Conditional kappa 0.001675338871723469: whole-owner map at the packin…

[gvconly]

…g optimum of the seven-slot k8 word

New package research/gauge44-slots-arcflow on research/gauge44-slots (#1018 chain). Only the owner map changes: idle remainder {1:1214, 2:1, 32:1, 48:2443} over 34,560 whole-helper roles, found by an exactly liftable pattern/time MILP at the time-free LP bound and a Konig edge-colouring lift. Strict replay, admission and bit-supplier controls, independent price and a standard-library recount pass.

Prepared by G. V. Chudnovsky with Anthropic Claude assistance.

Claude-Session: https://claude.ai/code/session_01KBSZTxNtzYkGa6MNbKajRo

Change and mathematical scope

State the result, changed hypotheses and status: construction, proof, bounded
search, audit or tooling. Distinguish a scalar saving from a paid-profile or κ
improvement. List remaining obligations.

Task and overlap

Give the base commit, exact dependency heads, concrete object and bounded domain.
For donated-agent work, include the task ID/scope key or link the task card from
autoresearch.md. Link the nearest overlapping PRs and explain the added value;
different random seeds alone are not separate contributions.

Validation

Give reproduction commands, environment, exact evidence and meaningful failure
controls. Distinguish finite tests from proof arguments and list checks not run.
For a proposed bound, include the full paid budget and source bindings.

Credit

Credit predecessor and parallel work, retain source licenses, and disclose
substantial AI assistance. Useful independent or superseded work still merits credit.

---

Conditional κ = 0.001674083482880778: whole-owner map at the packing optimum of the seven-slot k8 word

New package research/gauge44-slots-arcflow, stacked on the #1005/#1018 chain (this branch contains that chain's packages unchanged, as in #1055). Only the owner map changes: data/OWNERS.json.gz, the profile of data/PLAN.json, the re-derived data/SPEC.json, RESULT.json, the regenerated independent price receipt, the κ pin and the search/recount scripts. The seven-slot k8 word c2b29430…, its 72-call schedule, the idle stock and windows, and PR589's bit supplier are unchanged.

	κ	idle remainder
this package (T = 8)	0.001674083482880778	{1:1214, 2:1, 32:1, 48:2443}
research/gauge44-slots-t16 (chain's best)	0.001672251604432734	{1:2249, 2:1, 3:406, 7:5, 16:36, 48:4853} per 16 replicas
research/gauge44-slots (#1018)	0.00167166528321877	{1:1099, 2:9, 3:179, 7:54, 16:23, 48:2419}
selected #942	0.001661960547713947

That is +0.1095% over the chain's best and +0.7294% over the selected result. The complex side binds (lower endpoint 0.001676890737967805674397).

Why this is the ceiling of the family at eight replicas. The windowed aggregate LP over bin patterns per (owner replica, idle child) class and per-call width counts has optimum 1215.57 rank-1 remainders, equal to the time-free relaxation, so the owner windows do not bind. The integer optimum of the time-free pattern model is exactly {1:1214, 2:1, 32:1} (HiGHS, proved optimal), which this map attains. At any replica count the time-free bound is at least 151.95 rank-1 remainders per replica, so T = 16 or 37 could add at most about 1.5·10⁻⁸.

How it was found. An exactly liftable MILP over pattern counts n(c,p) and per-pattern time incidences x(c,p,t,w) (each port takes at most one helper per call; Σ_t x = n·mult_p(w); every call's inventory is placed inside its windows) solves in seconds once the patterns are restricted to the LP support. Each pattern's times × positions multigraph has maximum degree n and is properly n-edge-coloured (König); every colour class is one literal port, offsets are the partial sums of the pattern. construction/owners/search_owners.py is a search input only; construction/owners/recount_owners.py (standard library) recounts the frozen map.

Verification (explicit reuse of a completed #1005 replay, as research/gauge44-slots does):

python -B research/gauge44-slots-arcflow/verify_manifest.py
python -B research/gauge44-slots-arcflow/construction/slots/verify_slots.py \
  --main . --pr635 /tmp/gauge44-pr635 --baseline /tmp/lvc --work /tmp/slots-arcflow
# PASS_ARCFLOW_OWNERS_ON_CHECKED_K8_SLOT_BASELINE 0.001674083482880778
python -I -B research/gauge44-slots-arcflow/construction/owners/recount_owners.py
# PASS_OWNERS_RECOUNT

All stages pass locally (Linux, Python 3.12): baseline pins, slot layer and its 34 controls, native locals, independent audit_local, and the strict global replay (restore-parent, bind, emit, audit with mutation controls, invoice, assembly, 69 admission controls, 12 bit-supplier controls). independent/price_profile.py gives the same κ (PASS_FROZEN_PROFILE_TWO_EXACT_ENGINE_AND_OUTER47_CROSSCHECK).

