diff --git a/research/complex-rank-one-block-budget/ENVIRONMENT.json b/research/complex-rank-one-block-budget/ENVIRONMENT.json
new file mode 100644
index 0000000000..d51c8a61d5
--- /dev/null
+++ b/research/complex-rank-one-block-budget/ENVIRONMENT.json
@@ -0,0 +1,20 @@
+{
+  "generation": {
+    "python": "CPython 3.12.14",
+    "platform": "macOS-26.3.1-arm64-arm-64bit-Mach-O",
+    "command": "python3.12 -I -B research/complex-rank-one-block-budget/verify.py --write"
+  },
+  "dependencies": "Python standard library only",
+  "not_run": [
+    "full selected-frontier cold replay",
+    "native compiler or hosted supplier admission",
+    "Lean",
+    "hosted CI",
+    "independent human review"
+  ],
+  "executed": [
+    "python3.12 -I -B scripts/research_package.py --package research/complex-rank-one-block-budget",
+    "python3 -I -O -B research/complex-rank-one-block-budget/verify.py (CPython 3.14.7)",
+    "git diff --check"
+  ]
+}
diff --git a/research/complex-rank-one-block-budget/README.md b/research/complex-rank-one-block-budget/README.md
new file mode 100644
index 0000000000..b6986916ea
--- /dev/null
+++ b/research/complex-rank-one-block-budget/README.md
@@ -0,0 +1,172 @@
+# Sharp integer block budgets for the selected complex rank-one payload
+
+**Cost-alphabet screening result; no physical decoder or new κ.** At block rank at most 16,
+at least **363 of the selected word's 2,412 rank-one children per call** must be reblocked
+to pass the optimistic primitive-saving target below. Reblocking all 2,412 permits at most
+**10 additional integer rank units** of paid work. In particular, one additional rank-16
+call already fails. Charging one extra rank-one call per nontrivial output block erases
+every block's saving for every allowed rank 2..51 at this target and below.
+
+This supplies a concrete gathering/restoration budget for the revision-3 rank-shape card.
+It adds partial integer thresholds and additional-work pricing to Douglas Colkitt's
+retained all-mass ideal lever calculations. A decoder that passes these necessary cost
+conditions still needs its actual chronological frame and dirty-state realization.
+
+## Model and results
+
+The pinned selected local profile is `{1:2412, 2:418, 3:432, 4:142, 5:82, 8:8}`,
+with rank mass 5,586. Keep every non-rank-one child, the 64-per-call centre charge,
+width 73, 896 live families per nine-stage replica, and old idle mass 1,646 per call.
+For a chosen `k` in 0..2412, replace `k` unit-rank payload calls by integer ranks in
+`1..B`, preserving mass `k`, where `B` ranges from 2 to 51. All compatibility and
+scheduling restrictions are optimistically removed. Idle work is fractionally packed
+at rank 51. Additional copies, gathering or restoration are paid **on top** of this
+payload and idle bill; they receive no old idle credit.
+
+The exact target is `a*=35491136290869/20000000000000000`, or about 0.001774556815:
+21/20 times the baseline's certified *upper* primitive-ceiling bracket
+`1690054109089/10^15`. Acceptance here is a strict relaxation moment below one at
+`a*`. Thus a passing relaxation has more than 5% primitive-ceiling headroom relative
+to that upper bracket. This is not a 5% κ improvement or a realized supplier.
+The selected bit budget would already bind a full construction at this complex target.
+
+| Maximum block rank B | Minimum replaced k | Maximum extra rank-one calls after replacing all 2412 |
+|---:|---:|---:|
+| 2 | 1440 | 1 |
+| 4 | 722 | 4 |
+| 8 | 482 | 7 |
+| 16 | 363 | 10 |
+| 32 | 291 | 12 |
+| 51 | 255 | 14 |
+
+The full 50-cap table, adjacent strict signs and exact rational intervals are in
+[RESULT.json](RESULT.json). For B=16, the first passing partition is 22 rank-16
+blocks plus one rank-11 block. Its remaining budget is less than 0.004311 of a
+rank-one call, so it cannot pay even one additional rank unit. Full replacement
+is 150 rank-16 blocks plus one rank-12 block. Its maximal same-rank overhead bills
+are 10 rank-one, 5 rank-two, 2 rank-four, or 1 rank-eight call; these are alternatives,
+not simultaneous allowances. Any integer-rank overhead of mass at least 11 fails.
+
+Even putting all rank-one payload at the largest allowed block rank 51 gives a
+relaxation moment above **1.01133** at `a=1/99`. So changes to the rank-one payload
+alone cannot meet that planning target under this model. The predecessor already
+excludes 1/99 even after deleting all rank-one costs; this reblocking check is a
+corroborating, mass-preserving boundary, not a new exclusion of that target.
+
+## Proof of optimality and the block tax
+
+Write `q=1-a`, with `0<q<1`, and `c(r)=(r/73)^q`. Concavity gives a decreasing
+increment `c(x+1)-c(x)`. For two ranks `1<=x<=y<B`, transferring one unit from
+the smaller to the larger decreases total cost (remove the smaller at zero).
+Iterate until at most one non-full rank remains. Hence the minimum at total
+integer payload mass `k` is attained uniquely by `floor(k/B)` blocks of rank B
+and, when nonzero, a block of rank `k mod B`. This is optimal in the relaxed cost
+alphabet; no existence of a physical block decoder is inferred.
+
+With `P(k,B)` the resulting complete local profile, the lower moment is
+
+`F(k,B,a) = [sum_r P(k,B)_r c(r) + (1646/51)c(51)] / (896/9)`.
