Mister-Meeseeks · Oct 28, 2026 · Oct 27, 2026
diff --git a/research/selected-invoice-bit-cap-thresholds/INPUT.json b/research/selected-invoice-bit-cap-thresholds/INPUT.json
@@ -0,0 +1,60 @@
+{
+ "base_commit": "19fdb086ea96af9a9f87936a2b888e10af4ed810",
+ "components": {
+  "idle_paid": {
+   "16": 416,
+   "24": 416,
+   "32": 416,
+   "40": 416,
+   "48": 416,
+   "56": 416,
+   "64": 416,
+   "72": 208
+  },
+  "local_helpers": {
+   "1": 2,
+   "2": 5,
+   "4": 89,
+   "5": 11,
+   "7": 5,
+   "8": 16,
+   "9": 412
+  },
+  "local_paid": {
+   "1": 2203,
+   "2": 459,
+   "3": 493,
+   "4": 92,
+   "5": 168,
+   "6": 18,
+   "7": 14
+  },
+  "main_targets": {
+   "145": 208,
+   "17": 416,
+   "25": 416,
+   "33": 416,
+   "41": 416,
+   "49": 416,
+   "57": 416,
+   "65": 416,
+   "73": 208
+  },
+  "query_paid": {
+   "1": 1560
+  }
+ },
+ "exporter_inputs_sha256": {
+  "research/p104-helper-subspace-splits/RESULT.json": "1d92e587915052110dfb3aca8890df5afdaf496164efa5a4a52e69ceeb3c70f3",
+  "research/p104-helper-subspace-splits/WORD.json.gz": "bbb072bced5824a8d9094c49be2fc51fbfdd05fb2b7dab8106e1275053075099",
+  "research/p104-productive-controller/CONTROLLER.json": "a84f6bbd547379597464e56c420cc665f96bc43a9a1351eed4b7cd0fce3cc4b9"
+ },
+ "full_calls": 18,
+ "generated_by": "python3 -B scripts/export_selected_invoice.py --evidence EVIDENCE (stdout is the invoice)",
+ "invoice_file": "INVOICE.json",
+ "invoice_sha256": "ad826c829c11ae3193736ad2df3310081809ab6cdcd194346065bd97f943e4c1",
+ "schema": "selected-invoice-bit-cap-input/v1",
+ "selected_kappa": "62009/20000000",
+ "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5",
+ "source_head": "09dd0965711ca7eeb169057a420abab1c9052137"
+}
diff --git a/research/selected-invoice-bit-cap-thresholds/INVOICE.json b/research/selected-invoice-bit-cap-thresholds/INVOICE.json
@@ -0,0 +1,44 @@
+{
+  "schema": "integer-mult-recursive-invoice/v1",
+  "contract": {
+    "model": "mixed-target-unnormalized-power-v1",
+    "rank_semantics": "H_r costs r^(1-C); debit surrendered productive fields",
+    "ambient_rank": 145,
+    "boundary": "complete unit before the same mixed-target compiler"
+  },
+  "paid": {
+    "1": 41214,
+    "2": 8262,
+    "3": 8874,
+    "4": 1656,
+    "5": 3024,
+    "6": 324,
+    "7": 252,
+    "16": 416,
+    "24": 416,
+    "32": 416,
+    "40": 416,
+    "48": 416,
+    "56": 416,
+    "64": 416,
+    "72": 208
+  },
+  "targets": {
+    "1": 36,
+    "2": 90,
+    "4": 1602,
+    "5": 198,
+    "7": 90,
+    "8": 288,
+    "9": 7416,
+    "17": 416,
+    "25": 416,
+    "33": 416,
+    "41": 416,
+    "49": 416,
+    "57": 416,
+    "65": 416,
+    "73": 208,
+    "145": 208
+  }
+}
diff --git a/research/selected-invoice-bit-cap-thresholds/README.md b/research/selected-invoice-bit-cap-thresholds/README.md
@@ -0,0 +1,102 @@
+# Local thresholds for the selected complex invoice to reach the BIT saving
+
+**Finite exact certificate on an accounting relaxation. No new κ.** This
+result is produced by Nick Petalas. The selected frontier is unchanged.
