Finite bit construction: five-helper cycle saves 12 R439 ADDs (no new kappa)

research/r439-five-cycle-seed9f01/CERTIFICATE.json

@@ -0,0 +1,120 @@
+{
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "controls": [
+    "missing helper preparation edge rejected by D boundary",
+    "missing inverse restoration edge rejected by D boundary",
+    "stale injection rejected by D boundary",
+    "wrong inverse order rejected by D boundary",
+    "mutated injection coefficient rejected by D boundary"
+  ],
+  "exact_precision": {
+    "local": {
+      "lhs": "44468459235010318794652647424",
+      "rhs": "20282409603651670423947251286016"
+    },
+    "nine_stage": {
+      "lhs": "4676708453903757010070537011688772133559165966010548224",
+      "rhs": "6277101735386680763835789423207666416102355444464034512896"
+    }
+  },
+  "new_kappa": false,
+  "paid_profile": {
+    "1": 1323,
+    "2": 1342,
+    "3": 79,
+    "4": 44,
+    "5": 102,
+    "6": 11,
+    "7": 19
+  },
+  "paid_rank_mass": 5129,
+  "search": {
+    "candidates": 1860480,
+    "optimistic_max_net_ADD_saving": 109,
+    "positive_saving_candidates": 56,
+    "selected": {
+      "edges": [
+        [
+          59,
+          148
+        ],
+        [
+          148,
+          21
+        ],
+        [
+          21,
+          129
+        ],
+        [
+          129,
+          25
+        ],
+        [
+          25,
+          59
+        ]
+      ],
+      "injection_weight": 4442,
+      "new_support_weight": 70,
+      "old_support_weight": 92,
+      "permutation": [
+        59,
+        148,
+        21,
+        129,
+        25
+      ],
+      "saving": 12,
+      "support": [
+        21,
+        25,
+        59,
+        129,
+        148
+      ]
+    },
+    "supports": 15504,
+    "sweeps_per_support": 120
+  },
+  "selected_record_sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
+  "status": "PASS_R439_FIVE_CYCLE",
+  "unrun": [
+    "native C++ replay",
+    "supplier adoption",
+    "outer assembly replay",
+    "all-size/compiler and analytic proof obligations"
+  ],
+  "word": {
+    "columns": 661,
+    "copy_ranks": [
+      7,
+      7,
+      7,
+      7,
+      7,
+      7,
+      7,
+      7,
+      7
+    ],
+    "directions": 2,
+    "exporter": {
+      "adds": 11488,
+      "copies": 9,
+      "erasures": 9,
+      "forward": 27061,
+      "payload_bits": 96,
+      "record_sha256": "10de6f93ba0dcf14aaeb0bfeaea0e35f65ad5ba22de6284fe2828a7ee0504930",
+      "reverse": 5921344
+    },
+    "literal": {
+      "B": 5921344,
+      "D_adds": 5536,
+      "F": 27061,
+      "local_bits": 96,
+      "native_ADDs": 11488,
+      "nine_stage_bits": 182
+    }
+  }
+}