+
+Its value is nonincreasing in `k`: a partial block's next marginal cost is at
+most `c(1)`, and starting a new singleton leaves the bill unchanged. This
+justifies bisection over all `k=0..2412`. Both engines prove strict rejection
+at `k_min-1` and strict acceptance at `k_min`; this proves the threshold for
+every partition, not merely the partitions visited by bisection.
+
+An additional profile `J` contributes `sum_r J_r c(r)/(896/9)` to F. For a
+same-rank overhead call, solve this strict inequality and certify the adjacent
+integer counts. For arbitrary integer overhead mass, the same concavity lemma
+minimizes its cost by full rank-51 blocks plus a remainder. In the full B=16
+case, one rank-10 overhead call passes and one rank-11 fails; monotonicity of
+the optimal overhead cost excludes every mass at least 11.
+
+For one rank-r payload block, the saving in rank-one call units is `r-r^(1-a)`.
+Every r=2..51 has certified saving below one at a*. The largest is at r=51,
+below 0.354602. This expression increases with a, so the inequality holds
+for every `0<=a<=a*`. Paying one extra rank-one call per nontrivial block
+therefore makes each block strictly more expensive than its original singles.
+This proof applies to any partition or replacement count. It is conditional
+on that overhead tariff; the package does not prove a compiler must incur it.
+
+## Arithmetic and verification
+
+The retained `ceiling.py` uses atanh/logarithm and exponential Taylor intervals.
+The separately implemented check in `verify.py` uses no logarithm or exponential
+code: for `z=1-r/73` and `q=1-a`, expand
+`(1-z)^q=1-sum_{n>=1} t_n`, where `t_1=q*z` and
+`t_(n+1)=t_n*z*(n-q)/(n+1)`. All terms are positive, and every successive ratio
+is less than z. After a partial sum, the remaining tail is at most
+`t_(n+1)/(1-z)`. Integer interval arithmetic rounds outward at 160 bits and
+stops with tail at most 2^-90. Both methods certify the baseline bracket,
+all thresholds, all overhead decisions, and all 50 block-tax inequalities.
+
+Replay also compares 10,274 complete integer-partition fixtures at two rational
+savings and runs eight domain/mathematical controls. A deliberately incorrect
+invoice which subtracts extra work from old idle mass gives a false pass at
+the first rejected rank-one overhead count; the correct additional bill rejects
+it. `test_package.py` checks source drift/inventory omission, numeric and bool/int receipt drift,
+an incorrect arithmetic oracle, all 72 square-root oracle bounds by exact squaring,
+and optimized replay in a read-only minimal
+source relocation. These checks support the written proof; they are not a
+formalization or independent human review.
+
+## Reproduce
+
+From the repository root, Python 3.12 and standard library only:
+
+```sh
+python3.12 scripts/research_package.py --package research/complex-rank-one-block-budget
+```
+
+Expected markers: `PASS_COMPLEX_RANK_ONE_BLOCK_BUDGET` and
+`PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed`. The direct checker
+supports `-I -O -B`; `verify.py --write` regenerates RESULT.json. Source inputs
+are pinned by Git blob and SHA256 in [SOURCES.json](SOURCES.json); generic replay
+also checks historical task/selected-record provenance. [VERIFY.json](VERIFY.json)
+declares all four live inputs. Recorded environments and unrun checks are in
+[ENVIRONMENT.json](ENVIRONMENT.json).
+
+## Limits
+
+This is an exact cost-alphabet relaxation and a conditional per-block tariff
+obstruction. It supplies no decoder identity, support realization, physical
+gathering, dirty-state restoration construction, actual frame grouping,
+compiler/controller admission, hosting or new outer witness. Changed coverage,
+centre sharing, other local children, ranks above 51, different widths or
+templates, cross-call amortization and changed outer guards require their own
+model. Full selected cold replay, native execution, precision/tape obligations,
+Lean, hosted CI and independent human review were not supplied. The selected
+record, frozen sources and upstream inputs are unchanged.
+
+## Open next questions
+
+- Can a concrete rank-16 decoder meet this narrow gathering/restoration budget?
+  Pin one chronological climb group and exhibit its dirty-state identity and
+  complete additional paid profile. The blocker is physical realization; the
+  arithmetic alphabet alone supplies no free batching contract.
+- Can changed rank-two/rank-three delivery or centre sharing add enough budget
+  for 1/99? Derive one replacement local profile and its applicable coverage
+  law, then include every additional cost before screening. The blocker is the
+  missing new supplier interface, rather than an unexplored packing seed.
+
+Structured questions and the changed scope declaration are in [TASK.json](TASK.json).
+
+## Credit
+
+Started from current main `adc34159e0d6dc3dd2af2bdba3f5ca7376ef957e`, chooser nonce
+`3ab0bbddb657f850e7894cfc57f8d29e`, priority revision 3. The scope changed from
+physical decoder synthesis to its optimistic screening experiment. Coordination:
+[Discussion1596](https://github.com/CrocSwap/integer-mult-bounds/discussions/1596).
+
+Credit Douglas Colkitt / #1240 for the selected local-word ceiling, profile and
+ideal lever model; Sagar Thakrar / #1012 for the coverage law; Rohan Arun / #1094
+and #1042, Jeremy Kothe's mutable-leaver/packing lineage, Hao Yan / FanPUYen's
+controller, Josie Elliston's owner audit, and the retained construction sources
+named in the selected review. #1264 surveys pending packing ceilings; #1334
+prices ideal bit-side levers. Neither partial integer reblocking nor its extra
+work tariff is inferred from those different scopes. Existing Apache-2.0 source
+notices are retained; no third-party package is copied.
