Scoped negative result: R420 split target reads cannot lower the paid moment

research/r420-split-target-reads/CERTIFICATE.json

@@ -0,0 +1,70 @@
+{
+  "all_early_profile": {
+    "1": 1200,
+    "2": 1560,
+    "4": 180,
+    "8": 9
+  },
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "baseline_equivalent_per_target": 6,
+  "baseline_profile": {
+    "1": 1320,
+    "2": 1440,
+    "4": 60,
+    "5": 120,
+    "8": 9
+  },
+  "dirty_input_columns_per_replay": 660,
+  "distinct_sources": 360,
+  "geometry_census_sha256": "980ce1be0c746a3e4e960b72078fe4ab1d1802a788293747a360c707394712f0",
+  "legal_schedules_per_target": 8,
+  "majorization": {
+    "baseline": [
+      5,
+      1
+    ],
+    "early_read": [
+      4,
+      2
+    ]
+  },
+  "moment_difference": "2^p + 4^p - 1 - 5^p > 0 for 0 < p < 1",
+  "negative_controls": [
+    "wrong source at rank six",
+    "missing target read",
+    "shared source invalidates independent choices",
+    "omitted rank-two source cleanup"
+  ],
+  "paid_calls": 2949,
+  "payload_sha256": "e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2",
+  "profile_delta_per_early_read": {
+    "1": -1,
+    "2": 1,
+    "4": 1,
+    "5": -1
+  },
+  "rank_mass": 5112,
+  "simultaneous_schedules": "2348542582773833227889480596789337027375682548908319870707290971532209025114608443463698998384768703031934976",
+  "source_commit": "2065fe4f72a914d907f1046603f5a36f17018535",
+  "source_orders_per_target": 6,
+  "source_pr": 543,
+  "status": "PASS_R420_SPLIT_TARGET_READ_BOUND",
+  "strictly_worse_per_target": 2,
+  "targets": 120,
+  "worked_example": {
+    "cell": 145,
+    "early_source": 10,
+    "source_frames": [
+      380,
+      383,
+      384
+    ],
+    "sources": [
+      10,
+      290,
+      340
+    ],
+    "target": 3,
+    "top": 555
+  }
+}

research/r420-split-target-reads/CLAIM.json

@@ -0,0 +1,28 @@
+{
+  "track": "Bit supplier",
+  "assignment_seed": 3,
+  "randomness": "Node.js crypto.randomInt(0, 100), operating-system CSPRNG, drawn once",
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "claimed_at_utc": "2026-10-13T04:30:00Z",
+  "expected_finish_utc": "2026-10-13T05:03:00Z",
+  "coordination": "No pinned issues existed at claim time; local claim per sprint instructions.",
+  "object": "PR543 R420 final 120 phase-B three-source target gates",
+  "method": "Split each gate into separately framed single-source reads, all six source orders, arbitrary nested real subspaces, with all earlier gates and literal full-frame cleanup fixed. Include source cleanup costs. Seed 3 selects target 3 as the worked example; coverage includes every target.",
+  "deliverable": "Exact finite geometry census, complete local cost comparison, reproducible checker and scoped research note; no presumed kappa improvement.",
+  "candidates": [
+    "One block XOR rewrite: rejected because PR670, PR674 and PR688 already cover nearby circuit work.",
+    "First target-read frame promotion: rejected because PR661 already proves all fixed-gate phase-B frame reassignments optimal.",
+    "Split final target reads: selected because PR661 freezes operand lists and excludes circuit resynthesis."
+  ],
+  "overlap_checked_through_pr": 699,
+  "predecessor": {
+    "pr": 661,
+    "head": "9ca7aa3ab0a3260a2f36ac86b9911a31fd88bd89",
+    "distinction": "Changes gate grouping and allows different frames for reads within one original gate; the fixed-gate result does not cover this family."
+  },
+  "status": "complete",
+  "finished_at_utc": "2026-10-13T04:45:00Z",
+  "result": "Scoped negative result: each target has six equal-cost and two strictly worse split schedules for every 0 < p < 1, covering all 8^120 simultaneous choices. Exact primary replay and independent exhaustive rational enumeration pass; no new kappa.",
+  "final_overlap_checked_through_pr": 717,
+  "pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/718"
+}

