[petalas]

Result

Finite certificate with a scoped proof. No κ is claimed. This answers the open question in #4596's PROOF.md, "Whether 511 is the unique removable direction", for the check #4596 ships. The answer depends on the condition:

• Under the shipped literal replay, 32 of the 511 directions pass. They are exactly the directions outside the eight centre hyperplanes of the base word. Those hyperplanes have normals {7, 25, 42, 52, 76, 82, 97, 127}, of GF(2) rank 4, so the passing set is a coset of a 5-dimensional subspace. The other 479 directions all fail the same shipped check, "Paid rank and source frame".
• 511 is not the best of them. v = 153 has main widths {7: 168, 8: 56} against {7: 112, 8: 112}, local deficit 1688 against 1744, and rank surplus 3776 against 2768. Its complex supplier saving is 0.016567169219 against 0.012091850234, 37% higher. Conditional κ = 0.0034 (BIT-limited, +12.58%): complex supplier to 0.011417749 via the ambient quotient by span{511} #4596's invoice and screen are applied unchanged.
• No kernel of dimension 2 or more passes. The passing coset does not contain 0, so it contains no subspace.
• Under the geometric condition, 511 is unique. 511 is the only direction outside all 112 dim-8 main finals, whose normals have full rank 9. That is the condition Conditional κ = 0.0034 (BIT-limited, +12.58%): complex supplier to 0.011417749 via the ambient quotient by span{511} #4596's construction meets; the shipped replay does not check it.

Conditions

• Conditional κ = 0.0034 (BIT-limited, +12.58%): complex supplier to 0.011417749 via the ambient quotient by span{511} #4596 is open. The quotient, the helper re-padding and the invoice are DaysSky's. If Conditional κ = 0.0034 (BIT-limited, +12.58%): complex supplier to 0.011417749 via the ambient quotient by span{511} #4596 is not accepted, this classifies an unaccepted construction.
• Endpoint-frame audit of #4596: every main output chart changes under the span{511} quotient; replay checks only the word's own finals (no new κ) #4604 applies here too. pinterbanget's open Endpoint-frame audit of #4596: every main output chart changes under the span{511} quotient; replay checks only the word's own finals (no new κ) #4604 shows that the shipped replay compares final frames with the word's own record, not with required output charts. The 32 directions pass that same replay. If output charts are part of the supplier contract, none of the 32 words, 511 included, meets it.
• The outer κ does not move. Conditional κ = 0.0034 (BIT-limited, +12.58%): complex supplier to 0.011417749 via the ambient quotient by span{511} #4596's outer κ is limited by the BIT side, so a larger complex saving changes nothing until the BIT admission chain moves.

Evidence and reproduction

research/quotient-kernel-family/. quotient.py builds the PR4470 base word and the quotient words from pinned main packages; verify.py checks them with the shipped evaluators. From the repository root:

~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/quotient-kernel-family/verify.py                  # 31 s
~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/quotient-kernel-family/verify.py --scan 1 255     # 106 s
~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/quotient-kernel-family/verify.py --scan 256 511   # 105 s
~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/quotient-kernel-family/verify.py --physical 153   # 136 s
~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/quotient-kernel-family/verify.py --physical 511   # 120 s

Times are local (macOS, CPython 3.12).

• Default run: derives the centre and main-final hyperplanes from the word. It checks the rank-4 coset, its closed form {h + 128i + 256j} and the pair condition. It then replays all 32 predicted directions (632/632 endpoints, both orientations, owned C cleared), checks the declared 153 and 511 profiles, and confirms that four controls are rejected.
• Scan shards: replay every direction and require exactly the predicted set to pass.
• --physical: lowers every realised transport exactly. It requires Fourier rank to equal the absolute dimension difference, the convention of native_extensions.py:127 and verify_buffers.py:151. Result: 2102 edges for 153 and 1865 for 511, 0 mismatches.

Local: scripts/research_package.py --package research/quotient-kernel-family and scripts/check_research_commit.py --package research/quotient-kernel-family --check-source-history both pass.

Limitations

• One fixed base word and one quotient map family.
• The saving uses Conditional κ = 0.0034 (BIT-limited, +12.58%): complex supplier to 0.011417749 via the ambient quotient by span{511} #4596's float bisection. A directed-rounding recomputation agreed to 2·10⁻¹⁷ (153) and 7·10⁻¹⁷ (511); it is not shipped.
• The 479 rejections all come from synthesize charging every centre transport rank 8. A pipeline that recomputes centre ranks would accept a different set; that is not studied.
• Lowering was checked for 153 and 511 only. There is no formal proof, Lean or selected-construction replay.

Open next questions

• Does an operator-level endpoint check (Endpoint-frame audit of #4596: every main output chart changes under the span{511} quotient; replay checks only the word's own finals (no new κ) #4604's open question) accept any of the 32 words?
• Which kernels pass if centre ranks are recomputed, and does any beat 153?
• Does Conditional κ = 0.0034 (BIT-limited, +12.58%): complex supplier to 0.011417749 via the ambient quotient by span{511} #4596's controller strictification hold with controller targets from a quotient word? An unverified local substitution gave a negative delta-row margin for all 32 directions, 511 included. This needs an independent check before any outer κ is claimed for the family.

Related open work

• Conditional κ = 0.0034 (BIT-limited, +12.58%): complex supplier to 0.011417749 via the ambient quotient by span{511} #4596 (DaysSky): the construction this extends. It uses 511, which is among the 32 here; its published deficit 1744, surplus 2768 and saving 0.012091850234 reproduce exactly.
• Endpoint-frame audit of #4596: every main output chart changes under the span{511} quotient; replay checks only the word's own finals (no new κ) #4604 (pinterbanget): an endpoint-frame audit of Conditional κ = 0.0034 (BIT-limited, +12.58%): complex supplier to 0.011417749 via the ambient quotient by span{511} #4596. It is a different check that applies to all 32 directions; there are no overlapping numbers.

Credit

This package was produced by Nick Petalas. The quotient construction, re-padding and invoice are DaysSky's (#4596). The base word is Cinna Davis's: #4441, #4447 and #4470, on zxv's native supplier. The endpoint question is pinterbanget's (#4604).

CI notes

Run 39769931034 on head b5d71c724: plan failed and replay was skipped, so hosted CI has not certified this package.

The failure is not in this package. The "Onboarding helper tests" step fails 4 tests in tests/test_research_current_status.py:

• test_current_ci_section_matches_selected_record
• test_current_section_matches_selected_record
• test_history_banner_precedes_historical_sections
• test_only_the_first_heading_claims_current_or_selected

The same 4 fail locally on origin/main f626db5c3 (python3.12 -B -m unittest tests.test_research_current_status: 6 run, 4 failures). They also fail on other contributors' PR runs since at least 08:19 UTC today, e.g. runs 39754840618 and 39746873103.

The local results in the Evidence section stand.