docs(.arxiv): QR256 elimination, Four Russians, egglog and rotation research paths

.arxiv/2026-10_four_russians_m4ri_reverse_lookup_excerpts.md

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+# Four Russians / M4RI and reverse lookup: lookup tables versus row copying
+
+**Sources:** [M4RI repository](https://github.com/malb/m4ri) and Martin R. Albrecht, Gregory Bard, Clément Pernet, *Efficient Dense Gaussian Elimination over the Field with Two Elements* ([PDF](https://martinralbrecht.wordpress.com/wp-content/uploads/2011/11/ple.pdf)); plus Y. Lou and Y. Xu, *An Improved Method of the Four Russians for Inversion using Reverse Lookup Table*, IEEE ISIT 2026, [DOI:10.1109/ISIT62367.2026.11653817](https://doi.org/10.1109/ISIT62367.2026.11653817).  
+**Research status (2026-10-21):** Abstract/readme-grounded extracts. IEEE work is **not an arXiv article**. No implementation or timing evidence for Lance-graph.
+
+## Source-grounded excerpts
+
+The classic Four Russians idea precomputes combinations for a **block** of pivot rows so a group of elementary operations may be applied through one indexed combination. In M4RI, those combinations use **GF(2) row XOR**. Blocking limits repeated scanning at the cost of a lookup table. Original M4RI gives an O(n^3/log n)-style reduction for GF(2) elimination.
+
+Lou–Xu (2026) describe a **reverse lookup table** recording a *row index* for a given row value, rather than a cached *row vector*, and report retaining asymptotic time O(n^3/log n) with auxiliary storage O(n log n) instead of O(n^2). Treat those as the authors' algorithmic claims: no independent local verification and no inference that a QR256 truth matrix needs only 256 B.
+
+M4RI (library) is GPLv2+; keep it as an **external oracle/process** or follow a clean-room procedure rather than copying source into a differently licensed production crate.
+
+## Hypothetical transfer to QR256
+
+A valuable **index-not-payload** pattern matches the user's zero-copy SoA aim: keep row combinations, codebook references or replay offsets, not duplicated payload vectors. But there are two distinct domains:
+
+1. **GF(2) M4RI:** XOR cancellations, rank, inversion and parity constraints.
+2. **Boolean-semiring blocking:** OR of intersections, existential relational composition, bounded reachability.
+
+Substituting XOR for OR is forbidden. The second case needs its own small-table scheme or independent bit-matrix oracle; never call that M4RI correctness.
+
+## Falsifiers / test path F
+
+- **F1, GF(2) reference:** Compare scalar XOR Gaussian, blocked Four Russians and reverse-index experiment at dimensions 16,64,128,256. Require equal rank and row-equivalent echelon results, not byte-identical pivot order. Measure table construction, bytes, cache misses, XOR count and overall time.
+- **F2, Boolean lookup:** For random/sparse/dense QR256 source×target tiles, compare blocked Boolean composition with scalar triple loop and `tile_gemm::BitMat::compose`. Deliberately include duplicate two-hop paths; OR must remain 1 while XOR may cancel to 0.
+- **F3, 4-bit vs 8-bit block:** Measure lookup size and build/execute crossover including 1,8,64,1,000,000 subsequent folds. Preregister termination if LUT creation or cache misses outweigh saved Boolean operations.
+- **F4, reverse index / zero-copy:** Hold stable borrowed Arrow lanes and compare a stored row vector with an index into the original row. Verify exact value equality, source generation invalidation and no hidden `to_vec` or payload copy. Count index arrays and metadata separately.
+- **Silence twin:** Matrix with no profitable reuse chooses ordinary bit-masks or simple scalar path, rather than building a speculative LUT.
+
+**Note:** A 256×256 Boolean matrix is 65,536 bits = 8 KiB, while 256 u8 selectors are only 256 B. The latter can *address* the former but cannot encode arbitrary 65,536 independent truth bits.

.arxiv/2026-10_qr256_gaussian_egglog_rotation_falsifier_paths.md

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+# QR256 / Inception: pre-registered hypothetical test paths
+
+**Date:** 2026-10-21. **Status:** RESEARCH-ONLY TEST DESIGN; not executed. **Owners to review:** relalg, contract, R2IL, mask-risc, Quack, mork-pathmap-probe, crepe-plan-probe, Thought Git.  