+
+Prepared by Arham Tahir with substantial OpenAI Codex assistance in the research,
+proof, both arithmetic implementations, tests and documentation. No subagents.
+New material is Apache-2.0.
diff --git a/research/complex-rank-one-block-budget/RESULT.json b/research/complex-rank-one-block-budget/RESULT.json
new file mode 100644
index 0000000000..7bff1e841d
--- /dev/null
+++ b/research/complex-rank-one-block-budget/RESULT.json
@@ -0,0 +1,1789 @@
+{
+  "idle_mass": 1646,
+  "local_profile": {
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+    "2": 418,
+    "3": 432,
+    "4": 142,
+    "5": 82,
+    "8": 8
+  },
+  "mathematical_controls": 8,
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+        "31495447098747428246945503293214096379333128239878225646735233707070810906758869/31495448272550005155211307922363110936089435829054233418732462850152371262062592",
+        "1763745037529855981828948184419989397242655181433180636217173087595965410779003025/1763745103262800288691833243652334212421008406427037071449017919608532790675505152"
+      ]
+    },
+    {
+      "all_rank1_reblocked_extra_rank1_max": 14,
+      "cap": 51,
+      "extra_rank1_budget": [
+        "3660343550136192383367697791369490328227037596044984972001737997055133643775164/244542322739835195082498933098109800269601777876201508039131932361215092402755",
+        "3660343550136192383367697791369490328227037596044984972001737997055133643965121/244542322739835195082498933098109800269601777876201508039131932361215092402602"
+      ],
+      "minimum_replaced": 255,
+      "partition_at_threshold": {
+        "51": 5
+      },
+      "previous_moment": [
+        "110234156335427264706158414226381373333985521692271971611693169192798535411741485/110234068953925018043239577728270888276313025401689816965563619975533299417219072",
+        "881873250683418117649267313811050986671884173538175772893545353542388283294185443/881872551631400144345916621826167106210504203213518535724508959804266395337752576"
+      ],
+      "threshold_moment": [
+        "1763744374381280421960667052772358237022307099038218964654550382663929443289968223/1763745103262800288691833243652334212421008406427037071449017919608532790675505152",
+        "440936093595320105490166763193089559255576774759554741163637595665982360822618799/440936275815700072172958310913083553105252101606759267862254479902133197668876288"
+      ]
+    }
+  ],
+  "scope": "Optimal cost-alphabet relaxation; no decoder, frame legality or supplier admission.",
+  "status": "PASS_COMPLEX_RANK_ONE_BLOCK_BUDGET",
+  "target": "35491136290869/20000000000000000",
+  "target_definition": "21/20 times the certified baseline primitive ceiling upper bound"
+}
diff --git a/research/complex-rank-one-block-budget/SOURCES.json b/research/complex-rank-one-block-budget/SOURCES.json
new file mode 100644
index 0000000000..a6cdfba742
--- /dev/null
+++ b/research/complex-rank-one-block-budget/SOURCES.json
@@ -0,0 +1,42 @@
+{
+  "schema_version": 1,
+  "base_commit": "adc34159e0d6dc3dd2af2bdba3f5ca7376ef957e",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1",
+      "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075"
+    },
+    {
+      "path": "research/community-frontier1094-audit/ASSEMBLY.json",
+      "usage": "input",
+      "git_blob": "5fd676df060f3b450782fdf5ff5f6beb63a5eeb0",
+      "sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045"
+    },
+    {
+      "path": "research/community-frontier1094-audit/INDEPENDENT1094.json",
+      "usage": "input",
+      "git_blob": "009a385e26858058a18188364078d162aa6f9519",
+      "sha256": "301fb8a0e4ab7e3c383ff974c54c08178283b2a3453c5282757f692a3c7a4bfd"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/RESULT.json",
+      "usage": "input",
+      "git_blob": "77f55043a679dd722330f1e2499863e36376d9f9",
+      "sha256": "d91d6c46509875abf1f61de67c2609656d3dc685451299f83caa6f922493692d"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/ceiling.py",
+      "usage": "input",
+      "git_blob": "6c0f54527d879396f689523bfde19928a9b6157c",
+      "sha256": "86a1432870b3b21d8454106ec9162fddc0471472d9214c590a1420f2426cedfc"
+    }
+  ]
+}
diff --git a/research/complex-rank-one-block-budget/TASK.json b/research/complex-rank-one-block-budget/TASK.json
new file mode 100644
index 0000000000..bb4f8ba9de
--- /dev/null
+++ b/research/complex-rank-one-block-budget/TASK.json
@@ -0,0 +1,151 @@
+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 3,
+  "base_commit": "adc34159e0d6dc3dd2af2bdba3f5ca7376ef957e",
+  "original_card": {
+    "id": "complex-rank-shape-decoder",
+    "question": "Can a decoder deliver the local paid mass in fewer, larger-rank children (block frame climbs instead of rank-1 to rank-3 steps)?",
+    "starting_points": [
+      "research/complex-local-word-ceiling/README.md",
+      "research/complex-decoder-field-extension-seed44/README.md",
+      "research/complex-frame-pair-decoders/README.md"
+    ],
+    "methods": [
+      "Find one exact small decoding identity whose frame path climbs in blocks, and price its complete per-call profile.",
+      "Bound the best effective child rank achievable for one fixed support alphabet, including copies, frames and restoration."