Scope. A finite conditional witness on a pending construction; every assumption of the #1005/#1018 chain is inherited (all-size compiler, common-label/mutable-source realization, whole-call hosting, precision/tape, exact recovery, analytic transfer, bit H1–H5). No optimality beyond the stated T = 8 whole-helper family is claimed.

Credit. Prepared by G. V. Chudnovsky with Anthropic Claude assistance (Apache-2.0). The word and everything below the owner map are the #1005/#1018 chain's: icekylinx's #645 gauge44 chain rebuilt by PartialK, G44Stack, NoTriReopen, BlockInvert, BRoute, chafreaky's #741 frames, JointXF, NewBArch, LeaverSAT/LeaverSAT2, rohanarun's #987 slots, G44Pack/G44Pack3's plan, Hao Yan's #858 T16 machinery (comparison only), and the PR589 bit supplier; see NOTICE.md.

🤖 Generated with Claude Code

https://claude.ai/code/session_01KBSZTxNtzYkGa6MNbKajRo

## Conditional κ = 0.001674083482880778: whole-owner map at the packing optimum of the seven-slot k8 word

New package research/gauge44-slots-arcflow, stacked on the #1005/#1018 chain (this branch contains that chain's packages unchanged, as in #1055). Only the owner map changes: data/OWNERS.json.gz, the profile of data/PLAN.json, the re-derived data/SPEC.json, RESULT.json, the regenerated independent price receipt, the κ pin and the search/recount scripts. The seven-slot k8 word c2b29430…, its 72-call schedule, the idle stock and windows, and PR589's bit supplier are unchanged.

	κ	idle remainder
this package (T = 8)	0.001674083482880778	{1:1214, 2:1, 32:1, 48:2443}
research/gauge44-slots-t16 (chain's best)	0.001672251604432734	{1:2249, 2:1, 3:406, 7:5, 16:36, 48:4853} per 16 replicas
research/gauge44-slots (#1018)	0.00167166528321877	{1:1099, 2:9, 3:179, 7:54, 16:23, 48:2419}
selected #942	0.001661960547713947

That is +0.1095% over the chain's best and +0.7294% over the selected result. The complex side binds (lower endpoint 0.001676890737967805674397).

Why this is the ceiling of the family at eight replicas. The windowed aggregate LP over bin patterns per (owner replica, idle child) class and per-call width counts has optimum 1215.57 rank-1 remainders, equal to the time-free relaxation, so the owner windows do not bind. The integer optimum of the time-free pattern model is exactly {1:1214, 2:1, 32:1} (HiGHS, proved optimal), which this map attains. At any replica count the time-free bound is at least 151.95 rank-1 remainders per replica, so T = 16 or 37 could add at most about 1.5·10⁻⁸.

How it was found. An exactly liftable MILP over pattern counts n(c,p) and per-pattern time incidences x(c,p,t,w) (each port takes at most one helper per call; Σ_t x = n·mult_p(w); every call's inventory is placed inside its windows) solves in seconds once the patterns are restricted to the LP support. Each pattern's times × positions multigraph has maximum degree n and is properly n-edge-coloured (König); every colour class is one literal port, offsets are the partial sums of the pattern. construction/owners/search_owners.py is a search input only; construction/owners/recount_owners.py (standard library) recounts the frozen map.

Verification (explicit reuse of a completed #1005 replay, as research/gauge44-slots does):

python -B research/gauge44-slots-arcflow/verify_manifest.py
python -B research/gauge44-slots-arcflow/construction/slots/verify_slots.py \
  --main . --pr635 /tmp/gauge44-pr635 --baseline /tmp/lvc --work /tmp/slots-arcflow
# PASS_ARCFLOW_OWNERS_ON_CHECKED_K8_SLOT_BASELINE 0.001674083482880778
python -I -B research/gauge44-slots-arcflow/construction/owners/recount_owners.py
# PASS_OWNERS_RECOUNT

All stages pass locally (Linux, Python 3.12): baseline pins, slot layer and its 34 controls, native locals, independent audit_local, and the strict global replay (restore-parent, bind, emit, audit with mutation controls, invoice, assembly, 69 admission controls, 12 bit-supplier controls). independent/price_profile.py gives the same κ (PASS_FROZEN_PROFILE_TWO_EXACT_ENGINE_AND_OUTER47_CROSSCHECK).

Scope. A finite conditional witness on a pending construction; every assumption of the #1005/#1018 chain is inherited (all-size compiler, common-label/mutable-source realization, whole-call hosting, precision/tape, exact recovery, analytic transfer, bit H1–H5). No optimality beyond the stated T = 8 whole-helper family is claimed.