research/r439-five-cycle-seed9f01/README.md

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+# Five-helper cyclic basis change on the R439 D map
+
+**Finite local circuit construction; no new κ.** A five-edge directed cycle on
+five helpers reduces the pinned R439 D-stage cost by 12 ADDs after paying for
+both preparation and restoration. The selected result, accepted bit supplier,
+paid rank profile and `upstream/` are unchanged. The selected complex side still
+binds.
+
+## Exact construction
+
+The source is the source-pinned 11,500-ADD word in
+[`research/r439-bsection-stars-seed01/`](../r439-bsection-stars-seed01/README.md),
+which combines the R439 helper-star D word with its ten phase-B block programs.
+The cycle is applied **after** the old helper preparation. Using local helper
+indices, its chronological word is:
+
+```text
+helper[59]  ^= helper[148]
+helper[148] ^= helper[21]
+helper[21]  ^= helper[129]
+helper[129] ^= helper[25]
+helper[25]  ^= helper[59]
+```
+
+The corresponding absolute wire IDs are 281, 370, 243, 351 and 247. Write
+`H0` for the original helper basis change, `H1` for this five-operation cycle,
+and `J0` for the original 111-by-439 injection. The old helper state at
+injection is `H0 s`. The new state is `H1 H0 s`; use the injection
+`J0 H1^-1`, then restore `H1^-1` in reverse operation order before the old
+`H0^-1`. Thus
+
+```text
+J0 H1^-1 H1 H0 s = J0 H0 s
+```
+
+for every helper vector `s`, including arbitrary dirty values. The target and
+helper D boundary is identical. The existing target preparation/restoration,
+other stages, COPY/ERASE records and frame schedule remain in place.
+
+The five selected injection columns have total weight 92 before the cycle and
+70 after it. Their 111-row injection therefore drops from 4,464 to 4,442 ADDs.
+The cycle adds five preparation and five restoration ADDs, giving:
+
+| Quantity | Pinned word | Cyclic word |
+|---|---:|---:|
+| Injection ADDs | 4,464 | 4,442 |
+| Helper cycle preparation and restoration | 0 | 10 |
+| Complete D-stage ADDs | 5,548 | 5,536 |
+| Complete R439 word ADDs | 11,500 | **11,488** |
+| Paid rank mass | 5,129 | 5,129 |
+
+The complete-word bounds are `F=27,061` and `B=5,921,344`. Exact integer
+arithmetic gives
+
+```text
+64 F^3 B^2 = 44468459235010318794652647424
+             < 20282409603651670423947251286016 = 2^104
+
+2^18 F^5 B^4 = 4676708453903757010070537011688772133559165966010548224
+               < 6277101735386680763835789423207666416102355444464034512896 = 2^192
+```
+
+## Bounded search and optimistic price
+
+Rank all 439 injection columns by their total pairwise support overlap, breaking
+ties by column weight (larger first) and local helper index (smaller first).
+Take the top 20 helpers. For every one of the 15,504 five-element supports,
+test all 120 permutations, where each permutation gives the five chronological
+cyclic edges `p[i] ^= p[i+1]` with indices modulo five. The search tests exactly
+**1,860,480** words and stops there. It finds 56 words with positive complete
+ADD savings; four tie at the maximum of 12. The candidate above is the
+lexicographically first maximum.
+
+Before enumeration, a favorable basis bound gave at most 109 net ADDs of
+savings in this domain: the highest-ranked five columns have weight 124 and
+rank five, while preparation and restoration cost ten. For any invertible
+change on five columns, the new injection has at least that rank in nonzero
+columns. The cycle search attains 12 of the 109-ADD ceiling.
+
+The paid profile is unchanged, so the recursive moment and κ gain are exactly
+zero. This construction does not enter the selected assembly or establish a
+smaller integer-multiplication exponent. It is scoped to the frozen R439 word
+and the stated top-20 cyclic-sweep domain; it is not a minimum over other helper
+supports, longer cycles, target bases or complete circuits.
+
+## Verification
+
+The source-pinned replay uses the standard library and the retained R439
+boundary, owner and exporter checks:
+
+```sh
+python3 -B research/r439-five-cycle-seed9f01/verify.py
+python3 scripts/research_package.py --package research/r439-five-cycle-seed9f01 --check-only
+python3.12 scripts/research_package.py --package research/r439-five-cycle-seed9f01
+```
+
+The first command repeats the full finite search, checks all 661 formal columns
+in both directions, verifies the dirty D boundary, replays H1–H5 and the paid
+profile, and checks the complete exporter invoice and precision expressions.
+It also rejects five controls: a missing preparation edge, a missing inverse
+edge, the stale injection, a wrong inverse order, and a changed injection
+coefficient. The expected final line is:
+
+```text
+PASS_R439_FIVE_CYCLE candidates=1860480 best_saving=12 ADDs=11500->11488 columns=661 controls=5 no_new_kappa
+```
+
+The exact candidate, search receipt and replay receipt are
+[`CANDIDATE.json.gz`](CANDIDATE.json.gz),
+[`SEARCH-RESULT.json`](SEARCH-RESULT.json) and
+[`CERTIFICATE.json`](CERTIFICATE.json). `SOURCES.json` pins the task catalogue,
+selected result, review, source manifest, assembly certificate and every live
+input used by the predecessor replay. The conditional result remains subject
+to the inherited all-size/compiler, common-label/mutable-source realization,
+whole-call hosting, precision/tape and analytic-transfer assumptions. Native
+C++ replay, supplier adoption and outer assembly replay were not run here.
+
+## Open next questions
+
+- **Can a six-helper cyclic change save more on the same D map?** A longer cycle
+  could realize multi-column cancellations unavailable to a five-cycle. First
+  compute its exact rank-based optimistic ceiling and full six-edge
+  preparation/restoration bill on a bounded overlap-ranked support family.
+  Continue only if that ceiling is positive and the precision budget is
+  favorable.
+
+## Credit
+
+The R439 word, helper-star basis, phase-B programs and retained checkers come
+from the predecessor packages and PR lineage named in `SOURCES.json` and the
+linked README. Those inputs are reused unchanged. Prepared by Cinna Davis
+(@1337isnot1337) with substantial OpenAI Codex assistance in the derivation,
+search, implementation, checking and writing. No independent human review is
+claimed. This package is Apache-2.0.