Research: prove a chronological packing ceiling for every uniform shift of PR #2571's T74 order

research/t74-diagonal-window-bound/INPUTS.json

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+{
+  "schema_version": 1,
+  "files": [
+    {
+      "path": "inputs/PR2530-PROVIDERS.json",
+      "commit": "b4012098048e12500b2d33d2e84be281915a27ac",
+      "source_path": "research/source-partner-lifetimes/evidence/eight-providers.json",
+      "git_blob": "a678492aece195bf9b3e35a72cc518dd1e2ada31",
+      "sha256": "13d5abb0f5dccf32703a04334209d0d47516e985db4a51809a593538f45c1f45"
+    },
+    {
+      "path": "inputs/PR2530-ORDER.json",
+      "commit": "b4012098048e12500b2d33d2e84be281915a27ac",
+      "source_path": "research/source-partner-lifetimes/evidence/eight-order.json",
+      "git_blob": "a7de9f93b6b95a4f8b0b65459e135cd73836a9dd",
+      "sha256": "e48b51a7d6a315971fff8b8e51748251750ff4ed8fe30b8b023de1a1a8019783"
+    },
+    {
+      "path": "inputs/PR2571-ORDER.json",
+      "commit": "faff03cf5c29dadd6e0eb51e406efdc921bfa683",
+      "source_path": "research/source-partner-repack/evidence/repack-order.json",
+      "git_blob": "eebac3de3bd91def724db70a8a0dff6d58afc606",
+      "sha256": "e3702fbcaff7491bbeca6b0a5a9301778a816fa69329761b09b013e289c71c2b"
+    },
+    {
+      "path": "inputs/PR2571-CLAIM.json",
+      "commit": "faff03cf5c29dadd6e0eb51e406efdc921bfa683",
+      "source_path": "research/source-partner-repack/CLAIM.json",
+      "git_blob": "359a2f186ca874a8424f325994cc95356b5dc0ae",
+      "sha256": "a57f5f27c4fb250f4aabde29bbb738f16e05bd73940b381918de2eb928dbec65"
+    },
+    {
+      "path": "inputs/PR2435-TEMPLATE.json.gz",
+      "commit": "b4012098048e12500b2d33d2e84be281915a27ac",
+      "source_path": "research/t74-owner-consolidation/source-pr2435.tar.gz",
+      "archive_git_blob": "94878b26080479788d0870b4e536a2f2e111a996",
+      "archive_sha256": "f8937eabaa59667831f21ecbcb68eb163de02dd3154c5c51dd3f814a949ae0ff",
+      "member": "research/gmx-t74-borrow43-handoff/inputs/GLOBAL-TEMPLATE.json.gz",
+      "sha256": "21297b20f1f2dc76eb9dd8124416068c3cb096bab9df17fa4da19bc68dfe082e",
+      "source_credit": "Jeremy Kothe, PR2435, retained by zxv in PR2530",
+      "archive_origin_pr": 2435,
+      "archive_origin_commit": "324d7d3590dc21003940a44cb0cbdc32b3620f25"
+    }
+  ]
+}

research/t74-diagonal-window-bound/README.md

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+# A chronological packing bound for every uniform initial shift of the source-partner T74 schedule
+
+Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.
+
+**Scoped proof and exact finite certificate; no new κ.** For PR2530's fixed
+422-helper words and the uniform initial-shift order family containing PR2571,
+every legal whole-helper fill has complex primitive endpoint
+
+\[
+ b < \boxed{0.001714127696}.
+\]
+
+The source's window-free estimate is about 0.00171422836. The new bound comes
+from an unavoidable shortage of width-eight roles in a named collection of
+windows. It applies to **every nonnegative integer uniform shift**, including
+the original order (`s=0`) and PR2571's shifted order (`s=4`). It does not claim
+an attainable optimum, a better supplier, or a bound on other local words.
+
+The exact rational certificate is [CERTIFICATE.json](CERTIFICATE.json).
+The code uses the standard library, exact rational arithmetic, explicit
+logarithm/exponential remainder bounds, and no LP solver or predecessor checker.
