Research: rooted R420 target forests save 4,784 XORs with exact class floor

research/r420-target-forest/CERTIFICATE.json

@@ -0,0 +1,95 @@
+{
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "labels": {
+    "labels": 120,
+    "lattice_rank": 8,
+    "gram_det": 1,
+    "frames": 672
+  },
+  "minimum_forest_certificate": {
+    "cost": 14921,
+    "cycle_inequalities": 7260
+  },
+  "minimum_nominated_tree": {
+    "D_XORs": 14921,
+    "ADDs": 21915,
+    "records": 24873,
+    "COPYs": 9,
+    "ERASEs": 9,
+    "profile": {
+      "1": 1320,
+      "2": 1440,
+      "4": 60,
+      "5": 120,
+      "8": 9
+    },
+    "rank_mass": 5112,
+    "prefix": {
+      "forward": 144651,
+      "reverse": 19231132,
+      "payload": "71639738627497752220685905038336",
+      "payload_bits": 106
+    },
+    "word_sha256": "ee77ccdf2dcf10f4d33f5b26ccdb28523a41d797efbfe7c144cb1aad94d67c10"
+  },
+  "baseline": {
+    "D_XORs": 19922,
+    "ADDs": 26916,
+    "records": 29874,
+    "COPYs": 9,
+    "ERASEs": 9,
+    "profile": {
+      "1": 1320,
+      "2": 1440,
+      "4": 60,
+      "5": 120,
+      "8": 9
+    },
+    "rank_mass": 5112,
+    "prefix": {
+      "forward": 144651,
+      "reverse": 2270064,
+      "payload": "998206739997057013039187410944",
+      "payload_bits": 100
+    },
+    "word_sha256": "e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2"
+  },
+  "feasible_forest": {
+    "D_XORs": 15138,
+    "ADDs": 22132,
+    "records": 25090,
+    "COPYs": 9,
+    "ERASEs": 9,
+    "profile": {
+      "1": 1320,
+      "2": 1440,
+      "4": 60,
+      "5": 120,
+      "8": 9
+    },
+    "rank_mass": 5112,
+    "prefix": {
+      "forward": 144651,
+      "reverse": 9640706,
+      "payload": "18003708514079901815901709480704",
+      "payload_bits": 104
+    },
+    "word_sha256": "d7fbc12b50b35c39c10a8f416d5ad19b2486b7b87bd064f11bdda0fdd6bbb345"
+  },
+  "roots": 12,
+  "preparation_XORs": 108,
+  "columns_per_direction": 660,
+  "negative_controls": [
+    "cyclic parent",
+    "nonminimum tree",
+    "missing preparation",
+    "missing restoration",
+    "missing correction",
+    "nonzero D frame",
+    "helper write",
+    "minimum-cost nominated tree exceeds payload cap"
+  ],
+  "positive_control": "F2-neutral target pair costs two ADDs",
+  "interface": "Semantic H2-prime; literal H2 and literal K extractor reject. All other H and C predicates pass.",
+  "scope": "Finite F2 circuit and exact rooted-forest cost optimum; feasible cut is not claimed precision-constrained optimal; no new kappa."
+}

