Mister-Meeseeks · Oct 22, 2026 · Oct 21, 2026
diff --git a/research/direct-flow-chain-receipt/README.md b/research/direct-flow-chain-receipt/README.md
@@ -0,0 +1,62 @@
+# Disposition receipt: the #4436 direct-flow chain
+
+**Claim type:** a source-bound overlap and disposition receipt (the overlap-receipt card). **No new κ.**
+It makes no closures and no merge recommendations. It covers zxv's #4436 and the five follow-ups around
+it, all open on base `2a13eb18`. `RECEIPT.json` records heads and package blobs.
+
+## Disposition
+
+Nothing is retained on main. Only #4436 edits files outside its package:
+`docs/ci-verification.md` and `docs/research/current-status.md`. These conflict textually with the p17
+chain's edits of the same files (see my #4474).
+
+| PR | Head | What it establishes |
+| --- | --- | --- |
+| #4436 (zxv) | `56003680` | Direct overlapping flow (§2) and nested-width size bounds (§3). The exponent gate is open. |
+| #4456 | `bbef80cd` | Independent check of §2: 9 recorded flows, the potential identity, exact `L(1000)`. |
+| #4459 | `488a75a9` | Mirror identity (exact), and the rigorous boundary-share bound `β ≤ 0.321`. **It corrects its own first version.** |
+| #4465 | `36d9c475` | The refined-induction ceiling `s*·g/(g + 1 + ln m)`. The growth of `g` comes from leaf exits alone. |
+| #4467 | `30ff97c3` | Leaf walk: exact `√G` regime to depth 60; saturation observed (overshoot tends to `1/λ* = 133.3`); single-step cap about 0.00257 < `s*`; depth-variant diagnostics. |
+| #4476 | `4e7022aa` | Independent check of §3: width factor, the exact slot identity `S + X + 2·matched = U(W+P)`, and `N log N` growth. |
+
+## Withdrawn claim
+
+The first version of #4459 (head `04da76eb`) claimed that the direct exponent gate reduces to a uniform
+boundary-exit share. That claim is **withdrawn**. Leaf exits reach about `0.397N` and carry 85–92% of
+the `t`-potential. The correction is at `488a75a9`.
+
+## Where the chain stands
+
+Under #4405's single-step refined induction, #4436's direct flow certifies about 0.0008 by
+`N ≈ 3·10⁵`. By the leaf-walk saturation heuristic, it reaches at most about 0.00257 asymptotically,
+and 0.0023 only near `N ≈ 10^{940}`.
+
+Open:
+- a proof of the saturation and of the stability of the shares;
+- a multi-step exit test;
+- the admissibility of a shallower lockstep depth, given its tiny low-source mass;
+- a tighter constructive width.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/direct-flow-chain-receipt --check-only
+python3.12 -B scripts/research_package.py --package research/direct-flow-chain-receipt
+```
+
+The verifier prints `PASS_DIRECT_FLOW_CHAIN_RECEIPT`.
+
+## Open next questions
+
+1. **A multi-step exit test for #4436's flow.**
+   - *Why it matters:* the single-step test is capped below `s*`.
+   - *First step:* substitute each exit's own flow once, and remeasure `κ` against `η`.
+
+## Credit
+
+- zxv's #4436.
+- 1337isnot1337's #4405.
+- Jacob Alan Sussman's Theorem 1G.
+
+Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.
+Apache-2.0.
