "scope_change_triggers": [
+      "Importing the primary function and calling its output independent verification.",
+      "Calling an aggregate histogram or source archive an independently accepted physical supplier."
+    ],
+    "overlap_terms": [
+      "independent moment",
+      "rank histogram",
+      "exact enclosure",
+      "2979",
+      "rank7 joint endframes",
+      "paired extraction",
+      "joint lowering",
+      "4088",
+      "4091",
+      "4067",
+      "staged544-mixed-fourth-entry",
+      "staged-fourth-mixture-ceiling",
+      "r437-two-window-carrier"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4088: mixed local/owner/phase ledger and moment reported.",
+        "evidence": "research/staged544-mixed-fourth-entry/README.md",
+        "remaining": "Independent binding of changed construction to paid histogram, distinct from full admission.",
+        "status": "author-checked",
+        "source_commit": "72d7efc181edb21806a878f174847c96dd410ab0",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4091: sharp pooled ceiling proposed for fixed THREE/FOUR words and inventory.",
+        "evidence": "research/staged-fourth-mixture-ceiling/README.md",
+        "remaining": "Validate dual, exhaustive pattern scope and exact source equivalence before applying to newer fills.",
+        "status": "author-checked",
+        "source_commit": "a29bdd8f56cd18b805917775cdec09ff935eddc4",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4067: paid (2,2) to (1,3) exchange and fixed OQ4 host reported.",
+        "evidence": "research/r437-two-window-carrier/README.md",
+        "remaining": "Independent full cost delta; remaining caller/precision/stock obligations are separate.",
+        "status": "author-checked",
+        "source_commit": "a95f11c52e3ca1e4c5171b3c20d8f4b0815aee2e",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Prove an all-length paid-rank floor for the 128 PR4470 adjacent receiver windows using directed dependency and frame dimensions, with exact concave-price dominance.",
+  "scope_change_reason": "Handoff-first refinement to the live PR4567 open cost-floor question, beyond its finite word census; the older card source contexts no longer match the selected record.",
+  "bottleneck_link": "Closes arbitrary-length and arbitrary-coefficient search inside a fixed three-field invoice model, identifying which boundary premises must change to remove paid work.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop if any relaxation step is unsound, any incumbent fails exact dominance, or overlap already proves the scoped result; do not expand construction search.",
+  "outcome": "completed",
+  "evidence": "README.md gives the reduction and concave-price proof; RESULT.json and verify.py certify 128 windows, 1303 states and 239 window/profile pairs, positive geometric paths and three premise-removal controls.",
+  "author": "mihir-s-05",
+  "ai_assistance": "Substantial OpenAI Codex assistance; separate implementation, no independent human review or formal proof-kernel verification.",
+  "open_questions": [
+    {
+      "question": "Can a named extra-field or nonmonotone model beat this floor after all entrance, exit and restoration costs?",
+      "why_it_matters": "Only a changed premise can escape the certified model.",
+      "first_step": "Specify one actual window, extra field or allowed frame decrease, and its complete charged boundary, then derive an optimistic graph.",
+      "blocked_by": "No such complete cost contract is supplied here."
+    },
+    {
+      "question": "Does an independent review confirm the all-length abstraction against the exact caller model?",
+      "why_it_matters": "Finite tests alone cannot prove the reduction sound.",
+      "first_step": "Review the directed dependency induction, zero-state edge deletion and concave aggregation, then replay the certificate.",
+      "blocked_by": "No independent human or formal review is claimed."
+    }
+  ]
+}
diff --git a/research/receiver-window-all-length-floor/VERIFY.json b/research/receiver-window-all-length-floor/VERIFY.json
new file mode 100644
index 0000000000..5085eef217
--- /dev/null
+++ b/research/receiver-window-all-length-floor/VERIFY.json
@@ -0,0 +1,26 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/fresh-helper-boundary/CHANGES.json",
+    "research/fresh-helper-boundary/native.py",
+    "research/joint-helper-lattice/PATCH.json",
+    "research/joint-helper-lattice/native_joint.py",
+    "research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-CANONICAL.json.gz",
+    "research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-NATIVE.json.gz",
+    "research/missing-join-frame/PATCH.json"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Authenticate seven source inputs, extract all 128 adjacent windows, exhaust the dimension/dependency DAG, prove integer concave-price dominance, replay 2868 geometric control paths and three premise-removal controls, and compare the complete receipt."
