Research: screen selected rank-two macro fusion

research/complex-rank2-macro-fusion-screen/README.md

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+# Selected complex rank-two macro-fusion screen
+
+**A source-bound negative screen for direct pair fusion. No new supplier or κ claim.**
+
+## Result
+
+The selected complex trace has 2,412 rank-one child climbs per orientation. A direct pair-to-rank-two macro would be attractive in the recursive cost model: if it preserved the exact operator, replacing two rank-one calls by one rank-two call would multiply that pair's recursive charge by `2^-a`, for the provider saving exponent `a > 0`.
+
+This screen checks the narrowest concrete version of that idea against the pinned selected source. It walks each register's climb history and looks for two successive rank-one climbs whose combined endpoint transition also has an exact rank-two child recipe. Each orientation has 1,441 such rank-one successor links, of which 35 have a matching direct rank-two recipe. **All 35 candidates are interrupted by scalar operations on the same register**: 20 have five such touches, five have two, and ten have one. No candidate can be substituted directly while preserving the recorded scalar schedule.
+
+The forward phase-B witness on `L195` has a genuine exact identity. Its calls at events 12806 and 13402 use the recipes `0→2, rank 1` and `2→252, rank 1`; their 512-by-512 Gaussian-dyadic operator product equals the direct `0→252, rank 2` recipe on **all 512 basis vectors**. The exact operator hashes and the full intervening-window receipt are in [`RESULT.json`](RESULT.json). This confirms the child-map composition, rather than inferring it from matching endpoint frames.
+
+The intervening ADD still prevents a direct replacement. In the forward trace it changes the required output to `L195_out = D x + (1/2) B y`, where `D=BA` and `y=L521`; in the backward trace it also writes `L521_out = y - (1/2) A x`, so the caller consumes the intermediate `A x`. The one-output rank-two recipe supplies `D x` but neither cross-register block nor the intermediate tap. Every basis input has a nonzero missing contribution. Computing the tap separately needs a rank-one-equivalent call: the separated rank-two-plus-rank-one profile costs `2^(1-a)+1 > 2` for `0<a<1`, versus two rank-one calls, before adapter and restoration costs. A multi-output rank-two child that emits the tap would be a changed provider contract; this PR does not posit or price one. The selected κ remains `0.001672660197076918`.
+
+All 100 candidate/add-operand instances still have no existing same-key rank-one call to absorb the moved work. Thus the selected-source direct-fusion route reaches a concrete wall: exact child composition succeeds, but the required observable intermediate makes the available single-output replacement non-improving. This does not rule out changed-basis or multi-output synthesis.
+
+## Task and overlap
+
+Base: canonical main `20ecf2891d80b5acf18d980b2164fb44ea7faa90`. The original task is `complex-rank-shape-decoder`; this package narrows it to direct fusion of two rank-one frame climbs into an available rank-two endpoint recipe in the selected complex schedule. It does not satisfy the task's construction or five-percent acceptance gate.
+
+The refreshed overlap query covered 734 open PRs and 2,596 Discussions with complete pagination. It surfaced [PR #3998](https://github.com/CrocSwap/integer-mult-bounds/pull/3998), which removes two scalar ADDs in a staged544 graph; that is a different provider, graph and identity. The closest higher-rank antecedent remains [PR #1836](https://github.com/CrocSwap/integer-mult-bounds/pull/1836) and its follow-up [`complex-heterogeneous-child-interface` audit](../complex-heterogeneous-child-interface/README.md): those study simultaneous unequal-rank climbs, unlike this package's successive same-register pair and its observable intermediate ADD. The [two-invocation stage-pair audit](../../docs/research/stage-pair-audit.md) studies another bit-side graph. This is a scoped complement, not a broad novelty claim. Exact source archive, canonical word and child recipes are pinned; the verifier has no pending-PR dependencies.
+
+## Reproduction
+
+From the repository root:
+
+```sh
+python3 scripts/research_package.py --package research/complex-rank2-macro-fusion-screen --check-only
+python3.12 scripts/research_package.py --package research/complex-rank2-macro-fusion-screen
+```
+
+The registered normal and optimized replays pin the selected trace, canonical word and child recipes; replay climb and scalar-frame legality in both orientations; enumerate direct endpoint-recipe candidates; test same-key call reuse; reconstruct the `L195` candidate window; compare both rank-one operators' product with the direct rank-two operator on all 512 basis columns over `Q(i)`; bind the nonzero ADD tap; and run wrong-rank and source-removal controls. No full supplier replay or new κ is claimed.
+
+## Open next questions
+
+The direct single-output rank-two route is closed for this witness: its child composition is exact, but the ADD requires either the missing `B y` contribution (forward) or the intermediate `A x` tap (backward). The next distinct question belongs to vector/tap synthesis: can a changed-basis multi-output rank-two provider emit both `D x` and the required tap for less than two rank-one calls after all interface and restoration costs? No such provider is present in the pinned source, so this package stops before expanding that construction round. `TASK.json` records the changed-contract question and its gate.
+
+## Credit
+
+Credit Cinna Davis / PR #1836 and Adam Gall's `complex-heterogeneous-child-interface` package for the simultaneous unequal-rank census and reciprocal-encoding classification; credit the selected PR #1094 provider lineage and its pinned source audit. Prepared by Gonzalo Mier with substantial OpenAI Codex assistance. Apache-2.0; no independent human review claimed.

