Record that monotone selected entrances miss the rank-8/u2 16-child c…

research/rank8-u2-entrance-obstruction/README.md

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+# Monotone entrances miss the rank-8/u2 16-child cell
+
+**Negative result. No new κ.** The selected conditional exponent stays
+`60403/20000000` (`0.00302015`). This package does not touch that record.
+
+The integer cell in ToikY's `research/lever-integer-twins` puts paid mass
+5586 at `{8:698, 2:1}` on every call, with two unpaid coordinates and centre
+charge 64. On nine calls its pooled idle is `{51:272, 46:1}`, and the moment
+at `a = 1/98` still passes after 16 added rank-1 children and fails at 17.
+That 16-child margin is a budget for a physical producer, not a producer.
+The question left open by `research/rank8-u2-split-boundary` is whether any
+literal local word has this paid profile. On the selected register file, no
+monotone nested frame word of mass 5586 and rank at most 8 meets the moment
+at all.
+
+## Contract
+
+Frames are subspaces of \(\mathbb{F}_2^9\). A monotone register chain climbs
+from its pinned entrance to an exit inside the legal terminal, and the climb
+ranks are positive integers summing to at most \(\dim(\mathrm{exit}) -
+\dim(\mathrm{entrance})\). One chain therefore holds at most one part of size
+8, because the room is at most 9. The selected local word fixes the entrances
+and the register file:
+
+| Registers | Count | Entrance | Maximum exit | Room |
+|---|---:|---|---|---:|
+| Sources | 112 | the port line, dimension 1 | FULL, dimension 9 | 8 |
+| Targets | 112 | 0 | the port parity hyperplane, dimension 8 | 8 |
+| Helpers with entrance 0 | 288 | 0 | FULL | 9 |
+| Helpers with entrance 1 | 136 | dimension 1 | FULL | 8 |
+| Helpers with entrance 4 | 32 | dimension 4 | FULL | 5 |
+| Helpers with entrance 5 | 24 | dimension 5 | FULL | 4 |
+
+There are 648 rooms of size at least 8, 32 of size 5 and 24 of size 4.
+Raising a target exit from the parity hyperplane to FULL adds residual room
+but no eighth rank-8 slot, and it does not change the histogram below.
+
+## Result
+
+The minimum-cost histogram of mass 5586 with parts at most 8 that fits these
+rooms is
+
+```text
+{8:648, 5:32, 4:24, 1:146}.
+```
+
+Its nine-call moment at `a = 1/98`, with the optimal pooled idle
+`{51:272, 46:1}`, has certified bounds strictly above 1. The ideal cell is
+not among the feasible histograms: it needs 698 rank-8 children and the file
+has 648 monotone slots. The same checker reproduces the published control on
+`{8:698, 2:1}`: 16 added rank-1 children still pass and 17 fail.
+
+A comparison that is not a monotone word grants each of the eight centre
+registers three rank-8 steps (room 24 instead of 8 or 9). The cheapest
+histogram in that larger model is `{8:664, 5:32, 4:24, 1:18}`. Its bare
+pooled moment is below 1, and one added rank-1 child on the nine-call total
+pushes it above 1. The 16-child margin is not available there either.
+
+## Why this histogram is the monotone minimum
+
+Let \(f(r) = (r/73)^{97/98}\) for \(r \ge 1\) and \(f(0) = 0\). The child cost
+at \(a = 1/98\) is \(f(r)\). The pinned enclosures separate two strict
+inequalities on \(\{1,\ldots,8\}\):
+
+- merging: \(f(x) + f(y) > f(x+y)\) whenever \(x, y \ge 1\) and \(x+y \le 8\);
+- transfer: \(f(x) + f(y) > f(x-1) + f(y+1)\) whenever \(1 \le x \le y\) and
+  \(y+1 \le 8\), with \(f(0) = 0\) when \(x = 1\).
+
+Every composition of 8 into at least two positive parts is therefore strictly
+more expensive than one part of size 8. Any feasible assignment that uses
+fewer than 648 eights still has at least 8 units of mass in smaller parts and
+a large register that is not holding an 8. Gathering those units into one 8
+strictly lowers the cost. So every minimum uses exactly 648 eights, and the
+remaining mass is 402.