research/r420-split-target-reads/README.md

@@ -0,0 +1,166 @@
+# R420 split target reads cannot lower the paid moment
+
+**Scoped negative result, no new κ.** In the selected PR543 R420 bit word,
+splitting the final three-source target gates into separately framed reads
+does not reduce the local paid recursive moment. This covers all six source
+orders and every admissible real subspace at each read, with the earlier
+schedule and full-frame cleanup fixed.
+
+Each target has eight legal schedules. Six have the original profile. Two
+have strictly greater moment for every exponent `0 < p < 1`. The 120 targets
+use disjoint source registers, so the result covers all `8^120` simultaneous
+choices, without enumerating that Cartesian product. It also permits
+interleaving reads of different targets within this final segment.
+
+## Reproduce
+
+From the repository root, with Python 3.9+ and no third-party packages:
+
+```sh
+python3 -B research/r420-split-target-reads/check.py
+python3 -B research/r420-split-target-reads/independent_check.py
+```
+
+The first command checks the source payload, reconstructs the certificate,
+replays all 660 symbolic dirty-input columns for both the original word and
+the word using every early read, and runs four negative controls. Expected:
+
+```text
+PASS_R420_SPLIT_TARGET_READ_BOUND targets=120 schedules=8^120 controls=4 kappa=unchanged
+```
+
+`independent_check.py` uses rational Gaussian elimination and exhausts all
+`6 * 2^3 = 48` candidate schedules per target. It does not import the primary
+checker or predecessor checkers. The primary checker uses integer row
+elimination and the two possible local profiles. Neither implementation
+samples floating-point exponents to establish the inequality below.
+
+## Scope and provenance
+
+The exact base is `66849e1475c930fd7732b1a0067ffee7e1e95f60`. Its selected
+conditional κ is `51110230636731/31250000000000000`, or
+`0.001635527380375392`, from PR626 with the PR543 bit supplier. The selected
+certificate points to `research/community-k492-audit/pr626/replay/ASSEMBLY.json`
+and `docs/research/community-k492-review.md`. This experiment follows the
+actual bit source, `research/e8-bit-r420/data/bitE8mut-r420.json.gz`, from
+PR543 commit `2065fe4f72a914d907f1046603f5a36f17018535`.
+
+PR661 by Tuomo Kiiskinen / yuj1r0, inspected at
+`9ca7aa3ab0a3260a2f36ac86b9911a31fd88bd89`, proves fixed-gate phase-B frame
+optimality. It preserves each gate's operand list. Here a three-source gate
+becomes three single-source gates with potentially different frames. This
+is a separate family outside that claim, and the present result closes
+only this additional case. No code or certificate from PR661 is imported.
+
+The experiment changes no scalar source value, helper count, decoder,
+COPY, earlier gate, final frame, or cleanup gate. It adds no source-access
+operation, helper or recursive child beyond the frame climbs counted below.
+The same three XORs execute in a different grouping. Every source still
+reaches the full frame in C or its final climb. The 420 helpers and nine
+retained-total COPYs are unchanged. No alternative owner map or assembly is
+claimed. See `SOURCES.json` for byte pins and `CLAIM.json` for the assignment,
+candidate selection and overlap checks.
+
+## Complete geometry argument
+
+The final 120 B gates are `B[717:837]`, one per target. The checker establishes
+the following facts from the exact eight-dimensional label vectors:
+
+1. A target enters this segment at a rank-two cell `C` and must end at its
+   fixed rank-seven hyperplane `H`.
+2. Its three sources enter at distinct rank-six subspaces `S0, S1, S2`
+   contained in `H`. Each pair spans `H`.
+3. Exactly one source space contains `C`.
+4. The 360 source registers are distinct, none is a target, and each appears
+   once in this segment. Their fixed endpoint is rank eight. Every C-phase
+   gate has the full frame.
+
+At a single-source read the common gate frame `F` must contain that source's
+rank-six space and the target's current frame. The target's monotone path
+requires `F` to lie inside `H`. Thus `F` is either the source space itself
+or `H`: a rank-six space has codimension one in `H`. This accounts for all
+real subspaces, including ones outside the retained frame catalogue.
+
+The first read can use rank six only when its source is the unique space
+containing `C`. There are two orders with that source first. After this
+read, either remaining source joins the target's rank-six frame to `H`.
+The remaining reads therefore use `H`. In every other schedule all reads
+use `H`. There are six all-`H` orders and two early-read orders per target.
+Disjoint register sets make these choices independent across targets.
+
+## Full cost of an early read
+
+A climb of dimension `d` contributes `d^p` to the unnormalized recursive
+moment. In the inherited price engine, the exponent is `p = 1 - a` and the
+common positive normalization is `m^(a-1) / W`. The child-count fallback
+term is also unchanged here. These statements concern the paid moment,
+not a new admission theorem for a physical implementation.
+
+Count the target and its three sources, including their later full-frame
+cleanup:
+
+| Schedule | Target climbs | Source climbs through cleanup | Combined profile |
+|---|---|---|---|
+| Original or all reads at `H` | `5` | three copies of `1, 1` | `{1: 6, 5: 1}` |
+| One early rank-six read | `4, 1` | one `2`, two copies of `1, 1` | `{1: 5, 2: 1, 4: 1}` |
+
+The exact profile change per early read is
+`{1: -1, 2: +1, 4: +1, 5: -1}`. Both profiles have seven paid calls and
+rank mass eleven. The changed pair of ranks is `(5, 1) -> (4, 2)`.
+Strict concavity gives
+
+```text
+2^p - 1 > 5^p - 4^p,             0 < p < 1,
+2^p + 4^p - 1 - 5^p > 0.
+```
+
+For example, integrate the strictly decreasing derivative `p*x^(p-1)`
+over `[1,2]` and `[4,5]`. Their equal lengths give the first inequality.
+This proves the sign for the full open interval; finite exponent samples
+are unnecessary. The difference is zero at `p=0` and `p=1`, which are
+outside the strict claim.
+
+The complete baseline local profile is
+`{1:1320, 2:1440, 4:60, 5:120, 8:9}`. Taking all 120 early reads gives
+`{1:1200, 2:1560, 4:180, 8:9}`. Both have 2,949 paid calls and rank mass
+5,112. All fixed A/D/scatter/C/final work and the nine rank-eight COPYs are
+included in this recount. If exactly `k` targets use the early read, the
+moment increases by `k * (2^p + 4^p - 1 - 5^p)`.
+
+Seed 3 selects target 3 as the worked example. Its register is 123, its cell
+is frame 145, and `H` is frame 555. Sources 10, 290 and 340 enter at frames
+380, 383 and 384; only source 10 can be read early. These identifiers and a
+digest of the complete geometry census are retained in `CERTIFICATE.json`.
+
+## Checks and limits
+
+The primary replay checks exact F2 identities with arbitrary initial bits
+in every register. It rejects an incompatible source read at rank six, a
+missing target read, source reuse that would invalidate independent choices,
+and a cost certificate omitting the rank-two cleanup. The independent
+checker tests the entire finite frame/order domain by a different method.
+`VALIDATION.json` records execution results and checks that were not run.
+
+This is a proof for the stated finite family, supported by exact geometry
+and scalar replays. It is not a formal Lean proof of the multiplication
+theorem. It does not cover changes to earlier frames, source circuits,
+decoder totals, helper allocation, non-monotone frames, cleanup, or the
+recursive interface. It does not claim global bit optimality or a new κ.
+All inherited all-size/compiler/common-label/mutable-basis, pooled hosting,
+residual, precision/tape/exact recovery and analytic assumptions remain.
+
+## Credit and license
+
+The R420 word and its interleaved cleanup are eumemic's PR543 contribution,
+building on the PR493 carrier skeleton. Dylan Crow's PR370 supplies the E8
+decoder; DreamingOfClouds' PR510 supplies the nine-stage pricing lineage.
+PR589 explores a different punctured bit word and is not a dependency of
+this experiment. Tuomo Kiiskinen's PR661 supplied the fixed-gate boundary
+that motivated this distinct split-read question. Concurrent PR670,
+PR674, PR686, PR688, PR691 and PR694 change scalar circuits or compensation;
+their constructions are not imported here.
+
+Prepared for Sagar Thakrar / thakrarsagar with substantial OpenAI Codex
+assistance, including a separate Codex worker for independent exact
+enumeration. New files are under the repository's Apache-2.0 license.
+Original supplier files and `upstream/` are unchanged.

