Bound the P104 rank-1 hosting threshold

research/p104-hosting-threshold/README.md

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+# Rank-1 hosting of 1127 meets the selected saving
+
+**Partial. No new exponent.**
+
+The native channel histogram is
+
+```text
+1074 H1 + 58 H2 + 86 H3 + 104 H4 + 48 H5 + 24 H6 + 8 H7 + 8 H8
+```
+
+with rank mass 2,368. The minimizing side correction is 208 unit-step
+helpers, nine rank-one steps each, adding 1,872. The local mass is 4,240.
+On the template `W=832/9`, centre 56 and total mass 6,712, the idle mass is
+2,472, which is `824/17` ideal rank-51 children.
+
+At the selected saving the moment of that bill is `0.9988962438816702`,
+below one. The same bill still beats the selected saving after 1,126 extra
+rank-1 hosting units. At 1,127 units the moment goes above one. Both counts
+fit inside the idle room.
+
+The geometric budget leaves native ADDs, owners, routing, precision and tape
+unpriced. This cut is the rank-1 hosting threshold for that omission.
+
+## What was executed
+
+The checker rebuilds the histogram and calls `ceiling.relax` at the selected
+saving. It does not rerun the 208-flag matching or the two-type optimizer.
+
+`scripts/research_baseline.py` needs
+`research/community-frontier1094-audit/ASSEMBLY.json`, which this checkout
+does not have. The orientation used here is the channel histogram and the
+geometric-budget sentences.
+
+## Scope and evidence
+
+Original card: `p104-side-correction`, priority revision 6, refined to this
+hosting cut. [TASK.json](TASK.json) records the scope change.
+[RESULT.json](RESULT.json) is the executed receipt and is not rewritten.
+[SOURCES.json](SOURCES.json) pins the two notes, the ceiling function and
+the selected record. No selected frontier file changes.
+
+Upstream checkpoint: `20ecf2891d80b5acf18d980b2164fb44ea7faa90`, priority
+revision 6, checked 2026-10-18 06:15 UTC. Catalogue and selected record match
+that commit. Handoff discussion #1166, through the PR3564 note at 2026-10-18
+06:15 UTC, was released work. Draw seed
+`b35d4e81a6bbaab3024f2cde36708828`, track `complex-mechanisms`, third card
+`p104-side-correction`.
+
+## Reproduce
+
+From the repository root, standard-library Python 3.12:
+
+```sh
+python3.12 -B research/p104-hosting-threshold/verify.py
+python3.12 -B research/p104-hosting-threshold/test_verify.py
+python3.12 -B scripts/research_package.py --package research/p104-hosting-threshold
+```
+
+The moment checks finish in about a second.
+
+## Open next questions
+
+The 1,126/1,127 rank-1 cut is this template's hosting decision.
+
+## Credit
+
+The channel histogram is Rohan Arun's PR1972, with the native recipes
+prepared for Chafik Boukhalfa. The two-history budget and the 208-unit
+minimum are Nick Petalas's. The interval engine is Douglas Colkitt's packing
+ceiling, with Claude (Anthropic). The selected kappa is the #1094 record.
+
+Prepared by jwr with substantial Cursor agent assistance. No independent human
+review is claimed. Apache-2.0.

research/p104-hosting-threshold/RESULT.json

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+{
+  "channel_mass": 2368,
+  "unit_steps": 1872,
+  "local_mass": 4240,
+  "idle_mass": 2472,
+  "idle_children": "824/17",
+  "moment_below_one": true,
+  "hosting_rank1_below": 1126,
+  "hosting_rank1_above": 1127,
+  "no_new_kappa": true
+}

research/p104-hosting-threshold/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "files": [
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "research/p104-endpoint-geometric-budget/README.md",
+      "usage": "input",
+      "git_blob": "9a0b7e19facaa63166075b4a4f347f3336538804",
+      "sha256": "00d11f4e80a2c571ceaf71a492fce3578b53a6ec2ea66809381d4782593db153"
+    },
+    {
+      "path": "research/p104-native-channel/README.md",
+      "usage": "input",
+      "git_blob": "f187d1ac0b7792d08d2dedf4561380b798e541b5",
+      "sha256": "ef874f86415c98b553bda9199d5f8f2f7a4dd4f6a23e53712c3c1f7268ec5b23"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/ceiling.py",
+      "usage": "input",
+      "git_blob": "6c0f54527d879396f689523bfde19928a9b6157c",
+      "sha256": "86a1432870b3b21d8454106ec9162fddc0471472d9214c590a1420f2426cedfc"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "input",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    }
+  ]
+}

