Mister-Meeseeks · Oct 21, 2026 · Oct 21, 2026
diff --git a/research/depth5-connector-mitm/EVIDENCE.json b/research/depth5-connector-mitm/EVIDENCE.json
@@ -0,0 +1,214 @@
+{
+ "schema": "depth5-connector-mitm/1",
+ "mitm_sha256": "2b88b27be8c398f31e327ef742bbcf76cff295cc1086d927e2e04c9857c9631a",
+ "runs": [
+  {
+   "m": 5,
+   "counts": [
+    5,
+    5,
+    5,
+    5,
+    5
+   ],
+   "residues": [
+    1
+   ],
+   "seconds": 15,
+   "command": "verify.py reruns this",
+   "environment": "live"
+  },
+  {
+   "m": 5,
+   "counts": [
+    10,
+    5,
+    5,
+    5,
+    5
+   ],
+   "residues": [
+    1,
+    2
+   ],
+   "seconds": 138.7,
+   "command": "python3.12 -B mitm.py 5 10,5,5,5,5 --residues",
+   "environment": "CPython 3.12.4, numpy 1.26.4, Windows 11, 32 GB"
+  },
+  {
+   "m": 5,
+   "counts": [
+    5,
+    10,
+    5,
+    5,
+    5
+   ],
+   "residues": [
+    1,
+    2
+   ],
+   "seconds": 148.9,
+   "command": "python3.12 -B mitm.py 5 5,10,5,5,5 --residues",
+   "environment": "CPython 3.12.4, numpy 1.26.4, Windows 11, 32 GB"
+  },
+  {
+   "m": 5,
+   "counts": [
+    5,
+    5,
+    10,
+    5,
+    5
+   ],
+   "residues": [
+    1,
+    2
+   ],
+   "seconds": 162.5,
+   "command": "python3.12 -B mitm.py 5 5,5,10,5,5 --residues",
+   "environment": "CPython 3.12.4, numpy 1.26.4, Windows 11, 32 GB"
+  },
+  {
+   "m": 5,
+   "counts": [
+    5,
+    5,
+    5,
+    10,
+    5
+   ],
+   "residues": [
+    1,
+    2
+   ],
+   "seconds": 159.9,
+   "command": "python3.12 -B mitm.py 5 5,5,5,10,5 --residues",
+   "environment": "CPython 3.12.4, numpy 1.26.4, Windows 11, 32 GB"
+  },
+  {
+   "m": 5,
+   "counts": [
+    5,
+    5,
+    5,
+    5,
+    10
+   ],
+   "residues": [
+    1,
+    2
+   ],
+   "seconds": 143.0,
+   "command": "python3.12 -B mitm.py 5 5,5,5,5,10 --residues",
+   "environment": "CPython 3.12.4, numpy 1.26.4, Windows 11, 32 GB"
+  },
+  {
+   "m": 6,
+   "counts": [
+    6,
+    6,
+    6,
+    6,
+    6
+   ],
+   "residues": [
+    1,
+    3
+   ],
+   "seconds": 205.0,
+   "command": "python3.12 -B mitm.py 6 6,6,6,6,6 --residues",
+   "environment": "CPython 3.12.4, numpy 1.26.4, Windows 11, 32 GB"
+  }
+ ],
+ "witnesses": [
+  {
+   "m": 5,
+   "c": 1,
+   "word": "4512341231215432154325435",
+   "counts": [
+    5,
+    5,
+    5,
+    5,
+    5
+   ]
+  },
+  {
+   "m": 5,
+   "c": 2,
+   "word": "112345112311211432115432543545",
+   "counts": [
+    10,
+    5,
+    5,
+    5,
+    5
+   ]
+  },
+  {
+   "m": 5,
+   "c": 3,
+   "word": "45234111231112111543254322212225435",
+   "counts": [
+    10,
+    10,
+    5,
+    5,
+    5
+   ]
+  },
+  {
+   "m": 5,
+   "c": 4,
+   "word": "11223411223112211543221154322543545",
+   "counts": [
+    10,
+    10,
+    5,
+    5,
+    5
+   ]
+  },
+  {
+   "m": 6,
+   "c": 1,
+   "word": "345212341231121354321554325454",
+   "counts": [
+    6,
+    6,
+    6,
+    6,
+    6
+   ]
+  },
+  {
+   "m": 6,
+   "c": 3,
+   "word": "121234543112121354322443555435",
+   "counts": [
+    6,
+    6,
+    6,
+    6,
+    6
+   ]
+  }
+ ],
+ "minima": {
+  "5": {
+   "1": 25,
+   "2": 30,
+   "3": 35,
+   "4": 35
+  },
+  "6": {
+   "1": 30,
+   "3": 30,
+   "2": ">=36",
+   "4": ">=36",
+   "5": ">=36"
+  }
+ },
+ "note": "Length 5m for m<=5 reruns in the verifier; the length-30 runs are local evidence; length-35 hits are certified by witnesses."