+    ],
+    "acceptance_gate": "A per-call local profile, with all copied information, frames and restoration counted, whose ceiling exceeds the selected word's by at least 5%; masked or overcomplete supports must change an assumption of the eight-term all-pairs bound.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/complex-local-word-ceiling/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+      "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.",
+      "scope": "Positive control and planning budget only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin. Ideal budget: rank-16 delivery is 2.31x (levers.py).",
+    "bottleneck_link": "Rank-1 children are about half of recursive cost; the selected word's effective child rank is about 3.9. Larger-rank delivery lowers recursive cost per unit of paid mass.",
+    "allowed_variations": [
+      "One masked, overcomplete or block decoder family with exact labels and supports.",
+      "A scoped lower bound on effective rank for one alphabet.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "scope_change_triggers": [
+      "Repacking idle remainders of an unchanged word (covered by the packing ceiling).",
+      "Repeating the eight-term all-pairs search inside its proved scope."
+    ],
+    "overlap_terms": [
+      "rank shape",
+      "block climb",
+      "masked decoder",
+      "overcomplete",
+      "effective rank"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Sharp integer-rank cost-alphabet screening for replacing k of the selected local word's 2412 rank-one children by blocks of rank 2..B, B=2..51, retaining all other local costs and old idle mass. Charge added work on top; determine the 5%-above-baseline primitive-ceiling threshold and the one-added-call-per-block obstruction.",
+  "scope_change_reason": "The session completes the card's first optimistic budget experiment rather than synthesizing a physical decoder or proving which block ranks it can realize. The arbitrary integer block alphabet is an explicitly favorable relaxation; passing it does not satisfy the construction acceptance gate.",
+  "bottleneck_link": "Rank-one children dominate much of the binding complex recursion cost. A sharp partial-reblocking and overhead budget tells a proposed block decoder how much extra copy/restoration work it can afford before synthesis.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after all 50 block caps, sharp adjacent threshold signs, overhead bills, the 1/99 obstruction and source/receipt controls. Do not synthesize a new supplier or extend the alphabet.",
+  "outcome": "completed",
+  "evidence": "RESULT.json: rank16 minimum 363/2412; full rank16 reblocking permits at most 10 added rank units; one added rank1 call per output block precludes any local payload-cost improvement at a<=target for every rank2..51; 10274 partition fixtures and 8 mathematical controls.",
+  "author": "Arham Tahir",
+  "ai_assistance": "Substantial OpenAI Codex assistance in choosing the scope, proving the cost inequalities, implementing both interval methods, verification and documentation; no subagents or independent human review.",
+  "open_questions": [
+    {
+      "question": "Can a concrete rank16 block decoder combine at least 363 rank-one payload units with fewer than one added paid call per output block?",
+      "why_it_matters": "The cost gate admits zero added rank1 calls at the first threshold and only ten added rank units after complete reblocking. Naive per-block overhead fails.",
+      "first_step": "Select one actual chronological climb group, pin its labels and source/target frames, and exhibit a dirty-state identity plus its gathering/restoration profile. Compare its complete additional invoice against the budget in RESULT.json.",
+      "blocked_by": "No physical block-decoder realization or free gathering contract is supplied here."
+    },
+    {
+      "question": "What overhead budget remains if rank2/rank3 children also change, or if the centre path is shared?",
+      "why_it_matters": "Rank-one reblocking alone cannot reach a=1/99 even at maximum block rank51. Another paid component must change.",
+      "first_step": "Derive the completed replacement profile and applicable coverage identity; add its full additional-work moment before checking an optimistic ceiling.",
+      "blocked_by": "New local-word/centre realization and inherited supplier interfaces require separate proofs."
+    }
+  ],
+  "assignment_seed": "3ab0bbddb657f850e7894cfc57f8d29e",
+  "assignment_scope_key": "cad3ba853d7f324f16fa",
+  "discussion": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1596",
+  "overlap": {
+    "entry": {
+      "searched_at": "2026-10-14T03:40:50.966483+00:00",
+      "coverage": {
+        "open_prs": {
+          "fetched_at": "2026-10-14T03:40:05.744122+00:00",
+          "complete_pagination": true,
+          "records": 728
+        },
+        "discussions": {
+          "fetched_at": "2026-10-14T03:40:50.895804+00:00",
+          "complete_pagination": true,
+          "records": 239
+        }
+      },
+      "warnings": [],
+      "limits": "Remote PR scope is open PRs including drafts. Closed work is searched through local READMEs, TASK metadata and ledger only. No code diffs or arbitrary package files are searched. Recheck relevant heads before submission."
+    },
+    "nearest_work": [
+      {
+        "pr": 1240,
+        "author": "Douglas Colkitt",
+        "retained_package": "research/complex-local-word-ceiling",
+        "difference": "Prior all-mass ideal levers; this package adds partial integer thresholds and fully additional overhead.",
+        "head": "1e136b8c4a552050820334ad113628775182f08c"
+      },
+      {
+        "pr": 1264,
+        "author": "Douglas Colkitt",
+        "difference": "Pending-profile packing-cap census; no partial rank-one block tax.",
+        "head": "1be64e9ef79fe11f24f05b62b0fd7d70be0b7b4c"
+      },
+      {
+        "pr": 1334,
+        "head": "5ad74a4a04d1c9264768c43762e130690918b043",
+        "difference": "Bit-side ideal lever budgets, a different supplier and domain.",
+        "author": "Sagar Thakrar / thakrarsagar"
+      }
+    ],
+    "coverage_note": "No matches is not novelty evidence. Search covers bodies, comments, top-level local notes and ledger, not all source/diffs.",
+    "final": {
+      "searched_at": "2026-10-14T03:53:03.256060+00:00",
+      "terms": [
+        "rank-one block",
+        "block tax",
+        "reblock",
+        "partial integer",
+        "complex-rank-shape-decoder",
+        "rank shape",
+        "block climb",
+        "masked decoder",
+        "overcomplete",
+        "effective rank",
+        "complex-local-word-ceiling",
+        "complex-decoder-field-extension-seed44",
+        "complex-frame-pair-decoders"
+      ],
+      "coverage": {
+        "open_prs": {
+          "fetched_at": "2026-10-14T03:52:14.117624+00:00",
+          "complete_pagination": true,
+          "records": 733
+        },
+        "discussions": {
+          "fetched_at": "2026-10-14T03:53:03.173893+00:00",
+          "complete_pagination": true,
+          "records": 244
+        }
+      },
+      "warnings": [],
+      "limits": "Remote PR scope is open PRs including drafts. Closed work is searched through local READMEs, TASK metadata and ledger only. No code diffs or arbitrary package files are searched. Recheck relevant heads before submission."