+Evidence is author-checked, with no independent human review claimed.
+
+## Statement and fixed inputs
+
+There are 74 replicas, 666 calls, 112 ports per idle-child group, ambient width
+73 and 66,304 live families. The per-call helper inventory is
+`{4:33, 5:9, 7:62, 8:78, 9:240}`, and the local paid profile in **both** providers
+is `{1:2369, 2:632, 3:320, 4:118, 5:26, 7:8}`. There are respectively
+49,728, 33,152, 33,152 and 33,152 bins of capacities 8, 16, 32 and 48.
+The resulting rank mass is 4,816,512 and deficit 23,680, independently recounted
+from these quantities.
+
+For integer `s >= 0`, sort calls `(stage, replica)` by
+
+```
+(4*stage + replica + s*[stage >= 1], stage, replica).
+```
+
+A helper occurrence must be placed whole into one bin inside its strict
+source-event window. The bin's unused capacity contributes its remaining rank
+to the global profile. These are necessary conditions of the physical
+contract. The proof drops family, port-assignment and other admission
+restrictions, so it applies to every fill satisfying the stronger contract.
+Adding nonnegative charges cannot weaken this obstruction.
+
+Source pins and provenance are in [INPUTS.json](INPUTS.json):
+
+- PR2530, zxv, `b4012098048e12500b2d33d2e84be281915a27ac`: provider inventories,
+  paid profiles and original call order.
+- PR2571, Rohan Arun, `faff03cf5c29dadd6e0eb51e406efdc921bfa683`: shifted call
+  order and comparison endpoint. Its actual README/build/order files use
+  `s=4`; this package binds the bytes, rather than the older PR description.
+- The immutable PR2435 source archive (`324d7d3590dc21003940a44cb0cbdc32b3620f25`) retained by PR2530, from Jeremy Kothe's
+  supplier: its `inputs/GLOBAL-TEMPLATE.json.gz` supplies the 30-event recipe,
+  idle-child capacities and 112 ports. Its historical replica count and bank
+  stock are not substituted for the declared 74-replica, 66,304-live model.
+  Both container and extracted-member hashes are recorded. The raw member is
+  retained verbatim so no network access or historical archive unpacking is
+  needed during verification.
+
+The new work starts at main `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`.
+The selected record is unchanged. The claim is in [discussion #2608](https://github.com/CrocSwap/integer-mult-bounds/discussions/2608).
+
+## 1. A cut forces 114 non-width-eight bins
+
+Take this fixed set of 102 rank-eight groups:
+
+| Idle children | Owner replicas, inclusive | Groups |
+| --- | --- | ---: |
+| 1, 3 | 0 through 24 | 50 |
+| 5, 7 | 0 through 16 | 34 |
+| 9, 11 | 0 through 8 | 18 |
+
+For children 1/3, a legal call is strictly after that replica's stage-1
+bridge and no later than its stage-2 call. For 5/7 replace stages 1/2 by 3/4;
+for 9/11 use 5/6. The event interpreter reconstructs these strict windows and
+a separate stage-position calculation checks all 444 rank-eight windows.
+
+The union of the 102 groups' eligible calls is the interval strictly after
+call `(1,0)` through call `(6,8)`, inclusive. For each pair of children,
+consecutive owner windows overlap: their opening keys increase by one while
+the closing key is four larger. The three resulting intervals also overlap.
+Their last closing keys all equal `32+s`, with stage 6 ordered last.
+
+Inside this union the counts from stages 1 through 8 are
+
+```
+28 + 25 + 21 + 17 + 13 + 9 + 4 + 0 = 117.
+```
+
+The stage-zero calls have replica indices from `s+5` through `s+32`,
+intersected with `0..73`. Thus the number of eligible calls is exactly
+
+\[
+ U_s=117+\max(0,\min(28,69-s))\le145.