+
+The selected #5599 complex supplier binds at `U = 0.0031004563072…`. The
+admitted BIT saving is `17/5000`. This package fixes, exactly, how much the
+local word of the selected P104 invoice must change before the complex bill
+passes at `C = 17/5000`. Every other paid job and target stays fixed. It also
+bounds what free local helper splitting can do.
+
+## Result
+
+The invoice is the authenticated output of `scripts/export_selected_invoice.py`
+at base `19fdb086e`. The packaged `INVOICE.json` is its stdout byte for byte,
+and its SHA-256 is the exporter's own `invoice_sha256`. The verifier checks
+that every local paid job and local helper target appears exactly 18 times,
+once per FULL. Paid work is `18·local + 1560 query + idle`, and targets are
+`18·helpers + main`.
+
+The bill passes at `C` when `Σ T_r r^(1-C) > Σ P_r r^(1-C)`. At `C = 17/5000`
+the selected invoice fails by **72.892623830207**, which is
+**4.049590212789 per FULL**.
+
+| Local change per FULL, all else fixed | Passes at 17/5000? |
+|---|---|
+| delete one paid job of rank 1, 2, 3 or 4 | no; rank 4 gives 3.98 < 4.05 |
+| delete one paid job of rank 5, 6 or 7 | yes |
+| fewest deletions using only rank ≤ 1 / ≤ 2 / ≤ 3 / ≤ 4 | 5 / 3 / 2 / 2 |
+| free full refinement of rank-9 helper targets into rank-1 targets | needs 61 of the 412 |
+| free balanced split of rank-9 helpers into 4+5 | needs 194 of the 412 |
+| refine every local helper target into rank-1 targets | passes; first crossing in [0.0094772838495, 0.0094772838496] |
+| the same, at `C = 1/100` | **fails** |
+
+For a helper target of rank `r`, every partition into positive ranks has
+`Σ p^(1-C) ≤ r` when `0 ≤ C < 1`, with equality for `r` parts of rank one.
+So full refinement maximizes the target moment at every `C`. Since that
+maximum fails at `C = 1/100`, **no split of local helper targets alone, with
+paid work fixed, lets the selected invoice pass at 1/100.** The split counts
+are minimal for the same reason. A rank-9 helper gives the largest gain, and
+the greedy count is optimal.
+
+For comparison, #5599 refines thirteen helpers. theon199's #5702 shows that
+fifteen fully refined H9 sources fund one added H1 per FULL. Reaching the BIT
+saving needs about four times that, which is 61.
+
+## Claim scope
+
+This is an optimistic accounting relaxation of the complete signed moment
+used by the selected invoice. A deletion or split here is free: no restoration,
+native alignment, precision, compiler, stock or outer cost is charged, and no
+physical word is supplied. A passing row keeps a route open. It does not make
+a supplier. Once complex passes at `17/5000`, the admitted BIT saving binds;
+this package does not prove the outer κ. Only the sign at each stated `C` and
+the stated root bracket are certified. Uniqueness of the refined crossing is
+not claimed beyond the separate failure at `1/100`.
+
+Every `r^(1-C)` uses Python's correctly rounded decimal `ln` and `exp` at 80
+digits. The total rounding error is far below `1e-50`, and every reported sign
+clears `1e-30`. A float evaluation corroborates the deficit. The boundary rows,
+including rank 4 versus 5, 4 versus 5 rank-one deletions, 60 versus 61 and
+193 versus 194 H9 splits, and the `1/100` failure, are rerun on the actual
+changed invoice rather than inferred from the table.