+}
diff --git a/research/depth5-connector-mitm/README.md b/research/depth5-connector-mitm/README.md
@@ -0,0 +1,94 @@
+# Exact depth-five connector minima at m = 5, and the length-5m residues at m = 6
+
+**Claim type:** an exhaustive finite computation by meet-in-the-middle, with explicit
+certificates. **No new κ.** It answers open question 1 of petalas's
+[#4281](https://github.com/CrocSwap/integer-mult-bounds/pull/4281) at m = 5 and m = 6, which is
+also open next question 1 of my #4341.
+
+## Model
+
+The model is #4281's Lemma 1. A positive word over calls `1..5` is a correct depth-five dirty chain
+connector for residue `c` exactly when the product of the generators `g_l = I + E[l][l−1]` in the
+lower unitriangular group `U₆(ℤ/m)` equals `I + c·E[5][0]`. The subdiagonal entries count the calls,
+so every call is used a positive multiple of `m` times. A word of length `5m` uses each call exactly
+`m` times.
+
+## Method
+
+Split a word with call counts `n` into a prefix of `⌊L/2⌋` letters, with counts `a`, and a suffix
+with counts `n − a`. Write `P` for the prefix product and `S` for the suffix product. Row 0 of `S⁻¹`
+is `e₀`, so
+
+    P·S = I + cE₅₀  ⟺  P = S⁻¹ + cE₅₀.
+
+So `P` and `S⁻¹` agree on the 14 coordinates other than `(5,0)`, and then `c = P₅₀ − (S⁻¹)₅₀`.
+`mitm.py` builds the deduplicated sets of prefix products and suffix inverses for every count vector:
+the prefixes by right multiplication with `g_l`, and the suffix inverses by right multiplication with
+`g_l⁻¹`, prepending letters. It matches them by sorted search, which decides every residue in one
+pass. On a hit it extracts an explicit word by backtracking through the stored layers and replays it.
+#4281's own `m = 5` witness confirms the product convention, and so do all 173 of its witnesses,
+checked locally.
+
+## Results
+
+| m | c | minimum length | how decided |
+| --- | --- | --- | --- |
+| 5 | 1 | **25 = 5m** | witness; also #4281 |
+| 5 | 2 | **30** | none at 25 (exhaustive); witness at 30, counts (10,5,5,5,5) |
+| 5 | 3 | **35** | none at 25 or 30 (all five count vectors); witness at 35, counts (10,10,5,5,5) |
+| 5 | 4 | **35** | as for `c = 3` |
+| 6 | 1 | **30 = 5m** | **new** witness; #4281's search had not found it |
+| 6 | 3 | **30 = 5m** | witness; also #4281 |
+| 6 | 2, 4, 5 | **≥ 36** | none at 30 (exhaustive) |
+
+**Controls.** The same search finds no length-`5m` word at `m = 2, 3, 4`. That agrees with #4281's
+exhaustive BFS minima of 16, 24, and 24/28 at those `m`.
+
+So #4281's "search failures" at `m = 5` are genuine: the premium over `5m` is 5 for `c = 2` and 10
+for `c = 3, 4`. At `m = 6`, residue 1 attains the floor after all, while 2, 4 and 5 do not.
+
+## Evidence
+
+| Check | Where it runs | Time |
+| --- | --- | --- |
+| `m = 2, 3, 4, 5` at length `5m` | reruns in the verifier | about 30 s |
+| all witness words, replayed | in the verifier | instant |
+| `m = 5` at length 30, all five count vectors | local, `evidence/` | about 150 s each |
+| `m = 6` at length 30 | local, `evidence/` | about 160 s |
+
+The local runs record the exact command and the `mitm.py` SHA256 in `EVIDENCE.json`.
+`python3.12 -B verify.py --full` reruns them. The length-35 hits are certified by their witnesses,
+so no exhaustive run is needed for them.
+
+## Not covered
+
+- `m = 7, 8`. The `m = 7` length-35 run exceeded this machine's 32 GB when run in parallel.
+- Exact minima of `c = 2, 4, 5` at `m = 6`, beyond the bound `≥ 36`.