Credit. Prepared by G. V. Chudnovsky with Anthropic Claude assistance (Apache-2.0). The word and everything below the owner map are the #1005/#1018 chain's: icekylinx's #645 gauge44 chain rebuilt by PartialK, G44Stack, NoTriReopen, BlockInvert, BRoute, chafreaky's #741 frames, JointXF, NewBArch, LeaverSAT/LeaverSAT2, rohanarun's #987 slots, G44Pack/G44Pack3's plan, Hao Yan's #858 T16 machinery (comparison only), and the PR589 bit supplier; see NOTICE.md.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

https://claude.ai/code/session_01KBSZTxNtzYkGa6MNbKajR

[gvconly]

Update: a second package, research/gauge44-slots-t16-arcflow, runs the #858/#870 T16 controller on the same word with a 16-replica owner map from the same method: idle remainder {1:2430, 2:1, 16:1, 48:4887} over 69,120 roles, conditional κ = 0.001674087289164567 (PASS_GAUGE44_T16_STRICT_GLOBAL_REPLAY). Verification: run research/gauge44-slots-arcflow’s replay first, then python -B research/gauge44-slots-t16-arcflow/verify_t16.py --main . --local /tmp/slots-arcflow/local --work /tmp/slots-t16-arcflow.

[gvconly]

Comment:
Fourth package, research/gauge44-slots-deepen-arcflow (commit 76b9f99): ten low-ended helpers of the un-promoted word get the seven-dimensional end their cones force, frames re-optimized exactly, and the new inventory {4:40, 5:48, 7:40, 8:112, 9:240} packs with no small remainder at all (idle {48:2460}). Strict replay κ = 0.001675338871723469 (+0.0455% over the previous layer, +0.805% over #942). Verifier: construction/deepen/verify_deepen.py, chained on the un-promotion replay; three controls rejected. The same tools, notices and conditional scope as the earlier layers.

[Mister-Meeseeks]

100-PR batch review at 76b9f99e177ecdea82877a092dc21e767035d3a2.

This remains open after scoped triage. Next review step: Whole-head supplier review remains open for research/gauge44-slots-arcflow, research/gauge44-slots-deepen-arcflow, research/gauge44-slots-t16-arcflow, research/gauge44-slots-unpromo-arcflow. Bind its exact native providers, physical owner allocation and admission evidence, then reconcile changed predecessor blobs before merge. Path overlap with a replayed descendant does not establish complete coverage; see SUPPLIER-FAMILIES.json.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.

[gvconly]

Follow-up to the third 100-PR round (whole-head supplier review open for the four packages). Head is now 5bc0dbb.

1. Rebased on main 0b1cccf: the branch differs from main in exactly the four packages. Predecessor blobs are main's, including research/gauge44-slots-t16 at its reviewed head (κ 0.001672188119205421); the README of the head package records this and the selected Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 (0.001672660197076918), against which research/gauge44-slots-deepen-arcflow is +0.1601%.

2. Whole-head driver: python3 -B research/gauge44-slots-deepen-arcflow/verify_chain.py --main . --work /tmp/wholehead1175 (cold, ~40 min; or --baseline1042 DIR on a completed scripts/verify_frontier1042.py replay). It rebuilds the k8 chain with research/gauge44-leaver8's scripts and the vendored PR635 supplier, runs the four packages' own verifiers one on the other (arcflow → unpromo → deepen; t16 on arcflow's locals), the three standard-library recounts, the four price receipts (byte-identical to the committed ones), and wholehead/check_independent.py, which applies Josie Elliston's owner/event-window audit (research/t16-owner-audit-seed67: R1–R7, X1, X2, controls) and Sagar Thakrar's pricing (research/pending-claims-independent-pricing) unchanged to every map.

3. Receipt of the cold run: wholehead/RESULT.json — 20 stages with script/log SHA256, 594 artifact digests, the rebuilt native provider identities of the three words (c2b29430…, 4ffb07de…, 6ace902a…) against each PLAN's provider_registry, owner inventories (34,560 roles/72 calls per T8 map; 69,120/144 for T16; idle {1:1214,2:1,32:1,48:2443}, {1:72,2:17,3:2,32:1,48:2442}, {48:2460}, {1:2430,2:1,16:1,48:4887}) and the exact κ of each package. On the deepened map the auditor's prefix_gap control has no instance (no partially filled bin), which the receipt records.

The three earlier packages' blobs are unchanged from the reviewed head.

[gvconly]

Follow-up (October 15): packing caps and realised fractions, in the census field names. Head is now 979ea2d.

