Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa)

research/complex-local-word-ceiling/README.md

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+# Packing ceiling and target budget for the selected complex local word
+
+**Scoped obstruction and planning tool. No new κ; the selected record is unchanged.**
+
+The selected complex supplier (#1094) fixes a local word with this paid profile per call:
+`{1:2412, 2:418, 3:432, 4:142, 5:82, 8:8}`, local mass 5,586. Everything else in the record is
+idle-remainder packing on 896 live data families per nine-stage replica (2 × 4 pairs × 112 ports,
+all 492 helpers hosted).
+
+Under PR1012's coverage identity, paid mass is exactly `72·W + 64·N`, so each call carries 1,646 units
+of idle mass. Every child of rank `r` costs `(r/73)^(1−a) = r·r^(−a)·73^(a−1)`, which is non-increasing
+per unit of mass in `r`. So **no** idle histogram beats putting all idle mass into (fractional) children
+of the largest rank allowed by the outer halving guard, `R = 51` (73² > 2·51²).
+
+## Results (exact rational intervals, `verify.py`)
+
+| Quantity | Value |
+|---|---|
+| Ceiling for any hosting, owner packing or idle-remainder change of this word | **a < 0.001690054109089** |
+| Implied κ bound (`κ < a/(1+a)` when the complex side binds) | **κ < 0.0016872026453254** |
+| Maximum gain over the selected κ 0.001672660197076918 | **+0.869%** |
+| Selected endpoint / relaxation at the endpoint | below the ceiling; actual moment ≥ relaxation |
+| Previous record #1042 | same local word plus an admissible idle histogram (derived idle `{1:2249, 2:1, 3:400, 7:3, 16:47, 48:4850}` matches its README); endpoint below the same ceiling |
+
+This updates the fixed-word ceilings computed for earlier words
+([static44](../static44-packing-ceiling/README.md), PR525, PR643) to the selected hosted T16/T37 line. It
+uses the coverage law instead of a dual certificate. All of #1042→#1094's gain was packing at this
+fixed word, and less than 0.87% of such headroom remains.
+
+### What a local word must pay for the planning target a = 1/99
+
+Rewrite `relax(a) ≤ 1` as `Σ n_r p_r(a) ≤ B(a)`. Here `p_r(a) = (r/73)^(1−a) − r(51/73)^(1−a)/51` is the
+excess of a local rank-r child over ideal idle mass, and `B(a) = w − (72w+64)(51/73)^(1−a)/51` is the
+budget per call.
+
+| a | Budget B | Selected word's excess spend | Ratio |
+|---|---:|---:|---:|
+| selected endpoint 0.0016755 | 0.42752 | 0.42330 | 0.990 |
+| target 1/99 | 0.12753 | 2.59681 | **20.4** |
+
+At a = 1/99, the selected word's excess spend is 20.4 times the budget. **Even deleting all 2,412 rank-1
+children per call, with everything else fixed, cannot reach it.** The allowed rank-1 count is negative.
+With 112 ports and this nine-stage template, a target-scale gain needs a local word whose fully priced
+excess spend is about 95% lower. Otherwise the coverage law itself must change: more unpaid coordinates
+per family, a smaller 64-per-call centre charge, or more families per call. Helper-inventory proposals
+can be screened against these numbers before any network is built: `ceiling.ceiling(local, 73, 896, 9, 64)`
+takes a per-call local profile.
+
+### Ideal lever budgets
+
+`levers.py` changes one premise at a time and reports the packing ceiling. These are favorable relaxations
+that omit the costs of realizing each lever.
+
+| Lever | Ceiling × | κ cap |
+|---|---:|---:|
+| Centre path free (no rank-8 copies, no 64-per-call charge) | 2.80 | 0.00472 |
+| Centre path shared by two calls | 1.90 | 0.00320 |
+| Local mass as rank-16 children | 2.31 | 0.00389 |
+| Local mass as rank-8 children | 1.62 | 0.00273 |
+| Two unpaid coordinates per live family | 3.77 | 0.00634 |
+| Centre free and rank-16 children | 6.51 | 0.01089 |
+
+## Scope and limits
+
+- The bound holds for one local word (paid profile per call), 112 ports, the nine-stage four-pair
+  template, the coverage law and the outer halving guard. It allows fractional counts and every repacking,
+  without checking physical realizability.
+- It does **not** bound other local words. Many open claims above the selected κ (for example #1148's
+  joint-entrance word and its descendants, all ≤ 0.0016759) change the local word, so they are outside it.
+  Changed port counts, templates, coverage laws, centre charges or outer recipes are also outside it.
+- `κ < a/(1+a)` is the retained balanced reduction's planning ceiling. The actual seven-step recipe gives
+  slightly less.
+- Four controls run:
+  - an idle-rank cap of 52 is rejected;
+  - a local mass beyond the coverage law is rejected;
+  - one extra rank-1 child lowers the ceiling;
+  - 25 sampled integer idle histograms all have exact moment above one at the ceiling.
+
+## Reproduce
+
+```sh
+python3 -I -B research/complex-local-word-ceiling/verify.py          # ~10 s, PASS_COMPLEX_LOCAL_WORD_CEILING
+python3 scripts/research_package.py --package research/complex-local-word-ceiling
+python3 -I -B research/complex-local-word-ceiling/levers.py          # ~5 s, planning budgets
+```
+
+`verify.py --write` regenerates `RESULT.json`. The exact arithmetic in `ceiling.py` uses the standard
+library only: atanh logarithm chains, Taylor exponentials with remainder, and outward rounding to 2⁻²⁵⁶.
+
+## Open next questions
+
+- **What is the ceiling of each pending local word?** The ceiling separates packing headroom from
+  word gains in the review queue. A first step is to run `ceiling()` on the per-call local profiles of
+  #1148/#1169/#1218, after subtracting their idle histograms. The blocker is that each claim's idle
+  histogram must be extracted from its package.
+- **How does the target budget scale with the port count V?** `B(a)` grows linearly with families per call.
+  Whether a decoder's excess spend grows more slowly decides if wider words can help. A first step is to
+  fit `Σ n_r p_r` against V across the recorded complex words (R439–R560 lineage). The blocker is per-call
+  local profiles for different V on a comparable template.
+
+## Credit
+
+The coverage identity is PR1012's. The fixed-word packing-ceiling idea comes from the static44, PR525 and
+PR643 lineage, and the catalogue mixture bound from Ali Romman's
+[heterogeneous-owner-assembly](../heterogeneous-owner-assembly/README.md). The selected construction
+credits are in its review. Prepared by Douglas Colkitt with substantial Claude (Anthropic) assistance.
+Apache-2.0.

