Prove the boundary-arithmetic row-field pad is removed before the nex…

research/pr4100-row-field-movement/PROOF.md

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+# Boundary-arithmetic row-field movement
+
+Scoped lemma for one layout step of the #4100 Gaussian caller. No new κ.
+The selected conditional result stays `0.001672660197076918`.
+
+## Caller
+
+The sentence under audit is the boundary-arithmetic movement, not the union
+schedule's head travel.
+
+- `research/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md`, "Batched right-hand
+  sides do not add a large transpose", and the earlier exposure step: pad only
+  the exposed field to its next power of two, at a factor below two, and remove
+  that padding before the next axis. Move only that polynomial-width row field,
+  at `O(log p)` times the boundary volume, not a transpose of the long spectator
+  field. Scan each factor page once per right-hand-side batch.
+- `research/unconditional-batched-crt/GAUSSIAN.md` repeats the removal claim:
+  temporary padding of the one exposed polynomial row field is removed before
+  the next axis and never accumulates as `2^g`.
+- `research/unconditional-batched-crt/vendor/UNION-SCHEDULE.md` step 5 charges
+  the same movement to the boundary-algebra row and says it is not part of the
+  union movement.
+
+PR #4327 audits the union schedule's tape-head bound and leaves
+boundary-arithmetic batching uncovered. PR #1731's rectangular review says
+partial-batch padding is negligible. It does not prove the factor, the level
+count, or the removal alternative. This note proves those three layout claims.
+It does not prove the Strassen or rectangular exponent, the carving identity
+that turns the movement into `O(V d W^{τ-1}/B)`, conditioning, or any ledger
+row of #4100.
+
+## Lemma 1 — the row-field pad factor is below two
+
+For an integer `a ≥ 1`, let `M(a) = 2^{ceil(log2 a)}`. Then `a ≤ M(a) < 2a`,
+and `M(a)` is a power of two.
+
+If `a` is a power of two, `M(a) = a < 2a`. Otherwise `2^{k-1} < a < 2^k` for
+`k = ceil(log2 a)`, so `M(a) = 2^k = 2 · 2^{k-1} < 2a`.
+
+Thus padding the exposed field from `a` real positions to `M(a)` multiplies
+that field by a factor strictly less than two. The factor `2` is not attained
+for any finite `a`.
+
+## Lemma 2 — batches in `[W, 2W]` add no cells when `R ≥ W`
+
+Let `W ≥ 1` and `R ≥ 0` be integers.
+
+- If `R = 0`, there are no batches and no added cells.
+- If `R ≥ W`, write `R = qW + r` with `q ≥ 1` and `0 ≤ r < W`. When `r = 0`,
+  use `q` batches of width `W`. When `r > 0`, use `q − 1` batches of width `W`
+  and one batch of width `W + r`. Then `W < W + r ≤ 2W − 1`, every batch lies
+  in `[W, 2W]`, the widths sum to `R`, and the number of added cells is `0`.
+- If `0 < R < W`, the only batch is padded from `R` to `W`. The number of added
+  cells is `W − R < W`. The ratio `W/R` may exceed `2`.
+
+PHASE-LAYOUT calls `R` superpolynomial and uses a polynomial bandwidth `W`.
+Under `R ≥ W`, which is the asymptotic regime that hypothesis names, the final
+batch does not need padding. "Padded once at constant cost" is not an `O(1)`
+record-move bound. The proved added volume in the small-`R` case is less than
+one full batch.
+
+## Lemma 3 — moving only the short field costs `O(log a)` passes
+
+Let the exposed field have length `a ≥ 1` and let the orthogonal index have
+length `N ≥ 1`. The orthogonal index may contain the batch coordinate and any
+spectator coordinates. The boundary volume of real records is `V = a N`.
+Pad the exposed field to `M = M(a)` and let `V+ = M N`. Store the padded
+records row-major: row `r`, then the orthogonal index.