+    },
+    "final_conclusion": "Specific integer reblocking/tax matches were the local package and its own Discussion1596 claim. No competing specific claim was located in the searched coverage; this is not proof of novelty."
+  },
+  "publication_upstream_checked": "81e81af67714bdc8d1693b0bf5c29468b92f62ff",
+  "upstream_refresh_note": "The current upstream main changes only the retained packing-claim census; guidance, selected record and all four live input blobs are identical to the starting main commit."
+}
diff --git a/research/complex-rank-one-block-budget/VERIFY.json b/research/complex-rank-one-block-budget/VERIFY.json
new file mode 100644
index 0000000000..cdbbcf0877
--- /dev/null
+++ b/research/complex-rank-one-block-budget/VERIFY.json
@@ -0,0 +1,34 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/community-frontier1094-audit/ASSEMBLY.json",
+    "research/community-frontier1094-audit/INDEPENDENT1094.json",
+    "research/complex-local-word-ceiling/RESULT.json",
+    "research/complex-local-word-ceiling/ceiling.py"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact adjacent signs for all 50 block caps, independent binomial crosschecks, integer partition fixtures, complete overhead budgets and mathematical controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/test_package.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Source and result drift controls, interval oracle mutation, and read-only minimal-source relocation."
+    }
+  ]
+}
diff --git a/research/complex-rank-one-block-budget/test_package.py b/research/complex-rank-one-block-budget/test_package.py
new file mode 100644
index 0000000000..2ee99eb064
--- /dev/null
+++ b/research/complex-rank-one-block-budget/test_package.py
@@ -0,0 +1,109 @@
+#!/usr/bin/env python3
+"""Scoped corruption, arithmetic-mutation and relocated-source controls.
+
+Arham Tahir with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+import importlib.util
+import json
+from pathlib import Path
+import shutil
+import subprocess
+import sys
+import tempfile
+import unittest
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+
+
+class PackageControls(unittest.TestCase):
+    def setUp(self):
+        self.tmp = tempfile.TemporaryDirectory(prefix='rank block control ')
+        self.addCleanup(self.tmp.cleanup)
+        self.root = Path(self.tmp.name)
+        self.package = self.root/'research'/HERE.name
+        self.package.mkdir(parents=True)
+        for name in ('verify.py','SOURCES.json','RESULT.json'):
+            shutil.copyfile(HERE/name,self.package/name)
+        for row in json.loads((HERE/'SOURCES.json').read_text())['files']:
+            if row['usage']=='input':
+                destination = self.root/row['path']
+                destination.parent.mkdir(parents=True,exist_ok=True)
+                shutil.copyfile(ROOT/row['path'],destination)
+
+    def replay(self, optimized=False):
+        command = [sys.executable,'-I','-B']
+        if optimized:
+            command.append('-O')
+        command.append(str(self.package/'verify.py'))
+        return subprocess.run(command,cwd=self.root,capture_output=True,text=True,timeout=45)
+
+    def test_readonly_relocation(self):
+        for path in self.root.rglob('*'):
+            if path.is_file():
+                path.chmod(0o444)
+        result = self.replay(optimized=True)
+        self.assertEqual(result.returncode,0,result.stderr)
+        self.assertIn('PASS_COMPLEX_RANK_ONE_BLOCK_BUDGET',result.stdout)
+
+    def test_source_drift(self):
+        source = self.root/'research/complex-local-word-ceiling/ceiling.py'
+        source.write_text(source.read_text()+'\n# corruption control\n')
+        result = self.replay()
+        self.assertNotEqual(result.returncode,0)
+        self.assertIn('source drift:',result.stderr)
+
+    def test_source_inventory_omission(self):
+        path = self.package/'SOURCES.json'
+        sources = json.loads(path.read_text())
+        sources['files'] = [row for row in sources['files']
+                            if row['path'] != 'research/complex-local-word-ceiling/ceiling.py']
+        path.write_text(json.dumps(sources))
+        result = self.replay()
+        self.assertNotEqual(result.returncode,0)
+        self.assertIn('source inventory',result.stderr)
+
+    def test_receipt_value_and_type_drift(self):
+        path = self.package/'RESULT.json'
+        original = json.loads(path.read_text())
+        changed = json.loads(json.dumps(original))
+        changed['rows'][14]['minimum_replaced'] -= 1
+        path.write_text(json.dumps(changed))
+        result = self.replay()
+        self.assertNotEqual(result.returncode,0)
+        self.assertIn('result drift',result.stderr)
+        changed = json.loads(json.dumps(original))
+        changed['rank16_complete_overhead_by_rank'][3]['max_calls'] = True
+        path.write_text(json.dumps(changed))
+        result = self.replay(optimized=True)
+        self.assertNotEqual(result.returncode,0)
+        self.assertIn('result drift',result.stderr)
+
+    def test_independent_oracle_mutation(self):
+        spec = importlib.util.spec_from_file_location('block_budget_control',self.package/'verify.py')
+        module = importlib.util.module_from_spec(spec)
+        spec.loader.exec_module(module)
+        original = module.binomial_cost
+
+        def wrong(r,a):
+            bounds = original(r,a)
+            return tuple(x+module.Q(1,10) for x in bounds)
+
+        module.binomial_cost = wrong
+        with self.assertRaisesRegex(ValueError,'baseline lower sign'):
+            module.verify()
+
+    def test_binomial_square_root_enclosures(self):
+        spec = importlib.util.spec_from_file_location('block_budget_sqrt',self.package/'verify.py')
+        module = importlib.util.module_from_spec(spec)
+        spec.loader.exec_module(module)
+        # At a=1/2, squaring checks the interval without either moment engine.