research/complex-rank2-macro-fusion-screen/RESULT.json

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+{
+  "basis_columns_checked": 512,
+  "candidate": {
+    "child_rank": 1,
+    "event_indices": {
+      "add": 12807,
+      "first": 12806,
+      "second": 13402
+    },
+    "path": [
+      0,
+      2,
+      252
+    ],
+    "windows_per_orientation": {
+      "backward": {
+        "climbs": 139,
+        "events": 597,
+        "first_child": 12806,
+        "intervening_add": 12807,
+        "scalar_events": 458,
+        "second_child": 13402
+      },
+      "forward": {
+        "climbs": 139,
+        "events": 597,
+        "first_child": 12806,
+        "intervening_add": 12807,
+        "scalar_events": 458,
+        "second_child": 13402
+      }
+    }
+  },
+  "composition_equals_direct_rank2": true,
+  "direct_recipe": [
+    0,
+    252,
+    2
+  ],
+  "first_recipe": [
+    0,
+    2,
+    1
+  ],
+  "ideal_charge_screen": {
+    "comparison_for_0_lt_a_lt_1": "2^(1-a)+1 > 2",
+    "multioutput_rank2_tap": "changed provider interface; not in pinned recipes or priced",
+    "original_rank_profile": [
+      1,
+      1
+    ],
+    "split_rank2_plus_rank1_tap_profile": [
+      2,
+      1
+    ]
+  },
+  "max_nonzeros_per_column": {
+    "direct_rank2": 4,
+    "first_rank1": 2,
+    "second_rank1": 2
+  },
+  "operator_dimension": 512,
+  "operator_sha256": {
+    "composed_rank1_pair": "9c0e0ded05b5fb99160c05fb27624f4ee6a76d2f0922ef76caafd0a4c6808255",
+    "direct_rank2": "9c0e0ded05b5fb99160c05fb27624f4ee6a76d2f0922ef76caafd0a4c6808255",
+    "first_rank1": "b9357c40f5f003d761452d4695b65663254e3e01ef2b506ff73a5ff3da8c3c1d",
+    "second_rank1": "7b076f5dbf1720c05e9dc0a942836a6e283a25c64ac5e5143c8d74be7174abf7"
+  },
+  "second_recipe": [
+    2,
+    252,
+    1
+  ],
+  "selected_kappa": "0.001672660197076918",
+  "source_archive_sha256": "f1da80610bf33c58527f18ad63b4350d0b864e1d4486141ee17ac9bed4d529d6",
+  "status": "PASS_RANK1_PAIR_EQUALS_RANK2_BUT_ADD_TAP_BLOCKS_DIRECT_REPLACEMENT",
+  "unabsorbed_intermediate": {
+    "backward": {
+      "emitted_by_direct_rank2_recipe": false,
+      "max_nonzeros_per_basis_column": 2,
+      "nonzero_basis_inputs": 512,
+      "required_tap": "(-1/2)*first_rank1*L195 into L521"
+    },
+    "forward": {
+      "absent_from_direct_rank2_input": true,
+      "max_nonzeros_per_basis_column": 2,
+      "nonzero_basis_inputs": 512,
+      "required_block": "(1/2)*second_rank1*L521"
+    }
+  }
+}

