Record that the selected k8 word emits no paid child above rank 8.

research/selected-k8-child-rank-bound/README.md

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+# Largest paid child rank of the selected k8 word
+
+**Scoped rank bound. No new κ.** The selected conditional result stays
+`0.001672660197076918`.
+
+The rank-cap handoff on the selected complex local word asks for the largest
+child rank a physical word on that template can emit, counting frame climbs
+and the cone restrictions that restoration imposes. On the selected k8 word,
+content hash `c2b294300760e1fd6eeab8c22c4d13e4633dca40e52cac8da0d68163a76e3ca8`,
+the answer is **8**.
+
+## Contract
+
+Frames are subspaces of `F2^9`. A legal assignment may change the frame of any
+A, B or early-cleanup gate inside its forward/backward interval. Frames stay
+nested along each register. Centre and target pins, start and end frames, and
+the low-end cone ceilings stay fixed. This is the frame contract of the
+selected word's native auditor, enlarged only by charging a climb into a pin
+instead of requiring that climb to be zero.
+
+A paid child is one of two things:
+
+- a strict climb of dimension `d = dim(F') - dim(F)` on one register edge;
+- one of the eight centre descents, read off the fixed centre frames and the
+  cut.
+
+Each register pays its own climb. Idle packing at rank up to 51 is not a child
+of this word.
+
+## Result
+
+Every legal assignment obeys the following, recomputed from the pinned word:
+
+| Fact | Value |
+|---|---|
+| Largest graph climb | **5** |
+| Centre descents | eight children, each of rank **8** |
+| Largest paid child rank | **8**, attained by those descents |
+| Rank 44 emitted | no |
+| Free / forced gates | 1,741 / 3,672 |
+| Interval sizes | 1,098 gates with 2 frames, 613 with 5, 30 with 16 |
+| Edges whose interval allows a jump above the recorded jump | 3,222 (multiplicity) |
+| Same climb cap with cone ceilings removed | still 5 |
+
+The 82 register-edges that can jump by 5 are exactly the support of rank 5.
+Their interval minima are 5 (32 edges), 4 (26) and 3 (24). The recorded word
+uses jump 5 on all 82, so rank 5 is attained and no legal assignment adds
+another rank-5 climb. Fifty of those edges can drop. Other edges can rise
+only as far as 4. No edge interval contains a jump of 6 or more. Reassignment
+can move mass among ranks 1 through 5. It cannot create a paid child of rank
+9, 16, 44 or 51.
+
+A second walk, charging the recorded sections the way the native auditor does,
+reproduces the published profile `{1:2412, 2:418, 3:432, 4:142, 5:82, 8:8}`.
+The two routes share no climb loop. The free-gate and interval counts match
+the census already published for this word's moment bound.
+
+The ambient lemma is separate and weaker: any nested pair of subspaces of
+`F2^9` jumps by at most 9. The edge intervals give 5. A vector with bit 9 set
+is rejected. A claimed cap of 4 is rejected because 82 edges allow 5.
+
+## What this does not say
+
+This is the native climb alphabet of one fixed scalar word. It does not price
+a different child identity that bundles several rank-1 climbs into one block
+child, and it does not rerun the moment duals. A block whose rank is the
+number of gathered payloads would be a changed tariff. The recorded word has
+no single-gate incoming climb mass above 8, but that observation is not a
+census of multi-gate co-frame groups. Changed gates, entrances or end frames
+can leave this interval data.
+
+No supplier, invoice, native admission or selected-frontier change is claimed.
+
+## Reproduce
+
+Python 3.12, standard library only, from the repository root. About two
+seconds.
+
+```sh
+python3.12 -I -B research/selected-k8-child-rank-bound/verify.py
+python3.12 -B scripts/research_package.py --package research/selected-k8-child-rank-bound
+```
+
+Expected final line:
+
+```text
+PASS_SELECTED_K8_CHILD_RANK_BOUND max_paid_rank=8 max_climb=5 centre=8x8 rank44=0 free_gates=1741 increasable_edges=3222 no_new_kappa
+```
+
+`verify.py --write` regenerates `RESULT.json`. Without `--write` the checker
+requires that receipt byte for byte.
+
+## Open next questions
+
+- **Can one co-frame block on this same word have rank above 8?** The native
+  alphabet charges each climb separately, so a block child is a different
+  identity. The frames needed for that census are the intervals built here.
+  Owned temporary lifetimes are still not in the word archive, which is the
+  remaining blocker named by the co-frame gate screen.
+- **Does one legal scalar edit open a register edge of jump 6 or more?** The
+  cap of 5 is a property of this gate list and these pins. A rewrite has to
+  change an interval before the rank bound moves.
+
+## Credit
+
+The word, the `F2^9` frame contract and the published interval census are the
+selected k8 frame-ceiling package (PR #1671, Gareth Davidson, using navinor's
+frame method and the gauge44 auditor). The rank-44 threshold this bound is
+compared with is the open question in Discussion #1166 comment 19021222,
+recorded from PR #3701. The separate block-reblocking tariff is PR #3039,
+PR #3066 and `research/rank-shape-coframe-bill-seed850bc3`. This package does
+not import those programs.
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. No independent
+human or formal-kernel review is claimed. Apache-2.0.

