From f7d00eddd675920a61c554198171517325e58046 Mon Sep 17 00:00:00 2001
From: Mihir Srivastava <mihir2k5@gmail.com>
Date: Thu, 22 Oct 2026 01:05:34 -0700
Subject: [PATCH 1/2] Prove all-length transport floor for 128 fixed receiver
 windows

---
 .../.gitattributes                            |   1 +
 .../README.md                                 | 233 +++++++++++
 .../RESULT.json                               | 162 ++++++++
 .../SOURCES.json                              |  60 +++
 .../TASK.json                                 | 137 +++++++
 .../VERIFY.json                               |  26 ++
 .../verify.py                                 | 385 ++++++++++++++++++
 7 files changed, 1004 insertions(+)
 create mode 100644 research/receiver-window-all-length-floor/.gitattributes
 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

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diff --git a/research/receiver-window-all-length-floor/README.md b/research/receiver-window-all-length-floor/README.md
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+# An all-length transport floor for the 128 PR4470 receiver windows
+
+**Result.** In each of the 128 windows defined below, no finite word of shears
+on the same three persistent fields can lower the incumbent's paid transport
+cost, provided frames grow monotonically and the entrance/exit frames stay
+fixed. There is **no word-length or scalar-coefficient bound**. The result holds
+for every nonnegative concave rank price `f` with `f(0)=0`, in particular
+`f(r)=r^(1-C)` for every `0<C<1`. The incumbent attains the floor, so the
+optimistic gain ceiling is exactly zero.
+
+This is a source-bound, computer-assisted scoped obstruction with a written
+containment argument. It is author-checked, not formal-kernel acceptance,
+independent human review, a global supplier optimum or a new kappa.
+
+## What this adds
+
+[Chris R / Clear1nsight's #4546](https://github.com/CrocSwap/integer-mult-bounds/pull/4546)
+at `55b8d34c82fc4df6499c5037e156db6e990b96fe` introduced the fixed-boundary
+three-field relaxation and excluded length-at-most-four unit-shear rewrites in
+twelve windows at `C=0.00302015`.
+[Ryanson Jonathan's #4567](https://github.com/CrocSwap/integer-mult-bounds/pull/4567)
+at `50fedd98cda709707e4291c542e20ead96bcc786` extended it to all 128 windows,
+unit/nonunit finite alphabets and lengths five/six. That package explicitly
+left its all-length explanation unproved. Its exact nonunit coverage is
+`|c|<=2` through length six and `|c|<=3` through length five.
+
+This package replaces word enumeration by a finite-state closure which forgets
+coefficients, directions and most endpoint constraints. An exact concave-cost
+comparison proves the floor for every accepted relaxed path. No search,
+relaxation, price or acceptance function from either predecessor is imported.
+Their model and interpretation are credited, not claimed as newly invented.
+
+Canonical base is `f626db5c3a3d781d8f7695038776c621651f75e1`, priority revision 7.
+The selected record there is still `23/10000`. The constructor source is
+[Cinna Davis's #4470](https://github.com/CrocSwap/integer-mult-bounds/pull/4470),
+head `b9a13c1673e0e9697de410de16e8bbf3207a7363`, retained on main.
+Seven exact constructor inputs are authenticated by Git blob and SHA256 in
+`SOURCES.json` and declared in `VERIFY.json`. The constructor only supplies
+the instance; all window extraction and cost reasoning are implemented here.
+
+## Contract
+
+Each window replaces consecutive persistent-field additions
+`a += ca*x; b += cb*x`, where `ca,cb` are nonzero unit coefficients and
+`a,b,x` are distinct arbitrary-input fields. Eligibility is exactly the #4546
+contract: forward phase `B` or `cleanup_tail`, adjacency after removing emitted
+`climb` records, and no intervening scalar or warp. There are 123 windows in
+phase B and five in cleanup_tail.
+
+The entrance frame is the field's frame before the transports into the first
+contact. Its exit is the frame required by its next scalar contact or warp
+entry, or its final frame if it has no later use. `RESULT.json` records each
+window's index, roles, signs, entrance/exit bases, original contacts, paid gaps
+and full relaxed profile set. The inherited native endpoint/physical validity
+of the whole #4470 supplier is not replayed here.
+
+A competitor is any finite word `y_i += c*y_j`, `i!=j`, on these three fields,
+with the same arbitrary-input endpoint. Coefficients can be arbitrary rational
+numbers (indeed the dependence argument does not need a bound on their size).
+Every contact uses one common binary frame for its two operands. Along each
+field, frames only grow by inclusion from its fixed entrance `S_i` to its
+fixed exit `E_i`. Each positive frame-dimension increase `r` pays `f(r)`;
+terminal moves into the required exits are included. Scalar-height, scalar
+execution and other nonnegative charges are omitted in the competitor's favour.
+The conclusion is about this transport invoice; it does not price a complete
+changed supplier or rule out savings in other cost categories.
+
+## Proof of the relaxation and the all-length conclusion
+
+**1. Static contact bounds.** At a contact between fields `i,j`, its actual
+frame `F` contains `S_i+S_j` and lies inside `E_i intersect E_j`. If the former
+is not contained in the latter, that pair cannot contact at all. Otherwise put
+`l_ij=dim(S_i+S_j)` and `u_ij=dim(E_i intersect E_j)`. With current dimensions
+`d_i,d_j`, the contact dimension lies between `max(l_ij,d_i,d_j)` and `u_ij`.
+The relaxation enforces only these dimensions and static inclusions, discarding
+the stronger simultaneous geometric conditions. Thus it can admit spurious
+paths, but cannot exclude a legal contact sequence.
+
+**2. Chronological source dependence.** Initially only field `x` depends on
+the arbitrary initial value of `x`. Mark it reached. At every contact, mark
+both operands reached if either was reached before it. Actual propagation is
+directed and can cancel; this rule deliberately overestimates it. Inductively,
+every field with an actual nonzero dependence on the initial `x` is marked.
+The required endpoint has nonzero dependence on `x` in all three fields, so
+every valid final state must mark all three. An undirected connected contact
+graph alone is insufficient: contacting `a,b` before `b` obtains `x` does not
+retroactively deliver `x` to `a`.
+
+**3. States and charges.** A state consists of the three current dimensions,
+the reached mask, and the sorted multiset of positive transport gaps already
+paid. It starts at `(dim(S), {x}, empty)`. A contact of height `h` replaces
+both operand dimensions by `h`, adds their positive gaps to the multiset and
+updates the reached mask. Any state with all three fields reached can finish
+by adding the three positive gaps from its current dimensions to `dim(E)`.
+
+Every actual contact sequence maps to this graph. If an actual implementation
+splits a move between consecutive contacts into several monotone moves, the
+graph merges these gaps. Concavity and `f(0)=0` give
+`f(r_1+...+r_t) <= f(r_1)+...+f(r_t)`, so this merging only lowers the relaxed
+cost. Scalar steps without a frame increase are retained as zero-cost edges.
+The relaxed completed profile therefore costs no more than the actual path.
+
+**4. Finite closure covers every finite length.** Every dimension is an
+integer in its fixed entrance/exit interval; the reached mask has only eight
+values. At every state the sum of paid gaps is exactly the increase in the
+sum of dimensions, bounded by `sum_i(dim(E_i)-dim(S_i))`. Only finitely many
+integer multisets have such mass. Hence the state space is finite, including
+zero-cost cycles. The worklist starts at the initial state and continues until
+there is no unseen successor. A separate pass verifies closure under every
+allowed transition. Induction on word length then puts every finite path in
+the computed set. There is no depth limit, numerical stopping tolerance or
+coefficient alphabet in this computation.
+
+**5. Exact comparison for all concave costs.** Let `v` be a completed rank
+multiset and `T_k(v)=sum_{r in v} min(r,k)`. Every completed profile has the same
+total rank mass. For every relaxed profile `v`, the verifier checks exactly
+
+`T_k(incumbent) <= T_k(v)` for all integer `1 <= k < max_rank`.
