Independent verification: stdlib re-certification of PR1615's eight-transfer threshold, with all 2,502 eight-transfer schedules (only port splits r=1, r=5) (no new kappa)

research/two-connector-eight-transfer-census-seed58/EXPECTED.json

@@ -0,0 +1,161 @@
+{
+  "zero": {
+    "restored": 30240,
+    "max_by_split": {
+      "0": 31,
+      "1": 17,
+      "2": 11,
+      "3": 11,
+      "4": 17,
+      "5": 31
+    },
+    "histogram_sha256": "ea0992346256fcf4d50a3307274013c796e46338ed47396c332c34dd6e09703f"
+  },
+  "four": {
+    "max_by_split": {
+      "0": 31,
+      "1": 31,
+      "2": 21,
+      "3": 17,
+      "4": 19,
+      "5": 31
+    },
+    "pr1498_split_bounds": {
+      "0": 31,
+      "1": 31,
+      "2": 25,
+      "3": 17,
+      "4": 19,
+      "5": 31
+    },
+    "restored": 8994240,
+    "histogram_sha256": "81342993c98b860dac1e9120ae549dfdbb87f89893fba501574b1e62eeaefa5f"
+  },
+  "six": {
+    "restored": 44936640,
+    "max_by_split": {
+      "0": 31,
+      "1": 31,
+      "2": 28,
+      "3": 22,
+      "4": 23,
+      "5": 31
+    },
+    "histogram_sha256": "4d33ab99d91ef7a8f70f203dc5bbba20592aafb57dbaa93e1de9579d21fcfcd5"
+  },
+  "eight": {
+    "restored": 124284240,
+    "max_by_split": {
+      "0": 31,
+      "1": 32,
+      "2": 31,
+      "3": 27,
+      "4": 24,
+      "5": 32
+    },
+    "universal": 2502,
+    "universal_by_split": {
+      "1": 2376,
+      "5": 126
+    },
+    "universal_sha256": "ba04f2775171c9c32ae45406415c97a0440338a510266d9b758bf82da3ea7da0",
+    "discussion1412_pilot": {
+      "word": [
+        0,
+        1,
+        2,
+        0,
+        3,
+        1,
+        4,
+        2,
+        3,
+        4
+      ],
+      "ports": 31,
+      "gaps": [
+        0,
+        1,
+        1,
+        2,
+        1,
+        0,
+        1,
+        0,
+        1,
+        1,
+        0
+      ]
+    }
+  },
+  "independent_full_placement_receipts": {
+    "engine": "independent/census.c (every C placement, no gap reduction)",
+    "four": {
+      "candidates": 30270240,
+      "restored": 10402560,
+      "universal": 0,
+      "max_by_split": {
+        "0": 31,
+        "1": 31,
+        "2": 21,
+        "3": 17,
+        "4": 19,
+        "5": 31
+      }
+    },
+    "six": {
+      "candidates": 242161920,
+      "restored": 74692800,
+      "universal": 0,
+      "max_by_split": {
+        "0": 31,
+        "1": 31,
+        "2": 28,
+        "3": 22,
+        "4": 23,
+        "5": 31
+      }
+    },
+    "eight": {
+      "candidates": 1323241920,
+      "restored": 384199200,
+      "universal": 2502,
+      "max_by_split": {
+        "0": 31,
+        "1": 32,
+        "2": 31,
+        "3": 27,
+        "4": 24,
+        "5": 32
+      }
+    }
+  },
+  "independent_per_occurrence_receipts": {
+    "engine": "independent/census_numpy.py (each occurrence chooses its port; exact z1 restoration filter)",
+    "four": {
+      "legal": 10402560,
+      "universal": 0,
+      "max_by_split": {
+        "0": 31,
+        "1": 31,
+        "2": 21,
+        "3": 17,
+        "4": 19,
+        "5": 31
+      }
+    },
+    "six": {
+      "legal": 74692800,
+      "universal": 0,
+      "max_by_split": {
+        "0": 31,
+        "1": 31,
+        "2": 28,
+        "3": 22,
+        "4": 23,
+        "5": 31
+      },
+      "seconds_on_one_core": 842
+    }
+  }
+}