diff --git a/research/direct-flow-chain-receipt/RECEIPT.json b/research/direct-flow-chain-receipt/RECEIPT.json
@@ -0,0 +1,311 @@
+{
+ "schema": "direct-flow-chain-receipt/1",
+ "base_commit": "2a13eb1827784206202db90e5dc8327cf8a250db",
+ "prs": {
+  "4436": {
+   "author": "zxv",
+   "role": "Direct overlapping flow (finite chronological integer flow) and nested-width size bounds; exponent gate open",
+   "head": "56003680aee2871773de408e5f01cc5dd46b144c",
+   "merge_base": "2a13eb1827784206202db90e5dc8327cf8a250db",
+   "package": "research/direct-mixed-flow",
+   "files": [
+    {
+     "path": "research/direct-mixed-flow/.gitattributes",
+     "blob": "2435bfdf636aef506604b8cbf6f1158aa8f94471"
+    },
+    {
+     "path": "research/direct-mixed-flow/README.md",
+     "blob": "3be00f754b9ffa23e64b3eafe7706e3ac139993a"
+    },
+    {
+     "path": "research/direct-mixed-flow/RESULT.json",
+     "blob": "e657f58b5a276b57b472a814a8559aede853f55a"
+    },
+    {
+     "path": "research/direct-mixed-flow/SOURCES.json",
+     "blob": "fcfba8aa31e2b15be7abeb2191c9e35fa0d7919b"
+    },
+    {
+     "path": "research/direct-mixed-flow/TASK.json",
+     "blob": "e232fd2674d6f10fd5f5f9258e6fb33f95667c3b"
+    },
+    {
+     "path": "research/direct-mixed-flow/VENDOR.json",
+     "blob": "d95b1124e8418a4e92a99f4eec125177e1c01c18"
+    },
+    {
+     "path": "research/direct-mixed-flow/VERIFY.json",
+     "blob": "de075b7c20b2ada3384ad389022f68ed3a8a548f"
+    },
+    {
+     "path": "research/direct-mixed-flow/flow.py",
+     "blob": "07175cd6e1f2c2367c2e78eca7f163fef18d301d"
+    },
+    {
+     "path": "research/direct-mixed-flow/vendor/FINITE-PRECOMPILATION.md",
+     "blob": "4badb40f0919c515823cb07e44c6470d4d6469ca"
+    },
+    {
+     "path": "research/direct-mixed-flow/vendor/P7-FACTOR.json",
+     "blob": "04c7ef620e760cc8f1764bbf0ccd7f32c6a5bd86"
+    },
+    {
+     "path": "research/direct-mixed-flow/vendor/SUSSMAN-LICENSE",
+     "blob": "261eeb9e9f8b2b4b0d119366dda99c6fd7d35c64"
+    },
+    {
+     "path": "research/direct-mixed-flow/vendor/SUSSMAN-NOTICE",
+     "blob": "0b06967886fab2628afc0b881c884161bbf5e430"
+    },
+    {
+     "path": "research/direct-mixed-flow/vendor/THREE-base-FULL-IDENTITY-NATIVE.json.gz",
+     "blob": "b46d44c6ee624968dd57be928b0c15d9d45f3b2c"
+    },
+    {
+     "path": "research/direct-mixed-flow/vendor/higher-moment-README.md",
+     "blob": "0a736ca89e1eafb73a30e1f2589cc45e6b5f76c3"
+    },
+    {
+     "path": "research/direct-mixed-flow/vendor/higher-moment-RESULT.json",
+     "blob": "e66b39c674028adfd22f0de3170555dccda52f6a"
+    },
+    {
+     "path": "research/direct-mixed-flow/vendor/p7-NATIVE.md",
+     "blob": "394d9a03c165336d1cfbcd41921c93741042393b"
+    },
+    {
+     "path": "research/direct-mixed-flow/vendor/p7-RESULT.json",
+     "blob": "ebb30a9b3b94fb159a316350f7da57bde4a30c25"
+    },
+    {
+     "path": "research/direct-mixed-flow/vendor/theorem-1g.md",
+     "blob": "36d2399f3538c34e4b1567a15b4a5dfe77ffc7ad"
+    },
+    {
+     "path": "research/direct-mixed-flow/vendor/tree_flow.py",
+     "blob": "31a9c1ae4d116b12e8db7d8e06f723f10425068d"
+    },
+    {
+     "path": "research/direct-mixed-flow/verify.py",
+     "blob": "df9ca12ad0dae5437ed7d65688e962b9abda5af4"
+    }
+   ],
+   "paths_outside_package": [
+    "docs/ci-verification.md",
+    "docs/research/current-status.md"
+   ]
+  },
+  "4456": {
+   "author": "pinterbanget",
+   "role": "Independent check of #4436 section 2: 9 recorded flows, potential identity, exact L(1000)",
+   "head": "bbef80cd12c66afa31cb347c7393fafc873a0c88",
+   "merge_base": "2a13eb1827784206202db90e5dc8327cf8a250db",
+   "package": "research/pr4436-direct-flow-check",
+   "files": [
+    {
+     "path": "research/pr4436-direct-flow-check/README.md",
