Mister-Meeseeks · Oct 22, 2026 · Oct 22, 2026
diff --git a/research/quotient-kernel-family/README.md b/research/quotient-kernel-family/README.md
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+# Admissible quotient kernels of the #4596 word
+
+**Claim type: finite certificate with a scoped proof.** No κ is claimed. The
+result is conditional on the open pull request #4596 and on the endpoint
+question raised in #4604 (see [Conditions](#conditions)).
+
+DaysSky's open #4596 quotients the nine-axis ambient of the PR4470 complex word
+by `span{511}`, re-pads one helper so the word still spans nine axes, and
+reports a complex supplier saving of 0.012091850234. Its `PROOF.md` leaves
+open "Whether `511` is the unique removable direction". This package answers that
+question for the check #4596 itself ships.
+
+## Result
+
+Over all 511 nonzero directions `v` of F₂⁹, build the #4596 quotient word for
+`span{v}` and replay it with the shipped literal replay
+(`native_extensions.literal_replay` with the `verify_buffers.expected`
+endpoint table, both orientations, owned C cleared).
+
+1. **Exactly 32 directions are accepted.** The other 479 all fail the same
+   shipped check, "Paid rank and source frame".
+2. **The accepted set has an exact description.** The base word has eight centre
+   transports, each between the ground frame and a dim-8 frame, charged rank 8.
+   Those eight frames are hyperplanes of F₂⁹ with normals
+   {7, 25, 42, 52, 76, 82, 97, 127}, of GF(2) rank 4. A direction is accepted
+   exactly when it lies outside all eight hyperplanes, that is
+   `{v : λᵢ(v) = 1 for all i}`. This is a coset of a 5-dimensional subspace, so
+   it has 2⁵ = 32 points, and it does not contain 0. The normals use only bits
+   0–6, and each normal is itself outside all eight hyperplanes, so the set is
+   `{h + 128i + 256j : h a normal, i, j ∈ {0, 1}}`.
+3. **511 is not the best direction.** The best is `v = 153`:
+
+   | Kernel | Main widths | Paid rank mass | Local deficit | Rank surplus | Complex saving |
+   |---|---|---|---|---|---|
+   | 511 (#4596) | {7: 112, 8: 112} | 4834 | 1744 | 2768 | 0.012091850234 |
+   | 153 | {7: 168, 8: 56} | 4778 | 1688 | 3776 | 0.016567169219 |
+
+   The next two are 338 (0.016555611423) and 180 (0.016552449914), both with
+   deficit 1688 and surplus 3776. Twenty-four accepted directions, 153 and 511
+   among them, keep the helper profile {3: 2, 4: 26, 6: 7, 7: 47, 8: 325, 9: 1};
+   the other eight (7, 52, 82, 97, 391, 436, 466, 481) widen both rank-3
+   helpers to rank 4. The saving is #4596's invoice and dual-moment screen
+   applied unchanged.
+4. **No kernel of dimension 2 or more is accepted.** A 2-dimensional kernel
+   needs three nonzero vectors outside every centre hyperplane. The accepted
+   set is a coset that does not contain 0, so the sum of two accepted
+   directions lies in the difference subspace, which meets the coset nowhere.
+   The verifier checks this over all pairs.
+5. **Under the stronger geometric condition, 511 is unique.** Require that
+   every dim-8 main final keeps dimension 8 under the quotient, that is, that
+   `v` lies outside all of them. The 112 distinct dim-8 main finals have
+   normals of full rank 9, and the only direction outside all of them is 511.
+   This is the condition #4596's construction meets; the shipped replay does
+   not check it.
+
+So the answer to #4596's question depends on the condition. 511 is the unique
+direction that keeps every dim-8 main final at dimension 8, but 32 directions
+pass the shipped replay, and 153 prices 37% higher on the complex side.
+
+## What this does not change
+
+The outer κ in #4596 is limited by the BIT side (0.0034), not by the complex
+supplier. A larger complex saving does not move it until the BIT admission
+chain moves. This package claims no κ.
+
+## Conditions
+
+- **#4596 is open.** The quotient construction, the helper re-padding and the
+  invoice are #4596's. This package re-uses them unchanged; if #4596 is not
+  accepted, this classification describes an unaccepted construction.
+- **#4604's endpoint question applies here too.** pinterbanget's open #4604
+  shows that under the 511 quotient every main output frame changes, and that
+  the shipped replay compares final frames with the word's own record, not
+  with required output charts. The 32 accepted directions pass that same
+  replay, so they inherit the same question. If output charts are part of the
+  supplier contract, none of the 32 quotient words, including 511, meets it.
+- **The 479 rejections come from one hard-coded rank.** `synthesize` charges
+  every centre transport rank 8. A quotient is accepted only if each centre
+  frame keeps dimension 8, which is the hyperplane condition in item 2. A
+  pipeline that recomputed centre ranks would accept a different set; that
+  variant is not studied here.
+
+## Reproduce
+
+From the repository root, with CPython 3.12:
+
+```sh
+python3.12 -W error -B research/quotient-kernel-family/verify.py                  # about 30 s
+python3.12 -W error -B research/quotient-kernel-family/verify.py --scan 1 255     # about 100 s
+python3.12 -W error -B research/quotient-kernel-family/verify.py --scan 256 511   # about 100 s
+python3.12 -W error -B research/quotient-kernel-family/verify.py --physical 153   # exact lowering
+python3.12 -W error -B research/quotient-kernel-family/verify.py --physical 511
+```
+
+The default run checks items 2 to 5: it derives the eight centre hyperplanes
+from the base word, checks the normals, rank, coset and pair conditions, checks
+the 112 main-final hyperplanes, replays all 32 predicted directions, checks the
+declared profiles for 153 and 511 and the two helper profiles, and replays
+four rejected controls. The two `--scan` shards replay every
+direction and require the shipped replay to accept exactly the predicted set
+(item 1). `--physical` lowers every realised transport of one quotient word
+exactly and requires Fourier rank equal to the absolute dimension difference,
+the convention of the shipped evaluators (`native_extensions.py` line 127,
+`verify_buffers.py` line 151).