+
+To see why this suffices without evaluating a real power, put
+`s_j=f(j)-f(j-1)`. Concavity says `s_j-s_(j+1)>=0`, and for ranks at most `m`,
+
+`sum f(r) = s_m * sum r + sum_{k=1}^{m-1} (s_k-s_(k+1)) * T_k(v)`.
+
+The common mass cancels. Each remaining coefficient is nonnegative, giving
+`cost(incumbent) <= cost(v) <= cost(actual competitor)`. The original two
+contacts form a legal path and their exact profile is admitted in every
+window, proving attainment and a zero gain ceiling. The zero-mass case is
+immediate; absent ranks can be padded by zero since `f(0)=0`.
+
+## Evidence and reproduction
+
+Run from the repository root, Python 3.12, standard library only:
+
+```sh
+python -B scripts/research_package.py --package research/receiver-window-all-length-floor --check-only
+python -B scripts/research_package.py --package research/receiver-window-all-length-floor
+python -I -O -B research/receiver-window-all-length-floor/verify.py --self-test
+```
+
+The registered replay authenticates the source closure, runs only the inherited
+constructor in an isolated subprocess with a fresh bytecode-cache namespace,
+extracts the windows, computes every closure and compares every committed
+receipt field. It does not write artifacts. `--write` is a separate authoring
+option. The calculation uses integer arithmetic and exact rational sign checks;
+no floating-point logarithms, solver or root enclosure is needed.
+
+The receipt contains **128 windows, 1,496 reachable states, 4,730 transition
+edges and 239 terminal profiles** (summed separately per window). The count of
+profiles per window is `1:93, 2:9, 3:6, 4:2, 5:13, 7:4, 9:1`. This agrees with
+the finite-profile histogram reported by #4567, but that agreement is not an
+input or premise of the proof.
+
+A separate Dijkstra implementation discards the entire paid-profile history,
+rebuilds the scalar transition costs, and minimizes `sum min(r,k)` directly.
+All 248 such shortest-path computations attain exactly the incumbent values.
+This corroborates the profile enumeration by a second algorithm; it is not an
+independently launched agent or an external review.
+
+Controls check 894 actual geometric prefixes (496 accepting) over every
+monotone boundary triple in F2^1 and all contact/frame sequences through length
+three. They also test zero-cost self-loops, incompatible exits, a genuinely
+cheaper rank-coalescing example, reversal of the concave order, omitted terminal
+charges and the chronological-dependence mutation. In the last example,
+entrance dimensions `(0,0,1)` and exits `(2,1,1)` have valid profile `(1,1,1)`;
+omitting the reach requirement admits the false cheaper profile `(2,1)`.
+These small tests support implementation correctness; the arbitrary-length
+argument is steps 1–5, not an extrapolation from the tests.
+Another 22,621 exact directed unit-shear prefixes through length four check
+that scalar cancellation never creates an unmarked dependence on `x`.
+
+The registered replay and optimized self-tests passed locally on Windows with
+CPython 3.12.0, using only the standard library. The registered computation
+takes a few seconds here. Maintainer scientific review and hosted CI are
+separate from this local evidence.
+
+## Scope, coordination and limitations
+
+The result is local to each fixed three-field window. It does not cover extra
+owned/borrowed fields, frame decreases or resets, changed boundaries, changed
+warp words, larger/nonadjacent windows, or joint rewrites. Overlapping window
+results cannot be added. A negative result in this model is not a universal
+barrier for integer multiplication. No complete supplier, precision, stock,
+physical endpoint or selected cold replay was run.
+
+Handoff-first selection followed #4567's explicitly remaining all-length
+question. The `independent-moment` card is refined to that obligation;
+`TASK.json` preserves its original card and the rationale. The optional family
+decision is **narrow**: prove or falsify this missing implication, then stop;
+do not expand another finite-word census. All required transport and exit costs
+are present; actual geometry is relaxed, and scalar/physical/compiler costs
+remain outside this local invoice rather than being declared free in a supplier.
+
+[Discussion #4572](https://github.com/CrocSwap/integer-mult-bounds/discussions/4572)
+is the canonical claim/result thread, with a short reply to the #1166 handoff.
+The initial overlap helper paginated 355 open PRs and 3,470 Discussion records;
+it reported incomplete local notes in this sparse checkout. Relevant #4470,
+#4546/#4567 source, README and ledger context were inspected separately. This
+does not establish exhaustive novelty. The generic selected baseline was
+attempted but could not inspect omitted `research/kappa-0023-productive-jobs`
+inputs in the sparse checkout; it is not reported as passing. This package
+authenticates its own complete seven-file input closure instead.
+
+Before submission, canonical main was fetched again and remained the base
+above. #4567 had advanced to `5ac594d1bc1e6d91f20ef8157f38bb9b29768bfd`, adding
+88 nonadjacent pairs with untouched intervening fields and single-contact
+retiming checks; its all-length explanation was still explicitly unproved.
+Those additions are distinct from the present closure of the 128 adjacent
+windows. The older head above is the proposal inspected when this task began;
+neither predecessor checker is a runtime input here.
+
+## Open next questions
+
+1. **Can an extra owned field change the floor with its complete allocation and
+   cleanup bill?** This changes the number of fields, outside the theorem, and
+   could create contacts unavailable in this graph. First specify one window's
+   fourth-field entrance, exit and allocation contract and derive the optimistic
+   transport/cleanup state space. A favourable relaxed profile would still need
+   exact endpoint, geometric and physical validation. No such allowance is
+   supplied here.
+2. **Does a joint window with a shared boundary admit a cheaper paid profile?**
+   Fixing each window's exits may prevent savings across them. First select two
+   overlapping windows, remove only their internal boundary requirement and
+   derive the combined arbitrary-input endpoint and full external invoice.
+   This is blocked on that changed caller contract, not on a longer three-field
+   search within the present boundaries.
+
+Credit Chris R / Clear1nsight (#4546 and #4543), Ryanson Jonathan (#4567),
+Cinna Davis (#4470 and its retained predecessors), zxv's productive-helper,
+receiver-sharing and native work, Jacob Alan Sussman, and the retained scalar,
+Clifford, geometry and outer-ledger authors. Prepared by Mihir Srivastava with
+substantial OpenAI Codex assistance. Apache-2.0. Frozen predecessors and the
+selected frontier are unchanged.
