Research: save 114 R420 XORs by sharing restored decoder operands (no new kappa)

research/r420-decoder-sharing/CERTIFICATE.json

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+{
+  "status": "PASS_R420_DECODER_SHARING",
+  "assignment_seed": 15,
+  "search": {
+    "operands": 80,
+    "oriented_pairs": 6320,
+    "positive_pairs": 463,
+    "matchings_tried": 1
+  },
+  "pairs": 27,
+  "columns": 660,
+  "decoder_additions_before": 321,
+  "decoder_additions_after": 264,
+  "native_additions_before": 26916,
+  "native_additions_after": 26802,
+  "saved_with_cleanup": 114,
+  "baseline_canonical_sha256": "e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2",
+  "candidate_canonical_sha256": "9d54dae007fa9dd874b825a7614ee1e572db9ab6ff9d4f0f7da02443be4f75b9",
+  "original_H1_H5": true,
+  "independent": {
+    "status": "PASS",
+    "scope": "relative scalar-word rewrite; baseline geometry and hosted realization inherited",
+    "symbolic_columns": 660,
+    "checks": {
+      "only_A_and_C_changed": true,
+      "whole_A_operator_equal": true,
+      "full_operator_equal": true,
+      "baseline_full_operator_is_target_add": true,
+      "candidate_full_operator_is_target_add": true,
+      "all_register_frame_paths_equal": true,
+      "interface_checks_hold": true
+    },
+    "changed_immutable_keys": [],
+    "different_A_output_registers": [],
+    "different_frame_path_registers": [],
+    "interface": {
+      "baseline": {
+        "H1_no_target_feeds_z": true,
+        "H2_D_frame0_targets_from_helpers": true,
+        "H3_A_writes_z": true,
+        "H4_C_literal_reverse_at_full": true
+      },
+      "candidate": {
+        "H1_no_target_feeds_z": true,
+        "H2_D_frame0_targets_from_helpers": true,
+        "H3_A_writes_z": true,
+        "H4_C_literal_reverse_at_full": true
+      }
+    },
+    "counts": {
+      "baseline": {
+        "gates": {
+          "D": 120,
+          "A": 2064,
+          "B": 837,
+          "C": 2661
+        },
+        "adds": {
+          "D": 19922,
+          "A": 2376,
+          "B": 1077,
+          "C": 2973
+        },
+        "scatter_adds": 568
+      },
+      "candidate": {
+        "gates": {
+          "D": 120,
+          "A": 2118,
+          "B": 837,
+          "C": 2715
+        },
+        "adds": {
+          "D": 19922,
+          "A": 2319,
+          "B": 1077,
+          "C": 2916
+        },
+        "scatter_adds": 568
+      }
+    },
+    "scalar_ADD_saving_including_C": 114,
+    "positive_scalar_word_bounds": {
+      "baseline": {
+        "max_row_sum": 144651,
+        "max_column_sum": 40742,
+        "row_sum_bits": 18,
+        "column_sum_bits": 16
+      },
+      "candidate": {
+        "max_row_sum": 193441,
+        "max_column_sum": 53780,
+        "row_sum_bits": 18,
+        "column_sum_bits": 16
+      }
+    }
+  },
+  "negative_controls": [
+    "missing helper restoration",
+    "uncancelled decoder operand",
+    "scalar-valid nonliteral cleanup",
+    "illegal frame-zero preparation",
+    "retained register used as temporary",
+    "overlapping temporary pairs",
+    "missing dirty compensation"
+  ],
+  "kappa_claim": "none"
+}