research/complex-local-word-ceiling/RESULT.json

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+{
+ "at_selected_endpoint": {
+  "budget_B": [
+   "0.427516482255611063",
+   "0.427516482255611063"
+  ],
+  "excess_price_by_rank": {
+   "1": "9.0593779e-05",
+   "2": "0.00014915902",
+   "3": "0.00019566114",
+   "4": "0.00023433531",
+   "5": "0.0002671916",
+   "8": "0.00034085369"
+  },
+  "local_excess_spend": [
+   "0.423298432057444218",
+   "0.423298432057444218"
+  ],
+  "max_rank1_per_call_others_fixed": "2458.56",
+  "rank1_price": "9.05937788742848158e-05",
+  "spend_over_budget_lower": "0.99013359630975617"
+ },
+ "ceiling": {
+  "a_ceiling": "1690054109089/1000000000000000",
+  "a_ceiling_decimal": "0.00169005410908900006",
+  "a_relaxation_achievable": "52814190909/31250000000000",
+  "kappa_upper_bound": "0.0016872026453254019",
+  "max_kappa_gain_percent": "0.86942",
+  "selected_moment_at_ceiling_lower": "1.00004273855344605"
+ },
+ "controls": {
+  "idle_rank_52_violates_halving_guard": "rejected: idle rank cap violates the halving guard",
+  "local_mass_exceeds_coverage": "rejected: local word pays more mass than the coverage law allows",
+  "one_more_rank1_child_lowers_ceiling": "yes",
+  "sampled_idle_histograms_above_one_at_ceiling": "yes"
+ },
+ "model": {
+  "centre_per_call": 64,
+  "coverage_identity": "mass = 72 W + 64 N (checked on the selected profile)",
+  "idle_rank_cap": 51,
+  "live_families_per_call": "896/9",
+  "m": 73,
+  "pairs": 4,
+  "ports": 112,
+  "stages": 9
+ },
+ "previous_record_same_local_word": {
+  "below_ceiling": true,
+  "complex_lower": "418747253984418082117/250000000000000000000000",
+  "idle_histogram": {
+   "1": 2249,
+   "16": 47,
+   "2": 1,
+   "3": 400,
+   "48": 4850,
+   "7": 3
+  },
+  "kappa": "1672188119205421/1000000000000000000",
+  "record": "certificates/checkpoints/frontier1042.json"
+ },
+ "scope": "Bound for the selected local word (paid profile per call), 2*4*112 live data families per nine-stage replica, the coverage law and the outer halving guard. It is not a bound for other local words, port counts, templates, coverage laws or outer recipes, and allows fractional idle counts and every repacking without checking physical realizability.",
+ "selected": {
+  "calls": 144,
+  "complex_lower": "67018507072324140917/40000000000000000000000",
+  "idle_histogram": {
+   "1": 2276,
+   "3": 377,
+   "48": 4866,
+   "7": 7
+  },
+  "idle_mass_per_call": "1646",
+  "kappa": "836330098538459/500000000000000000",
+  "live": 14336,
+  "local_mass_per_call": "5586",
+  "local_profile_per_call": {
+   "1": 2412,
+   "2": 418,
+   "3": 432,
+   "4": 142,
+   "5": 82,
+   "8": 8
+  },
+  "rank_mass": 1041408
+ },
+ "status": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+ "target": {
+  "a": "1/99",
+  "budget_B": [
+   "0.127529653671111354",
+   "0.127529653671111354"
+  ],
+  "excess_price_by_rank": {
+   "1": "0.00055700884",
+   "2": "0.00091439983",
+   "3": "0.0011974157",
+   "4": "0.0014323494",
+   "5": "0.0016316381",
+   "8": "0.0020773305"
+  },
+  "local_excess_spend": [
+   "2.59681461996869167",
+   "2.59681461996869167"
+  ],
+  "max_rank1_per_call_others_fixed": "-2021.1163",
+  "rank1_price": "0.000557008840686534388",
+  "spend_over_budget_lower": "20.3624376387445238"
+ }
+}