research/complex-rank2-macro-fusion-screen/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "input",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "research/community-frontier1094-audit/selected-replay.tar.gz",
+      "usage": "input",
+      "git_blob": "c24daf067f9c543421d89ca2eeaa4eb78a57fb6f",
+      "sha256": "f1da80610bf33c58527f18ad63b4350d0b864e1d4486141ee17ac9bed4d529d6"
+    }
+  ]
+}

research/complex-rank2-macro-fusion-screen/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 6,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "original_card": {
+    "id": "complex-rank-shape-decoder",
+    "question": "Can a decoder deliver the local paid mass in fewer, larger-rank children (block frame climbs instead of rank-1 to rank-3 steps)?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/complex-local-word-ceiling/README.md",
+      "research/complex-decoder-field-extension-seed44/README.md",
+      "research/complex-frame-pair-decoders/README.md",
+      "research/selected-k8-unrestricted-ends/README.md",
+      "research/eisenstein-reduction-copy-floor/README.md"
+    ],
+    "methods": [
+      "Find one exact small decoding identity whose frame path climbs in blocks, and price its complete per-call profile.",
+      "Bound the best effective child rank achievable for one fixed support alphabet, including copies, frames and restoration."
+    ],
+    "acceptance_gate": "A per-call local profile, with all copied information, frames and restoration counted, whose ceiling exceeds the selected word's by at least 5%; masked or overcomplete supports must change an assumption of the eight-term all-pairs bound. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/complex-local-word-ceiling/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+      "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.",
+      "scope": "Positive control and planning budget only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin. Ideal budget: rank-16 delivery is 2.31x (levers.py).",
+    "bottleneck_link": "Rank-1 children are about half of recursive cost; the selected word's effective child rank is about 3.9. Larger-rank delivery lowers recursive cost per unit of paid mass.",
+    "allowed_variations": [
+      "One masked, overcomplete or block decoder family with exact labels and supports.",
+      "A scoped lower bound on effective rank for one alphabet.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "scope_change_triggers": [
+      "Repacking idle remainders of an unchanged word (covered by the packing ceiling).",
+      "Repeating the eight-term all-pairs search inside its proved scope.",
+      "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013.",
+      "Searching below C=40 in the #2017 Eisenstein binary scalar-rectangle or fixed-support block-linear additive pairing model; specify a changed premise instead. This bound does not cover every complex decoder.",
+      "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts."
+    ],
+    "overlap_terms": [
+      "rank shape",
+      "block climb",
+      "masked decoder",
+      "overcomplete",
+      "effective rank"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Screen the selected complex source for direct fusion of successive rank-one frame climbs into an available rank-two endpoint recipe, including exact scalar-interference windows in both orientations.",
+  "scope_change_reason": "The task card asks for a new block decoder and fully charged profile. This experiment screens a sequential same-register macro contract from the attachment: two rank-one climbs separated by scalar work. PR #1836 and complex-heterogeneous-child-interface instead study simultaneous unequal-rank climbs and reciprocal one-child encodings; this screen complements that work without repeating its census or claiming a profile improvement.",
+  "bottleneck_link": "The selected complex word contains 2,412 rank-one children, and the recursive cost is concave in child rank when a > 0. An exact two-rank-one to one-rank-two replacement would lower that pair's recursive charge by a factor 2^-a, but only if the composed operator and all intervening work are preserved.",
+  "budget_minutes": 60,