research/two-connector-eight-transfer-census-seed58/README.md

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+# Independent verification of the eight-transfer threshold, with all 2,502 solutions
+
+**Independent verification; no new κ.** Seed 58 (recursive interfaces), card
+`connector-alphabet-change`, refined scope recorded in [TASK.json](TASK.json).
+Base `0b1cccf8`; the catalogue is at priority revision 3, which leaves this card
+unchanged.
+
+The threshold theorem is **not new here**. Hanson Sun's (bbMisk)
+[PR1615](https://github.com/CrocSwap/integer-mult-bounds/pull/1615), at head
+`96916897`, answered first:
+
+- **PR1498's question.** It answers open question 1 of bitplane's
+  [PR1498](https://github.com/CrocSwap/integer-mult-bounds/pull/1498): six
+  transfers fail for every source-port split.
+- **The threshold.** Cinna Davis's word from Discussion #1412 reaches eight, so
+  the threshold is eight.
+
+This package was built concurrently, without reading PR1615's code. It agrees
+with every PR1615 number it overlaps, and adds three things:
+
+1. **Independent re-certification in the generic CI hook.** PR1615's
+   registered hook is witness-only, and its exclusion needs a native C++
+   workflow. Here a gap-reduction lemma (below) brings the exhaustive `t ≤ 7`
+   exclusion and the `t = 8` count into a pure-Python census. It takes about
+   20 s and 50 s on 10 cores and runs in the generic Python 3.12 package hook.
+   It is cross-checked by two further code-disjoint engines:
+   - a C engine that enumerates every placement;
+   - a numpy engine with per-occurrence ports.
+2. **Every eight-transfer solution.**
+   - Exactly **2,502** canonical schedules realize all 31 masks with eight
+     transfers.
+   - **r = 1** (one source on `z1`) gives 2,376; **r = 5** (all on `z1`) gives
+     126.
+   - No schedule with r = 0, 2, 3 or 4 is universal at eight transfers. The
+     best coverages are 31, 31, 27 and 24 of 32.
+   - The r = 5 class is Discussion #1412's restricted-port alphabet, up to
+     swapping `z0` and `z1`. The r = 1 schedules lie outside it. Free port choice
+     enlarges the solution set but saves no transfer.
+   - Every one of the 2,502 is replayed as a complete 80-call program.
+   - PR1615's census tool would yield these counts with a small change; they
+     are recorded here with replay evidence.
+3. **PR1498's four-transfer split bounds.** PR1498 gives bounds
+   31, 31, 25, 17, 19, 31 for r = 0…5 and notes they need not be sharp. The
+   r = 2 bound of 25 is **not attained**: the exact maximum is 21. The other
+   five bounds are exact.
+
+## Interface
+
+The interface is unchanged from
+[woklerx's PR1438](https://github.com/CrocSwap/integer-mult-bounds/pull/1438)
+and PR1498. PR1438 is merged in-repo as
+[two-connector-transfer-boundary](../two-connector-transfer-boundary/README.md).
+PR1498 is pinned in [REMOTE-SOURCES.json](REMOTE-SOURCES.json).
+
+**Rows.**
+- Five read-only source rows `x_j`.
+- 31 output rows `y_m`, one per nonzero mask `m` in F2⁵.
+- Two independent arbitrary-dirty connector rows `z0` and `z1`.
+
+**Complete operations.**
+- `B_j(p): z_p += B_j x_j`.
+- `A_m(p): y_m += A_m z_p`.
+- One paid one-way transfer `C: z0 += z1`.
+
+**Constraints.**
+- Ports are chosen freely at every occurrence, and are charged.
+- Every `B_j` and every `A_m` occurs exactly twice: 10 B and 62 A calls.
+- Every source and both connectors are restored.
+- The maps are generic, non-commuting F2-linear maps, and the dirt is
+  arbitrary.
+
+PR1498 step 1 shows that restoration forces both calls of a source onto one
+port. A mask is realizable exactly when two cuts on the same connector have
+equal dirt and source parts that differ by that mask.
+
+## Census
+
+| `t` | Universal schedules | Maximum masks covered (of 32) by port split r = 0…5 |
+| --- | --- | --- |
+| 1, 3, 5 | none: no schedule restores the connectors | — |
+| 0 | 0 | 31, 17, 11, 11, 17, 31 |
+| 4 | 0 | 31, 31, **21**, 17, 19, 31 |
+| 6 | 0 | 31, 31, 28, 22, 23, 31 (as in PR1615) |
+| 8 | **2,502** (r = 1: 2,376; r = 5: 126) | 31, 32, 31, 27, 24, 32 |
+
+Counts use canonical words, with labels by first occurrence, and one port per
+source.
+
+## Gap reduction
+
+Between two consecutive B calls `z1` is fixed. Further transfers there only
+alternate `z0` between two states of opposite dirt parity. So three transfers
+in one gap give the same readout states and parities as one, and four give the
+same as two.
+
+A schedule with three or more transfers in some gap therefore reduces to a
+schedule with `t − 2` transfers, identical coverage and unchanged restoration
+parities. This has three consequences:
+
+- censuses with at most two transfers per gap, run at `t = 0, 4, 6`, together
+  with PR1438's full-placement `t = 2` census (no 32-mask bin), exclude every
+  universal schedule with `t ≤ 7`;
+- odd `t` cannot restore the dirt;
+- at `t = 8`, a universal schedule with a long gap would reduce to a universal
+  `t = 6` schedule, so the reduced count of 2,502 is the exact count.