research/r420-decoder-sharing/CLAIM.json

@@ -0,0 +1,26 @@
+{
+  "track": "bit supplier",
+  "assignment_seed": 15,
+  "seed_source": "Python secrets.randbelow(100), operating-system randomness",
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "started_utc": "2026-10-13T04:24:07Z",
+  "expected_finish_utc": "2026-10-13T05:04:07Z",
+  "coordination": "No pinned issues returned by GitHub; local claim, overlap refreshed through PR700 before selection.",
+  "candidates": [
+    "Restored sharing among non-retained operands of the nine phase-A decoder-total gates",
+    "Rechoose decoder-total hosts and their order with fixed retained dimension",
+    "Reassign phase-A D6 frames at fixed scalar gate order"
+  ],
+  "chosen": "Restored disjoint-pair sharing in PR543 R420's nine final phase-A decoder-total gates, preserving their complete dirty-input operator and literal reverse cleanup",
+  "method": "Enumerate oriented pairs of non-retained operands, price preparation plus restoration and matching cleanup, then select a positive-saving disjoint matching with seed-15 tie ordering. Exact full-column and frame checks; recompute positive-prefix bounds.",
+  "expected_deliverable": "One finite circuit improvement with frozen recipe, exact checks, negative controls, proof and source pins; or a scoped negative result if the complete budget fails. No new kappa.",
+  "overlap": "PR670/674/688 change phase-B blocks; PR691 changes phase-A quad 13; PR686/694 factor frame-zero compensation and change H2. This task factors decoder totals at FULL under the original H1-H5 and does not import those changes. PR661 fixes phase-B gates and excludes resynthesis. PR620's decoder rank/floor research does not synthesize this fixed nine-total circuit.",
+  "status": "completed",
+  "result": {
+    "pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/708",
+    "summary": "27 restored operand pairs reduce the nine decoder totals from 321 to 264 ADDs; matching cleanup gives 114 fewer native ADDs. All 660 symbolic columns, all frame paths and H1-H5 pass. Unchanged native nine-stage replay passes 1380 physical columns and 18 controls per word; payload rises from 100 to 102 bits, below 104. No new kappa.",
+    "final_overlap_refresh_through": 704,
+    "main_at_refresh": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+    "scope_remaining": "Full hosted-parent construction and changed global invoice; inherited all-size and analytic assumptions."
+  }
+}