research/p104-hosting-threshold/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "p104-side-correction",
+  "priority_revision": 6,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "original_card": {
+    "id": "p104-side-correction",
+    "question": "Can the proposed P104 joint endpoint flags be realized by a chronological correction program with coefficients, dirty restoration and an affordable complete bill?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/p104-native-channel/README.md",
+      "research/p104-native-channel/PROOF.md",
+      "docs/research/community-efficient1000-review.md",
+      "research/p104-joint-unit-flags/README.md"
+    ],
+    "methods": [
+      "Construct one side-correction word at an explicit chronological cut with the needed frames still available.",
+      "Derive a lower bound on the side correction and hosting costs that decides whether this channel can beat the selected budget."
+    ],
+    "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_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read the revision-6 family record and #2981 static joint flags, alongside the reviewed #1991 channel. #2981 is an author-checked source archive, not an independently validated executable correction or a refutation of program-level lifetime floors.",
+      "scope": "Source-bound orientation only. Read the named package and review before relying on its local evidence; this command is not its replay or supplier admission."
+    },
+    "first_experiment": "After checking live handoffs, pin one named collection from #2981 and attach acquisition/delivery times, source dependencies and coefficient requirements. Test legal access and arbitrary dirty restoration; price any completion against the full channel/endpoint/hosting allowance. Stop with the exact missing implication or a scoped counterexample rather than expanding the static cover census.",
+    "bottleneck_link": "Completing an already realized changed channel tests a concrete construction opportunity, instead of refining the incumbent packing. A positive local result still needs global and all-size admission.",
+    "allowed_variations": [
+      "One correction factorization, one chronological cut, or one explicit hosting obstruction.",
+      "A changed correction schedule with an updated full bill and native frame contract."
+    ],
+    "scope_change_triggers": [
+      "Recompiling the already certified low-rank channel without closing a new obligation.",
+      "Treating a pending ancestor, local histogram or unpriced side correction as an admitted global supplier.",
+      "Repeating a correction or physical conversion already supplied by a pending follow-up without adding independent coverage or a different unresolved obligation.",
+      "Repeating the 208-flag static matching or calling a static nested cover a chronological correction program. Independent audit of that certificate is distinct and remains useful."
+    ],
+    "overlap_terms": [
+      "P104",
+      "I-G/2",
+      "side correction",
+      "native channel",
+      "1972",
+      "1991",
+      "2586",
+      "endpoint transport",
+      "physical dual",
+      "relay composition",
+      "2981",
+      "joint unit flags",
+      "acquisition delivery",
+      "static nesting"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #2981 reports static joint endpoint flags and a matching of 208 departure and arrival flags; archival intake is not independent verification.",
+        "evidence": "research/p104-joint-unit-flags/README.md",
+        "remaining": "Independent flag check and a chronological coefficient program with dirty restoration and a complete cost.",
+        "status": "author-checked",
+        "source_commit": "3d24b92d8ba057b57cb2d480170ec407520e6882",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Price the native channel plus 208 unit-step children at the selected saving, and find the extra rank-1 hosting count that meets moment one.",
+  "scope_change_reason": "The card allows one hosting obstruction. This package uses the published channel histogram and the 208-unit side correction. It does not rerun the flag matching.",
+  "bottleneck_link": "That side-correction bill stays below moment one at the selected saving. Rank-1 hosting of 1127 units meets moment one inside the same template.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop at this rank-1 hosting cut. Do not replay the 208-flag matching or the two-type optimizer.",
+  "outcome": "partial",
+  "evidence": "The channel mass is 2368 and the unit steps add 1872. At the selected saving the moment stays below one. Extra rank-1 hosting of 1126 stays below one, and 1127 goes above.",
+  "author": "jwr",
+  "ai_assistance": "Substantial assistance from the Cursor coding agent in the execution, checker and writeup.",
+  "open_questions": [],
+  "no_open_questions_reason": "The 1126/1127 rank-1 cut is this template's hosting decision. A native ADD invoice would leave this count."
+}

research/p104-hosting-threshold/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/p104-endpoint-geometric-budget/README.md",
+    "research/p104-native-channel/README.md",
+    "research/complex-local-word-ceiling/ceiling.py",
+    "certificates/selected-result.json"
+  ],
+  "commands": [
+    {
+      "argv": ["{python}", "-B", "{package}/verify.py"],
+      "timeout_seconds": 60,
+      "checks": "Price the channel plus unit steps and the 1126/1127 rank-1 hosting cut."
+    },
+    {
+      "argv": ["{python}", "-B", "{package}/test_verify.py"],
+      "timeout_seconds": 30,
+      "checks": "Check the channel mass, the unit steps and the idle remainder."
+    }
+  ]
+}

research/p104-hosting-threshold/test_verify.py

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+#!/usr/bin/env python3
+"""The channel histogram plus 208 nine-step helpers is the published local mass."""
+
+import importlib.util
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+spec = importlib.util.spec_from_file_location("p104_hosting_threshold", HERE / "verify.py")
+verify = importlib.util.module_from_spec(spec)
+spec.loader.exec_module(verify)
+
+
+def main():
+    if verify.mass(verify.CHANNEL) != 2368:
+        raise SystemExit("the channel mass moved")
+    if verify.CHANNEL[1] + verify.UNITS * verify.STEP != 2946:
+        raise SystemExit("the unit steps moved")
+    if 6712 - (2368 + 1872) != 2472:
+        raise SystemExit("the idle mass moved")
+    print("PASS_P104_HOSTING_THRESHOLD_TESTS groups=1")
+
+
+if __name__ == "__main__":
+    main()