+
+Merge adjacent row blocks by the lockstep schedule. At a stage whose current
+block holds `span` original padded rows, a node is two adjacent blocks. Pass 1
+copies the two children apart. Pass 2 writes, for each orthogonal position,
+the first child's `span` records and then the second child's `span` records.
+Each pass moves every padded record once. There are `log2(M)` stages because
+`M` is a power of two and `span` doubles from `1` to `M`. The forward movement
+is therefore exactly
+
+```text
+2 · log2(M) · V+
+```
+
+record-moves. The reverse schedule, splitting each suffix-major chunk back
+into the two children, has the same count and restores row-major order. The
+number of stages is `ceil(log2 a)`. It depends on `a` and not on `N`, the
+batch width, the spectator length, or the number of axes `g`.
+
+By Lemma 1, `V+ < 2V`, so the forward count is strictly less than
+`4 · ceil(log2 a) · V` when `a ≥ 2`, and it is `0` when `a = 1`. In all cases
+the forward-plus-undo count is strictly less than `8 · max(1, ceil(log2 a)) · V`.
+When `a` is polynomial in `p`, `ceil(log2 a) = O(log p)`, which is the stated
+movement bound: `O(log p)` times the boundary volume.
+
+A transpose whose stages instead follow the orthogonal index has
+`ceil(log2 M(N))` stages. Whenever `N > M(a)`, that is strictly more stages
+than the short-field schedule. A spectator of length `2^g` is the long index
+this schedule does not transpose.
+
+After the solve, drop the `M − a` pad rows. Undo uses the real field length
+only in the sense that the pad rows are not part of the tensor address: they
+are tape cells deleted before the next axis. The real record count returns
+to `V`.
+
+## Lemma 4 — removal does not accumulate across axes
+
+Process axes of lengths `a_1, ..., a_g` one at a time. While axis `i` is
+exposed, the live volume multiplier is `M(a_i) / a_i < 2`. After its pad cells
+are deleted, the multiplier returns to `1`. The peak multiplier over the whole
+sequence is therefore below `2`, and the final multiplier is `1`, for every
+`g` and every tuple of lengths.
+
+If the pad coordinate is retained, the final multiplier is
+`∏_i M(a_i) / a_i`. For `a_i = 3`, this is `(4/3)^g`, which is unbounded in
+`g` and already greater than `2` at `g = 3`. Lemma 1 says each factor is
+below `2`, so the retained product is below `2^g`, but it still grows
+exponentially whenever every factor is at least `4/3`. The sentence "never
+accumulates as `2^g`" is the removal schedule. It is false for the retained
+schedule.
+
+## Lemma 5 — one factor-page scan per batch
+
+Under the partition in Lemma 2, the number of factor-page scans is the number
+of batches. It is not the right-hand-side count `R`, except in the special
+case where every batch has width `1`. It is not multiplied by `g`: each axis
+is a separate window, and this count is inside one axis.
+
+## What remains inherited
+
+The identity "Strassen `W^ω` per `W` right-hand sides gives
+`O(V W^{ω−1} polylog p)`", and #4100's later form
+`O(V d W^{τ−1}/B polylog p)`, need the carving, the relation between the
+exposed length and `W`, and the exponent `τ`. PHASE-LAYOUT already assigns
+stability and the no-pivot factor theorem to the phase-assembly branch.
+#4100's Gaussian paragraph uses `τ = 45/16`, while its union-schedule ledger
+uses `τ_eff = 2 + 10^{-64}`. This layout lemma does not choose between them.
+
+A finite execution of the schedule corroborates the exact move count, the
+padding inequality, and the removal alternative. The statements above are
+integer identities for every positive size named in their hypotheses, not
+sample checks.