+
+That remainder has to sit in the 32 rooms of size 5, the 24 rooms of size 4,
+and the single leftover slot of a room that already holds an 8. No leftover
+slot has size greater than 1, and a room of size 5 or 4 cannot hold a 7.
+Filling the size-5 and size-4 rooms with one part each, then placing the last
+146 units as ones, is the unique minimum: splitting a 5 or a 4 raises the
+cost by the merge inequality. The resulting moment is already above 1, before
+any additive rank-1 surcharge and before integer idle is replaced by anything
+worse than the pooled histogram.
+
+## What was not shown
+
+Non-monotone paths, extra registers, changed entrances and a different
+template are outside this certificate. Unknown restoration, precision and
+consumer costs are not zero. The three-step centre comparison is only an
+upper model on the same eight centre indices; it is not a scalar word. The
+obstruction is about this entrance file and the older \(u = 2\) planning cell.
+It is not a statement about the #4470 productive-helper invoice, and it does
+not move the selected exponent.
+
+The family split in `research/rank8-u2-split-boundary` still applies to the
+ideal histogram. It does not create a 649th monotone rank-8 slot.
+
+## Scope
+
+Card `complex-new-helper-inventory`, priority revision 7, base
+`ff3cc58e370d0f678e529246f8c73e8094e18fcf`. Scope status is **refined**. The
+card asks for a new inventory with a favorable full bill. The bounded question
+here is the literal rank-at-most-8 producer inside the 16-child slack. The
+answer on these entrances is that the monotone minimum already fails
+\(a = 1/98\).
+
+Ordered eligible list for session r31, rebuilt on 2026-10-23 against
+`ff3cc58e370d0f678e529246f8c73e8094e18fcf`. The order is the one recorded in
+`research/f2-six-difference-basis`: open handoffs from the 19 October index,
+then priority-table rows whose stated next decision is still open, then
+catalogue cards that are not already one of those rows. Dropped before the
+cut: work this account already delivered, questions closed on current main,
+questions covered by an open pull request, and topics already on a fork
+`cursor/*` branch.
+
+1. Consumer of `(w, q, v_N)` (discussion comment 19030045), distinct from the
+   encoded-exit and `(N-P)⊗M` consumers.
+2. Fresh T385 wrapper completion (comment 19030047), distinct from the reverse
+   coefficient and from the merged compiler integration.
+3. Six-tape head-move bill for the `L546` addition (comment 19030049),
+   distinct from the F0 magnitude ledger and from #4162.
+4. `BIT.md` weighted-leaf and fixed-tape induction. The `2^{-4p}` sentence,
+   the row-field pad, the uniform evaluator constant and the shared-arena axis
+   are already delivered.
+5. Four-provider ordinary/outer rebind. The idle-cost block is recorded; the
+   four-id rebind is not.
+6. Matched park/restore bill for the #4166 caller. Open #4167 does not price
+   that comparison.
+7. R437 fixed-tape and parking bill. The reverse coefficient is excluded.
+8. `#4091` model equivalence on the later 1090-call fill.
+9. This package.
+
+The retiming handoff (comment 19030053) stays dropped because #4122, #4128 and
+open #4567 already address it. `p104-side-correction` is not a separate row:
+item 1 is that consumer, and the slot, chart, encoding and fiber gaps are
+certified. `formal-interface-lemma`, `bit-helper-contract`,
+`bit-new-register-model`, `complex-rank-shape-decoder`, `centre-path-sharing`,
+`connector-alphabet-change`, and the stock, census, replay, packing and
+scalar-itinerary cards stay dropped for the same reasons already recorded on
+the difference-basis package, plus this account's later deliveries through
+#4651.
+
+This is open question 2 of `research/rank8-u2-split-boundary`. Question 1 of
+that package, the census of family splits that still pass 16 extra rank-1
+children, is a different refinement of #4151 and is not answered here.