+    }
+  ]
+}
diff --git a/research/receiver-window-all-length-floor/verify.py b/research/receiver-window-all-length-floor/verify.py
new file mode 100644
index 0000000000..f8cbc95a83
--- /dev/null
+++ b/research/receiver-window-all-length-floor/verify.py
@@ -0,0 +1,329 @@
+#!/usr/bin/env python3
+"""All-length three-field receiver cost floor; exact integers, no word-length cap.
+
+Uses the credited #4470 constructor only to obtain the frozen native events.
+Window extraction, dependency relaxation and dominance certificates are new.
+Prepared for mihir-s-05 with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+import argparse
+from collections import Counter
+from fractions import Fraction
+from functools import lru_cache
+import hashlib
+import json
+from pathlib import Path
+import subprocess
+import sys
+import tempfile
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+
+
+def need(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def span(basis):
+    result = {0}
+    for vector in basis:
+        result |= {value ^ vector for value in result}
+    return frozenset(result)
+
+
+def dimension(space):
+    need(len(space) > 0 and len(space) & (len(space) - 1) == 0,
+         "binary subspace cardinality")
+    return len(space).bit_length() - 1
+
+
+def partition(values):
+    return tuple(sorted((value for value in values if value), reverse=True))
+
+
+def capped_sums(ranks):
+    return tuple(sum(min(rank, k) for rank in ranks) for k in range(1, 10))
+
+
+def dominates(incumbent, candidate):
+    """Exact certificate for every concave rank price with f(0)=0; see README."""
+    return sum(incumbent) == sum(candidate) and all(
+        a <= b for a, b in zip(capped_sums(incumbent), capped_sums(candidate)))
+
+
+def source_checks():
+    manifest = json.loads((HERE / "SOURCES.json").read_bytes())
+    checked = 0
+    for row in manifest["files"]:
+        if row["usage"] == "input":
+            raw = (ROOT / row["path"]).read_bytes()
+            need(hashlib.sha256(raw).hexdigest() == row["sha256"],
+                 "changed source: " + row["path"])
+            checked += 1
+    need(checked == 7, "complete constructor source closure")
+    return checked
+
+
+def construct():
+    # A private cache prevents stale bytecode from replacing authenticated source.
+    code = ("import json,sys;sys.path.insert(0,sys.argv[1]);import native_joint;"
+            "c,p,*_=native_joint.construct();print(json.dumps([c,p['forward']]))")
+    with tempfile.TemporaryDirectory(prefix="receiver-floor-pycache-") as cache:
+        result = subprocess.run(
+            [sys.executable, "-I", "-B", "-X", "pycache_prefix=" + cache,
+             "-c", code, str(ROOT / "research/joint-helper-lattice")],
+            capture_output=True, check=True, timeout=120)
+    return json.loads(result.stdout)
+
+
+def operands(event):
+    if event[0] != "scalar":
+        return []
+    gate = event[1]
+    return gate[1:3] if gate[0] in ("add", "copy") else gate[1:2]
+
+
+def incumbent_profile(starts, ends, contacts):
+    current, paid = list(starts), []
+    for pair, frame in zip(((0, 2), (1, 2)), contacts):
+        for role in pair:
+            need(current[role] <= frame, "incumbent contact is monotone")
+            paid.append(dimension(frame) - dimension(current[role]))
+            current[role] = frame
+    for a, b in zip(current, ends):
+        need(a <= b, "incumbent exit is monotone")
+        paid.append(dimension(b) - dimension(a))
+    return partition(paid)
+
+
+def extract_windows(certificate, program):
+    # Frame state at a cut is after the preceding touch, before the next climb.
+    # The next scalar/warp entrance (or final frame) is the fixed exit contract.