+**Sources:** companion notes in this .arxiv shelf: [2201.06508](2201.06508_gaussian_greedy_reversible_circuit_excerpt.md), [2610.17934](2610.17934_chordal_codes_elimination_order_excerpt.md), [2610.08701](2610.08701_gaussian_partial_pivot_growth_warning_excerpt.md), [2609.28901](2609.28901_quantum_gaussian_reversible_proof_excerpt.md), [2304.04332](2304.04332_egglog_equality_saturation_datalog_excerpt.md), [Four Russians](2026-10_four_russians_m4ri_reverse_lookup_excerpts.md).
+
+## 0. Representation and authorization contract
+
+- The **8 KiB 256×256 packed Boolean matrix** is the hot, resident execution state. Its axes must be *typed* (e.g. method×entity membership or source×target adjacency). One does **not** represent a general binary relation by a method×entity unary plane alone.
+- A `u8` method palette supplies up to 256 **method selectors**, not 256 matrix-valued truths. A pair of u8 axes forms a 16-bit local coordinate, but its bits require storage or a separately verified virtual backing.
+- The **cold sidecar** records global↔local mapping, source generations, borrowed SoA row ordinals, BEFORE/AFTER snapshots or hashes plus reconstructible snapshots, dirty deltas, typed rewrite-law IDs, proof references, and authorized writeback positions. It is not an inner-loop operand.
+- During PREPARE, values may be read from the **same** source table and resolved by an attested frozen codebook; counted/witnessed same-table seeks are permitted under [ADMISSION B / #1640](https://github.com/AdaWorldAPI/lance-graph/pull/1640). **Cross-table foreign-value-directed position reads remain forbidden**, including preparation. A preallocated map does not automatically confer admission. Future exceptions demand an explicit new decision.
+- Existing zero-copy index map is lab-only [#1659](https://github.com/AdaWorldAPI/lance-graph/pull/1659), co-addressed multi-side join [#1658](https://github.com/AdaWorldAPI/lance-graph/pull/1658), 256-rule CSE [#1653](https://github.com/AdaWorldAPI/lance-graph/pull/1653), Boolean tile [D-RMAP-2](../.claude/plans/d-rmap-2-context-tile-algebra-probe-v1.md). Do not substitute GraphBLAS, C/MKL, or a new runtime for already-probed native kernels.
+
+**Cost accounting:** cold bind/build/index; source bytes touched; hot Boolean folds; cache misses; output tuple materialization; dirty update; proof DAG bytes and checker time; writeback/commit separately. The motivating 20 ms preparation + 2 s operations is a user hypothesis, **not a measurement**. For 1,000,000 operations, amortized preparation is 20 ns/operation if bind is 20 ms; for 4 billion operations it is 5 ps/operation.
+
+## 1. Paths / hypotheses / independent oracles
+
+| ID | Hypothesis / path | Oracle | Can-fire (required) | Can-stay-silent / kill |
+|---|---|---|---|---|
+| P1 | **Same-ordinal AD/Entra/EXO:** witnessed shared identity means A[i],B[i] borrow exact source rows, without joins | independent immutable GUID→row map | distinct fields across 3 sources bind correctly and reused | same row number but different external ID MUST refuse; separate deletion/reorder/stale generation |
+| P2 | **Many-to-many group membership:** bound CSR/offset index plus Boolean QR register answers EXISTS without tuple enumeration | scalar multiset join and transitive memberships fixture | preserved duplicates for COUNT/BAG via ordinal sidecar | set truth does not imply multiplicity; group cycle must terminate |
+| P3 | **Cross-ontology:** Mouse→Cheese→Milk→Cow and Cheese→Käserei use explicitly typed identity bridges | scalar graph path with source lineage | no repeated global resolution after bind | mismatched ontology homonym and out-of-focus escape MUST refuse/continue; no value-steered foreign read |
+| P4 | **Four Russians GF(2):** block-pivot XOR LUT or reverse-index lookup amortizes linear elimination | independent GF(2) Gaussian reference | equal rank, row space and reconstructible pivot schedule | all-zero/singular inputs and tiny matrices must not claim inversion or speculative speedup |
+| P5 | **Boolean tile:** bitwise OR/AND relational composition beats repeated raw joins after reuse | scalar Boolean matrix multiply and existing BitMat | 1,8,64,256 focus runs and densities 0.1%,1%,10%,50%,100%; exact tuple/mask equality | two distinct paths to same target: OR=1, XOR=0; bag answer differs from set answer |