+\]
+
+Each eligible call supplies only 78 width-eight roles. Each of the 102 groups
+has 112 distinct bins, and a rank-eight bin can contain at most one
+width-eight role. Therefore at least
+
+\[
+ 102\cdot112-145\cdot78 = \boxed{114}
+\]
+
+of these bins must contain **no width-eight role**. The exact bound at shift
+`s` is `11424 - 78*U_s`; it is 114 for `0 <= s <= 41` and only grows later.
+For `s=0` the interval is call indices `[6,150]`; for `s=4` it is `[10,154]`.
+
+The checker enumerates shifts 0 through 70. For `s >= 70`, every stage-zero
+key is smaller than every later-stage key, and later stages all receive the
+same shift. The entire call order is consequently constant. This proves the
+infinite tail; the large-shift replay is an additional control.
+
+This is **not a claim of 114 empty bins**. A bin without a width-eight role
+may, for example, contain two width-four roles. Its cost is charged through
+the full width-inventory dual below.
+
+## 2. Convert the cut into a complete moment lower bound
+
+Let `h_a(r)=(r/73)^(1-a)` for positive `r`, and `h_a(0)=0`. Enumerate every
+whole-helper pattern `p=(R,n4,n5,n7,n8,n9)` with total width at most `R`.
+There are exactly 1,163 patterns. Let `rem(p)` be the remaining capacity.
+Find bin prices `y_R`, width prices `z_w`, and a positive `delta` such that
+
+\[
+ y_R+\sum_w n_wz_w+
+ \delta\,\mathbf1_{R=8,\ n_8=0}\le h_a(\operatorname{rem}(p))
+\]
+
+for **all** 1,163 patterns. Sum over all physical bins. All helper roles are
+used exactly once, and the cut supplies at least 114 indicator terms. Hence
+
+\[
+ 66304\,\mu(a)\ \ge\
+ 666\sum_r P_rh_a(r)+\sum_R N_Ry_R+
+ 666\sum_w I_wz_w+114\delta.
+\]
+
+This includes the complete fixed local profile and every residual bin.
+It does not infer a moment gain from scalar operation counts.
+
+A convenient dual nomination has `t=(h_a(3)-h_a(48))/5` and
+`Y=h_a(5)-3t`, with
+
+```
+(y8,y16,y32,y48) = (Y/4,Y/2,Y,h_a(48))
+(z4,z5,z7,z8,z9) = (h_a(1)-h_a(5),-h_a(5),(t-h_a(5))/2,-Y/4,t).
+```
+
+The checker derives this both by exact elimination and by these formulas.
+It then uses rational **lower** enclosures for the costs, repairs each bin
+price downwards if needed, and checks every inequality. `delta` is the
+smallest remaining slack of a rank-eight pattern without width eight; no
+floating-point optimizer is trusted. Exact prices and repairs are saved in
+the certificate.
+
+At `a=0.001714127696`, the following displayed decimals are only summaries
+of the exact rational comparisons:
+
+| Certified quantity | Value, approximately |
+| --- | ---: |
+| Per-defect penalty `delta` | 0.000167626935270 |
+| Window-free dual lower bill | 66303.9808907017 |
+| Chronological lower bill | 66304.0000001723 |
+| Strict excess above 66304 | 0.000000172320646 |
+
+Thus `mu(a)>1`. Every positive-rank cost is increasing in `a`, so no fill in
+the stated family can have endpoint `b >= a`.
+
+The strengthening is genuine even against the window-free **fractional**
+pattern LP. A separate exact feasible primal, on the nine patterns used in
+the dual nomination, has counts
+
+```
+49728, 33152, 555, 1110, 3515, 5994, 21978, 132793/5, 32967/5.
+```
+
+The checker verifies all bin and width equations. Using **upper** cost
+enclosures, its total bill is below 66304 at the excluded threshold. This
+primal is an LP witness, not a physical fill. It violates the chronological
+cut because every rank-eight bin uses a width-eight role.
+
+The pinned PR2571 complex endpoint is
+`428508074646920477229/250000000000000000000000`. It is below the new cap by
+less than `9.54e-8`, and the checker confirms that this bound does not reject
+that endpoint. Its construction and native admission are not reverified here.