research/r420-decoder-sharing/README.md

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+# Restored sharing in R420's nine decoder totals
+
+This finite bit-circuit construction saves **114 binary XOR additions** in
+eumemic's PR543 R420 word: **26,916 to 26,802**. It changes the last nine
+phase-A decoder-total gates and their literal reverse cleanup. It preserves
+the complete phase-A operator on arbitrary source and helper inputs, every
+register's ordered frame path, 420 helpers, nine retained totals, and the
+original H1-H5 interface. **No new kappa is claimed.**
+
+The exact base is `66849e1475c930fd7732b1a0067ffee7e1e95f60`. At that base the
+selected conditional result is #626 with #543,
+`kappa = 51110230636731/31250000000000000 = 0.001635527380375392`.
+The source is followed through `certificates/selected-result.json` and
+`docs/research/community-k492-review.md`, rather than historical frontiers
+in the bit supplier's README. [SOURCES.json](SOURCES.json) pins the reviewed
+PR543 head, canonical word content, checker, generator and supplier README proof.
+No existing construction or selected-frontier file is edited.
+
+## Reproduce
+
+Python 3.11+ and its standard library suffice for the exact local check:
+
+```sh
+python3 -B research/r420-decoder-sharing/verify.py
+```
+
+Expected final line:
+
+```text
+PASS_R420_DECODER_SHARING decoder=321->264 native_ADDs=26916->26802 saved=114 columns=660 controls=7 no_new_kappa
+```
+
+This recomputes the bounded nomination, reconstructs the candidate, checks
+source pins, runs the unchanged PR543 checker and the separate set-valued
+interpreter, rejects seven controls, and compares every field of
+[CERTIFICATE.json](CERTIFICATE.json). Run without `-O`, because the inherited
+checker contains assertions. `--show-report` prints the full certificate;
+`--output /absolute/new/directory` also writes both words and the report.
+Existing output directories are refused.
+
+For native nine-stage replay, choose two new output directories:
+
+```sh
+python3 -B research/r420-decoder-sharing/verify.py --output /tmp/decoder-words
+BOOST_INCLUDE=/path/to/boost CXX=c++ python3 -B research/r420-decoder-sharing/native.py \
+  /tmp/decoder-words/baseline.json.gz /tmp/decoder-words/candidate.json.gz /tmp/decoder-native
+```
+
+The optional [native.py](native.py) helper binds both inputs to the canonical
+word hashes in the local certificate.
+It uses the unchanged record exporter, legality checker and C++ nine-stage
+column checker. A C++17 compiler and Boost headers are required. Native
+results and environment details are recorded separately in
+[VALIDATION.json](VALIDATION.json).
+
+## Circuit and full cost
+
+At the start of the nine-total suffix, choose two non-retained operand
+registers `a,b`. Temporarily apply `a <- a XOR b`. In each total's source
+set containing `a`, toggle membership of `b`. After all nine totals,
+apply `a <- a XOR b` again. The register `b` is unchanged, so this restores
+the exact incoming value of `a`, including arbitrary dirty-input terms.
+
+To check a total algebraically, let its old coefficients of `a,b` be `p,q`
+in F2. Its new expression is `p(a+b)+(q+p)b = pa+qb`. All other operands
+are unchanged. The 27 pairs are disjoint and exclude every retained total,
+so their substitutions commute and no total gate writes a temporary
+operand. Each retained total's value agrees after its gate, and the complete
+phase-A operator agrees after restoration, with no zero-scratch assumption.
+The same scatter, B and D then
+remain valid. Rebuilding C as the literal reverse of all A/B z-gates
+retains H4 and restores every source and helper.
+
+For one oriented pair, write `u` for the number of rows containing both
+operands and `v` for rows containing `a` but not `b`. The net phase-A
+saving is `u-v-2`; the two charged operations are preparation and
+restoration. The exact phase-A saving over the 27 pairs is 57. Cleanup
+saves the same amount.
+
+| Charged binary additions | Original | New |
+|---|---:|---:|
+| Decoder-total source reads | 321 | 210 |
+| Temporary preparation and restoration | 0 | 54 |
+| Complete decoder suffix | 321 | 264 |
+| All phase A | 2,376 | 2,319 |
+| All cleanup C | 2,973 | 2,916 |
+| D, scatter and B together | 21,567 | 21,567 |
+| Complete native word | 26,916 | 26,802 |
+
+Every temporary operand already occurs at FULL in the old suffix. All
+new operations are at FULL, after the same earlier frame visits. The
+checker compares every register's entire frame-ID sequence, including
+final climbs. All operand reads, including preparation and restoration,
+are charged in the ADD counts. No helper, copy, erase, frame transition or
+recursive child is added. The inherited paid profile therefore remains
+`{1:1320, 2:1440, 4:60, 5:120, 8:9}`, with rank mass 5,112.
+The JSON multi-source gate count increases by 108 across A and C; the
+binary ADD count above is the physical scalar count, not that JSON count.
+
+Positive arithmetic is priced separately. Cancellation-free scalar-word
+maximum row sums rise from 144,651 to 193,441, and maximum column sums
+from 40,742 to 53,780. These are exact integer norms, not bounds for the
+complete hosted program. They do not justify inheriting its prefix bill.
+The unchanged native checker gives forward bounds 144,651 to 193,441 and
+reverse bounds 2,270,064 to 2,702,178. Its nine-stage payload charge rises
+from 100 to 102 bits, still below its 104-bit budget. Both words pass all
+1,380 formal physical columns and all 18 omitted-bridge controls. The
+complete hosted invoice remains an integration obligation.
+
+## Search coverage and controls
+
+The operating-system assignment seed is 15. Three candidate tasks and the
+local claim are recorded in [CLAIM.json](CLAIM.json); GitHub returned no
+pinned coordination issue. The chosen search fixes the nine totals, their
+hosts, their order, the original retained basis and the rest of the word.
+It covers all **6,320 oriented pairs** of the 80 non-retained operands;
+463 have positive complete savings. One greedy disjoint matching is taken
+in descending saving order, after a seed-15 shuffle to break ties. The
+stopping rule is the end of that single list. The frozen 27-pair recipe is
+in [recipe.json](recipe.json). There is no claim of a minimum circuit or
+optimal matching, and no closed decoder-rank search is reopened.
+
+The controls remove restoration, corrupt a decoder operand, retain a
+scalar-valid but nonliteral old cleanup, move preparation illegally to
+frame zero, attempt a retained-register temporary, overlap pairs, and
+remove dirty compensation. The first two also fail the separate
+set-valued checker. The old cleanup is deliberately classified as
+scalar-valid but outside literal H4, not as a false endpoint identity.
+
+## Scope, prior work and attribution
+
+This stops at a finite native bit circuit. The selected assembly is
+complex-limited; unchanged recursive profiles do not create a new bit
+exponent. Full parent construction, whole-owner hosting and the changed
+global positive-prefix invoice still need replay before integrating this
+word into an accepted supplier. The all-size compiler, common-label and
+mutable-basis realization, pooled-cell hosting, precision/tape, residual
+and analytic-transfer assumptions remain inherited. Finite symbolic
+execution is not a formal proof of the full multiplication theorem.
+
+Credit belongs to eumemic for PR543/493's R420 construction, decoder-total
+schedule and interleaved-source interface; Dylan Crow for the earlier
+bit-word/exporter line; DreamingOfClouds for retained-total and supplier
+pricing work; Zhihao Chen for the mutable-basis scalar interface; and the
+full realization, geometry, hosting and assembly lineage retained by
+the original package and selected review.
+
+Concurrent PR686 and PR694 are credited for restored pair sharing in
+frame-zero dirt compensation. This contribution applies that algebraic
+mechanism to the separate nine-total suffix at FULL, with complete A
+operator equality and the original H2 unchanged. It imports no code or
+circuits from them. PR670/674/688 alter phase-B blocks, PR691 alters
+phase-A quad 13, and PR661 fixes phase-B gates and excludes resynthesis.
+Composition with those changes is not claimed. PR620's decoder-floor
+research concerns a different question from synthesizing these fixed
+retained totals.
+
+Prepared for **Sagar Thakrar (@thakrarsagar)** with substantial **OpenAI
+Codex assistance** in nomination, proof, implementation, verification and
+packaging, including separate Codex workers for independent replay and
+native checks. No independent human review is claimed. This contribution
+is under Apache-2.0, retaining the repository and predecessor licenses.