+| P6 | **Gaussian substitution / E-graph:** a=2b,b=2c,z=2a → z=8c; 256 templates × 4096 frames = 1,048,576 applications | exact symbolic substitution evaluator, Datafrog for typed relational subset | shared proof DAG and lower source-read count; exact equivalence under declared laws | f64 reassociation, u8 saturation, contradictory cycles, nonsharing source demand MUST NOT produce false exactness/compression |
+| P7 | **Chordal ordering:** PEO on proven chordal primal graphs reduces intermediate fill | independent PEO/triangulation checker and GF(2) row operations | identical results, reduced peak fill, count cost | 4-cycle without chord; cyclic join that survives semijoins; arbitrary ontology not assumed chordal |
+| P8 | **GEPP robustness:** f64 elimination correctness requires conditioning/growth checks | high-precision linear solver | well-conditioned solved LGS agrees | fixed absolute 1e-12 cutoff fails scale invariance; failure cannot contaminate exact QR semiring |
+| P9 | **Reversible proof sidecar:** elementary GF(2) row steps invert; lossy Boolean OR needs extra BEFORE evidence | independent replay checker over raw 8 KiB state | before→rule schedule→after, inverse when legally reversible | sidecar holding only final OR matrix cannot reconstruct originals; stale source or no write acknowledgement cannot mark committed |
+| P10 | **Dirty algebra:** `CandidateDirty`, `ActualDirty = BEFORE XOR AFTER`, `CommittedDirty` distinct | full recompute from clean snapshot | changed input updates only affected legal dependencies | multiple supports + single deletion must NOT delete surviving reachability; no-op delta remains silent |
+| P11 | **Rotation as indexed view:** 3×3 CCW 90° through coordinate arithmetic; 256×256 QR without physical copy | explicit scalar permutation and four-rotation inverse | rotated view matches oracle; 4×90° returns original bytes | rotation of source×target directed relation may change semantics; demand axis/type witness |
+| P12 | **Amortization:** reused frame over 1,000,000 folds outperforms rebind each time | identical answers and independent timing harness | break-even observed with cold binding separated from repeated folds | if cold index, staging or proof grows with every fold, reject speedup claim |
+
+## 2. Rotation (user's exact fixture)
+
+Input:
+```text
+1 2 3
+4 5 6
+7 8 9
+```
+Requested CCW 90° rotation:
+```text
+3 6 9
+2 5 8
+1 4 7
+```
+For N×N matrix in row-major coordinates (zero-based): `new[r,c] = old[c,N-1-r]`. Equivalently an old cell `(r,c)` moves to `(N-1-c,r)`. Check: old(0,2)=3 → new(0,0); old(2,0)=7 → new(2,2)? **No:** under this formula old(2,0)→new(2,2)=7, while the actual fixture has new(2,0)=1 and new(2,2)=7. This explicitly guards against mixing lookup and movement directions.
+
+For a **logical/view rotation**, do not move bytes. Bind `QRView {base, axis_types, transform: RotateCCW90, generation}`; emit transformed address arithmetic at access time. Composing two quarter-turns yields 180°; four returns identity. Only physicalize if the layout economics warrant it. Track dirty positions through the *same bijection*. A lazy view is zero payload copies, not zero address arithmetic.
+
+**Semantic hazard:** a binary directed adjacency matrix has both axes drawn from the same entity domain, but rotation is transpose plus reversal, **not generally a graph isomorphism relabeling**. A true consistent node relabeling must apply the **same permutation to both endpoints**, `R'[P(i),P(j)] = R[i,j]`; this is not a 90° image rotation. For method×entity, swapping axes changes their types, so a rotated view is only admissible as a physical/presentation transform with explicit swapped axis metadata and inverse mapping. Never interpret rotated bit positions as new facts.
+
+## 3. Schedule / proof shape (illustrative, not ABI)
+
+```text
+BIND  frame_id, source_seals[], codebook_digest, palette_version
+EXEC  method8, domain_kind, axes, QR256, operands, focus_version
+RULE  proof_node_id, law_id, child_refs[], guard_ids[]
+DELTA actual_dirty256x256, before_root, after_root
+REPLAY inverse_coordinate_map, row_ordinal_refs, deterministic_schedule
+COMMIT target_seal, authorization, store_ack  // separate from a proof of computation
+```
+
+A proof DAG compresses repeated **structure**; arbitrary independent output facts still require proportional data/entropy. Hashes prove identity *relative to retained data*, not mathematical validity. Verify each rewrite and independent execution oracle; the checker must never trust the producer's claimed `success`.