+     "blob": "53f9980b213b48dca636a253effadf5fc99614a2"
+    },
+    {
+     "path": "research/pr4436-direct-flow-check/SOURCES.json",
+     "blob": "9124bb10fe889a583982b0f5228776c779f092bf"
+    },
+    {
+     "path": "research/pr4436-direct-flow-check/TASK.json",
+     "blob": "255f54a627f2a2f90026a5de5848243596af305b"
+    },
+    {
+     "path": "research/pr4436-direct-flow-check/VECTORS.json",
+     "blob": "d64806c78f375187d77aa4851a80cf8eda60d23b"
+    },
+    {
+     "path": "research/pr4436-direct-flow-check/VERIFY.json",
+     "blob": "5b6023d6edcc2489779b79dfe49d721888535828"
+    },
+    {
+     "path": "research/pr4436-direct-flow-check/trend_local.py",
+     "blob": "c9c735fd365dce5527e89a9c2075209a6b820a6c"
+    },
+    {
+     "path": "research/pr4436-direct-flow-check/verify.py",
+     "blob": "f6f3816e86ae0febd09809f996ea58b98676539b"
+    }
+   ],
+   "paths_outside_package": []
+  },
+  "4459": {
+   "author": "pinterbanget",
+   "role": "Mirror identity and rigorous boundary-share bound 0.321; CORRECTS its first version (leaf exits dominate; the reduction to the boundary share is withdrawn)",
+   "head": "488a75a9e95e14448b6301719249f876f3999931",
+   "merge_base": "2a13eb1827784206202db90e5dc8327cf8a250db",
+   "package": "research/direct-flow-boundary-share",
+   "files": [
+    {
+     "path": "research/direct-flow-boundary-share/README.md",
+     "blob": "a9c718b8e33752846d7a618c763d0d8f4a62aa13"
+    },
+    {
+     "path": "research/direct-flow-boundary-share/SOURCES.json",
+     "blob": "9124bb10fe889a583982b0f5228776c779f092bf"
+    },
+    {
+     "path": "research/direct-flow-boundary-share/TASK.json",
+     "blob": "4b41ba78d6a74cb6808420ca9ee4e73caa5f2e21"
+    },
+    {
+     "path": "research/direct-flow-boundary-share/VERIFY.json",
+     "blob": "b2b4795ff570219d963644e932996ad9844f8f63"
+    },
+    {
+     "path": "research/direct-flow-boundary-share/verify.py",
+     "blob": "44db6538ade1f715d847f6a77351f94343e4727d"
+    }
+   ],
+   "paths_outside_package": []
+  },
+  "4465": {
+   "author": "pinterbanget",
+   "role": "Refined-induction ceiling via the log-gap g; growth comes from leaf exits alone",
+   "head": "36d9c475d4bbc99e93952e0ad86a802e1bc581e8",
+   "merge_base": "2a13eb1827784206202db90e5dc8327cf8a250db",
+   "package": "research/direct-flow-log-gap",
+   "files": [
+    {
+     "path": "research/direct-flow-log-gap/README.md",
+     "blob": "3b71834adb961327da05c602b2ead7d98857dfa2"
+    },
+    {
+     "path": "research/direct-flow-log-gap/SOURCES.json",
+     "blob": "9124bb10fe889a583982b0f5228776c779f092bf"
+    },
+    {
+     "path": "research/direct-flow-log-gap/TASK.json",
+     "blob": "51276fd201392a05da98050683b39dbdd84dc6b8"
+    },
+    {
+     "path": "research/direct-flow-log-gap/VERIFY.json",
+     "blob": "45b34b1178ff1763e5986a618ca2c6b9548a7966"
+    },
+    {
+     "path": "research/direct-flow-log-gap/evidence/LOGGAP-LARGE.txt",
+     "blob": "28ec5db893235073eb131c557fd17f2ed924dfae"
+    },
+    {
+     "path": "research/direct-flow-log-gap/evidence/gap_large.py",
+     "blob": "0dd296ecc72b6786c4e577d3c29f55b49e344ecc"
+    },
+    {
+     "path": "research/direct-flow-log-gap/verify.py",
+     "blob": "563548374264d24260dacd845c0a017d2b04cb7b"
+    }
+   ],
+   "paths_outside_package": []
+  },
+  "4467": {
+   "author": "pinterbanget",
+   "role": "Leaf walk: exact sqrt(G) regime to depth 60, saturation observed (overshoot -> 1/lambda* = 133.3); single-step cap ~0.00257 < s*; depth-variant diagnostics",
+   "head": "30ff97c34f0a0e01ad40770fb6d21ddc11ea071b",
+   "merge_base": "2a13eb1827784206202db90e5dc8327cf8a250db",
+   "package": "research/direct-flow-leaf-walk",
+   "files": [
+    {
+     "path": "research/direct-flow-leaf-walk/README.md",
+     "blob": "02ddde948f2e94874850eadd70d7d32559916848"