+  "stop_rule": "For the selected direct single-output rank-two contract, stop after the 512-column exact operator composition and the complete ADD tap are checked. Do not count the equality of the child product as a replacement of the full two-register map. Reopen only for a changed multi-output/tap provider with a complete interface and cost below the two rank-one calls.",
+  "outcome": "negative-result",
+  "evidence": "Both orientations contain 2,412 rank-one climbs, 1,441 successive rank-one climb links per register history, and 35 pairs with an available rank-two recipe on the combined endpoint frames. Every candidate is interrupted by same-register scalar work (histogram: 10 with one touch, 5 with two, and 20 with five); there are zero direct schedule-preserving substitutions. For the concrete L195 path 0->2->252, exact Gaussian-dyadic matrices verify on all 512 basis columns that the two rank-one child maps compose to the pinned rank-two recipe. The 597-event window contains 139 climbs and 458 scalar events; its ADD requires a nonzero cross-register block in the forward orientation and a nonzero intermediate A*x tap in the backward orientation. The single-output rank-two recipe emits neither. A separate rank-one-equivalent tap makes the ideal profile [2,1], whose charge 2^(1-a)+1 exceeds the original [1,1] charge 2 for 0<a<1. Across 100 candidate/add-operand instances, no same-key rank-one call can absorb the moved work. This is a scoped negative result for the selected direct-replacement contract, with an exact positive composition identity; no new supplier or kappa improvement is claimed.",
+  "author": "Gonzalo Mier",
+  "ai_assistance": "Substantial OpenAI Codex assistance in deriving and running the source-bound schedule screen, implementing the exact verifier and controls, and documenting scope and limits.",
+  "open_questions": [
+    {
+      "question": "Can a changed-basis multi-output rank-two provider emit the direct composition and the ADD-consumed intermediate tap for less than two rank-one calls after all interfaces and restoration are charged?",
+      "why_it_matters": "The pinned rank-two operator is exactly the product of the two rank-one maps, but the existing ADD observes or injects an intermediate value. One-output fusion therefore loses part of the full two-register linear map; a cheap tap would be the only path past this witness's wall.",
+      "first_step": "Use the forward/backward L195 witness and the exact matrices in RESULT.json. Form the complete block maps (x,y)->(D*x+(1/2)*B*y,y) and (x,y)->(D*x,y-(1/2)*A*x); search for an explicit output encoding whose rank-two child returns the required pair of outputs, then compute the full copy, frame, precision, restoration and recursion invoice.",
+      "blocked_by": "The pinned CLIFFORD-RECIPES provide one 512-vector output per child. No multi-output rank-two contract, emitted tap, or physical input/output ownership for such a provider is present. A separated rank-two plus rank-one implementation is already more expensive in the recursive profile. This changed-contract question overlaps the vector-decoder proposal and is not an extension of the closed direct-replacement screen."
+    }
+  ]
+}

research/complex-rank2-macro-fusion-screen/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "certificates/selected-result.json",
+    "research/community-frontier1094-audit/selected-replay.tar.gz"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Pinned selected record, replay archive, canonical word and exact member bytes; both orientations' climb and scalar-frame replay; successive rank-one links; exact rank-two endpoint-recipe matching; same-register event-touch histogram and candidate phase pairs; no add operand has a reusable same-key rank-one transition; reconstruct the 597-event L195 window; compare exact Gaussian-dyadic child matrices on all 512 basis columns and bind the product to the direct rank-two recipe; verify nonzero forward source block/backward intermediate tap and stored operator digests; wrong-rank and removed-recipe controls; scoped no-kappa conclusion."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Repeat the complete source-pinned frame replay, exact operator and ADD-tap screen, call-reuse screen, stored result-digest comparison, and controls under optimized Python."
+    }
+  ]
+}