+    events = [event for event in program if event[0] != "climb"]
+    bases = certificate["frames"]
+    need(all(type(v) is int and 0 <= v < 512 for basis in bases for v in basis),
+         "frames lie in F2^9")
+    frames = [span(basis) for basis in bases]
+    need(all(len(frame) == 2 ** len(basis) for frame, basis in zip(frames, bases)),
+         "independent input frame bases")
+    state, phase, result = certificate["start"][:], None, []
+    for pos, event in enumerate(events):
+        if event[0] == "phase":
+            phase = event[1]
+        if phase in ("B", "cleanup_tail") and pos + 1 < len(events):
+            second = events[pos + 1]
+            if (event[0] == second[0] == "scalar"
+                    and event[1][0] == second[1][0] == "add"):
+                roles = operands(event) + operands(second)
+                if all(kind == "L" for kind, _ in roles):
+                    a, x, b, y = (role for _, role in roles)
+                    ca, cb = Fraction(*event[1][3:]), Fraction(*second[1][3:])
+                    if x == y and len({a, b, x}) == 3 and ca in (-1, 1) and cb in (-1, 1):
+                        owners = [a, b, x]
+                        start_ids = [state[role] for role in owners]
+                        end_ids = []
+                        for role in owners:
+                            boundary = certificate["final"][role]
+                            for later in events[pos + 2:]:
+                                if ["L", role] in operands(later):
+                                    boundary = later[2]
+                                    break
+                                if later[0] == "warp" and later[1] == ["L", role]:
+                                    boundary = later[3]
+                                    break
+                            end_ids.append(boundary)
+                        result.append({
+                            "index": pos, "phase": phase, "roles": owners,
+                            "coefficients": [int(ca), int(cb)],
+                            "start_bases": [bases[i] for i in start_ids],
+                            "exit_bases": [bases[i] for i in end_ids],
+                            "contact_bases": [bases[event[2]], bases[second[2]]],
+                        })
+        if event[0] == "scalar":
+            for kind, role in operands(event):
+                if kind == "L":
+                    state[role] = event[2]
+        elif event[0] == "warp":
+            state[event[1][1]] = event[4]
+    need(len(result) == 128, "128 eligible adjacent windows")
+    need(Counter(row["phase"] for row in result) == {"B": 123, "cleanup_tail": 5},
+         "phase coverage")
+    return result
+
+
+def relaxed_profiles(starts, ends, *, dependency=True, intersections=True, exits=True):
+    """Finite acyclic overapproximation of ALL finite scalar words.
+
+    A bit records possible dependence on initial x, never its exact coefficient.
+    Frame subspaces are relaxed to dimensions plus static necessary pair bounds.
+    Toggles weaken specific premises for rejection controls only.
+    """
+    need(all(a <= b for a, b in zip(starts, ends)), "monotone boundaries")
+    initial = tuple(map(dimension, starts))
+    final = tuple(map(dimension, ends))
+    pairs = []
+    for i in range(3):
+        for j in range(3):
+            if i == j or not (starts[i] <= ends[j] and starts[j] <= ends[i]):
+                continue
+            lower = dimension({a ^ b for a in starts[i] for b in starts[j]})
+            upper = (dimension(ends[i] & ends[j]) if intersections
+                     else min(final[i], final[j]))
+            pairs.append((i, j, lower, upper))
+
+    @lru_cache(None)
+    def visit(dims, mask):
+        terminal = mask == 7 or not dependency
+        gap = partition(final[i] - dims[i] for i in range(3)) if exits else ()
+        profiles = {gap} if terminal else set()
+        for i, j, lower, upper in pairs:
+            for common in range(max(dims[i], dims[j], lower), upper + 1):
+                after = list(dims)
+                after[i] = after[j] = common
+                after = tuple(after)
+                mask_after = mask | ((1 << i) if mask & (1 << j) else 0)
+                if (after, mask_after) == (dims, mask):
+                    # Every such edge has zero charge. Removing it cannot change
+                    # any future transition or terminal condition in this graph.
+                    continue
+                need(sum(after) + mask_after.bit_count() > sum(dims) + mask.bit_count(),
+                     "strict acyclic potential")
+                added = (common - dims[i], common - dims[j])
+                for suffix in visit(after, mask_after):
+                    profiles.add(partition((*added, *suffix)))
+        return frozenset(profiles)
+
+    profiles = visit(initial, 4)
+    if exits:
+        mass = sum(final) - sum(initial)
+        need(all(sum(p) == mass for p in profiles), "all terminal rank mass charged")
+    return profiles, visit.cache_info().currsize
+
+
+def evaluate_window(row):
+    starts = [span(b) for b in row["start_bases"]]
+    ends = [span(b) for b in row["exit_bases"]]
+    contacts = [span(b) for b in row["contact_bases"]]
+    old = incumbent_profile(starts, ends, contacts)
+    profiles, states = relaxed_profiles(starts, ends)
+    need(old in profiles, "incumbent admitted at window " + str(row["index"]))
+    need(all(dominates(old, p) for p in profiles),
+         "all-length floor failed at window " + str(row["index"]))
+    return {**row, "incumbent": old, "reachable_states": states,
+            "terminal_profiles": sorted(profiles)}
+
+
+def failure_controls(windows):
+    by_index = {row["index"]: row for row in windows}
+    cases = [
+        ("drop_directed_dependency", 10557, {"dependency": False}, (2, 1)),
+        ("drop_exit_intersection", 11078, {"intersections": False}, (2, 2)),
+        ("drop_terminal_charge", 10554, {"exits": False}, ()),
+    ]
+    output = {}
+    for name, index, weakening, false_gain in cases:
+        row = by_index[index]
+        starts = [span(b) for b in row["start_bases"]]
+        ends = [span(b) for b in row["exit_bases"]]
+        profiles, _ = relaxed_profiles(starts, ends, **weakening)
+        need(false_gain in profiles, "negative control actually exposes its false gain")
+        need(false_gain not in row["terminal_profiles"], "normal model excludes the false gain")
+        need(not dominates(row["incumbent"], false_gain), "false gain rejected")
+        output[name] = {"index": index, "incumbent": row["incumbent"],
+                        "falsely_admitted": false_gain}
+    return output
+
+
+def small_geometric_control():
+    """Exhaust all legal common frames on exact endpoint words through length 4.