+        for rank in range(1,73):
+            lo,hi = module.binomial_cost(rank,module.Q(1,2))
+            self.assertGreater(lo,0)
+            self.assertLessEqual(lo*lo,module.Q(rank,73))
+            self.assertGreaterEqual(hi*hi,module.Q(rank,73))
+
+
+if __name__=='__main__':
+    unittest.main()
diff --git a/research/complex-rank-one-block-budget/verify.py b/research/complex-rank-one-block-budget/verify.py
new file mode 100644
index 0000000000..732d065d2f
--- /dev/null
+++ b/research/complex-rank-one-block-budget/verify.py
@@ -0,0 +1,302 @@
+#!/usr/bin/env python3
+"""Integer block-delivery screening, not supplier admission.
+
+Arham Tahir with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+import argparse
+from collections import Counter
+from fractions import Fraction as Q
+from functools import lru_cache
+import hashlib
+import importlib.util
+import json
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+WORD = {1: 2412, 2: 418, 3: 432, 4: 142, 5: 82, 8: 8}
+M, W, IDLE = 73, Q(896, 9), 1646
+BASE_HI = Q(1690054109089, 10**15)
+TARGET = Q(21, 20) * BASE_HI
+INPUTS = {
+    'research/complex-local-word-ceiling/ceiling.py',
+    'research/complex-local-word-ceiling/RESULT.json',
+    'research/community-frontier1094-audit/ASSEMBLY.json',
+    'research/community-frontier1094-audit/INDEPENDENT1094.json',
+}
+
+
+def require(ok, why):
+    if not ok:
+        raise ValueError(why)
+
+
+def sources():
+    pins = json.loads((HERE / 'SOURCES.json').read_text())
+    inputs = [pin['path'] for pin in pins['files'] if pin['usage']=='input']
+    require(len(inputs)==len(INPUTS) and set(inputs)==INPUTS, 'source inventory')
+    for pin in pins['files']:
+        if pin['usage'] == 'input':
+            require(hashlib.sha256((ROOT / pin['path']).read_bytes()).hexdigest() == pin['sha256'],
+                    'source drift: ' + pin['path'])
+    path = ROOT / 'research/complex-local-word-ceiling/ceiling.py'
+    spec = importlib.util.spec_from_file_location('retained_ceiling', path)
+    engine = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(engine)
+    return engine
+
+
+@lru_cache(None)
+def binomial_cost(r, a):
+    """Independent enclosure of (r/73)^(1-a), using the negative binomial tail.
+
+    For q=1-a and z=1-r/73, (1-z)^q=1-sum(t_n), all t_n>0;
+    t_1=q*z, t_(n+1)=t_n*z*(n-q)/(n+1). Every tail ratio is <z.
+    Outward integer rounding also encloses accumulated roundoff.
+    """
+    require(type(r) is int and 1 <= r < M and 0 < a < 1, 'binomial domain')
+    scale = 1 << 160
+    q, z = 1-a, Q(M-r, M)
+
+    def floor(x):
+        return x.numerator // x.denominator
+
+    def ceil(x):
+        return -((-x.numerator) // x.denominator)
+
+    lo, hi = floor(q*z*scale), ceil(q*z*scale)
+    sum_lo, sum_hi, n = lo, hi, 1
+    while True:
+        ratio = z*(n-q)/(n+1)
+        lo, hi = floor(lo*ratio), ceil(hi*ratio)
+        tail = ceil(hi/(1-z))
+        if tail <= 1 << 70:  # absolute remainder at most 2^-90
+            return Q(scale-sum_hi-tail, scale), Q(scale-sum_lo, scale)
+        sum_lo, sum_hi, n = sum_lo+lo, sum_hi+hi, n+1
+
+
+def partition(k, cap):
+    require(type(k) is int and 0 <= k <= WORD[1], 'replacement count')
+    require(type(cap) is int and 2 <= cap <= 51, 'block cap')
+    count, remainder = divmod(k, cap)
+    out = Counter({cap: count}) if count else Counter()
+    if remainder:
+        out[remainder] += 1
+    return dict(sorted(out.items()))
+
+
+def profile(k, cap):
+    out = Counter(WORD)
+    out[1] -= k
+    out.update(partition(k, cap))
+    return {r: n for r, n in sorted(out.items()) if n}
+
+
+def interval(profile_, a, cost, overhead=None):
+    """Old payload mass/idle are fixed; overhead is additional paid work."""