+- Any physical price, admission or caller.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/depth5-connector-mitm --check-only
+python3.12 -B scripts/research_package.py --package research/depth5-connector-mitm
+python3.12 -B research/depth5-connector-mitm/mitm.py 5 10,10,5,5,5      # the length-35 witnesses, ~25 min
+```
+
+The registered command prints `PASS_DEPTH5_CONNECTOR_MITM`. It needs `numpy`.
+
+## Open next questions
+
+1. **What are the `m = 7, 8` length-`5m` residues?**
+   - *Why it matters:* they would settle the rest of #4281's eleven search failures.
+   - *First step:* run the decision-only `residues` mode one count vector at a time on a machine
+     with 64 GB of RAM, or reduce memory by streaming prefix groups.
+2. **What are the exact `m = 6` minima for `c = 2, 4, 5`?**
+   - *First step:* the length-36 runs, three count vectors up to reversal.
+
+## Credit
+
+- The connector model, Lemma 1, the witnesses and the question are Nick Petalas's (#4281,
+  building on #4249 and #4263).
+
+Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.
+Apache-2.0.
diff --git a/research/depth5-connector-mitm/SOURCES.json b/research/depth5-connector-mitm/SOURCES.json
@@ -0,0 +1,18 @@
+{
+  "schema_version": 1,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65",
+      "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96"
+    }
+  ]
+}
diff --git a/research/depth5-connector-mitm/TASK.json b/research/depth5-connector-mitm/TASK.json
@@ -0,0 +1,68 @@
+{
+  "schema_version": 1,
+  "task_id": "connector-alphabet-change",
+  "priority_revision": 7,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "original_card": {
+    "id": "connector-alphabet-change",
+    "question": "Does one explicit extension of the connector alphabet offer a favorable completed budget?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/universal-connector-boundary/README.md",
+      "certificates/selected-result.json"
+    ],
+    "methods": [
+      "Construct and charge a legal extra operation on a minimal instance.",
+      "Prove a target-specific bound after adding one specified operation family."
+    ],
+    "acceptance_gate": "State which fixed-alphabet premise changes and charge its implementation; do not declare it free. 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/universal-connector-boundary/verify.py"
+      ],
+      "expected": "PASS_UNIVERSAL_CONNECTOR_BOUNDARY",
+      "inspect": "Reproduce the 945-pairing control, then read Exact interface to list the allowed calls.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Add exactly one legally priced operation and test a tiny dirty-input instance before enumerating larger cases.",
+    "bottleneck_link": "A new legal connector operation could change recursive interface cost; a larger census in the same alphabet is a separate side result.",
+    "allowed_variations": [
+      "Choose one additional operation and a small source count for its first counterexample."
+    ],
+    "scope_change_triggers": [
+      "Keeping the alphabet and only increasing the source count; record this as a changed assignment.",
+      "Extending a source-count census without naming a supplier/interface decision it could change; audit #2973 pruning concerns before relying on the affected predecessor exclusions."
+    ],
+    "overlap_terms": [
+      "universal connector",
+      "source-toggle",
+      "connector alphabet"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Answer #4281's open question 1 at m=5 (exact minima for every residue) and decide the length-5m residues at m=6, by exact meet-in-the-middle.",
+  "scope_change_reason": "#4281 and #4341 left the m=5..8 residues as search failures; a meet-in-the-middle avoids the 30 GB table.",
+  "bottleneck_link": "Depth-five connector premiums price the chain-connector alphabet change.",
+  "budget_minutes": 90,
+  "stop_rule": "Stop at m=5 exact and m=6 length 30; m=7,8 exceed local memory.",
+  "outcome": "completed",
+  "evidence": "m=5 minima 25,30,35,35 for c=1..4 with witnesses and exhaustive exclusions; m=6 length 30 attained exactly for c=1 (new witness) and c=3; controls m=2,3,4 none at 5m.",
+  "author": "Ryanson Jonathan, with Claude Code (Claude Opus 5.5) assistance",
+  "ai_assistance": "Substantial Claude Code (Claude Opus 5.5) assistance; no independent human review.",
+  "open_questions": [
+    {
+      "question": "What are the m=7, 8 length-5m residues?",
+      "why_it_matters": "They settle the rest of #4281's search failures.",
+      "first_step": "Decision-only runs one count vector at a time on a 64 GB machine.",
+      "blocked_by": "Local memory."
+    },
+    {
+      "question": "Exact m=6 minima for c=2,4,5?",
+      "why_it_matters": "Completes the m=6 table.",
+      "first_step": "Length-36 runs, three count vectors up to reversal.",
+      "blocked_by": "None."
+    }
+  ]
+}