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..3e13f2b641
--- /dev/null
+++ b/research/receiver-window-all-length-floor/RESULT.json
@@ -0,0 +1,162 @@
+{
+  "base_commit": "f626db5c3a3d781d8f7695038776c621651f75e1",
+  "controls": {
+    "accepted_geometric_prefixes": 496,
+    "actual_geometric_prefixes": 894,
+    "chronology_mutation_rejected": true,
+    "cost_order_mutation_rejected": true,
+    "directed_coefficient_prefixes": 22621,
+    "incompatible_contact_rejected": true,
+    "positive_improvement_fixture": true,
+    "terminal_mass_mutation_rejected": true,
+    "zero_rank_loop_checked": true
+  },
+  "dijkstra_cost_checks": 248,
+  "new_kappa": false,
+  "profile_count_histogram": {
+    "1": 93,
+    "2": 9,
+    "3": 6,
+    "4": 2,
+    "5": 13,
+    "7": 4,
+    "9": 1
+  },
+  "reachable_states": 1496,
+  "schema": "receiver-window-all-length-floor/v1",
+  "source_pr4470_head": "b9a13c1673e0e9697de410de16e8bbf3207a7363",
+  "status": "PASS_RECEIVER_WINDOW_ALL_LENGTH_FLOOR",
+  "terminal_profiles_checked": 239,
+  "transition_edges": 4730,
+  "windows_checked": 128,
+  "windows": [
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [-1, 1], "contact_bases": [[], []], "dijkstra_truncated_cost_minima": [1, 2], "entrance_bases": [[], [], []], "entrance_dimensions": [0, 0, 0], "exit_bases": [[], [224, 30], []], "exit_dimensions": [0, 2, 0], "incumbent_gaps_by_field": [[], [2], []], "incumbent_profile": [2], "index": 10554, "pair_bounds_ab_ax_bx": [[0, 0], [0, 0], [0, 0]], "phase": "B", "reachable_states": 4, "relaxed_profiles": [[2]], "roles": [162, 172, 513], "transition_edges": 12},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [1, 1], "contact_bases": [[], []], "dijkstra_truncated_cost_minima": [1, 2], "entrance_bases": [[], [], []], "entrance_dimensions": [0, 0, 0], "exit_bases": [[224, 30], [], []], "exit_dimensions": [2, 0, 0], "incumbent_gaps_by_field": [[2], [], []], "incumbent_profile": [2], "index": 10555, "pair_bounds_ab_ax_bx": [[0, 0], [0, 0], [0, 0]], "phase": "B", "reachable_states": 4, "relaxed_profiles": [[2]], "roles": [172, 196, 513], "transition_edges": 12},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [1, -1], "contact_bases": [[30], [30]], "dijkstra_truncated_cost_minima": [3], "entrance_bases": [[], [], [30]], "entrance_dimensions": [0, 0, 1], "exit_bases": [[416, 30], [30], [30]], "exit_dimensions": [2, 1, 1], "incumbent_gaps_by_field": [[1, 1], [1], []], "incumbent_profile": [1, 1, 1], "index": 10557, "pair_bounds_ab_ax_bx": [[0, 1], [1, 1], [1, 1]], "phase": "B", "reachable_states": 7, "relaxed_profiles": [[1, 1, 1]], "roles": [140, 144, 502], "transition_edges": 22},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [-1, 1], "contact_bases": [[30], [30]], "dijkstra_truncated_cost_minima": [2], "entrance_bases": [[], [], [30]], "entrance_dimensions": [0, 0, 1], "exit_bases": [[30], [30], [30]], "exit_dimensions": [1, 1, 1], "incumbent_gaps_by_field": [[1], [1], []], "incumbent_profile": [1, 1], "index": 10558, "pair_bounds_ab_ax_bx": [[0, 1], [1, 1], [1, 1]], "phase": "B", "reachable_states": 7, "relaxed_profiles": [[1, 1]], "roles": [144, 187, 502], "transition_edges": 22},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [1, 1], "contact_bases": [[51], [51]], "dijkstra_truncated_cost_minima": [3], "entrance_bases": [[], [], [51]], "entrance_dimensions": [0, 0, 1], "exit_bases": [[51], [388, 51], [51]], "exit_dimensions": [1, 2, 1], "incumbent_gaps_by_field": [[1], [1, 1], []], "incumbent_profile": [1, 1, 1], "index": 10560, "pair_bounds_ab_ax_bx": [[0, 1], [1, 1], [1, 1]], "phase": "B", "reachable_states": 7, "relaxed_profiles": [[1, 1, 1]], "roles": [168, 150, 548], "transition_edges": 22},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [1, -1], "contact_bases": [[51], [51]], "dijkstra_truncated_cost_minima": [3], "entrance_bases": [[], [], [51]], "entrance_dimensions": [0, 0, 1], "exit_bases": [[388, 51], [51], [51]], "exit_dimensions": [2, 1, 1], "incumbent_gaps_by_field": [[1, 1], [1], []], "incumbent_profile": [1, 1, 1], "index": 10561, "pair_bounds_ab_ax_bx": [[0, 1], [1, 1], [1, 1]], "phase": "B", "reachable_states": 7, "relaxed_profiles": [[1, 1, 1]], "roles": [150, 115, 548], "transition_edges": 22},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [1, -1], "contact_bases": [[51], [51]], "dijkstra_truncated_cost_minima": [3], "entrance_bases": [[], [], [51]], "entrance_dimensions": [0, 0, 1], "exit_bases": [[196, 51], [51], [51]], "exit_dimensions": [2, 1, 1], "incumbent_gaps_by_field": [[1, 1], [1], []], "incumbent_profile": [1, 1, 1], "index": 10563, "pair_bounds_ab_ax_bx": [[0, 1], [1, 1], [1, 1]], "phase": "B", "reachable_states": 7, "relaxed_profiles": [[1, 1, 1]], "roles": [131, 117, 560], "transition_edges": 22},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [-1, -1], "contact_bases": [[51], [51]], "dijkstra_truncated_cost_minima": [1], "entrance_bases": [[], [51], [51]], "entrance_dimensions": [0, 1, 1], "exit_bases": [[51], [51], [51]], "exit_dimensions": [1, 1, 1], "incumbent_gaps_by_field": [[1], [], []], "incumbent_profile": [1], "index": 10564, "pair_bounds_ab_ax_bx": [[1, 1], [1, 1], [1, 1]], "phase": "B", "reachable_states": 6, "relaxed_profiles": [[1]], "roles": [117, 584, 560], "transition_edges": 18},
+    {"accepting_states": 5, "all_concave_costs_certified": true, "coefficients": [-1, -1], "contact_bases": [[], []], "dijkstra_truncated_cost_minima": [3, 5, 6], "entrance_bases": [[], [], []], "entrance_dimensions": [0, 0, 0], "exit_bases": [[30], [491, 30], [459, 38, 30]], "exit_dimensions": [1, 2, 3], "incumbent_gaps_by_field": [[1], [2], [3]], "incumbent_profile": [3, 2, 1], "index": 10575, "pair_bounds_ab_ax_bx": [[0, 1], [0, 1], [0, 1]], "phase": "B", "reachable_states": 13, "relaxed_profiles": [[2, 1, 1, 1, 1], [2, 2, 1, 1], [3, 1, 1, 1], [3, 2, 1]], "roles": [205, 170, 452], "transition_edges": 51},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [-1, -1], "contact_bases": [[30], [30]], "dijkstra_truncated_cost_minima": [3, 4], "entrance_bases": [[], [], [30]], "entrance_dimensions": [0, 0, 1], "exit_bases": [[30], [30], [448, 21, 11]], "exit_dimensions": [1, 1, 3], "incumbent_gaps_by_field": [[1], [1], [2]], "incumbent_profile": [2, 1, 1], "index": 10577, "pair_bounds_ab_ax_bx": [[0, 1], [1, 1], [1, 1]], "phase": "B", "reachable_states": 7, "relaxed_profiles": [[2, 1, 1]], "roles": [118, 194, 453], "transition_edges": 22},
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+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [-1, -1], "contact_bases": [[51], [51]], "dijkstra_truncated_cost_minima": [3, 4], "entrance_bases": [[], [], [51]], "entrance_dimensions": [0, 0, 1], "exit_bases": [[51], [51], [469, 51, 11]], "exit_dimensions": [1, 1, 3], "incumbent_gaps_by_field": [[1], [1], [2]], "incumbent_profile": [2, 1, 1], "index": 10581, "pair_bounds_ab_ax_bx": [[0, 1], [1, 1], [1, 1]], "phase": "B", "reachable_states": 7, "relaxed_profiles": [[2, 1, 1]], "roles": [192, 179, 455], "transition_edges": 22},
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+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [-1, -1], "contact_bases": [[258, 130, 85, 45], [258, 130, 85, 45]], "dijkstra_truncated_cost_minima": [1], "entrance_bases": [[258, 130, 85, 45], [258, 130, 85, 45], [258, 130, 85, 45]], "entrance_dimensions": [4, 4, 4], "exit_bases": [[258, 130, 85, 45], [258, 130, 85, 45], [258, 130, 84, 44, 1]], "exit_dimensions": [4, 4, 5], "incumbent_gaps_by_field": [[], [], [1]], "incumbent_profile": [1], "index": 11768, "pair_bounds_ab_ax_bx": [[4, 4], [4, 4], [4, 4]], "phase": "B", "reachable_states": 4, "relaxed_profiles": [[1]], "roles": [88, 31, 381], "transition_edges": 12},