+    terms = dict(profile_)
+    terms[51] = terms.get(51, 0) + Q(IDLE, 51)
+    for r, n in (overhead or {}).items():
+        require(type(r) is int and 1 <= r <= 51 and Q(n) >= 0, 'overhead alphabet')
+        terms[r] = terms.get(r, 0) + Q(n)
+    lo = hi = Q(0)
+    for r, n in terms.items():
+        c = cost(r, a)
+        lo, hi = lo+n*c[0], hi+n*c[1]
+    return lo/W, hi/W
+
+
+def sign(bounds):
+    if bounds[1] < 1:
+        return -1
+    if bounds[0] > 1:
+        return 1
+    raise ValueError('unresolved strict moment sign')
+
+
+def threshold(cap, a, cost):
+    require(sign(interval(profile(0, cap), a, cost)) == 1, 'baseline must fail target')
+    if sign(interval(profile(WORD[1], cap), a, cost)) == 1:
+        return None
+    lo, hi = 0, WORD[1]
+    while hi-lo > 1:
+        mid = (lo+hi)//2
+        if sign(interval(profile(mid, cap), a, cost)) < 0:
+            hi = mid
+        else:
+            lo = mid
+    return hi
+
+
+def max_extra_ones(k, cap, a, cost):
+    bounds = interval(profile(k, cap), a, cost)
+    require(bounds[1] < 1, 'overhead needs a feasible relaxation')
+    c = cost(1, a)
+    lo = (1-bounds[1])*W/c[1]
+    hi = (1-bounds[0])*W/c[0]
+    # h must be STRICTLY below the affordable real number.
+    h = (lo.numerator-1)//lo.denominator
+    require(Q(h+1) > hi, 'overhead integer sign unresolved')
+    require(sign(interval(profile(k, cap), a, cost, {1:h})) < 0, 'overhead accepted sign')
+    require(sign(interval(profile(k, cap), a, cost, {1:h+1})) > 0, 'overhead rejected sign')
+    return h, (lo, hi)
+
+
+def small_partitions(total, cap, lower=1):
+    if total == 0:
+        yield ()
+    for first in range(lower, min(total, cap)+1):
+        for rest in small_partitions(total-first, cap, first):
+            yield (first,)+rest
+
+
+def verify():
+    C = sources()
+    @lru_cache(None)
+    def retained(r, a):
+        return C.child_cost(r, M, a)
+    baseline = json.loads((ROOT / 'research/complex-local-word-ceiling/RESULT.json').read_text())
+    asm = json.loads((ROOT / 'research/community-frontier1094-audit/ASSEMBLY.json').read_text())
+    owner = json.loads((ROOT / 'research/community-frontier1094-audit/INDEPENDENT1094.json').read_text())
+    H = {int(r):n for r,n in asm['complex_profile'].items()}
+    idle = {int(r):n for r,n in owner['owner']['idle'].items()}
+    calls, live = owner['price']['calls'], owner['price']['live']
+    require(calls == 144 and live == 14336 and Q(live,calls) == W, 'selected template')
+    require(H == {int(r):n for r,n in owner['owner']['profile'].items()}, 'selected profile')
+    require(all(H.get(r,0)-idle.get(r,0) == calls*WORD.get(r,0) for r in set(H)|set(idle)|set(WORD)),
+            'source-bound local profile')
+    require(sum(r*n for r,n in idle.items()) == calls*IDLE, 'idle mass')
+    require(sum(r*n for r,n in H.items()) == 72*live+64*calls, 'coverage mass')
+    require(BASE_HI == Q(baseline['ceiling']['a_ceiling']), 'baseline ceiling reference')
+    for cost in (retained, binomial_cost):
+        require(sign(interval(WORD, BASE_HI, cost)) > 0, 'baseline upper sign')
+        require(sign(interval(WORD, Q(baseline['ceiling']['a_relaxation_achievable']), cost)) < 0,
+                'baseline lower sign')
+
+    rows = []
+    for cap in range(2,52):
+        k = threshold(cap,TARGET,retained)
+        require(k == threshold(cap,TARGET,binomial_cost), 'independent threshold')
+        require(sum(r*n for r,n in profile(WORD[1],cap).items()) == 5586, 'reblocked mass')
+        row = {'cap':cap,'minimum_replaced':k}
+        if k is not None:
+            h, budget = max_extra_ones(WORD[1],cap,TARGET,retained)
+            h2, _ = max_extra_ones(WORD[1],cap,TARGET,binomial_cost)
+            require(h == h2, 'independent overhead count')
+            row.update(partition_at_threshold=partition(k,cap),
+                       threshold_moment=[str(x) for x in interval(profile(k,cap),TARGET,retained)],
+                       previous_moment=[str(x) for x in interval(profile(k-1,cap),TARGET,retained)],
+                       all_rank1_reblocked_extra_rank1_max=h,
+                       extra_rank1_budget=[str(x) for x in budget])
+        rows.append(row)
+
+    # Exhaustive positive fixtures for the concavity partition theorem.
+    fixtures = 0
+    for a in (TARGET,Q(1,99)):
+        for cap in range(2,9):
+            for total in range(1,19):
+                optimum = partition(total,cap)
+                best_hi = sum(n*binomial_cost(r,a)[1] for r,n in optimum.items())
+                for parts in small_partitions(total,cap):
+                    if Counter(parts) != Counter(optimum):
+                        alternative_lo = sum(binomial_cost(r,a)[0] for r in parts)
+                        require(alternative_lo > best_hi, 'integer-partition dominance')
+                    fixtures += 1
+
+    rank16 = next(row for row in rows if row['cap']==16)
+    k = rank16['minimum_replaced']
+    minimal_h, minimal_budget = max_extra_ones(k,16,TARGET,retained)
+    require(minimal_h == max_extra_ones(k,16,TARGET,binomial_cost)[0], 'minimal overhead crosscheck')
+    rank16['at_threshold_extra_rank1_max'] = minimal_h
+    rank16['at_threshold_extra_rank1_budget'] = [str(x) for x in minimal_budget]
+
+    # One added rank-one call per nontrivial output block erases its saving.