research/complex-local-word-ceiling/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "e3bb23b4c6bae16ba277f9443e11257a0a37780a",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "input",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "c05fafaa25ba95b2d5af86b1800427198e3d20eb",
+      "sha256": "89b5928bc92c81e2647b6d702a7ad85a6795b8f44e2eb267b2ff3145d8f2b2ce"
+    },
+    {
+      "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"
+    }
+  ]
+}

research/complex-local-word-ceiling/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-new-helper-inventory",
+  "priority_revision": 2,
+  "base_commit": "e3bb23b4c6bae16ba277f9443e11257a0a37780a",
+  "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": [
+      "research/heterogeneous-owner-assembly/README.md",
+      "certificates/selected-result.json"
+    ],
+    "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, not another mixture already covered by the frozen catalogue.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read the frozen catalogue bound exclusions and select one genuinely different helper endpoint inventory.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Write the complete optimistic child histogram and reject it immediately if even the relaxation has insufficient headroom.",
+    "bottleneck_link": "Helper inventory changes can improve the moment; repacking a covered cost interface cannot evade its scoped bound.",
+    "allowed_variations": [
+      "Narrow to one changed boundary or helper type with explicit counts."
+    ],
+    "scope_change_triggers": [
+      "Mixing only providers already inside the pinned catalogue."
+    ],
+    "overlap_terms": [
+      "helper inventory",
+      "heterogeneous-owner",
+      "mutable leaver"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "For the selected #1094 complex local word, certify the best saving any hosting/owner/idle repacking can reach under the coverage law and halving guard. Then compute the local-word excess-spend budget a changed helper inventory needs for a = 1/99.",
+  "scope_change_reason": "This is the card's first experiment (an optimistic exact relaxation that rejects routes with insufficient headroom), applied to packing of the incumbent word and turned into a screening calculator for proposed inventories. No new inventory was synthesized.",
+  "bottleneck_link": "The complex side binds. This shows packing of the incumbent word has under 0.87% headroom, and that a target-scale gain needs about 95% less local excess spend or a changed coverage law.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after the selected word, the previous record and the 1/99 budget. Pending words are left as an open question.",
+  "outcome": "completed",
+  "evidence": "RESULT.json: a ceiling 0.001690054109089 (kappa < 0.0016872026453254, +0.869%); at a=1/99 the excess spend is 20.4 times the budget; 4 controls.",
+  "author": "Douglas Colkitt",
+  "ai_assistance": "Substantial: Claude (Anthropic, Claude Code) chose the relaxation, wrote ceiling.py and verify.py, and drafted the README.",
+  "open_questions": [
+    {
+      "question": "What is the packing ceiling of each pending local word (#1148/#1169/#1218 lineage)?",
+      "why_it_matters": "It separates packing headroom from local-word gains in the review queue.",
+      "first_step": "Run ceiling() on each claim's per-call local profile after subtracting its idle histogram.",
+      "blocked_by": "Each claim's idle histogram must be extracted from its package."
+    },
+    {
+      "question": "How does the a=1/99 budget scale with the port count V relative to a decoder's excess spend?",
+      "why_it_matters": "It decides whether wider words can reach target-scale gains.",
+      "first_step": "Fit sum n_r p_r against V across recorded complex words.",
+      "blocked_by": "Per-call local profiles for different V on a comparable template."
+    }
+  ]
+}

research/complex-local-word-ceiling/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "certificates/checkpoints/frontier1042.json",
+    "certificates/selected-result.json",
+    "research/community-frontier1042-audit/ASSEMBLY.json",
+    "research/community-frontier1094-audit/ASSEMBLY.json",
+    "research/community-frontier1094-audit/INDEPENDENT1094.json"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Coverage identity and local-word split of the selected and previous records; exact ceiling bracket; 1/99 budget; RESULT.json byte match; four controls"
+    }
+  ]
+}