+    {"accepting_states": 2, "all_concave_costs_certified": true, "coefficients": [-1, 1], "contact_bases": [[258, 130, 85, 45], [258, 130, 85, 45]], "dijkstra_truncated_cost_minima": [2, 4], "entrance_bases": [[258, 130, 85, 45], [258, 130, 85, 45], [258, 130, 85, 45]], "entrance_dimensions": [4, 4, 4], "exit_bases": [[258, 130, 72, 41, 25, 4], [258, 130, 70, 39, 19, 10], [258, 130, 85, 45]], "exit_dimensions": [6, 6, 4], "incumbent_gaps_by_field": [[2], [2], []], "incumbent_profile": [2, 2], "index": 11770, "pair_bounds_ab_ax_bx": [[4, 5], [4, 4], [4, 4]], "phase": "B", "reachable_states": 6, "relaxed_profiles": [[1, 1, 1, 1], [2, 2]], "roles": [80, 88, 345], "transition_edges": 18},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [1, -1], "contact_bases": [[258, 130, 85, 45], [258, 130, 85, 45]], "dijkstra_truncated_cost_minima": [3, 4], "entrance_bases": [[258, 130, 85, 45], [258, 130, 85, 45], [258, 130, 85, 45]], "entrance_dimensions": [4, 4, 4], "exit_bases": [[258, 130, 70, 39, 19, 10], [258, 130, 85, 37, 8], [258, 130, 74, 45, 31]], "exit_dimensions": [6, 5, 5], "incumbent_gaps_by_field": [[2], [1], [1]], "incumbent_profile": [2, 1, 1], "index": 11771, "pair_bounds_ab_ax_bx": [[4, 4], [4, 4], [4, 4]], "phase": "B", "reachable_states": 4, "relaxed_profiles": [[2, 1, 1]], "roles": [88, 31, 345], "transition_edges": 12},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [1, -1], "contact_bases": [[455, 33, 18, 12], [455, 33, 18, 12]], "dijkstra_truncated_cost_minima": [0], "entrance_bases": [[455, 33, 18, 12], [455, 33, 18, 12], [455, 33, 18, 12]], "entrance_dimensions": [4, 4, 4], "exit_bases": [[455, 33, 18, 12], [455, 33, 18, 12], [455, 33, 18, 12]], "exit_dimensions": [4, 4, 4], "incumbent_gaps_by_field": [[], [], []], "incumbent_profile": [], "index": 11791, "pair_bounds_ab_ax_bx": [[4, 4], [4, 4], [4, 4]], "phase": "B", "reachable_states": 4, "relaxed_profiles": [[]], "roles": [33, 55, 329], "transition_edges": 12},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [1, -1], "contact_bases": [[455, 33, 18, 12], [455, 33, 18, 12]], "dijkstra_truncated_cost_minima": [1, 2], "entrance_bases": [[455, 33, 18, 12], [455, 33, 18, 12], [455, 33, 18, 12]], "entrance_dimensions": [4, 4, 4], "exit_bases": [[388, 64, 33, 17, 12, 3], [455, 33, 18, 12], [455, 33, 18, 12]], "exit_dimensions": [6, 4, 4], "incumbent_gaps_by_field": [[2], [], []], "incumbent_profile": [2], "index": 11795, "pair_bounds_ab_ax_bx": [[4, 4], [4, 4], [4, 4]], "phase": "B", "reachable_states": 4, "relaxed_profiles": [[2]], "roles": [33, 74, 297], "transition_edges": 12},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [-1, -1], "contact_bases": [[455, 33, 18, 12], [455, 33, 18, 12]], "dijkstra_truncated_cost_minima": [1, 2], "entrance_bases": [[455, 33, 18, 12], [455, 33, 18, 12], [455, 33, 18, 12]], "entrance_dimensions": [4, 4, 4], "exit_bases": [[455, 33, 18, 12], [455, 33, 18, 12], [259, 132, 64, 33, 18, 12]], "exit_dimensions": [4, 4, 6], "incumbent_gaps_by_field": [[], [], [2]], "incumbent_profile": [2], "index": 11796, "pair_bounds_ab_ax_bx": [[4, 4], [4, 4], [4, 4]], "phase": "B", "reachable_states": 4, "relaxed_profiles": [[2]], "roles": [74, 85, 297], "transition_edges": 12},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [-1, 1], "contact_bases": [[455, 33, 18, 12], [455, 33, 18, 12]], "dijkstra_truncated_cost_minima": [2, 3], "entrance_bases": [[455, 33, 18, 12], [455, 33, 18, 12], [455, 33, 18, 12]], "entrance_dimensions": [4, 4, 4], "exit_bases": [[387, 64, 33, 18, 8, 4], [385, 70, 33, 18, 12], [455, 33, 18, 12]], "exit_dimensions": [6, 5, 4], "incumbent_gaps_by_field": [[2], [1], []], "incumbent_profile": [2, 1], "index": 11798, "pair_bounds_ab_ax_bx": [[4, 4], [4, 4], [4, 4]], "phase": "B", "reachable_states": 4, "relaxed_profiles": [[2, 1]], "roles": [55, 74, 313], "transition_edges": 12},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [1, -1], "contact_bases": [[455, 33, 18, 12], [455, 33, 18, 12]], "dijkstra_truncated_cost_minima": [3, 4], "entrance_bases": [[455, 33, 18, 12], [455, 33, 18, 12], [455, 33, 18, 12]], "entrance_dimensions": [4, 4, 4], "exit_bases": [[385, 70, 33, 18, 12], [262, 193, 33, 18, 12], [260, 131, 64, 33, 18, 12]], "exit_dimensions": [5, 5, 6], "incumbent_gaps_by_field": [[1], [1], [2]], "incumbent_profile": [2, 1, 1], "index": 11799, "pair_bounds_ab_ax_bx": [[4, 4], [4, 4], [4, 4]], "phase": "B", "reachable_states": 4, "relaxed_profiles": [[2, 1, 1]], "roles": [74, 85, 313], "transition_edges": 12},
+    {"accepting_states": 4, "all_concave_costs_certified": true, "coefficients": [1, -1], "contact_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1]], "dijkstra_truncated_cost_minima": [2, 4, 6], "entrance_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [1, 6, 50, 82, 136, 264], [257, 129, 68, 34, 22, 8]], "entrance_dimensions": [9, 6, 6], "exit_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1]], "exit_dimensions": [9, 9, 9], "incumbent_gaps_by_field": [[], [3], [3]], "incumbent_profile": [3, 3], "index": 12491, "pair_bounds_ab_ax_bx": [[9, 9], [9, 9], [8, 9]], "phase": "cleanup_tail", "reachable_states": 11, "relaxed_profiles": [[2, 2, 1, 1], [3, 3]], "roles": [408, 567, 105], "transition_edges": 35},
+    {"accepting_states": 4, "all_concave_costs_certified": true, "coefficients": [-1, -1], "contact_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1]], "dijkstra_truncated_cost_minima": [2, 4, 6], "entrance_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [3, 4, 48, 72, 144, 265], [10, 20, 32, 65, 132, 259]], "entrance_dimensions": [9, 6, 6], "exit_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1]], "exit_dimensions": [9, 9, 9], "incumbent_gaps_by_field": [[], [3], [3]], "incumbent_profile": [3, 3], "index": 12494, "pair_bounds_ab_ax_bx": [[9, 9], [9, 9], [8, 9]], "phase": "cleanup_tail", "reachable_states": 11, "relaxed_profiles": [[2, 2, 1, 1], [3, 3]], "roles": [412, 569, 414], "transition_edges": 35},
+    {"accepting_states": 4, "all_concave_costs_certified": true, "coefficients": [1, -1], "contact_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1]], "dijkstra_truncated_cost_minima": [2, 4, 6], "entrance_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [7, 27, 40, 73, 136, 264], [2, 8, 20, 32, 65, 389]], "entrance_dimensions": [9, 6, 6], "exit_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1]], "exit_dimensions": [9, 9, 9], "incumbent_gaps_by_field": [[], [3], [3]], "incumbent_profile": [3, 3], "index": 12497, "pair_bounds_ab_ax_bx": [[9, 9], [9, 9], [8, 9]], "phase": "cleanup_tail", "reachable_states": 11, "relaxed_profiles": [[2, 2, 1, 1], [3, 3]], "roles": [424, 575, 44], "transition_edges": 35},