+    },
+    {
+     "path": "research/direct-flow-leaf-walk/SOURCES.json",
+     "blob": "9124bb10fe889a583982b0f5228776c779f092bf"
+    },
+    {
+     "path": "research/direct-flow-leaf-walk/TASK.json",
+     "blob": "ac1d757ef46de2effd51c99c4ea84d38c5d581ba"
+    },
+    {
+     "path": "research/direct-flow-leaf-walk/VERIFY.json",
+     "blob": "ef13be973c6e7895c330114642bf4600e38fc439"
+    },
+    {
+     "path": "research/direct-flow-leaf-walk/diagnostics/DEPTH-VARIANTS.md",
+     "blob": "a874789ab7397d20f5eec8d5069cefb0e8131de2"
+    },
+    {
+     "path": "research/direct-flow-leaf-walk/diagnostics/SATURATION.md",
+     "blob": "418c78f90902aabfaa8e06e8503ac56b83b66734"
+    },
+    {
+     "path": "research/direct-flow-leaf-walk/diagnostics/depth_mixed_scan.py",
+     "blob": "962f48bcad83e9b745f235baca55b2949c47ea3d"
+    },
+    {
+     "path": "research/direct-flow-leaf-walk/diagnostics/depth_variants.py",
+     "blob": "669f551c22927decb696f2bb8e280c5cee874e33"
+    },
+    {
+     "path": "research/direct-flow-leaf-walk/diagnostics/overlap_flow_depth.py",
+     "blob": "3bed7b0bea540797e08ae4fcd1531bcec9bc8f05"
+    },
+    {
+     "path": "research/direct-flow-leaf-walk/diagnostics/saturate.py",
+     "blob": "1dc6bbed6ca92dbdc13f036e780dca03e34a51cb"
+    },
+    {
+     "path": "research/direct-flow-leaf-walk/evidence/LEAF-DEPTH-G60.txt",
+     "blob": "342b57de041c86911ec73323b882648660f0780b"
+    },
+    {
+     "path": "research/direct-flow-leaf-walk/evidence/leafwalk.py",
+     "blob": "7485ee36e692cb69e08728a285e323b981865e06"
+    },
+    {
+     "path": "research/direct-flow-leaf-walk/verify.py",
+     "blob": "42440e630a46c6e581797f52bfb4c0c87bbdac3a"
+    }
+   ],
+   "paths_outside_package": []
+  },
+  "4476": {
+   "author": "pinterbanget",
+   "role": "Independent check of #4436 section 3: width factor, exact routing-slot identity, N log N growth",
+   "head": "4e7022aa9128c8673d3e2024052b2deaa18cb6e4",
+   "merge_base": "2a13eb1827784206202db90e5dc8327cf8a250db",
+   "package": "research/direct-flow-width-check",
+   "files": [
+    {
+     "path": "research/direct-flow-width-check/README.md",
+     "blob": "c6db026010b7e2ec8a4063065c4d054401cfc9cd"
+    },
+    {
+     "path": "research/direct-flow-width-check/SOURCES.json",
+     "blob": "9124bb10fe889a583982b0f5228776c779f092bf"
+    },
+    {
+     "path": "research/direct-flow-width-check/TASK.json",
+     "blob": "3186769cf01485eaad1f50e598396b7f0870b52c"
+    },
+    {
+     "path": "research/direct-flow-width-check/VERIFY.json",
+     "blob": "d9b91f131cc57ac8ccfb9c75205ed3a2e9c19c57"
+    },
+    {
+     "path": "research/direct-flow-width-check/verify.py",
+     "blob": "c014cd5079587585120d86115d75999a8965fcc2"
+    }
+   ],
+   "paths_outside_package": []
+  }
+ },
+ "withdrawn_claims": [
+  {
+   "pr": "4459",
+   "head": "04da76ebfb827cabb3560d3c65edee908f39df63",
+   "claim": "the direct exponent gate reduces to a uniform boundary-exit share",
+   "reason": "leaf exits reach ~0.397 N and carry 85-92% of the t-potential; corrected at 488a75a9"
+  }
+ ],
+ "open": [
+  "proof of leaf-walk saturation and share stability",
+  "multi-step exit test",
+  "admissibility of shallower lockstep depth (low-source mass)",
+  "tighter constructive width"
+ ]
+}
diff --git a/research/direct-flow-chain-receipt/SOURCES.json b/research/direct-flow-chain-receipt/SOURCES.json
@@ -0,0 +1,18 @@
+{
+  "schema_version": 1,
+  "base_commit": "2a13eb1827784206202db90e5dc8327cf8a250db",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ed33caef5c5827693e9572d252a66f99cbb73d40",
+      "sha256": "5732707cc1e460b090b53f2463162749ad860de878a18f7408938671d295a9d9"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65",
+      "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96"
+    }
+  ]
+}