research/r420-target-forest/CLAIM.json

@@ -0,0 +1,83 @@
+{
+  "track": "Bit supplier",
+  "assignment_seed": 5,
+  "random_source": "Python secrets.randbelow(100), operating-system randomness",
+  "started_utc": "2026-10-13T04:54:08.686716+00:00",
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "claim_utc": "2026-10-13T04:56:56.290190+00:00",
+  "expected_finish_utc": "2026-10-13T05:44:08Z",
+  "time_budget_minutes": 50,
+  "orientation_limit_minutes": 10,
+  "verification_and_handoff_reserved_minutes": 12.5,
+  "coordination": "No pinned issues returned by GitHub GraphQL; local claim per sprint instructions.",
+  "candidates": [
+    {
+      "object": "R420 phase-A cell or quad circuits",
+      "decision": "Existing PR691 already changes quad synthesis; simple cell gates are too narrow."
+    },
+    {
+      "object": "R420 phase-B block resynthesis",
+      "decision": "Existing PR670/674/688 cover circuit shortening and integration."
+    },
+    {
+      "object": "R420 frame-zero target forest basis",
+      "decision": "Selected: replace disjoint pairs with arbitrary rooted forests; exact minimum spanning tree certificate and full dirty restoration."
+    }
+  ],
+  "hypothesis": "Rooted target forests lower frame-zero compensation XOR cost beyond the disjoint-pair class without modifying helper contents or recursive frame paths.",
+  "method": "All 7260 edges on 120 targets plus zero root; seeded tie ordering from seed 5; one MST; independent path-max optimality witness; complete 660-column replay; positive-prefix accounting.",
+  "deliverable": "Finite bit-circuit construction and exact scoped optimum, reproducible verifier, proof, pins, controls; no new kappa.",
+  "scope_exclusions": [
+    "General linear basis synthesis",
+    "Helper basis changes",
+    "Composition with pending work",
+    "New recursive profile or kappa"
+  ],
+  "related_prs": {
+    "713": "Target-pair basis, independently developed; direct mechanism predecessor.",
+    "716": "Exact disjoint target-pair optimum; tree class strictly extends this domain.",
+    "694": "Helper-column basis; different domain and potentially stronger scalar reduction.",
+    "710": "Cleanup-only rewrite; distinct section.",
+    "742": "Concurrent shared-donor stars; credited. This contribution adds multiple levels, exact full-forest floor, and 1496 extra saved XORs.",
+    "733": "Two-sided helper/target basis with larger overall reduction; distinct domain, credited and not superseded.",
+    "731": "Independent compensation rank/kernel verification; distinct obligation."
+  },
+  "overlap_snapshot": {
+    "open_pr_count": 545,
+    "max_pr_number": 726,
+    "tree_or_forest_matches": []
+  },
+  "initial_complete_cost": {
+    "baseline_D_XORs": 19922,
+    "forest_D_XORs_including_restore": 14921,
+    "savings": 5001
+  },
+  "result": {
+    "updated_utc": "2026-10-13T05:09:53.117846+00:00",
+    "status": "Completed finite construction and verification; ready for focused PR",
+    "D_XORs": 15138,
+    "saved_native_XORs": 4784,
+    "exact_unconstrained_forest_floor": 14921,
+    "payload_bits": 104,
+    "native_stages_verified": [
+      5,
+      9,
+      11
+    ],
+    "kappa": "unchanged",
+    "result_link": "https://github.com/ofennell-quipex/integer-mult-bounds/tree/research/r420-target-forest-seed5/research/r420-target-forest"
+  },
+  "final_overlap": {
+    "checked_utc": "2026-10-13T05:09:53.117836+00:00",
+    "open_pr_count": 575,
+    "max_pr_number": 756,
+    "pinned_issues": [],
+    "main_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+    "outcome": "No competing general rooted-forest floor found; PR742 shared-donor overlap explicitly credited."
+  },
+  "experiment_update": {
+    "reason": "The nominated cheapest tree fails the inherited 104-bit budget.",
+    "bounded_search": "256 seed-ordered minimum trees, followed by 900 root-cut candidates over 8 rounds.",
+    "scope": "Same target-forest mechanism; no precision-constrained optimality claim."
+  }
+}