research/r420-split-target-reads/SOURCES.json

@@ -0,0 +1,37 @@
+{
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "selected_result": {
+    "record": "certificates/selected-result.json",
+    "sha256": "b3bf402f31a69bde0629936a9b569bed48281568d7b16d5fba120b5311ea1f53",
+    "kappa": "51110230636731/31250000000000000",
+    "source_pr": 626,
+    "source_commit": "660de893d27bb7abd1d5b01b1640207528a5c851",
+    "review": "docs/research/community-k492-review.md"
+  },
+  "bit_source": {
+    "pr": 543,
+    "author": "eumemic",
+    "commit": "2065fe4f72a914d907f1046603f5a36f17018535",
+    "path": "research/e8-bit-r420/data/bitE8mut-r420.json.gz",
+    "git_blob_at_source_and_base": "0c63ca62759dc63d302be6552aeeee1ede70350f",
+    "compressed_sha256_for_provenance_only": "9721a3819efd68307f66f00d84a952eb28eb7c3ad2ce79653ce295c98a764784",
+    "decompressed_sha256_checked": "e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2"
+  },
+  "reference_sha256": {
+    "research/e8-bit-r420/README.md": "9807be57c6adf26b30e4f54e2f8160075bbc05091f1a6d157e43d2a4620c1c9f",
+    "research/e8-bit-r420/code/gen_l3.py": "d5c802447c99255db1bdfcae62b2116bfc9612c9dc9000d41aa4f5f2fec9f2f2",
+    "research/e8-bit-r420/code/xcheck.py": "5737d18a7826fd51e131035f2d7bebba2e2868af91dd7672a5f11208b036d424",
+    "research/e8-bit-r420/EXPECTED.json": "06eccee9f6b77e140833df83407e7996dc948c17d49b0779ab171fbe463fd28d",
+    "research/e8-bit-carrier-skeleton/code/hub/e8-r740-replica32/code/engines/moment.py": "81cbc864e106418fc7854322746f72180cb8110ff39f74450a3b8d55a82c1482"
+  },
+  "predecessor_scope_reference": {
+    "pr": 661,
+    "author": "Tuomo Kiiskinen / yuj1r0",
+    "head": "9ca7aa3ab0a3260a2f36ac86b9911a31fd88bd89",
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/661",
+    "use": "Scope comparison only. No implementation or certificate imported. Its fixed-gate frame bound excludes the single-source factorization studied here."
+  },
+  "runtime": "Python >= 3.9, standard library only; no solver or third-party dependencies",
+  "attribution": "Prepared for Sagar Thakrar / thakrarsagar with substantial OpenAI Codex assistance, including independent computational checking by a separate Codex worker.",
+  "license": "Apache-2.0"
+}