+`research/rank8-u2-plane-port-slack` and `research/selected-k8-child-rank-bound`
+likewise leave the literal producer open. A cached overlap search on
+2026-10-23 (307 open pull requests, 4479 discussion records) did not find
+another note stating the histogram `{8:648, 5:32, 4:24, 1:146}`. Same-day
+fork branches for the L546 six-tape bill, the R437 parent blank-return, the
+shared chart-1 centre visit, the SOURCE87 fibre count and the #4091 1090-call
+ceiling price other rows of this list.
+
+## Reproduce
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3 -B research/rank8-u2-entrance-obstruction/verify.py
+python3 -B scripts/research_package.py --package research/rank8-u2-entrance-obstruction
+```
+
+Expected line:
+
+```text
+PASS_RANK8_U2_ENTRANCE_OBSTRUCTION slots=648 monotone=GT/0 centre3=LT/0/GT/1 ideal=LT/16/GT/17 no_new_kappa
+```
+
+The checker hashes the word gzip, the decompressed word and `ceiling.py`,
+rebuilds the 704 entrance rooms, packs both histograms, and separates every
+merge and transfer inequality on ranks `1..8` by the pinned outward
+enclosures. Not run: Lean, Docker, the selected cold replay, and any search
+for a non-monotone scalar word.
+
+## Open next questions
+
+1. Can one centre register legally emit three rank-8 steps without an added
+   rank-1 child? The arithmetic room of that comparison is zero: the bare
+   moment passes and one aggregate rank-1 child fails. A legal trace would
+   still need a changed path law. The first step is one explicit up, down, up
+   frame trace and its own paid histogram. The standing monotone contract is
+   the blocker.
+2. Do the #4470 productive-helper frames have at least as many monotone
+   rank-8 slots as that invoice demands? This certificate does not read that
+   word. The first step is the same entrance count on the #4470 complex
+   frames, and the blocker is that the two templates must not be mixed.
+
+## Credit
+
+The 16/17 integer boundary and the cell `{8:698, 2:1}` are ToikY's
+`research/lever-integer-twins`, following Nick Petalas's PR #3874. The
+entrance fields and the \(\mathbb{F}_2^9\) frame contract are the selected k8
+word in `research/r480-owner-optimum`, as read by the frame-ceiling line
+(PR #1671, Gareth Davidson, using navinor's frame method). The child-cost
+enclosures are Douglas Colkitt's `research/complex-local-word-ceiling`, with
+Claude assistance. The coverage identity is PR #1012. The split dependence of
+the ideal slack is `research/rank8-u2-split-boundary`; this note does not
+recompute those three family splits. The plane-port screen that already misses
+the same slack is `research/rank8-u2-plane-port-slack`.
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.
+No independent human review and no formal-kernel acceptance are claimed.

research/rank8-u2-entrance-obstruction/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "ff3cc58e370d0f678e529246f8c73e8094e18fcf",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395",
+      "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/ceiling.py",
+      "usage": "input",
+      "git_blob": "6c0f54527d879396f689523bfde19928a9b6157c",
+      "sha256": "86a1432870b3b21d8454106ec9162fddc0471472d9214c590a1420f2426cedfc"
+    },
+    {
+      "path": "research/r480-owner-optimum/inputs/local/FULL-WORD.json.gz",
+      "usage": "input",
+      "git_blob": "723decbca8995e916d6f4b783aa3e1ddd4eb47fe",
+      "sha256": "49596aa4ed4fd36cd4582fd0e8ddeb09fd80370a7d8aea1ede52a135b536d289"
+    }
+  ]
+}

research/rank8-u2-entrance-obstruction/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-new-helper-inventory",
+  "priority_revision": 7,
+  "base_commit": "ff3cc58e370d0f678e529246f8c73e8094e18fcf",
+  "original_card": {
+    "id": "complex-new-helper-inventory",
+    "question": "Can a new local cost interface change the deficit-to-recursive-cost ratio beyond current packing?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/complex-local-word-ceiling/README.md",
+      "research/heterogeneous-owner-assembly/README.md",
+      "certificates/selected-result.json",
+      "research/selected-k8-unrestricted-ends/README.md"
+    ],
+    "methods": [
+      "Propose a changed helper inventory and derive an optimistic exact moment.",
+      "Design a small completed instance and independently verify its endpoint and rank bill."