research/r420-target-forest/PROOF.md

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+# Proof and exact scope
+
+Work over F2. Write the 120 target registers as y and the 420 arbitrary helper
+registers as s. The pinned D matrix K has 120 rows and 420 columns. Its original
+implementation adds each nonzero coefficient separately, for |K| = 19,922 XORs.
+All sources, targets and helpers are independent symbolic inputs; source rows
+are not read by this replacement.
+
+## Rooted-forest identity, including restoration
+
+Fix a forest on the target vertices, orienting every component away from one
+root. For a nonroot t with parent p(t), prepare `y_t <- y_t + y_p(t)` in decreasing
+depth order. Since a parent has not yet been changed when its child executes,
+the prepared target vector is Py, where row t of P is e_t+e_p(t) for nonroots
+and e_t for roots. The product of these elementary transvections is invertible.
+The reversed preparation stream is its inverse over F2.
+
+At target t add `(K_t+K_p(t))s` for a nonroot, and `K_t s` for a root. This adds
+PKs in the prepared basis. Apply the reversed preparation stream:
+
+`P^-1 (Py + PKs) = y + Ks`.
+
+Sources and helpers have not changed at any time in D. This identity holds for
+arbitrary original y and s. In particular, preparation cannot be omitted merely
+because the zero-target case seems correct. All gates act on existing frame-zero
+target/helper registers. No descent, new frame, extra helper, source read, copy,
+or recursive child is introduced. Both the preparation and its inverse are
+included in the invoice.
+
+Replacing D by this operator leaves the input to A identical to the original
+word, on every input column. Thus the unchanged A/scatter/B/C program retains its
+full dirty-restored endpoint. Its original literal H2 syntax and literal K
+extractor are insufficient to recognize this implementation; the sufficient
+semantic contract is the full D endpoint just proved. The rest of the scalar
+interface is inherited without alteration. Same-frame target XOR operations
+use the original common-label conditional realization.
+
+## Exact optimal scalar cost within this class
+
+A forest F implemented by the preceding recipe costs
+
+`C(F) = sum_roots |K_t| + sum_nonroots (|K_t XOR K_p(t)| + 2)`.
+
+The two units per forest edge are its preparation and restoration. This defines
+the scoped class: one parent-difference helper row per target and no additional
+factoring among those rows. More general target bases, helper bases, shared
+helper expressions, cycles or arbitrary reversible circuits are outside it.
+
+Add a vertex z with K_z = 0. Give each edge z--t weight |K_t| and each target
+edge u--v weight |K_u XOR K_v|+2. Joining every component root to z maps a rooted
+forest to a spanning tree of the 121-vertex graph, at identical cost. Conversely,
+removing z from any spanning tree and orienting away from its neighbors gives a
+rooted forest with the same invoice. Therefore the exact class minimum is the
+minimum-spanning-tree weight.
+
+`RECIPE.json:minimum_parents` specifies an attaining tree. The verifier checks
+its connectivity and acyclicity and all 7,260 pair inequalities
+
+`w(u,v) >= max { w(e) : e lies on the tree path u to v }`.
+
+These inequalities are sufficient for minimality. To transform an arbitrary
+spanning tree S into the proposed tree T, add an edge e of T absent from S and
+consider its fundamental cut in T. The cycle in S+e contains an edge f of S not
+in T crossing that cut. The T-path of f contains e, so the verified inequality
+gives w(f) >= w(e). Replacing f by e does not increase cost and increases the
+number of shared edges. Repeating reaches T. Thus no S costs less than T.
+The verified sum is **14,921**, yielding the exact unconstrained saving ceiling
+**19,922 - 14,921 = 5,001** within this recipe class. The separate Prim
+implementation confirms the same integer minimum without using this certificate
+checker or the nomination algorithm.
+
+## Precision-limited submitted construction
+
+The certificate above concerns scalar counts; it contains no precision premise.
+The best reverse-bound tree among 256 nominated minimum-cost trees needs 106
+payload bits under the inherited bound. This is a failed budget check on those
+nominations, **not a proof that every optimum tree fails**.
+
+Deleting eight target edges, making their children roots, gives the submitted
+forest. Its 12 roots and 108 edges cost 14,922 helper XORs plus 216 target XORs,
+or **15,138** total. No precision-constrained optimality is asserted. The
+complete scalar stream costs **22,132**, a reduction of **4,784** from 26,916.
+
+The actual exported stream is used to recompute both positive prefix envelopes:
+initialize every local row bound to one; ADD replaces L_d by L_d+|c|L_s;
+COPY opens a temporary with its source bound; ERASE closes it. Reverse traversal
+interchanges COPY and ERASE. This is an entrywise triangle-inequality bound for
+the signed lift, independently of F2 cancellations. The largest forward and
+reverse bounds are respectively **144,651** and **9,640,706**. The inherited
+local invoice is therefore
+
+`64 * 144651^3 * 9640706^2 = 18003708514079901815901709480704 < 2^104`.
+
+The unchanged C++ checker independently verifies this bound. The nominated
+minimum-cost tree fails that same native budget check. Missing target restoration
+fails the native full-column check at five, nine and eleven stages.
+
+No gate outside D changes. All D gates remain at frame zero. Consequently every
+frame transition, COPY and ERASE is identical to the source. Exact comparison of
+the entire exported non-ADD subsequence checks this directly, including source
+and helper endpoints. The paid histogram remains
+`{1:1320, 2:1440, 4:60, 5:120, 8:9}`, with rank mass **5,112**.
+
+This proves a finite F2 circuit identity, a forest-class scalar floor, and the
+stated local invoices. It does not prove a new κ, a precision-constrained
+optimum, equality of the signed integer endpoint, or the uniform/physical/tape/
+analytic hypotheses of integer multiplication. Global hosting and invoice
+integration remain separate obligations; an increased local prefix must not
+silently inherit an old global receipt.
+
+Oscar Fennell, with substantial OpenAI Codex assistance; Apache-2.0. Predecessor
+credit and exact source pins are in README.md and SOURCES.json.