research/r420-split-target-reads/VALIDATION.json

@@ -0,0 +1,61 @@
+{
+  "date_utc": "2026-10-13",
+  "environment": "macOS Darwin 25.5.0 arm64, CPython 3.14.8",
+  "status": "PASS scoped finite check and independent enumeration",
+  "commands": [
+    {
+      "command": "python3 -B research/r420-split-target-reads/check.py",
+      "exit_code": 0,
+      "observed_seconds": 0.198,
+      "result": "PASS_R420_SPLIT_TARGET_READ_BOUND targets=120 schedules=8^120 controls=4 kappa=unchanged"
+    },
+    {
+      "command": "python3 -B research/r420-split-target-reads/independent_check.py",
+      "exit_code": 0,
+      "observed_seconds": 1.172,
+      "result": "PASS_INDEPENDENT_SPLIT_READS targets=120 candidates=5760 legal=960 baseline=720 promoted=240 rejected_first_D6=240"
+    },
+    {
+      "command": "python3 -O -B research/r420-split-target-reads/check.py",
+      "exit_code": 0,
+      "result": "same PASS; checks do not depend on assert"
+    },
+    {
+      "command": "python3 -O -B research/r420-split-target-reads/independent_check.py",
+      "exit_code": 0,
+      "result": "same PASS; checks do not depend on assert"
+    },
+    {
+      "command": "python3 -B scripts/verify_selected_result.py",
+      "exit_code": 0,
+      "result": "PASS selected record, exact kappa, original source inventory and proof supplement: 51110230636731/31250000000000000"
+    }
+  ],
+  "additional_checks": [
+    "Both scripts pass with python3 -I -B using absolute script paths from /tmp, with isolated imports and no dependency on the current directory.",
+    "A separate Codex worker independently checked the proof, both implementations, certificate and all SOURCES.json reference hashes; no concrete defect found.",
+    "The source word's Git blob is identical at PR543 and the recorded base.",
+    "In-memory external controls reject a modified decompressed source and a modified certificate; no repository files were changed by those controls.",
+    "git diff --cached --check passes. Only research/r420-split-target-reads/ is added; upstream/ and the selected frontier are unchanged."
+  ],
+  "coverage": {
+    "primary_scalar_replays": "all 660 symbolic dirty-input columns, original and all-early words",
+    "primary_negative_controls": 4,
+    "independent_candidate_schedules": 5760,
+    "independent_legal_schedules": 960,
+    "independent_illegal_first_rank_six_reads": 240,
+    "simultaneous_coverage": "All 8^120 combinations by disjoint register sets and additive cost; not brute-force replay of the Cartesian product.",
+    "exponent_coverage": "All real 0 < p < 1 by the written strict-concavity proof; no numerical exponent sweep."
+  },
+  "not_run": [
+    "Repository-wide make verify and make frontier626-verify: no selected construction, generator, interface or assembly was changed; the standalone additions were checked directly and selected source binding passed.",
+    "Remote CI, Linux execution and older Python versions.",
+    "Full physical/compiler/precision/tape/analytic admission or Lean multiplication proof."
+  ],
+  "overlap_refresh": {
+    "through_pr": 717,
+    "pr661_head": "9ca7aa3ab0a3260a2f36ac86b9911a31fd88bd89",
+    "main_head": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+    "result": "No competing split-target-read result found. PR661 freezes gate grouping; newer R420 proposals concern scalar circuits, compensation, changed cleanup, adjacent A swaps or checker performance."
+  }
+}