research/selected-k8-child-rank-bound/RESULT.json

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+{
+  "ambient_dimension": 9,
+  "centre_descent_count": 8,
+  "centre_descent_rank": 8,
+  "edge_max_histogram": {
+    "0": 5529,
+    "1": 2712,
+    "2": 1790,
+    "3": 738,
+    "4": 254,
+    "5": 82
+  },
+  "edges_whose_jump_can_increase": 3222,
+  "forced_gates": 3672,
+  "free_gates": 1741,
+  "given_profile": {
+    "1": 2412,
+    "2": 418,
+    "3": 432,
+    "4": 142,
+    "5": 82,
+    "8": 8
+  },
+  "interval_sizes": {
+    "2": 1098,
+    "5": 613,
+    "16": 30
+  },
+  "largest_graph_climb": 5,
+  "largest_paid_child_rank": 8,
+  "max_climb_without_cone_ceilings": 5,
+  "no_new_kappa": true,
+  "rank5_capable_edges_by_minimum": {
+    "3": 24,
+    "4": 26,
+    "5": 32
+  },
+  "rank_44_emitted": false,
+  "schema": "selected-k8-child-rank-bound-v1",
+  "section_walk_agrees": true,
+  "word_sha256": "c2b294300760e1fd6eeab8c22c4d13e4633dca40e52cac8da0d68163a76e3ca8"
+}

research/selected-k8-child-rank-bound/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "bb4bccbf42844d2801487450727c2f63e10d1d0f",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "research/r480-owner-optimum/inputs/local/FULL-WORD.json.gz",
+      "usage": "input",
+      "git_blob": "723decbca8995e916d6f4b783aa3e1ddd4eb47fe",
+      "sha256": "49596aa4ed4fd36cd4582fd0e8ddeb09fd80370a7d8aea1ede52a135b536d289"
+    }
+  ]
+}

research/selected-k8-child-rank-bound/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 6,
+  "base_commit": "bb4bccbf42844d2801487450727c2f63e10d1d0f",
+  "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": [
+      "docs/research/research-family-status.md",
+      "research/complex-local-word-ceiling/README.md",
+      "research/complex-decoder-field-extension-seed44/README.md",
+      "research/complex-frame-pair-decoders/README.md",
+      "research/selected-k8-unrestricted-ends/README.md",
+      "research/eisenstein-reduction-copy-floor/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. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "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.",
+      "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013.",
+      "Searching below C=40 in the #2017 Eisenstein binary scalar-rectangle or fixed-support block-linear additive pairing model; specify a changed premise instead. This bound does not cover every complex decoder.",
+      "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts."
+    ],
+    "overlap_terms": [
+      "rank shape",
+      "block climb",
+      "masked decoder",
+      "overcomplete",
+      "effective rank"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "On the selected k8 word, what is the largest paid child rank any legal monotone F2^9 frame assignment can emit?",
+  "scope_change_reason": "The card's second method asks for the best effective child rank on one fixed support alphabet, with copies, frames and restoration counted. This package answers that bound for the selected k8 native frame alphabet. It does not synthesize a block decoder or a profile aimed at the +5% gate.",
+  "bottleneck_link": "Rank-1 children are about half of recursive cost. The rank-cap handoff says a target-scale local saving on this template needs a paid child of rank at least 44. This word's frame alphabet never emits a paid child above rank 8, so that histogram is not a child of the selected word.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the edge-interval census and the independent section walk agree on the published profile and on a climb cap of 5. Do not search a different scalar word or a block-bundling identity.",
+  "outcome": "completed",
+  "evidence": "RESULT.json: largest paid child rank 8, attained by eight fixed centre descents; every register-edge interval has dimension jump at most 5 (82 edges can reach 5; their minima are 5, 4 and 3 on 32, 26 and 24 edges); 1741 free and 3672 forced gates with interval sizes 2/5/16 matching the published census; cone ceilings do not change the cap; section walk reproduces the published profile; rank 44 does not occur.",
+  "author": "unpaidmoon",
+  "ai_assistance": "Substantial Cursor Grok assistance. No independent human or formal-kernel review claimed. The checker does not import the predecessor frame model.",
+  "open_questions": [
+    {
+      "question": "Can one co-frame block on this same scalar word have rank above 8?",
+      "why_it_matters": "The native alphabet charges each climb separately. A block child that replaces several rank-1 climbs by one higher-rank child is a different tariff, and it is the identity the rank-cap handoff would need in order to emit rank 44.",
+      "first_step": "Partition the 2412 recorded rank-1 climbs into co-frame groups using the intervals this census builds, then price one group with the existing per-block restoration tax.",
+      "blocked_by": "Owned temporary lifetimes are still not recorded in the word archive."
+    },
+    {
+      "question": "Does one legal scalar edit open a register edge whose interval jump exceeds 5?",
+      "why_it_matters": "The climb cap is a property of this fixed gate list, pins and end frames. A different word can leave the interval data.",
+      "first_step": "Name one legal rewrite that changes an active frame interval and recompute the edge-maximum census on that word.",
+      "blocked_by": "No such rewrite is proposed in this package."
+    }
+  ]
+}

research/selected-k8-child-rank-bound/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/r480-owner-optimum/inputs/local/FULL-WORD.json.gz"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Selected-word hash, edge-interval climb cap 5, eight rank-8 centre descents, largest paid rank 8, free-gate tie-out 1741, rank-5 minima, cone-ceiling sensitivity, section-walk profile, outside-vector and false-cap-4 controls, RESULT.json byte match."
+    }
+  ]
+}