research/pr4100-row-field-movement/README.md

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+# Boundary-arithmetic row-field movement
+
+**Scoped layout lemma. No new κ.** The selected conditional result stays
+`0.001672660197076918`. This does not accept #4100.
+
+## Scope and evidence
+
+Card `uniform-transfer-audit`, refined to one caller. #4100's Gaussian text
+says the one exposed polynomial row field is padded by a factor below two,
+moved at `O(log p)` times the boundary volume, and removed before the next
+axis, so the pad does not accumulate as `2^g`. The same movement is
+PHASE-LAYOUT's short-field transpose, and UNION-SCHEDULE step 5 charges it
+to the boundary-algebra row rather than to the union head travel. PR #4327
+audits that head travel and leaves this batching row open.
+
+[PROOF.md](PROOF.md) proves, for every positive integer size in the
+hypotheses:
+
+- `M(a) = 2^{ceil(log2 a)}` satisfies `a ≤ M(a) < 2a`.
+- When the right-hand-side count is at least the batch width `W`, those
+  right-hand sides partition into batches in `[W, 2W]` with no added cells.
+  A shorter final batch adds fewer than `W` cells, not `O(1)` record moves.
+- The lockstep transpose of the padded field uses `ceil(log2 a)` stages and
+  two passes per stage. Forward movement is exactly `2 · ceil(log2 a) · M(a) · N`,
+  which is strictly less than `4 · ceil(log2 a)` times the real boundary
+  volume when `a ≥ 2`. The stage count does not depend on the spectator length.
+- Deleting the pad before the next axis keeps every axis multiplier below two
+  and returns the final multiplier to one. Retaining a length-3 pad on each
+  axis multiplies the volume by `(4/3)^g`.
+
+`verify.py` recomputes the padding inequality for `a = 1..65536`, the batch
+partition for `W = 1..64` and `R = 0..512`, and the schedule on twelve shapes.
+The worst executed forward ratio is `2048/144`. A three-axis tensor of length
+`3` peaks at `36` records and returns to `27`. Two rejection controls fail
+closed: a pad of length `3` is not `8`, and six retained length-3 pads are
+not a factor below two.
+
+The Strassen count, the carving that produces `O(V d W^{τ-1}/B)`, conditioning,
+and the #4100 ledger remain inherited. #4100 names both `τ = 45/16` and
+`τ_eff = 2 + 10^{-64}`; this lemma does not choose between them.
+
+## Reproduce
+
+From the repository root, on CPython 3.12:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/pr4100-row-field-movement --check-only
+python3.12 -B scripts/research_package.py --package research/pr4100-row-field-movement
+```
+
+The registered command is `verify.py`. It uses the standard library and reads
+the three pinned prose inputs. Expected first line:
+
+```text
+PASS_ROW_FIELD_MOVEMENT padding_strictly_below_two=true zero_added_cells_when_R_ge_W=true small_batch_added_cells_lt_W=true forward_passes_per_stage=2 max_forward_moves_over_real_volume=2048/144 stage_count_depends_only_on_row_length=true removal_peak_below_two=true retained_three_length_g6=(4/3)^6>2 page_scans_equal_batch_count=true no_new_kappa=true
+```
+
+## Open next questions
+
+1. **Carry the move count into the ledger term.** The layout bound is
+   `O(log a)` times the boundary volume. It becomes #4100's
+   `O(V d W^{τ-1}/B)` only after the exposed length is identified with the
+   carving parameters `W` and `B`. First pin `|E_i|` and `μ` at the #4100
+   balance and substitute Lemma 3. Blocked here because the carving program
+   is not replayed.
+2. **Fix the boundary exponent.** GAUSSIAN.md uses `τ = 45/16` for the
+   seven-product paragraph, while UNION-SCHEDULE section 4 uses
+   `τ_eff = 2 + 10^{-64}`. The movement lemma can be copied into the ledger
+   only after one of those is the stated exponent. First compare those two
+   sentences with the generated ledger at head
+   `337d04939571a637aeab54a0f58f54fde811663a`. Blocked here because that
+   ledger is a different program, and PR #1731's rectangular exponent is a
+   different caller.
+
+## Credit
+
+The layout sentences are from the capped phase-cell audit (Vlado Vojdanovski,
+PR #1673, and the contributors named there) and from Cinna Davis's #4100
+Gaussian and union-schedule text. PR #4327 (Ryanson Jonathan) audited the
+union head travel and marked this batching row as separate. PR #1731's
+review (the rectangular exponent and the operator identity) says partial-batch
+padding is negligible and is not this level count.
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.
+No independent human review and no formal-kernel acceptance are claimed.

research/pr4100-row-field-movement/SOURCES.json

@@ -0,0 +1,36 @@
+{
+  "schema_version": 1,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65",
+      "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96"
+    },
+    {
+      "path": "research/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md",
+      "usage": "input",
+      "git_blob": "bbeeee7431720817f031360f67b2186abac5cd23",
+      "sha256": "54aafc7afd09fe1a5312e69fad09c87422690543ab75c6ae35f6cbcc26b3eb8a"
+    },
+    {
+      "path": "research/unconditional-batched-crt/GAUSSIAN.md",
+      "usage": "input",
+      "git_blob": "adfb28c93e6b06178c901caaac39ede41a9e541b",
+      "sha256": "2f63c91c58f2d1cf31294812c75970ea390c61866a8da7ed700d4316a033a242"
+    },
+    {
+      "path": "research/unconditional-batched-crt/vendor/UNION-SCHEDULE.md",
+      "usage": "input",
+      "git_blob": "b5eb03e6745804f465a536fc9ffa87c2515d1bfa",
+      "sha256": "aabb28c343bcb9697a4cac7058060132617073df17ee18e591d2bf4defe9e921"
+    }
+  ]
+}