+
+## Reproduce
+
+From the repository root with CPython 3.12 and the standard library:
+
+```sh
+python3.12 -I -B -W error research/selected-invoice-bit-cap-thresholds/verify.py
+python3.12 -I -O -B -W error research/selected-invoice-bit-cap-thresholds/verify.py
+python3.12 -B scripts/research_package.py --package research/selected-invoice-bit-cap-thresholds
+python3.12 -B scripts/export_selected_invoice.py | shasum -a 256
+```
+
+Each verifier run takes about 0.2 s and prints
+`PASS_SELECTED_INVOICE_BIT_CAP_THRESHOLDS`. The last command reproduces the
+invoice hash in `INPUT.json` from a full checkout.
+
+## Open next questions
+
+1. Which physical window of the selected local word can delete one paid job of
+   rank 5, 6 or 7, or two of rank 3 or 4, while keeping every target and
+   restoration? Under this accounting, any one of them lets complex pass at the
+   admitted BIT saving. The first step is to list the 200 local paid jobs of
+   rank at least five and their producers and consumers. Closed rewrites of
+   older words do not transfer automatically.
+2. Splitting helper targets alone stops below `1/100`. Which changes to the
+   paid side or to the main targets, priced with this same moment, would let
+   the complex side pass at `1/100`? Any target above `.0034` also needs a BIT
+   improvement.
+
+## Credit and sources
+
+The selected P104 word is by Pierre-Loic Doulcet (hexapode), with Ali Romman's
+complete caller and helper accounting, Cinna Davis's controller, Jacob Alan
+Sussman's productive recursion and zxv's invoice interface. The invoice exporter
+is Ali Romman's. theon199's `helper-refinement-budget-s29` (#5702) and
+Clear1nsight's #5614 price related helper refinements, and the merged
+`productive-lever-reprice` package priced a similar BIT-level gap on the older
+#4470 invoice. They are cited, not re-checked. Pins are in
+[SOURCES.json](SOURCES.json). Apache-2.0.
diff --git a/research/selected-invoice-bit-cap-thresholds/SOURCES.json b/research/selected-invoice-bit-cap-thresholds/SOURCES.json
@@ -0,0 +1,48 @@
+{
+  "schema_version": 1,
+  "base_commit": "19fdb086ea96af9a9f87936a2b888e10af4ed810",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "scripts/export_selected_invoice.py",
+      "usage": "provenance",
+      "git_blob": "00224b1bc549b0bab3f87d62cf739e907f04bfe3",
+      "sha256": "a76767b8d02efa722a61ce3461e530246fd0d6785c376d6e58893b452f966013"
+    },
+    {
+      "path": "research/p104-helper-subspace-splits/RESULT.json",
+      "usage": "provenance",
+      "git_blob": "35f75c8555193c6e845cd52cc562ca23fd5924cb",
+      "sha256": "1d92e587915052110dfb3aca8890df5afdaf496164efa5a4a52e69ceeb3c70f3"
+    },
+    {
+      "path": "research/p104-productive-controller/CONTROLLER.json",
+      "usage": "provenance",
+      "git_blob": "cc539db709cf7a6c42872c4380902736ad453af3",
+      "sha256": "a84f6bbd547379597464e56c420cc665f96bc43a9a1351eed4b7cd0fce3cc4b9"
+    },
+    {
+      "path": "research/helper-refinement-budget-s29/README.md",
+      "usage": "provenance",
+      "git_blob": "e7b173fb96da2f45fa49ae49979486611cb13065",
+      "sha256": "87b0376c0156d1e87ae8a6e048b94806cbf9e718207fcd6de5798b720e639dd7"
+    },
+    {
+      "path": "research/productive-lever-reprice/README.md",
+      "usage": "provenance",
+      "git_blob": "4cb841dfb4399edcbf1394b918d0d038257b0b89",
+      "sha256": "57d2e2252eaf8844b7946f54aea5ecd386855d3220723acb350bfd34b2ab51d6"
+    }
+  ]
+}
diff --git a/research/selected-invoice-bit-cap-thresholds/TASK.json b/research/selected-invoice-bit-cap-thresholds/TASK.json
@@ -0,0 +1,129 @@
+{
+  "schema_version": 1,
+  "task_id": "p104-side-correction",
+  "priority_revision": 8,
+  "base_commit": "19fdb086ea96af9a9f87936a2b888e10af4ed810",
+  "original_card": {
+    "id": "p104-side-correction",
+    "question": "Can joint helper bases and a legal causal split improve the complete P104 productive/paid invoice?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+      "research/p104-helper-subspace-splits/README.md",