research/r420-decoder-sharing/SOURCES.json

@@ -0,0 +1,23 @@
+{
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "selected_kappa": "51110230636731/31250000000000000",
+  "selected_construction_pr": 626,
+  "selected_construction_head": "660de893d27bb7abd1d5b01b1640207528a5c851",
+  "bit_supplier_pr": 543,
+  "bit_supplier_author": "eumemic",
+  "bit_supplier_head": "2065fe4f72a914d907f1046603f5a36f17018535",
+  "word": "research/e8-bit-r420/data/bitE8mut-r420.json.gz",
+  "word_canonical_sha256": "e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2",
+  "word_hash_rule": "SHA-256 of parsed JSON reserialized with sorted keys and separators comma/colon; gzip container metadata is not mathematical identity",
+  "files_sha256": {
+    "research/e8-bit-r420/code/xcheck.py": "5737d18a7826fd51e131035f2d7bebba2e2868af91dd7672a5f11208b036d424",
+    "research/e8-bit-r420/README.md": "9807be57c6adf26b30e4f54e2f8160075bbc05091f1a6d157e43d2a4620c1c9f",
+    "research/e8-bit-r420/code/gen_l3.py": "d5c802447c99255db1bdfcae62b2116bfc9612c9dc9000d41aa4f5f2fec9f2f2",
+    "research/e8-bit-carrier-skeleton/code/bit/export_records.py": "8a5999601f917dbb492bb9ae9a344611f18cc7cedc53dc3c76496a369e530eb3"
+  },
+  "concurrent_context_not_dependencies": {
+    "686": "23dcb610c2b156b76b9d599ea296c3cbde224b2d",
+    "691": "ee6b9175fe006c11871802866c487b5ecc1a93a4",
+    "694": "9732ac127dd4bef98a846aaa904c20b5a84e72c1"
+  }
+}