+    ],
+    "acceptance_gate": "A new physically realizable contract outside the frozen catalogue whose per-call profile, screened with ceiling.ceiling(), beats the selected word's packing ceiling; a lower cap does not establish identical circuitry or order attainable optima, and is insufficient to pass the stronger-budget target under this model. 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.",
+    "bottleneck_link": "Packing of the selected word is capped at +0.87% (ceiling model); only a lower local excess spend or a changed coverage law raises the cap.",
+    "allowed_variations": [
+      "Narrow to one changed boundary or helper type with explicit counts.",
+      "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": [
+      "Mixing only providers already inside the pinned catalogue.",
+      "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013.",
+      "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": [
+      "helper inventory",
+      "heterogeneous-owner",
+      "mutable leaver"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "On the selected k8 register file, with pinned entrances and monotone nested frames in F_2^9, no rank-at-most-8 paid profile of mass 5586 meets the u=2 nine-call moment at a=1/98. The ideal cell {8:698, 2:1} needs 698 rank-8 children; these entrances supply 648 monotone slots. The minimum feasible profile is {8:648, 5:32, 4:24, 1:146}, and its pooled moment is strictly above 1.",
+  "scope_change_reason": "The card asks for a new helper inventory whose fully charged profile beats the packing ceiling. This package answers the still-open literal rank-at-most-8 producer inside the published 16-child u=2 slack, and stops at the entrance obstruction. It does not propose another ideal ceiling.",
+  "bottleneck_link": "The complex child-rank lever is what the selected-r8-u2 cell uses to aim at a=1/98. A monotone word on the pinned entrances cannot reach that histogram, so the 16-child slack is not a producer budget on this register file. Selected kappa is unchanged.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop when the monotone optimum and the three-step centre comparison are signed against the published 16/17 control. Do not search scalar rewrites, add registers, or rerun the selected replay.",
+  "outcome": "negative-result",
+  "evidence": "python3 -B research/rank8-u2-entrance-obstruction/verify.py prints PASS_RANK8_U2_ENTRANCE_OBSTRUCTION slots=648 monotone=GT/0 centre3=LT/0/GT/1 ideal=LT/16/GT/17 no_new_kappa.",
+  "author": "unpaidmoon",
+  "ai_assistance": "Substantial Cursor Grok assistance on the obstruction, the exchange checks and the verifier.",
+  "open_questions": [
+    {
+      "question": "Can one centre register legally emit three rank-8 steps, and can that word avoid every added rank-1 child?",
+      "why_it_matters": "Granting three rank-8 steps on each of the eight centres passes the bare u=2 moment and fails as soon as one aggregate rank-1 child is added. The 16-child margin is not available there either.",
+      "first_step": "Write one centre register as an explicit up, down, up frame trace with its scalar gates and the paid histogram of that trace alone.",
+      "blocked_by": "The standing frame contract keeps non-centre paths monotone. A descent plus two climbs is a changed path law and needs its own restoration bill before it can be counted."
+    },
+    {
+      "question": "Do the #4470 productive-helper frames still have fewer than 698 monotone rank-8 slots?",
+      "why_it_matters": "This obstruction is about the selected k8 entrance file and the older u=2 planning cell. It does not automatically transfer to the current selected invoice.",
+      "first_step": "Count dim(exit)-dim(entrance) on the #4470 complex word, if that word still uses nested F_2^9 frames, and compare the count with its own paid rank-8 demand.",
+      "blocked_by": "The #4470 word is a different bill. Reusing these 648 slots as if they were that invoice would mix two templates."
+    }
+  ]
+}

research/rank8-u2-entrance-obstruction/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/complex-local-word-ceiling/ceiling.py",
+    "research/r480-owner-optimum/inputs/local/FULL-WORD.json.gz"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Entrance census, monotone and centre-comparison moments, merge and exchange enclosures, the published 16/17 control, and the selected kappa pin."
+    }
+  ]
+}