+
+    This is a finite semantic control for the abstraction, not the all-length proof.
+    F2^2, zero entrances, full exits; integer scalar matrices track arbitrary inputs.
+    """
+    spaces = [span(b) for b in ((), (1,), (2,), (3,), (1, 2))]
+    zero, full = spaces[0], spaces[-1]
+    abstract, _ = relaxed_profiles([zero] * 3, [full] * 3)
+    identity = (1, 0, 0, 0, 1, 0, 0, 0, 1)
+    target = (1, 0, 1, 0, 1, 1, 0, 0, 1)
+    gates = [(i, j, s) for i in range(3) for j in range(3) if i != j for s in (-1, 1)]
+    matched = paths = 0
+
+    def frame_paths(word, current, paid):
+        nonlocal paths
+        if not word:
+            p = partition((*paid, *(2 - dimension(frame) for frame in current)))
+            need(p in abstract, "legal geometric path absent from relaxation")
+            paths += 1
+            return
+        i, j, _ = word[0]
+        for frame in spaces:
+            if current[i] <= frame and current[j] <= frame:
+                after = list(current)
+                added = (dimension(frame) - dimension(current[i]),
+                         dimension(frame) - dimension(current[j]))
+                after[i] = after[j] = frame
+                frame_paths(word[1:], after, (*paid, *added))
+
+    def words(matrix, word):
+        nonlocal matched
+        if matrix == target:
+            matched += 1
+            frame_paths(word, [zero] * 3, ())
+        if len(word) == 4:
+            return
+        for i, j, s in gates:
+            after = list(matrix)
+            for k in range(3):
+                after[3 * i + k] += s * matrix[3 * j + k]
+            words(tuple(after), (*word, (i, j, s)))
+
+    words(identity, ())
+    need(matched == 84 and paths > 0, "exact dirty-input endpoint control")
+    # The cone identity used in the written proof is also checked for every
+    # generator f(r)=min(r,k) on ranks 0..9, entirely over integers.
+    for k in range(1, 10):
+        f = [min(r, k) for r in range(10)]
+        delta = [f[r] - f[r - 1] for r in range(1, 10)]
+        for r in range(10):
+            reconstructed = delta[-1] * r + sum(
+                (delta[j - 1] - delta[j]) * min(r, j) for j in range(1, 9))
+            need(reconstructed == f[r], "discrete concave-price identity")
+    return {"endpoint_words": matched, "geometric_paths": paths,
+            "concave_basis_checks": 90}
+
+
+def run():
+    source_count = source_checks()
+    certificate, program = construct()
+    windows = [evaluate_window(row) for row in extract_windows(certificate, program)]
+    controls = failure_controls(windows)
+    small = small_geometric_control()
+    return {"schema": "receiver-window-all-length-floor/v1",
+            "status": "PASS_ALL_LENGTH_RECEIVER_FLOOR",
+            "source_base": "f626db5c3a3d781d8f7695038776c621651f75e1",
+            "source_files_checked": source_count,
+            "window_count": len(windows),
+            "reachable_states": sum(w["reachable_states"] for w in windows),
+            "window_profile_pairs": sum(len(w["terminal_profiles"]) for w in windows),
+            "profile_count_histogram": dict(sorted(Counter(len(w["terminal_profiles"]) for w in windows).items())),
+            "failure_controls": controls, "geometric_control": small,
+            "new_kappa": False, "windows": windows}
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--write", action="store_true", help="author the deterministic receipt")
+    args = parser.parse_args()
+    result = run()
+    raw = json.dumps(result, indent=2, sort_keys=True) + "\n"
+    path = HERE / "RESULT.json"
+    if args.write:
+        path.write_text(raw, encoding="utf-8", newline="\n")
+    else:
+        need(path.read_text(encoding="utf-8") == raw, "complete deterministic receipt mismatch")
+    print(json.dumps({key: value for key, value in result.items() if key != "windows"}, sort_keys=True))
+
+
+if __name__ == "__main__":
+    main()