+
+## Verification
+
+```sh
+python3 scripts/research_package.py --package research/two-connector-eight-transfer-census-seed58 --check-only
+python3.12 scripts/research_package.py --package research/two-connector-eight-transfer-census-seed58
+```
+
+`verify.py` uses only the standard library, parallelised over words. It has
+three registered parts.
+
+1. **`controls`.** The engine reproduces:
+   - PR943's in-repo 105- and 945-word censuses for 4 and 5 sources, bin for
+     bin;
+   - PR1438's in-repo full-placement `t = 2` censuses for 1 to 5 sources: every
+     restored count and coverage histogram, with no gap reduction;
+   - at `t = 0` and `t = 4`, every port split: no universal schedule, with the
+     complete per-split coverage histogram pinned;
+   - no restored schedule for `t = 1, 3, 5`.
+2. **`six`.** An exhaustive `t = 6` census over all 945 words and 32 port
+   splits: 44,936,640 restored reduced schedules, none universal, with the
+   per-split maxima and the complete coverage histogram pinned.
+3. **`eight`.** An exhaustive `t = 8` census:
+   - exactly 2,502 universal schedules, with a pinned SHA-256 of the sorted list
+     (`ba04f277…`) and the split counts;
+   - each becomes an 80-call program (62 A, 10 B, 8 C), run on dense random
+     non-commuting F2 maps with arbitrary dirt;
+   - Discussion #1412's pilot is among them;
+   - on three witnesses (the first r = 1 schedule, the first r = 5 schedule and
+     the pilot), deleting any `C` or `A`, or reading any `A` on the wrong port,
+     fails.
+
+**Independent engine (offline).**
+[independent/census.c](independent/census.c) enumerates every transfer
+placement, with no gap reduction. It agrees on every per-split maximum, and
+reproduces PR1615's `t = 6` totals:
+
+| `t` | Candidates | Restored | Universal |
+| --- | --- | --- | --- |
+| 4 | 30,270,240 | 10,402,560 | 0 |
+| 6 | 242,161,920 | 74,692,800 | 0 |
+| 8 | 1,323,241,920 | 384,199,200 | 2,502 |
+
+Run it with `cc -O2 -o census independent/census.c && ./census 8 0 1`, or
+slice it with `./census 8 <slice> <slices>`. The receipts are in
+[EXPECTED.json](EXPECTED.json).
+
+[independent/census_numpy.py](independent/census_numpy.py) lets every
+occurrence choose its own port, filtered by exact `z1` restoration.
+
+| `t` | Legal schedules | Universal | Run time |
+| --- | --- | --- | --- |
+| 4 | 10,402,560 | 0 | — |
+| 6 | 74,692,800 | 0 | 842 s on one core |
+
+Both match the per-split maxima above. So does PR1498's lemma that
+restoration forces one port per source.
+
+## Scope
+
+- **Family:** only the incidence-minimal 72-operator-call family. Extra or odd
+  A/B incidences, reverse or bidirectional transfers, combined queries and
+  source mutations are not covered.
+- **Proof:** a finite census with a short reduction lemma.
+- **No supplier claim and no κ change.** The completed-price condition is
+  PR1498's `t·σ` tariff, instantiated at `t = 8` by PR1615: `8σ + ΔH < 2c`
+  against the 74-call repair, under equal port prices.
+
+## Open next questions
+
+The structured entries are in [TASK.json](TASK.json).
+
+1. **Is there a short structural proof that six transfers fail?** It should
+   also explain why only r = 1 and r = 5 reach eight. A projection argument
+   like PR1498's step 2 might do both, and it would remove the census
+   dependency.
+   - *First step:* classify the 2,502 witnesses by their transfer phases `u_i`.
+   - *Blocker:* none.
+2. **Can extra incidences lower the threshold?** This is PR1615's question 1.
+   - *First step:* extend `engine.py` to 73 calls, with full dirt vectors per
+     readout.
+   - *Blocker:* the paired-readout reduction no longer applies.
+
+## Credit
+
+- **Hanson Sun (bbMisk), PR1615:** the six-transfer exclusion and the sharp
+  threshold, with priority.
+- **Gareth Davidson (bitplane), PR1498:** the four-transfer obstruction,
+  source-port lemma, tariff and open question.
+- **woklerx, PR1438:** the two-connector interface and full-placement censuses.
+- **Cinna Davis (1337isnot1337), Discussion #1412:** the eight-transfer
+  construction.
+- **Ross Richardson, PR943:** the universal-connector boundary and 945-word
+  census.
+
+Prepared by Sagar Thakrar with substantial Anthropic Claude assistance. That
+included explorer and judge subagents, two of which wrote the independent
+engines. New work is Apache-2.0.

research/two-connector-eight-transfer-census-seed58/REMOTE-SOURCES.json

@@ -0,0 +1,15 @@
+{
+  "schema_version": 1,
+  "note": "Unmerged predecessor package PR1498, pinned at its PR head; the copy is byte-identical. PR1438 is merged and read in-repo.",
+  "copies": [
+    {
+      "copy": "references/PR1498-README.md",
+      "pr": 1498,
+      "author": "bitplane (Gareth Davidson)",
+      "head": "b7baacc8048ab11d31751aee065206f2725a1a16",
+      "path": "research/two-connector-four-transfer-obstruction/README.md",
+      "git_blob": "abed92b09bde16fe47212b510086764944ef34bb",
+      "sha256": "5bbde91eb405150dce3073bf39419a7813d8f0529df3465ad97df66cb278a614"
+    }
+  ]
+}