+
+`quotient.py` builds the base word from the pinned main packages and the
+quotient words; it writes no files.
+
+## Limitations
+
+- Finite certificate over one word. The base word is fixed; other words or
+  other quotient maps are not covered.
+- The complex saving is computed with #4596's float bisection of the dual
+  screen. A directed-rounding recomputation during development agreed to
+  2·10⁻¹⁷ for 153 and 7·10⁻¹⁷ for 511; that recomputation is not shipped.
+- Physical lowering was checked for 153 and 511 only, not for the other 30
+  accepted directions.
+- No formal proof, no Lean, no selected-construction replay.
+
+## Open next questions
+
+- Does an operator-level endpoint check (#4604's open question) accept any of
+  the 32 quotient words, or does it reduce the admissible set to the empty set?
+- If centre ranks are recomputed instead of hard-coded, which kernels pass, and
+  is any of them better than 153?
+- Does a different base word have a centre-hyperplane family of lower rank,
+  and so a larger admissible coset?
+- Does #4596's controller strictification hold when its controller targets come
+  from a quotient word? An unverified local substitution gave a negative margin
+  on the delta row for all 32 directions, 511 included. This needs an
+  independent check before any outer κ is claimed for the family.
+
+## Credit
+
+This package was produced by Nick Petalas. The quotient construction, helper
+re-padding and invoice are DaysSky's (#4596). The base word is Cinna Davis's: query thinning
+and helper retiming (#4441), the missing join frame (#4447) and joint helper
+boundaries (#4470), on zxv's native supplier. The endpoint-contract question
+is pinterbanget's (#4604).
diff --git a/research/quotient-kernel-family/SOURCES.json b/research/quotient-kernel-family/SOURCES.json
@@ -0,0 +1,90 @@
+{
+  "schema_version": 1,
+  "base_commit": "f626db5c3a3d781d8f7695038776c621651f75e1",
+  "files": [
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+      "usage": "provenance",
+      "git_blob": "ed33caef5c5827693e9572d252a66f99cbb73d40",
+      "sha256": "5732707cc1e460b090b53f2463162749ad860de878a18f7408938671d295a9d9"
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+      "usage": "provenance",
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+      "usage": "input",
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+      "sha256": "9ab5328245ff02c9d0be21d39bcc375f50bc5d7909e9f7b9a3a9a6f812f5bd6e"
+    },
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+}
diff --git a/research/quotient-kernel-family/TASK.json b/research/quotient-kernel-family/TASK.json
@@ -0,0 +1,84 @@
+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 7,
+  "base_commit": "f626db5c3a3d781d8f7695038776c621651f75e1",
+  "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": "Classify the kernel directions v of F2^9 for which the ambient quotient of open PR #4596 passes the shipped literal replay, describe the admissible set exactly, and price its best member with #4596's invoice.",
+  "scope_change_reason": "PR #4596 answered this card with one quotient direction (511) and left open whether 511 is the unique removable direction. The classification answers that question for the shipped check; it is a complex-side result and does not change the outer kappa.",
+  "bottleneck_link": "Complex-side supplier result conditional on #4596. The outer kappa in #4596 is limited by the BIT side, so a better complex saving does not move it until the BIT admission chain moves; #4604's endpoint-chart question also applies.",
+  "budget_minutes": 120,
+  "stop_rule": "Stop at the exact one-dimensional classification under the shipped replay, its coset description, the dimension-two obstruction and exact lowering for 153 and 511. Do not study recomputed centre ranks or controller strictification.",
+  "outcome": "completed",
+  "evidence": "verify.py derives the eight centre hyperplanes and the 112 main-final hyperplanes from the pinned base word, checks the rank-4 coset and the pair condition, replays all 32 predicted directions and four rejected controls with the shipped literal replay; --scan shards replay all 511 directions; --physical lowers every transport of the 153 and 511 words exactly.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Does an operator-level endpoint check accept any of the 32 quotient words?",
+      "why_it_matters": "If output charts are part of the supplier contract (#4604), the whole family, 511 included, may be inadmissible.",
+      "first_step": "Run #4604's endpoint-frame comparison on the v = 153 quotient word.",
+      "blocked_by": "The required output charts are not an input to the shipped replay."
+    },
+    {
+      "question": "Which kernels pass if centre transport ranks are recomputed instead of hard-coded to 8?",
+      "why_it_matters": "The 479 rejections all come from the hard-coded centre rank; a recomputing pipeline would accept a different set.",
+      "first_step": "Recompute centre ranks in synthesize for one rejected direction and replay it.",
+      "blocked_by": "A changed synthesizer is a different certificate and needs its own review."
+    },
+    {
+      "question": "Does #4596's controller strictification hold when the controller targets come from a quotient word?",
+      "why_it_matters": "An unverified local substitution gave a negative delta-row margin for all 32 directions, 511 included; an outer kappa for the family depends on it.",
+      "first_step": "Recompute the delta row of #4596's controller with the v = 511 targets.",
+      "blocked_by": "Requires the controller contract #4596 uses."
+    }
+  ]
+}