+    {"accepting_states": 4, "all_concave_costs_certified": true, "coefficients": [-1, -1], "contact_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1]], "dijkstra_truncated_cost_minima": [2, 4, 6], "entrance_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [1, 2, 4, 48, 72, 408], [257, 129, 73, 42, 27, 5]], "entrance_dimensions": [9, 6, 6], "exit_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1]], "exit_dimensions": [9, 9, 9], "incumbent_gaps_by_field": [[], [3], [3]], "incumbent_profile": [3, 3], "index": 12500, "pair_bounds_ab_ax_bx": [[9, 9], [9, 9], [8, 9]], "phase": "cleanup_tail", "reachable_states": 11, "relaxed_profiles": [[2, 2, 1, 1], [3, 3]], "roles": [428, 577, 430], "transition_edges": 35},
+    {"accepting_states": 1, "all_concave_costs_certified": true, "coefficients": [-1, 1], "contact_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1]], "dijkstra_truncated_cost_minima": [1], "entrance_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 34, 16, 10, 4, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1]], "entrance_dimensions": [9, 8, 9], "exit_bases": [[256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1], [256, 128, 64, 32, 16, 8, 4, 2, 1]], "exit_dimensions": [9, 9, 9], "incumbent_gaps_by_field": [[], [1], []], "incumbent_profile": [1], "index": 12655, "pair_bounds_ab_ax_bx": [[9, 9], [9, 9], [9, 9]], "phase": "cleanup_tail", "reachable_states": 6, "relaxed_profiles": [[1]], "roles": [40, 18, 226], "transition_edges": 18}
+  ]
+}
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"
+    },
+    {
+      "path": "research/fresh-helper-boundary/CHANGES.json",
+      "usage": "input",
+      "git_blob": "45744fedd0e28967aa72586048b8bc0c72f419a9",
+      "sha256": "cde73b8ff5b3c0d06e69e2293b991b24fd14a16dd4ccab7ff6b3439074a94624"
+    },
+    {
+      "path": "research/fresh-helper-boundary/native.py",
+      "usage": "input",
+      "git_blob": "1b36f6cb5f205c33bf48811e0b1b3621c55e8154",
+      "sha256": "9b55a2d3c3972148fae9d112e745d9244968166bd89bd900c938f429fc76ab29"
+    },
+    {
+      "path": "research/joint-helper-lattice/PATCH.json",
+      "usage": "input",
+      "git_blob": "51304a6601061ed4e698f910b09135ef109555e3",
+      "sha256": "296d0e8c58cc70febe042c681ef30d6bb469818c0835fbcaac8aa8d330f5b727"
+    },
+    {
+      "path": "research/joint-helper-lattice/native_joint.py",
+      "usage": "input",
+      "git_blob": "0315f80e2fdef5e0eef13e3e9b8f72066238164d",
+      "sha256": "50869b1bcad7493e6b4cb2a4cd5d2fd9a193a5921100483fa2c3f2c999ddfd46"
+    },
+    {
+      "path": "research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-CANONICAL.json.gz",
+      "usage": "input",
+      "git_blob": "4a8c4ed26e53a72362d571c972e65aff546be784",
+      "sha256": "6418ec3da24d548a57f5c3fe5b9062c26c9478e638504a61c387f5cfb5f4ed61"
+    },
+    {
+      "path": "research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-NATIVE.json.gz",
+      "usage": "input",
+      "git_blob": "b46d44c6ee624968dd57be928b0c15d9d45f3b2c",
+      "sha256": "c54b0a3e129d28e6dbf74fa731fc3bb70fe3c6705f07256750fbecddcc842d39"
+    },
+    {
+      "path": "research/missing-join-frame/PATCH.json",
+      "usage": "input",
+      "git_blob": "f96d295d40014d4dd503f58eb3cdabcbb4b8e101",
+      "sha256": "9ab5328245ff02c9d0be21d39bcc375f50bc5d7909e9f7b9a3a9a6f812f5bd6e"
+    }
+  ]
+}
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..05bcdec156
--- /dev/null
+++ b/research/receiver-window-all-length-floor/TASK.json
@@ -0,0 +1,137 @@
+{
+  "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 or falsify an all-length concave transport-cost floor for all 128 fixed three-field PR4470 receiver windows, completing the missing implication explicitly left by PR4567.",
+  "scope_change_reason": "Handoff-first selection follows PR4567 at 50fedd98cda709707e4291c542e20ead96bcc786 rather than repeating the older card examples. Replace bounded shear-word enumeration with a finite-state relaxation and exact all-concave pricing. This remains an independent source-bound paid-invoice audit.",
+  "bottleneck_link": "A proof that longer words and larger scalar coefficients cannot improve these fixed windows prevents repeated finite searches and identifies changed boundary or field contracts as the remaining routes.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after source-bound closure of all 128 windows and containment/cost controls, or report any relaxed counterexample. No larger-window search, physical supplier replay or frontier change.",
+  "outcome": "negative-result",
+  "evidence": "README.md proves containment, finite closure and concave-order implication; RESULT.json records 128 windows, 1496 states, 4730 transitions and 239 profiles; verify.py reproduces every field with positive and failure controls.",
+  "author": "Mihir Srivastava",
+  "ai_assistance": "Substantial OpenAI Codex assistance for mathematical reasoning, implementation, verification, documentation and GitHub coordination. No independent human or formal-kernel review claimed.",
+  "open_questions": [
+    {
+      "question": "Can an extra owned field change the floor after allocation and cleanup are fully charged?",
+      "why_it_matters": "An additional field can admit contacts outside the present three-field graph.",
+      "first_step": "Specify one fourth-field entrance/exit and ownership contract, then derive its complete optimistic transport and cleanup state graph.",
+      "blocked_by": "No fourth-field allowance is supplied; a positive relaxation still requires endpoint, geometric and physical validation."
+    },
+    {
+      "question": "Can a joint window save paid work when its internal boundary is allowed to move?",
+      "why_it_matters": "Fixed separate exits may prevent savings available to a combined caller.",
+      "first_step": "Select two overlapping windows and derive their combined endpoint and external invoice without double counting.",
+      "blocked_by": "A changed caller contract and new source-bound geometry are required; single-window gains cannot be added."
+    }
+  ],
+  "canonical_discussion": "https://github.com/CrocSwap/integer-mult-bounds/discussions/4572",
+  "family_expansion_decision": "narrow: finish the missing all-length implication, then stop the fixed three-field search; extra-field or changed-boundary work needs a fresh complete contract and invoice.",
+  "overlap": {
+    "checked_at": "2026-10-22T07:50:00Z",
+    "open_pr_records": 355,
+    "discussion_records": 3470,
+    "remote_complete_pagination": true,
+    "local_notes_complete": false,
+    "limitation": "Sparse local notes; inspected relevant PR4470 primary inputs, root guidance, ledger and PR4546/4567 source and body directly. No claim of exhaustive novelty.",
+    "predecessors": [
+      {
+        "pr": 4470,
+        "head": "b9a13c1673e0e9697de410de16e8bbf3207a7363",
+        "author": "Cinna Davis",
+        "scope": "native source constructor"
+      },
+      {
+        "pr": 4546,
+        "head": "55b8d34c82fc4df6499c5037e156db6e990b96fe",
+        "author": "Chris R / Clear1nsight",
+        "scope": "fixed three-field model and twelve finite ceilings"
+      },
+      {
+        "pr": 4567,
+        "head": "50fedd98cda709707e4291c542e20ead96bcc786",
+        "author": "Ryanson Jonathan",
+        "scope": "all 128 finite ceilings and explicitly unproved all-length explanation"
+      }
+    ]
+  },
+  "baseline_orientation": "Attempted python -B scripts/research_baseline.py; failed because selected package roots are omitted in the sparse checkout. No selected baseline/cold-replay pass claimed. Read selected record/review and authenticate all seven inputs used by this scoped audit."