+
+## 4. Experiment stages and gates
+
+**S0** Documentation review: keep Gaussian REAL_F64, GF2_XOR and BOOL_OR_AND as separate contracts. Inventory the already-existing masked tile and semijoin paths; no duplicated engine.
+
+**S1** Low-risk independent fixtures: 3×3 rotation, 16/64/256 exact Boolean bit rows, scalar closure/compose, two-path XOR-vs-OR counterexample, stale witness refusals. Run these before benchmarks.
+
+**S2** Cold optimization research: egglog outside production, bounded fuel, rewrite/equality proof reconstruction on the exact subset, normalized DAG size and signature/provenance check.
+
+**S3** Optional GF(2) and chordal probes: clean-room method of Four Russians, reverse-index LUT, PEO order, numerical growth adversaries. GF(2) oracle remains external if GPL constraints apply.
+
+**S4** Million-fold amortization: 256×4096 shared-rule test plus nonsharing control; binding versus execution versus trace-check overhead, source bytes and RSS. Reject without wall-time parity and all refusal guards.
+
+**S5** Writeback only after contract ruling: do not reinterpret a preparatory gather as admitted; no writes based on hypothetical experiments.
+
+**Success =** correct typed operations; reproducible and verifiable BEFORE→AFTER; mutation/stale/refusal oracles; measured cost reduction net of preparation and trace. **Failure =** wrong truth bit, lost multiplicity/escape, illegal foreign address, nondeterministic witness, unverifiable derivation, silent coordinate semantic change.

.arxiv/2201.06508_gaussian_greedy_reversible_circuit_excerpt.md

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+# Gaussian elimination versus greedy synthesis: exact reversible linear transforms
+
+**Source:** Timothée Goubault de Brugière et al., *Gaussian Elimination versus Greedy Methods for the Synthesis of Linear Reversible Circuits*, [arXiv:2201.06508v1](https://arxiv.org/abs/2201.06508), [HTML](https://arxiv.org/html/2201.06508v1), abstract, §1, conclusion. Submitted 2022-01-17, journal reference 2021.  
+**Research status (2026-10-21):** Primary-source excerpt, research only. Paraphrases, not copied passages. **No Lance-graph benchmark or runtime admission.**
+
+## Source-grounded excerpt
+
+The paper synthesizes *linear reversible circuits*, representable by invertible matrices over GF(2). It compares Gaussian elimination coupled with tuned LU/greedy selection against purely greedy circuit synthesis. The claimed output improvements are CNOT **count** and **circuit depth**, not CPU speed for relational joins. Reported preference: an optimized Gaussian/greedy method on random operators above 150 wires; purely greedy methods are near-optimal in their small-domain trials below 30 wires. Both ranges are from this paper's circuits, not transferable performance thresholds.
+
+Elementary row additions over GF(2) are XORs. Reversibility matters because a sequence of elementary operations can be recorded or inverted. The representation is not the Boolean OR/AND semiring used for existential Datalog joins.
+
+## Hypothetical transfer to lance-graph
+
+Use **operation selection and schedule extraction** as an idea: given a GF(2)-typed 256-column matrix, produce a compact palette-coded sequence of row-XOR, swap, and pivot choices. Compare to naive row-by-row elimination, an M4RI-style blocked baseline and a scalar reference. Treat the schedule as a proof witness, not a record of subjective reasoning.
+
+For relalg existential joins, use separate AND/OR rules; an XOR elimination schedule is **not** interchangeable with relational existence.
+
+## Falsifiable tests
+
+- **G1, GF(2) parity:** For random and singular 8/64/256-column inputs, apply the extracted elementary-operation schedule and independently verify `E · A = R` under GF(2). If E is not preserved, replay the logged elementary steps; do not claim inversion for singular input.
+- **G2, no false semiring parity:** Select a two-path graph with two derivations of the same target. XOR yields 0, OR yields 1. Any attempt to reuse the XOR schedule as a Datalog EXISTS kernel must refuse or fail this negative oracle.
+- **G3, cost model:** Compare CNOT-oriented schedule length with actual scalar/SIMD wall time, cold plan cost, replay bytes and cache traffic. A shorter circuit is *not* automatically faster on the CPU.
+- **Silence twin:** A no-op row addition or a permuted list of independent operations must not emit a committed QR dirty bit.
+
+**Existing related work:** [D-RMAP-2 tile plan](../.claude/plans/d-rmap-2-context-tile-algebra-probe-v1.md), [D-RELALG-UNIFIED](../.claude/plans/d-relalg-unified-v1.md).