research/p104-hosting-threshold/verify.py

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+#!/usr/bin/env python3
+"""Price the P104 channel plus its unit-step side correction.
+
+The published histogram plus 208 nine-step helpers is the local bill.
+Extra rank-1 hosting is added until the moment at the selected saving
+crosses one.
+"""
+
+import hashlib
+import json
+import sys
+from fractions import Fraction
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[2]
+PACKAGE = Path(__file__).resolve().parent
+sys.path.insert(0, str(ROOT / "research/complex-local-word-ceiling"))
+import ceiling as ceiling_mod  # noqa: E402
+
+MARKER = "PASS_P104_HOSTING_THRESHOLD moment_below_one=true no_new_kappa"
+CHANNEL = {1: 1074, 2: 58, 3: 86, 4: 104, 5: 48, 6: 24, 7: 8, 8: 8}
+UNITS = 208
+STEP = 9
+TOTAL = 6712
+SHOWN = "0.9988962438816702"
+
+
+def fail(message):
+    raise SystemExit("FAIL_P104_HOSTING_THRESHOLD " + message)
+
+
+def require(condition, message):
+    if not condition:
+        fail(message)
+
+
+def sha256(path):
+    digest = hashlib.sha256()
+    with path.open("rb") as handle:
+        for block in iter(lambda: handle.read(1 << 20), b""):
+            digest.update(block)
+    return digest.hexdigest()
+
+
+def authenticate():
+    sources = json.loads((PACKAGE / "SOURCES.json").read_text(encoding="utf-8"))
+    pinned = {row["path"]: row for row in sources["files"] if row["usage"] == "input"}
+    expected = {
+        "research/p104-endpoint-geometric-budget/README.md",
+        "research/p104-native-channel/README.md",
+        "research/complex-local-word-ceiling/ceiling.py",
+        "certificates/selected-result.json",
+    }
+    require(set(pinned) == expected, "input inventory changed")
+    for path, row in pinned.items():
+        require(sha256(ROOT / path) == row["sha256"], "input drift " + path)
+
+
+def mass(profile):
+    return sum(rank * count for rank, count in profile.items())
+
+
+def bill():
+    paid = dict(CHANNEL)
+    paid[1] += UNITS * STEP
+    return paid
+
+
+def moment(profile, saving):
+    return ceiling_mod.relax(profile, 73, 832, 9, 56, saving, unpaid=1)
+
+
+def evaluate():
+    authenticate()
+    budget = (ROOT / "research/p104-endpoint-geometric-budget/README.md").read_text(encoding="utf-8")
+    native = (ROOT / "research/p104-native-channel/README.md").read_text(encoding="utf-8")
+    selected = json.loads((ROOT / "certificates/selected-result.json").read_text(encoding="utf-8"))
+    for sentence in (
+        "1074 H1 + 58 H2 + 86 H3 + 104 H4 + 48 H5 + 24 H6 + 8 H7 + 8 H8.",
+        "Its rank mass is 2368. Every added lifetime consumes nine more rank units,",
+        "The minimizing profile has `H1=2946`",
+        "local rank mass 4240 and idle rank mass 2472",
+        SHOWN,
+        "Native ADDs, owners, routing, precision, tape and\nall other physical charges are unknown and additional.",
+        "centre charge 56, total paid\nrank mass 6712",
+    ):
+        require(sentence in budget, "the budget sentence moved")
+    require("chronological global hosting, precision and all-size admission remain open." in native, "the hosting sentence moved")
+    require(mass(CHANNEL) == 2368, "the channel mass moved")
+    paid = bill()
+    require(paid[1] == 2946 and mass(paid) == 4240, "the side-correction bill moved")
+    idle = TOTAL - mass(paid)
+    require(idle == 2472 and Fraction(idle, 51) == Fraction(824, 17), "the idle children moved")
+    kappa = Fraction(selected["kappa"])
+    saving = kappa / (1 - kappa)
+    low, high = moment(paid, saving)
+    require(high < 1 and f"{float(low):.16f}" == SHOWN and f"{float(high):.16f}" == SHOWN, "the side-correction moment moved")
+    below = dict(paid)
+    below[1] += 1126
+    above = dict(paid)
+    above[1] += 1127
+    require(moment(below, saving)[1] < 1, "1126 rank-1 hosting met moment one")
+    require(moment(above, saving)[0] > 1, "1127 rank-1 hosting stayed under moment one")
+    require(1127 <= idle, "the hosting cut left the template")
+    return {
+        "channel_mass": 2368,
+        "unit_steps": UNITS * STEP,
+        "local_mass": 4240,
+        "idle_mass": idle,
+        "idle_children": "824/17",
+        "moment_below_one": True,
+        "hosting_rank1_below": 1126,
+        "hosting_rank1_above": 1127,
+        "no_new_kappa": True,
+    }
+
+
+def main():
+    observed = evaluate()
+    stored = json.loads((PACKAGE / "RESULT.json").read_text(encoding="utf-8"))
+    require(observed == stored, "RESULT.json does not match the execution: " + json.dumps(observed, sort_keys=True))
+    print(MARKER)
+
+
+if __name__ == "__main__":
+    main()