+}
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..a5f7e8c09a
--- /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": 90,
+      "checks": "Authenticate all seven constructor inputs; exact closure of every receiver window; concave-order proof predicates; positive geometry and chronology/terminal/cost-order controls; exact receipt comparison."
+    }
+  ]
+}
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..125736d4e3
--- /dev/null
+++ b/research/receiver-window-all-length-floor/verify.py
@@ -0,0 +1,385 @@
+#!/usr/bin/env python3
+"""Exact all-length receiver-window floor. Apache-2.0.
+
+Mihir Srivastava with substantial OpenAI Codex assistance. Model/source credit:
+Clear1nsight #4546, Ryanson Jonathan #4567, Cinna Davis #4470, zxv and the
+retained native/Clifford contributors. No predecessor acceptance code imported.
+"""
+import argparse
+from collections import Counter, deque
+from fractions import Fraction
+from itertools import combinations, product
+import hashlib
+import heapq
+import json
+from pathlib import Path
+import subprocess
+import sys
+import tempfile
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+PAIRS = tuple(combinations(range(3), 2))
+SOURCE_BIT, ALL_REACHED = 4, 7
+
+
+def need(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def span(basis):
+    values = {0}
+    for value in basis:
+        need(type(value) is int and 0 < value < 512 and value not in values,
+             "independent F2^9 basis")
+        values |= {x ^ value for x in tuple(values)}
+    return frozenset(values)
+
+
+def dim(space):
+    need(len(space) > 0 and len(space) & (len(space) - 1) == 0, "subspace size")
+    return len(space).bit_length() - 1
+
+
+def bounds(starts, ends):
+    need(all(s <= e for s, e in zip(starts, ends)), "monotone boundaries")
+    out = []
+    for i, j in PAIRS:
+        low = {x ^ y for x in starts[i] for y in starts[j]}
+        high = ends[i] & ends[j]
+        out.append((dim(low), dim(high)) if low <= high else None)
+    return tuple(out)
+
+
+def charge(paid, gaps):
+    need(all(type(g) is int and g >= 0 for g in gaps), "nonnegative rank gaps")
+    return tuple(sorted(paid + tuple(g for g in gaps if g), reverse=True))
+
+
+def successors(state, intervals):
+    dimensions, reached, paid = state
+    for (i, j), interval in zip(PAIRS, intervals):
+        if interval is None:
+            continue
+        for height in range(max(interval[0], dimensions[i], dimensions[j]), interval[1] + 1):
+            after = list(dimensions)
+            after[i] = after[j] = height
+            pair = (1 << i) | (1 << j)
+            reach = reached | pair if reached & pair else reached
+            yield (tuple(after), reach,
+                   charge(paid, (height - dimensions[i], height - dimensions[j])))
+
+
+def finish(state, ends):
+    return charge(state[2], tuple(e - d for e, d in zip(ends, state[0])))
+
+
+def closure(starts, ends, require_reach=True):
+    intervals = bounds(starts, ends)
+    initial_dims, final_dims = tuple(map(dim, starts)), tuple(map(dim, ends))
+    initial = (initial_dims, SOURCE_BIT, ())
+    seen, pending = {initial}, deque([initial])
+    while pending:
+        for nxt in successors(pending.popleft(), intervals):
+            need(sum(nxt[2]) == sum(nxt[0]) - sum(initial_dims), "intermediate paid mass")
+            if nxt not in seen:
+                seen.add(nxt)
+                pending.append(nxt)
+    # Independently check closure after the worklist terminates; no length cutoff.
+    edges, accepted, profiles = 0, 0, set()
+    for state in seen:
+        for nxt in successors(state, intervals):
+            need(nxt in seen, "transition-closed state set")
+            edges += 1
+        if not require_reach or state[1] == ALL_REACHED:
+            profile = finish(state, final_dims)
+            need(sum(profile) == sum(final_dims) - sum(initial_dims), "complete terminal mass")
+            profiles.add(profile)
+            accepted += 1
+    return profiles, seen, edges, accepted, intervals
+
+
+def no_dearer(old, candidate):
+    """Exact order for every concave f with f(0)=0 (README slope identity)."""
+    return sum(old) == sum(candidate) and all(
+        sum(min(r, k) for r in old) <= sum(min(r, k) for r in candidate)
+        for k in range(1, max(old + candidate, default=0)))
+
+
+def shortest_truncated_cost(starts, ends, cutoff):
+    """Second computation: scalar Dijkstra costs, with no paid-profile history.
+
+    Rebuild bounds and transitions directly instead of calling closure helpers.
+    All edge weights are nonnegative integers, including zero-rank loops.
+    """
+    begin = tuple(len(s).bit_length() - 1 for s in starts)
+    final = tuple(len(e).bit_length() - 1 for e in ends)
+    edges = []
+    for i in range(3):
+        for j in range(i + 1, 3):
+            required = {x ^ y for x in starts[i] for y in starts[j]}
+            allowed = ends[i] & ends[j]
+            if required <= allowed:
+                edges.append((i, j, len(required).bit_length() - 1,
+                              len(allowed).bit_length() - 1))
+    initial = (begin, (False, False, True))
+    distances = {initial: 0}
+    queue = [(0, initial)]
+    best = None
+    while queue:
+        cost, state = heapq.heappop(queue)
+        if cost != distances[state]:
+            continue
+        dimensions, reached = state
+        if all(reached):
+            complete = cost + sum(min(e - d, cutoff) for e, d in zip(final, dimensions))
+            best = complete if best is None else min(best, complete)
+        for i, j, low, high in edges:
+            for h in range(max(low, dimensions[i], dimensions[j]), high + 1):
+                ds, rs = list(dimensions), list(reached)
+                ds[i] = ds[j] = h
+                rs[i] = rs[j] = reached[i] or reached[j]
+                target = (tuple(ds), tuple(rs))
+                value = cost + min(h - dimensions[i], cutoff) + min(h - dimensions[j], cutoff)
+                if target not in distances or value < distances[target]:
+                    distances[target] = value
+                    heapq.heappush(queue, (value, target))
+    return best
+
+
+def operands(event):
+    if event[0] != "scalar":
+        return ()
+    gate = event[1]
+    need(gate[0] in ("add", "copy", "scale", "clear"), "known scalar event")
+    return gate[1:3] if gate[0] in ("add", "copy") else gate[1:2]
+
+
+def windows(c, emitted):
+    # Include transports into both contacts and the next-use exits, as in #4546.