research/pr4100-row-field-movement/TASK.json

@@ -0,0 +1,80 @@
+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 7,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "original_card": {
+    "id": "uniform-transfer-audit",
+    "question": "Does one quantified step of the proposed #4100 all-size construction hold at its actual caller?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/unconditional-batched-crt/README.md"
+    ],
+    "methods": [
+      "Prove one missing implication with explicit quantified hypotheses.",
+      "Find an exact counterexample at the claimed interface boundary."
+    ],
+    "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read PROOF.md and AUDIT.md in #4100. Select one distinct obligation: CRT total repair, native/free-algebra lifting, precision/conditioning, or complete fixed-tape cost. This is an author-submitted proof, not an accepted unconditional theorem.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Check live handoffs; name one proposition, its caller, quantified domain and dependency closure. Independently prove the implication or exhibit a precise gap/counterexample. Prioritize CRT.md total repair or COMPLEX.md free-algebra lift only if unclaimed. Stop at the chosen obligation; do not substitute finite samples or root arithmetic for an all-size proof, or launch the complete package by default.",
+    "bottleneck_link": "Finite supplier improvements are useful only under the retained transfer obligations.",
+    "allowed_variations": [
+      "One CRT repair, native lift, precision/conditioning or tape-cost proposition, with its exact caller."
+    ],
+    "scope_change_triggers": [
+      "Replacing a quantified obligation with another finite smoke test.",
+      "Treating archive intake, an author PASS receipt or a finite composed fixture as discharge of the universal hypotheses."
+    ],
+    "overlap_terms": [
+      "uniform transfer",
+      "all-size",
+      "precision",
+      "4100",
+      "unconditional-batched-crt"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4100: proposed unconditional construction and finite checks, without independent acceptance.",
+        "evidence": "research/unconditional-batched-crt/README.md",
+        "remaining": "Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.",
+        "status": "author-checked",
+        "source_commit": "337d04939571a637aeab54a0f58f54fde811663a",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Prove the boundary-arithmetic row-field movement at the #4100 Gaussian caller: the exposed field pads by a factor strictly below two, a balanced short-field transpose uses ceil(log2 a) stages and two passes each, and deleting the pad before the next axis keeps the live multiplier below two. This is the layout step PHASE-LAYOUT and GAUSSIAN.md state, which #4327 left uncovered. It is not a CRT repair, a free-algebra lift, or an acceptance of #4100.",
+  "scope_change_reason": "The card asks for one quantified implication at its actual caller. CRT total repair and the free-algebra lift already have open audits. This package takes the still-open precision/conditioning layout obligation: GAUSSIAN.md's boundary-arithmetic row-field movement, at the PHASE-LAYOUT caller those sentences cite.",
+  "bottleneck_link": "The boundary row is one of the rows that can bind a proposed all-size Gaussian supplier. A 2^g pad, or a transpose of the long spectator, would replace the stated O(log p) movement. The lemma shows the removal schedule does not do that. It does not by itself raise the selected kappa.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after the row-field factor, the short-field move count, and the removal-versus-retention alternative. Do not re-derive the union head-travel constant, the Strassen exponent, or the ledger balance.",
+  "outcome": "completed",
+  "evidence": "PROOF.md. verify.py recomputes the padding inequality for a = 1..65536, the batch partition for W = 1..64 and R = 0..512, and the lockstep schedule on twelve shapes plus a three-axis tensor of length 3. Worst executed forward ratio 2048/144.",
+  "author": "unpaidmoon",
+  "ai_assistance": "Substantial Cursor Grok assistance with the proof writeup and the verifier.",
+  "open_questions": [
+    {
+      "question": "Does the exposed length equal the bandwidth W in #4100's carving, so that this movement count is the ledger term O(V d W^{tau-1}/B)?",
+      "why_it_matters": "The layout bound is O(log a) times the boundary volume. The quoted Gaussian cost still needs the relation between that volume, W, B, and the exponent tau.",
+      "first_step": "Pin the carving identities for |E_i| and mu at the #4100 balance and substitute the move count from PROOF.md Lemma 3.",
+      "blocked_by": "This package does not replay the phase-cell carving or choose between tau = 45/16 and tau_eff = 2 + 10^{-64}."
+    },
+    {
+      "question": "Which boundary exponent does the #4100 ledger actually use, 45/16 from the seven-product paragraph or 2 + 10^{-64} from the union-schedule ledger?",
+      "why_it_matters": "The two statements name different tau, so a movement lemma cannot be copied into the ledger row until the exponent is fixed.",
+      "first_step": "Compare GAUSSIAN.md's tau = 45/16 sentence with UNION-SCHEDULE.md section 4 and the generated ledger checks at head 337d04939571a637aeab54a0f58f54fde811663a.",
+      "blocked_by": "The ledger program is outside this package, and PR #1731's rectangular exponent is a different caller."
+    }
+  ]
+}

research/pr4100-row-field-movement/VERIFY.json

@@ -0,0 +1,22 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md",
+    "research/unconditional-batched-crt/GAUSSIAN.md",
+    "research/unconditional-batched-crt/vendor/UNION-SCHEDULE.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Padding factor M(a) < 2a for a = 1..65536; batch partition; exact lockstep move count; removal versus retained padding; source-sentence binding; rejection controls."
+    }
+  ]
+}