+
+## Reproduction and controls
+
+From the repository root, with Python 3.12 and no third-party dependencies:
+
+```sh
+python3 -B scripts/research_package.py --package research/t74-diagonal-window-bound
+python3 -I -O -B research/t74-diagonal-window-bound/verify.py
+```
+
+Expected output includes:
+
+```
+PASS_T74_DIAGONAL_WINDOW_BOUND shifts=71+stable_tail groups=102 deficit>=114 patterns=1163 b<0.001714127696 controls=10 no_new_kappa
+```
+
+The default replay is read-only and compares every field of the regenerated
+certificate, including JSON types. `--write-certificate` explicitly regenerates
+it. Input pins check the retained bytes; direct Git-source inputs also carry
+Git blob identities.
+
+Positive checks include the two window derivations, the closed-form union
+count, the exact primal/dual equations, square-root/linear/constant identities
+for the transcendental bounds, and non-rejection of the predecessor endpoint.
+Ten failure controls reject changed source bytes, reversed replica stages,
+an inclusive bridge endpoint, an omitted source bridge, a duplicate cut group,
+a claimed defect of 115, unbound width-eight supply 79, an inflated bin dual,
+an inflated defect penalty, and an unbalanced fractional primal. The checks
+remain enabled under `python -O`.
+
+## Limits
+
+This is a lower bound for fixed whole-helper widths, the named source recipe,
+fixed live stock and this order family. It neither constructs nor optimizes
+a physical fill. Arbitrary nonuniform shifts, other interleavings, split
+helper reservations, different stocks and changed local profiles are outside
+its numerical theorem. In particular, PR2588 changes the local profile: its
+claimed endpoint is not contradicted by this bound. The reusable 114-bin cut
+can be repriced for a changed profile only after binding the new inputs.
+
+The provider profiles are pinned declared inputs, not an independent replay
+of their scalar/native words. No complete cold supplier replay, physical
+hosting admission, all-size/compiler proof, precision/exact-recovery audit,
+outer transfer audit, bit-supplier audit, Lean proof, or full multiplication
+theorem is claimed.
+
+## Open next questions
+
+- **Escape the cut with a nonuniform stage shift.** It matters because every
+  uniform initial delay retains the shortage. Start by giving the named 102
+  groups at least 147 eligible calls, or deriving a different inventory-aware
+  cut if the groups change. This only removes this necessary obstruction;
+  any candidate still needs a fully charged fill and native/global admission.
+- **Tighten the fixed-order bound with coupled width constraints.** Width-seven
+  shortages alone do not imply a positive reduced cost: a rank-16 bin can use
+  two width-eight roles at zero reduced cost. A feasible first step is to
+  couple that substitution to the rank-eight cut in one exact dual. The
+  blocker is a checked stronger inequality, not another moment root search.
+
+Structured versions of these questions are in [TASK.json](TASK.json).
+
+## Credit
+
+Rohan Arun's PR2571 supplies the shifted-order question; zxv's PR2530 supplies
+the source-partner words, diagonal order, interval search and owner pipeline.
+Jeremy Kothe's PR2435 and T74 supplier lineage supply the retained source
+recipe. The window-free pattern-dual method is due to Ali Romman (PR1129),
+Jeremy Diviney (PR2360), and Sagar Thakrar (PR2387/PR2442). This contribution
+adds the named chronological cut, its uniform-shift formula, the strengthened
+exact dual and a separately written standard-library checker.
+
+Jeremy Kothe/PR2575 and PR2588, Philippe Van Velzen/PR2559, Tekalign/PR2436,
+Vlado Vojdanovski/PR2322, Cinna Davis/PR2226, and the retained supplier authors
+are credited for the neighboring constructions and mechanisms. No such
+construction is imported as a newly proved result. Retained input bytes keep
+their repository Apache-2.0 provenance; the new work is also Apache-2.0.

research/t74-diagonal-window-bound/SOURCES.json

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+{
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+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "e8870e5917057d59b730eb14721a778639eca050",
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