+    word = [e for e in emitted if e[0] != "climb"]
+    state, phase, out = list(c["start"]), None, []
+    for index, event in enumerate(word):
+        if event[0] == "phase":
+            phase = event[1]
+        if phase in ("B", "cleanup_tail") and index + 1 < len(word):
+            following = word[index + 1]
+            if (event[0] == following[0] == "scalar"
+                    and event[1][0] == following[1][0] == "add"
+                    and all(r[0] == "L" for r in operands(event) + operands(following))):
+                a, x = event[1][1][1], event[1][2][1]
+                b, y = following[1][1][1], following[1][2][1]
+                ca, cb = Fraction(*event[1][3:]), Fraction(*following[1][3:])
+                if x == y and len({a, b, x}) == 3 and ca in (-1, 1) and cb in (-1, 1):
+                    roles, exits = (a, b, x), []
+                    for r in roles:
+                        end = c["final"][r]
+                        for later in word[index + 2:]:
+                            if ["L", r] in operands(later):
+                                end = later[2]
+                                break
+                            if later[0] == "warp" and later[1] == ["L", r]:
+                                end = later[3]
+                                break
+                        exits.append(end)
+                    out.append({"index": index, "phase": phase, "roles": list(roles),
+                                "coefficients": [int(ca), int(cb)],
+                                "entrance_bases": [c["frames"][state[r]] for r in roles],
+                                "exit_bases": [c["frames"][f] for f in exits],
+                                "contact_bases": [c["frames"][event[2]], c["frames"][following[2]]]})
+        for r in operands(event):
+            if r[0] == "L":
+                state[r[1]] = event[2]
+        if event[0] == "warp":
+            need(event[1][0] == "L", "persistent warp")
+            state[event[1][1]] = event[4]
+    return out
+
+
+def incumbent(starts, ends, contacts):
+    current, per_field = list(starts), [[], [], []]
+    for pair, contact in zip(((0, 2), (1, 2)), contacts):
+        for i in pair:
+            need(current[i] <= contact <= ends[i], "legal incumbent contact")
+            gap = dim(contact) - dim(current[i])
+            if gap:
+                per_field[i].append(gap)
+            current[i] = contact
+    for i in range(3):
+        need(current[i] <= ends[i], "legal incumbent terminal move")
+        gap = dim(ends[i]) - dim(current[i])
+        if gap:
+            per_field[i].append(gap)
+    return charge((), tuple(g for field in per_field for g in field)), per_field
+
+
+def authenticate():
+    count = 0
+    for row in json.loads((HERE / "SOURCES.json").read_bytes())["files"]:
+        if row["usage"] != "input":
+            continue
+        path = ROOT / row["path"]
+        need(path.is_file() and not path.is_symlink(), "regular source file")
+        raw = path.read_bytes()
+        need(hashlib.sha256(raw).hexdigest() == row["sha256"], "source SHA256: " + row["path"])
+        blob = hashlib.sha1(b"blob " + str(len(raw)).encode() + b"\0" + raw).hexdigest()
+        need(blob == row["git_blob"], "source Git blob: " + row["path"])
+        count += 1
+    need(count == 7, "complete seven-input constructor closure")
+
+
+def construct():
+    authenticate()
+    code = ("import json,sys;sys.path.insert(0,sys.argv[1]);import native_joint;"
+            "c,p,*_=native_joint.construct();print(json.dumps({'c':c,'p':p['forward']}))")
+    with tempfile.TemporaryDirectory(prefix="receiver-floor-cache-") 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=60)
+    return json.loads(result.stdout)
+
+
+def controls():
+    zero, line, plane = span(()), span((1,)), span((1, 2))
+    need(no_dearer((2,), (1, 1)), "concave merging positive control")
+    need(not no_dearer((1, 1), (2,)), "reversed cost order rejected")
+    need(not no_dearer((1,), (2,)), "missing mass rejected")
+    # A model with an actual cheaper coalesced path must not be falsely excluded.
+    positive, *_ = closure((zero,) * 3, (plane,) * 3)
+    need((2, 2, 2) in positive and not no_dearer((1,) * 6, (2, 2, 2)),
+         "positive improvement fixture")
+    starts, ends = (zero, zero, line), (plane, line, line)
+    valid, states, _, _, intervals = closure(starts, ends)
+    mutant, *_ = closure(starts, ends, require_reach=False)
+    need(valid == {(1, 1, 1)} and (2, 1) in mutant, "chronology omission caught")
+    early = ((0, 0, 1), SOURCE_BIT, ())
+    late = ((0, 1, 1), 6, (1,))
+    need(early in set(successors(early, intervals)), "a-b zero contact before x arrives")
+    need(late in set(successors(early, intervals)) and late[1] != ALL_REACHED,
+         "b-x contact cannot retroactively inform a")
+    need(finish(late, (2, 1, 1)) == (2, 1), "connected graph alone admits false cheap path")
+    accepted = [s for s in states if s[1] == ALL_REACHED]
+    need(any(sum(s[2]) < 3 for s in accepted)
+         and all(sum(finish(s, (2, 1, 1))) == 3 for s in accepted), "terminal-charge omission caught")
+    incompatible = bounds((line, span((2,)), zero), (line, span((2,)), zero))
+    need(incompatible[0] is None, "incompatible exits disable contact")
+    # Enumerate actual F2^1 geometries for all 27 monotone boundary triples and
+    # all contact/frame sequences through length three. Check relaxation inclusion.
+    actual_count = accepted_count = 0
+    for selected in product(((zero, zero), (zero, line), (line, line)), repeat=3):
+        starts, ends = tuple(s[0] for s in selected), tuple(s[1] for s in selected)
+        profiles, seen, *_ = closure(starts, ends)
+        layer = [(starts, SOURCE_BIT, ())]
+        for length in range(4):
+            next_layer = []
+            for current, reach, paid in layer:
+                abstract = (tuple(map(dim, current)), reach, paid)
+                need(abstract in seen, "actual geometric prefix belongs to closure")
+                actual_count += 1
+                if reach == ALL_REACHED:
+                    need(finish(abstract, tuple(map(dim, ends))) in profiles, "actual terminal profile included")
+                    accepted_count += 1
+                if length == 3:
+                    continue
+                for i, j in PAIRS:
+                    for contact in (zero, line):
+                        if current[i] <= contact <= ends[i] and current[j] <= contact <= ends[j]:
+                            nxt = list(current)
+                            nxt[i] = nxt[j] = contact
+                            pair = (1 << i) | (1 << j)
+                            new_reach = reach | pair if reach & pair else reach
+                            new_paid = charge(paid, (dim(contact) - dim(current[i]), dim(contact) - dim(current[j])))
+                            next_layer.append((tuple(nxt), new_reach, new_paid))
+            layer = next_layer
+    fixed = ((1, 1, 1), ALL_REACHED, (1, 1))
+    need(fixed in set(successors(fixed, bounds((zero, zero, line), (line,) * 3))),
+         "zero-rank loop retained without length cutoff")
+    # Separately execute every directed unit-shear word through length four on
+    # the exact x coefficient. Cancellations must never create unmarked support.
+    coefficient_checks = 0
+    layer = [((0, 0, 1), frozenset({2}))]
+    for length in range(5):
+        nxt = []
+        for coefficients, reached in layer:
+            need(all(not value or i in reached for i, value in enumerate(coefficients)),
+                 "exact directed coefficient support is covered by reached mask")
+            coefficient_checks += 1
+            if length == 4:
+                continue
+            for i in range(3):
+                for j in range(3):
+                    if i == j:
+                        continue
+                    for sign in (-1, 1):
+                        values = list(coefficients)
+                        values[i] += sign * coefficients[j]
+                        reach = reached | {i, j} if i in reached or j in reached else reached
+                        nxt.append((tuple(values), reach))
+        layer = nxt
+    return {"actual_geometric_prefixes": actual_count, "accepted_geometric_prefixes": accepted_count,
+            "directed_coefficient_prefixes": coefficient_checks,
+            "chronology_mutation_rejected": True, "terminal_mass_mutation_rejected": True,
+            "cost_order_mutation_rejected": True, "positive_improvement_fixture": True,
+            "incompatible_contact_rejected": True, "zero_rank_loop_checked": True}
+
+
+def run():
+    tested = controls()
+    data = construct()
+    extracted = windows(data["c"], data["p"])
+    need(len(extracted) == 128, "128 eligible windows")
+    need(Counter(w["phase"] for w in extracted) == {"B": 123, "cleanup_tail": 5}, "phase census")
+    rows = []
+    for w in extracted:
+        starts, ends = tuple(map(span, w["entrance_bases"])), tuple(map(span, w["exit_bases"]))
+        old, per_field = incumbent(starts, ends, tuple(map(span, w["contact_bases"])))
+        profiles, seen, edges, accepted, intervals = closure(starts, ends)
+        need(old in profiles, "incumbent admitted")
+        need(all(no_dearer(old, p) for p in profiles), "all-concave floor at " + str(w["index"]))
+        m = max((r for p in profiles for r in p), default=1)
+        independently_minimized = [shortest_truncated_cost(starts, ends, k) for k in range(1, m + 1)]
+        need(independently_minimized == [sum(min(r, k) for r in old) for k in range(1, m + 1)],
+             "Dijkstra corroborates every truncated-cost floor independently of profile history")
+        rows.append({**w, "entrance_dimensions": list(map(dim, starts)), "exit_dimensions": list(map(dim, ends)),
+                     "pair_bounds_ab_ax_bx": intervals, "incumbent_gaps_by_field": per_field,
+                     "incumbent_profile": old, "relaxed_profiles": sorted(profiles),
+                     "reachable_states": len(seen), "transition_edges": edges,
+                     "accepting_states": accepted, "all_concave_costs_certified": True,
+                     "dijkstra_truncated_cost_minima": independently_minimized})
+    return {"schema": "receiver-window-all-length-floor/v1", "status": "PASS_RECEIVER_WINDOW_ALL_LENGTH_FLOOR",
+            "base_commit": "f626db5c3a3d781d8f7695038776c621651f75e1", "new_kappa": False,
+            "source_pr4470_head": "b9a13c1673e0e9697de410de16e8bbf3207a7363",
+            "windows_checked": len(rows), "reachable_states": sum(w["reachable_states"] for w in rows),
+            "transition_edges": sum(w["transition_edges"] for w in rows),
+            "terminal_profiles_checked": sum(len(w["relaxed_profiles"]) for w in rows),
+            "dijkstra_cost_checks": sum(len(w["dijkstra_truncated_cost_minima"]) for w in rows),
+            "profile_count_histogram": dict(sorted(Counter(len(w["relaxed_profiles"]) for w in rows).items())),
+            "controls": tested, "windows": rows}
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--write", action="store_true", help="authoring only: replace receipt")
+    parser.add_argument("--self-test", action="store_true")
+    args = parser.parse_args()
+    if args.self_test:
+        print(json.dumps(controls(), sort_keys=True))
+        return
+    result = run()
+    if args.write:
+        # Keep the machine-readable window receipts at one record per line.