+      "research/dyadic-helper-channel-capacity/README.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "An exact arbitrary-input identity, source/target frame availability and restoration at the chosen cut, plus the complete channel+correction+idle/hosting bill or an explicit favorable bound. The native channel alone is not a supplier. 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",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Start with the selected #5599 emitted word and review pending #5718 distinct input/output bases plus #5717/#5719 coverage before claiming work. Pick one uncovered source word, helper group and cut region. Prove the arbitrary-input relation and legal entry/exit corrections, then compare full target and paid profiles. Reject if completion, owned lifetimes or consumer costs repay the gain. A static pairing rank alone is not a physical split.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "P104",
+      "helper basis",
+      "helper split",
+      "5718",
+      "dyadic capacity"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4133: literal-slot defect",
+        "evidence": "research/p104-literal-slot-defect/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "88666f1dee93fbd85b31fd5d6f91a219e111b5f4",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4136: uniform commutation gap",
+        "evidence": "research/p104-uniform-commutation-gap/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "281ce601a8be9db470de630759179c23e1dfaa88",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4140: uniform encoding gap",
+        "evidence": "research/p104-uniform-encoding-gap/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "de25c6525aa866962309df5d107b32d55e6e5628",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4153: source-chart shortcut exhausts its optimistic allowance",
+        "evidence": "research/p104-source-chart-shortcut/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "9123241fea53df77e51ffe30d9aca1625d4bf3a8",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4154: fiber-tag seam mismatch",
+        "evidence": "research/p104-fiber-tag-seam/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "4ffcdaee5d6255a9cea1fe53a5424435068089d0",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Exact accounting thresholds on the authenticated selected #5599 P104 invoice: the per-FULL deficit at C=17/5000, the fewest local paid deletions and free local helper splits that pass there, and the raw ceiling of free local helper refinement.",
+  "scope_change_reason": "The donation assignment asks for original frontier work. Before a helper group or causal cut is built, this prices how much local change the binding complex invoice needs before the admitted BIT saving binds instead.",
+  "bottleneck_link": "Complex binds at U=0.0031004563; the admitted BIT saving is 17/5000. This fixes the exact local per-FULL gap between them and shows that local helper refinement alone cannot pass at 1/100.",
+  "budget_minutes": 60,
+  "stop_rule": "Ship the first exact threshold table with changed-invoice controls; do not search constructions.",
+  "outcome": "completed",
+  "evidence": "verify.py authenticates the exported invoice bytes, checks the 18-fold local replication, evaluates every moment with correctly rounded 80-digit decimal ln/exp, and reruns each boundary case on the actual changed invoice.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Which physical local window can delete one paid rank-5, rank-6 or rank-7 job, or two rank-3/rank-4 jobs, while keeping every target and restoration?",
+      "why_it_matters": "Under this accounting, any one such deletion per FULL lets complex pass at the admitted BIT saving, after which BIT binds.",
+      "first_step": "List the 168+18+14 local paid jobs of rank at least five in the selected word and their producers and consumers.",
+      "blocked_by": "Earlier closed local rewrites of older words do not transfer automatically; no candidate window is checked here."
+    },
+    {
+      "question": "Can combined helper refinement and paid change raise both suppliers past 1/100?",
+      "why_it_matters": "Refining every local helper target alone fails at 1/100 with all paid work fixed, so a target near 0.01 needs paid or controller change on the complex side as well as BIT.",
+      "first_step": "Price the main-target and idle-paid families with the same exact moment at C=1/100.",
+      "blocked_by": "No changed controller or main-target contract is supplied."
+    }
+  ],
+  "claim_type": "finite exact certificate",
+  "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19227727",
+  "question_source": {
+    "origin": "original",
+    "note": "Original accounting question on the selected #5599 invoice."
+  }
+}