Conditional κ ≈ 0.0017291527560043: R1 words at their packing cap (T740), +1.55e-9 over #3021

[rohanarun]

Change and mathematical scope

Packing composed with #3015's R1 edit. The words are Jeremy Kothe's #3015 R1 words (#2979's words plus the declared restore edit R1), used unchanged. A literal T740 allocation attains their certified fixed-capacity packing cap.

	κ (headline; each − 10⁻⁴⁶ − 10⁻⁷⁰)	T	packing
#3015 (R1)	0.001729150973534224428695	74	#2979's fill, 1.78e-9 below the R1 cap
#3021 (R1 + #3018)	0.001729151203983018442998	74	1.55e-9 below
this PR	0.001729152756004330934535	740	attains the R1 cap

The exact gain over #3021's released headline is 1552021312491537/10^24, about 1.552e-9. Over #3015 it is about 1.782e-9.

Why T740. On the optimal face of the relaxation (12 tight patterns), exact row multipliers fix three counts per replica:

• [9,9,9]@32 at 93/4;
• [9,9,9,9,9]@48 at 1341/20;
• []@48 at 7619/20.

So an integral optimum needs 20 | T. T740 is the first multiple of #2917's T74 order that qualifies. At T74 the best integral mix is 1.55e-9 below the cap, which is exactly where #3021 sits.

Why it composes. R1 changes only the local profile (−2H₁ + H₂ per call). Widths, windows, order, dispatch and buffers are unchanged, and #3015's R1 cap certificate has the same basis and idle profile as #2979's.

Construction.

• Lift. Ten lockstep copies of Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979's literal T74 fill, which Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015 uses unchanged, in Compose normal-45 extraction and fourteen gate rewrites: candidate κ ≈ 0.0017240352247481 #2969's lift convention.
• Relocations. 332 whole-role relocations in allocation/RELOCATIONS.json change the mix by:
  • 60 × [4,8,9,9,9,9]@48 → 57 × [9⁵]@48 + 3 × []@48;
  • 75 × [9,9,9]@32 → 60 × [4,9,9,9]@32 + 15 × [8,8,8,8]@32.
• Exchange chains. The relocations move along same-width exchange chains, because the displaced 4s and 8s are late and almost all [9,9,9] bins close early.
• Result. The literal idle histogram {1:173160, 3:49617, 5:17205, 48:281903} is exactly 740 × the basic optimum. Physical stock is 663,549.

Remaining obligations.

• This is a moment nomination with literal T740 owners, the same status as Attain the joint rank-seven packing cap exactly at T740 #3024 and Compose normal-45 extraction and fourteen gate rewrites: candidate κ ≈ 0.0017240352247481 #2969. The complete fresh-native admission and modern outer release have not been rerun at T740; Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015's and feat(research): compose restore and packing gains #3021's releases are at T74.
• All of Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015's and Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979's word-level, native and inherited hypotheses are inherited unchanged. Their pending independent audits apply here too.
• No selected-frontier change is claimed.

Task and overlap

• Base and sources. Base main d8e0f83f, priority revision 6. Sources:

  • Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979 at be191346 (on main as an archive);
  • Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015 at e244296d;
  • feat(research): compose restore and packing gains #3021 at 217edac2 (comparator only).

  The vendored files carry exact heads, Git blob IDs and SHA-256 in VENDOR.json.

• Task. Card selected-packing, scope changed (reason in TASK.json).

• Overlap. Nick Petalas' Attain the joint rank-seven packing cap exactly at T740 #3024 (18:39 UTC) independently attains the analogous T740 cap for Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979's words without R1. It uses a 550-row journal on feat(research): lower the rank-seven packing moment #3018's fill. My 332 relocations were found separately from Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979's own fill and reach the same idle histogram. Attain the joint rank-seven packing cap exactly at T740 #3024 leaves the R1 composition as an explicit separate obligation, and this PR is that composition. feat(research): lower the rank-seven packing moment #3018, feat(research): compose restore and packing gains #3021 and feat(research): improve doubled rank-seven packing #3022 are the T74 and T148 packing lineage.

Validation

Standard library only, Python 3.12:

• verify.py (about 70 s):

  • live and vendored pins;
  • providers re-derived from the R1 words (Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979's widths, the R1 local profile, 91 buffers);
  • a deterministic T740 rebuild with pinned canonical payload SHA-256;
  • Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979's vendored owner engine over all 2,717,280 roles (windows, families, capacities, masks, compact prefixes, complete profile, rank mass);
  • Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015's R1 cap certificate re-verified;
  • the fixed face counts and the 20 | T condition;
  • every literal bin pattern tight;
  • the root bracket equal to the R1 cap bracket;
  • the dyadic moment auditor re-pinned to Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015, showing a strict gain;
  • an exact strict gain over feat(research): compose restore and packing gains #3021.

  Locally it prints PASS_RANK7_R1_T740_CAP 0.0017291527560043309345349999… in 68.5 s.

• controls.py: seven controls, PASS_RANK7_R1_T740_CAP_CONTROLS in 195 s.

  • The bare lift is valid, but its root is Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015's.
  • A dropped chain link, a window violation, a family collision and a stale profile are all rejected by the engine.
  • Tampered relocations fail the payload pin.
  • A wrong moment baseline is rejected.

python3 scripts/research_package.py --package research/rank7-r1-t740-cap --check-only
python3.12 scripts/research_package.py --package research/rank7-r1-t740-cap

Not run:

• the complete T740 admission/outer release;
• any independent human or formal review.

Hosted CI is separate.

Credit

• Jeremy Kothe (Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015): R1, its words and its cap certificate.
• Marius-Paul Dumitrean (Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979): the words, the T74 fill lifted here and the checkers.
• Nick Petalas (feat(research): lower the rank-seven packing moment #3018, feat(research): compose restore and packing gains #3021, feat(research): improve doubled rank-seven packing #3022, Attain the joint rank-seven packing cap exactly at T740 #3024): the integrality lineage and the concurrent T740 cap.
• Dugongue (Compose normal-45 extraction and fourteen gate rewrites: candidate κ ≈ 0.0017240352247481 #2969): the lift convention.
• themilkman216: the observation about the integral T.
• Mason Hale, Tekalign, Jeremy Diviney, Yiyang Lu and icekylinx: the retained lineage.

Prepared by Rohan Arun with substantial AI assistance (Anthropic Claude). Apache-2.0.

Open next questions

1. Run the complete T740 release: Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015's reproduce chain, using the ORDER/DISPATCH/FILL that verify.py --work DIR writes.
2. Find further declared edits beyond R1 that keep the same optimal basis. Any such profile inherits this allocation's cap attainment.

{
  "schema": "integer-mult-construction/v1",
  "word_sha256": null,
  "dependencies": [3015, 2979, 2969],
  "remaining_obligations": ["Complete T740 fresh-native admission and modern outer release", "Independent audits of #2979/#3015 (inherited)"],
  "profile": {
    "model": "one-unpaid-coordinate-halving-v1",
    "m": 73,
    "families": 896,
    "stages": 9,
    "centre": 64,
    "idle_rank_cap": 51,
    "local_profile_per_call": {"1": 2443, "2": 707, "3": 232, "4": 108, "5": 18, "8": 8}
  }
}

🤖 Generated with Claude Code

https://claude.ai/code/session_01TGoMCHAqctZe4wTb4Qr7VV