+        header = json.dumps({k: v for k, v in result.items() if k != "windows"},
+                            indent=2, sort_keys=True).rstrip()[:-1].rstrip()
+        records = ",\n".join("    " + json.dumps(w, sort_keys=True) for w in result["windows"])
+        encoded = header + ',\n  "windows": [\n' + records + "\n  ]\n}\n"
+        (HERE / "RESULT.json").write_text(encoded, encoding="utf-8", newline="\n")
+    else:
+        recorded = json.loads((HERE / "RESULT.json").read_bytes())
+        need(json.dumps(recorded, sort_keys=True) == json.dumps(result, sort_keys=True),
+             "all receipt fields and JSON types reproduce")
+    print(json.dumps({k: v for k, v in result.items() if k != "windows"}, sort_keys=True))
+
+
+if __name__ == "__main__":
+    main()

From 23c515a3c071a6bfdd1fe80bc67cea55f6d43643 Mon Sep 17 00:00:00 2001
From: Mihir Srivastava <mihir2k5@gmail.com>
Date: Thu, 22 Oct 2026 01:07:30 -0700
Subject: [PATCH 2/2] Credit concurrent all-length investigations and clarify
 independent evidence

---
 .../receiver-window-all-length-floor/README.md   | 16 ++++++++++++++++
 .../receiver-window-all-length-floor/TASK.json   | 14 ++++++++++----
 2 files changed, 26 insertions(+), 4 deletions(-)

diff --git a/research/receiver-window-all-length-floor/README.md b/research/receiver-window-all-length-floor/README.md
index b7ab09bef7..f3ae74d54e 100644
--- a/research/receiver-window-all-length-floor/README.md
+++ b/research/receiver-window-all-length-floor/README.md
@@ -30,6 +30,18 @@ comparison proves the floor for every accepted relaxed path. No search,
 relaxation, price or acceptance function from either predecessor is imported.
 Their model and interpretation are credited, not claimed as newly invented.
 
+The submission refresh found concurrent investigations
+[#4573](https://github.com/CrocSwap/integer-mult-bounds/discussions/4573),
+[#4574](https://github.com/CrocSwap/integer-mult-bounds/discussions/4574) and
+[#4577](https://github.com/CrocSwap/integer-mult-bounds/discussions/4577), all
+under the same donor account. This package is **independent corroboration and
+a separately reviewable certificate**, with no exclusive novelty or priority
+claim. Its code and proof were developed before reading those claims. #4577
+reports a directed-dependency graph with 1,303 states; the present more
+permissive bidirectional propagation has 1,496 states and the same 239 terminal
+profiles. The separate integer Dijkstra checks add an algorithmically distinct
+check of the cost floor. No concurrent artifact is imported or certified here.
+
 Canonical base is `f626db5c3a3d781d8f7695038776c621651f75e1`, priority revision 7.
 The selected record there is still `23/10000`. The constructor source is
 [Cinna Davis's #4470](https://github.com/CrocSwap/integer-mult-bounds/pull/4470),
@@ -208,6 +220,10 @@ retiming checks; its all-length explanation was still explicitly unproved.
 Those additions are distinct from the present closure of the 128 adjacent
 windows. The older head above is the proposal inspected when this task began;
 neither predecessor checker is a runtime input here.
+The refreshed overlap corpus completed 355 open PRs and 3,486 Discussion
+records at 08:04 UTC and exposed the concurrent claims credited above. Local
+note coverage remains incomplete. This is the basis for the independent
+corroboration framing, rather than an assertion of unique new coverage.
 
 ## Open next questions
 
diff --git a/research/receiver-window-all-length-floor/TASK.json b/research/receiver-window-all-length-floor/TASK.json
index 05bcdec156..99f5552873 100644
--- a/research/receiver-window-all-length-floor/TASK.json
+++ b/research/receiver-window-all-length-floor/TASK.json
@@ -86,7 +86,7 @@
   "budget_minutes": 60,
   "stop_rule": "Stop after source-bound closure of all 128 windows and containment/cost controls, or report any relaxed counterexample. No larger-window search, physical supplier replay or frontier change.",
   "outcome": "negative-result",
-  "evidence": "README.md proves containment, finite closure and concave-order implication; RESULT.json records 128 windows, 1496 states, 4730 transitions and 239 profiles; verify.py reproduces every field with positive and failure controls.",
+  "evidence": "README proof and RESULT receipt: 128 windows, 1496 states, 4730 transitions, 239 profiles, 248 independently minimized integer costs; 894 geometric and 22621 directed coefficient prefixes; source binding and semantic controls.",
   "author": "Mihir Srivastava",
   "ai_assistance": "Substantial OpenAI Codex assistance for mathematical reasoning, implementation, verification, documentation and GitHub coordination. No independent human or formal-kernel review claimed.",
   "open_questions": [
@@ -106,9 +106,9 @@
   "canonical_discussion": "https://github.com/CrocSwap/integer-mult-bounds/discussions/4572",
   "family_expansion_decision": "narrow: finish the missing all-length implication, then stop the fixed three-field search; extra-field or changed-boundary work needs a fresh complete contract and invoice.",
   "overlap": {
-    "checked_at": "2026-10-22T07:50:00Z",
+    "checked_at": "2026-10-22T08:04:34Z",
     "open_pr_records": 355,
-    "discussion_records": 3470,
+    "discussion_records": 3486,
     "remote_complete_pagination": true,
     "local_notes_complete": false,
     "limitation": "Sparse local notes; inspected relevant PR4470 primary inputs, root guidance, ledger and PR4546/4567 source and body directly. No claim of exhaustive novelty.",
@@ -131,7 +131,13 @@
         "author": "Ryanson Jonathan",
         "scope": "all 128 finite ceilings and explicitly unproved all-length explanation"
       }
+    ],
+    "concurrent_discussions": [
+      4573,
+      4574,
+      4577
     ]
   },
-  "baseline_orientation": "Attempted python -B scripts/research_baseline.py; failed because selected package roots are omitted in the sparse checkout. No selected baseline/cold-replay pass claimed. Read selected record/review and authenticate all seven inputs used by this scoped audit."
+  "baseline_orientation": "Attempted python -B scripts/research_baseline.py; failed because selected package roots are omitted in the sparse checkout. No selected baseline/cold-replay pass claimed. Read selected record/review and authenticate all seven inputs used by this scoped audit.",
+  "contribution_position": "Independent corroboration: concurrent investigations found on submission refresh are credited. Bidirectional 1496-state relaxation plus separate Dijkstra truncated-cost minima; no exclusive novelty or priority claim."
 }