+    # r*c(1)-c(r) < c(1) for every allowed r; monotonicity extends to a<=TARGET.
+    tax_rows = []
+    for r in range(2,52):
+        engines = []
+        for cost in (retained,binomial_cost):
+            one, block = cost(1,TARGET), cost(r,TARGET)
+            saving = (r-block[1]/one[0], r-block[0]/one[1])
+            require(0 < saving[0] <= saving[1] < 1, 'one-call-per-block obstruction')
+            engines.append([str(x) for x in saving])
+        tax_rows.append({'rank':r, 'saving_in_rank1_call_units':engines})
+
+    # Sharp total overhead rank mass for complete rank-16 payload reblocking.
+    # At any fixed mass, the cheapest overhead is itself optimally blocked.
+    overhead_rows = []
+    for r in (1,2,4,8,16,32,51):
+        bounds = interval(profile(2412,16),TARGET,retained)
+        c = retained(r,TARGET)
+        lo,hi = (1-bounds[1])*W/c[1], (1-bounds[0])*W/c[0]
+        h = (lo.numerator-1)//lo.denominator
+        require(h+1 > hi, 'overhead rank threshold')
+        for cost in (retained,binomial_cost):
+            require(sign(interval(profile(2412,16),TARGET,cost,{r:h})) < 0, 'rank overhead pass')
+            require(sign(interval(profile(2412,16),TARGET,cost,{r:h+1})) > 0, 'rank overhead fail')
+        overhead_rows.append({'rank':r, 'max_calls':h, 'real_budget':[str(lo),str(hi)]})
+    for cost in (retained,binomial_cost):
+        require(sign(interval(profile(2412,16),TARGET,cost,{10:1})) < 0, 'mass10 overhead pass')
+        require(sign(interval(profile(2412,16),TARGET,cost,{11:1})) > 0, 'mass11 overhead fail')
+        # For any mass >=51, even fractional rank51 delivery exceeds the bill.
+        require(sign(interval(profile(2412,16),TARGET,cost,{51:1})) > 0, 'large overhead fail')
+
+    # Even putting ALL selected rank-one mass at rank 51 cannot pass 1/99.
+    target_large = []
+    for cost in (retained,binomial_cost):
+        bound = interval(profile(2412,51),Q(1,99),cost)
+        require(sign(bound) > 0, 'target-scale rank-one-only obstruction')
+        target_large.append([str(x) for x in bound])
+
+    controls = 0
+    for bad in ((-1,16),(2413,16),(True,16),(16,1),(16,52)):
+        try:
+            partition(*bad)
+        except ValueError:
+            controls += 1
+        else:
+            raise ValueError('invalid domain accepted')
+    # Added work cannot borrow idle credit. One more than the maximal bill fails.
+    h = rank16['all_rank1_reblocked_extra_rank1_max']+1
+    bad_profile = Counter(profile(2412,16)); bad_profile[1] += h
+    wrong = C.relax(dict(bad_profile),M,896,9,64,TARGET)
+    right = interval(profile(2412,16),TARGET,retained,{1:h})
+    require(sign(wrong)<0 and sign(right)>0, 'idle-debit mutation must give a false pass')
+    controls += 1
+    # Non-greedy equal mass and deletion-without-accounting controls.
+    require(sum(binomial_cost(r,TARGET)[0] for r in (8,8)) > binomial_cost(16,TARGET)[1],
+            'split-block control')
+    require(sum(r*n for r,n in {r:n for r,n in WORD.items() if r!=1}.items()) != 5586,
+            'deletion is not mass-preserving reblocking')
+    controls += 2
+    return {'status':'PASS_COMPLEX_RANK_ONE_BLOCK_BUDGET', 'target':str(TARGET),
+            'target_definition':'21/20 times the certified baseline primitive ceiling upper bound',
+            'local_profile':WORD, 'idle_mass':IDLE, 'rows':rows,
+            'one_extra_rank1_per_block_obstruction':tax_rows,
+            'rank16_complete_overhead_by_rank':overhead_rows,
+            'rank16_complete_max_extra_integer_rank_mass':10,
+            'rank_one_only_at_1_over_99':target_large, 'partition_fixtures':fixtures,
+            'mathematical_controls':controls,
+            'scope':'Optimal cost-alphabet relaxation; no decoder, frame legality or supplier admission.'}
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__)
+    ap.add_argument('--write',action='store_true')
+    args = ap.parse_args()
+    result = verify()
+    text = json.dumps(result,sort_keys=True,indent=2)+'\n'
+    if args.write:
+        (HERE/'RESULT.json').write_text(text)
+    else:
+        saved = json.loads((HERE/'RESULT.json').read_text())
+        require(json.dumps(saved,sort_keys=True) == json.dumps(json.loads(text),sort_keys=True),
+                'result drift')  # Preserve integer/bool types as well as values.
+    r16 = next(row for row in result['rows'] if row['cap']==16)
+    print(result['status'], 'rank16_minimum=',r16['minimum_replaced'],
+          'extra_rank1_max=',r16['all_rank1_reblocked_extra_rank1_max'],
+          'partition_fixtures=',result['partition_fixtures'],'controls=',result['mathematical_controls'])
+
+
+if __name__ == '__main__':
+    main()