The queue-ceiling screens (#1264; research/complex-queue-ceiling-uncovered) read #1175's first package only, hence their "title κ differs from the package" note: the title κ 0.001675338871723469 is research/gauge44-slots-deepen-arcflow's recorded κ (RESULT.json, data/SPEC.json, construction/deepen/DEEPEN-PINS-k8.json). Per autoresearch.md, research/gauge44-slots-deepen-arcflow/CLAIM.json now records all four packages in the census's field names — local_profile_per_call, idle_histogram, global profile, live/calls, word SHA256, exact κ and endpoint — with each word's cap under research/complex-local-word-ceiling and the fraction realised; wholehead/claim_caps.py --main . recomputes it (standard library, ~20 s).

package	word	κ	model cap	realised
gauge44-slots-arcflow	c2b29430…	0.001674083482880778	0.0016872026	0.9922
gauge44-slots-t16-arcflow	c2b29430…	0.001674087289164567	0.0016872026	0.9922
gauge44-slots-unpromo-arcflow	4ffb07de…	0.001674576537247681	0.0016828484	0.9951
gauge44-slots-deepen-arcflow	6ace902a…	0.001675338871723469	0.0016832052	0.9953

On the deepened word the map attains the stock-feasible optimum (idle {48:2460}: every idle unit in a rank-48 child, the largest idle bin of the retained template); its residual 0.47% to the model cap is the rank-48/rank-51 idealisation, not unrealised packing. The slot word has the higher model cap but realises less of it because the cap ignores widths (its 240 width-9 helpers lack partners) — an instance of the census's own caution that caps alone do not order attainable optima. Within this model nothing further is available on these words.

wholehead/RESULT.json, STAGES.json and INDEPENDENT.json were regenerated by a fresh cold whole-head run on the final package state (20 stages, all four κ reproduced).

[gvconly]

Fifth package (head d3a5d69): research/gauge44-slots-balance-arcflow, κ = 0.001675784247432857 — fourteen restored entrances and ten deepened ends on the slot word c2b29430…, perfectly packed (idle {48:2463}).

Pricing the frame moves against perfect packing shows research/gauge44-slots-deepen-arcflow over-converted: a restored entrance (width 4→5) costs 0.0057 moment units per helper and call and removes one rank-1 remainder worth 0.0066, so once the packing is perfect every further restoration is a loss. A joint integer programme over conversion counts and time-free bin patterns gives fourteen restorations and ten deepenings; the exact frame re-optimisation prices the two entrances left promoted cheapest when they are two of {603, 605, 606, 607} (603 and 605 in the recipe). Details and the comparison table in the README.

+0.0266% over the deepened package, +0.1868% over the selected #1094, 0.0015% below the census's top claims (#1239/#1218/#1193). Cap under research/complex-local-word-ceiling 0.0016836644, realised 0.99532, stock-feasible optimum attained; CLAIM.json covers all five packages.

Verification: construction/balance/verify_balance.py on a completed research/gauge44-slots-arcflow replay (recipe, four rejected mutants, native locals, independent audit, strict replay); verify_chain.py now replays all five packages from one cold k8 reconstruction (wholehead/RESULT.json: 25 stages, 695 artifact digests, both independent auditors on every map). Nothing else in the branch changed; the four earlier packages are as reviewed.

[gvconly]

Execution notes for the pinned whole-head check (head 007a7c64).

Command, from the repository root:

python3 -B research/gauge44-slots-mixed-arcflow/verify_chain.py --main . --work <fresh dir>

It ends with PASS_WHOLE_HEAD_GAUGE44_SLOTS_ARCFLOW_CHAIN and the six κ values (arcflow 0.001674083482880778, unpromo 0.001674576537247681, deepen 0.001675338871723469, t16 0.001674087289164567, balance 0.001675784247432857, mixed 0.001675867534384272).

• Runtime: a fresh run from a clean checkout of 007a7c64 took about 38 minutes of stage time on 2 cores (Xeon 2.1 GHz, 8 GB RAM): 29 stages, 2,291 s. The longest are the cold chain (13 min) and the six native package replays (4–7 min each).
• Dependencies: Python 3 only. Keep assertions enabled (no -O).

Comparing with the committed receipts (research/gauge44-slots-mixed-arcflow/wholehead/{RESULT,STAGES,INDEPENDENT}.json):

• INDEPENDENT.json reproduces byte for byte.
• RESULT.json and STAGES.json also contain hashes of the stage logs and of each provider's CLIFFORD-REVIEW.json. Those files record timings, so these hashes change from run to run.
• Compare with timings, log hashes, the artifact-hash map and the CLIFFORD-REVIEW.json hashes removed:

python3 -c "import json,sys;D={'seconds','log_sha256','artifacts','CLIFFORD-REVIEW.json'};s=lambda x:{k:s(v) for k,v in x.items() if k not in D} if isinstance(x,dict) else [s(v) for v in x] if isinstance(x,list) else x;[print(n,'SAME' if s(json.load(open(sys.argv[1]+'/'+n)))==s(json.load(open(sys.argv[2]+'/'+n))) else 'DIFFERENT') for n in ('RESULT.json','STAGES.json','INDEPENDENT.json')]" <fresh dir> research/gauge44-slots-mixed-arcflow/wholehead

On my fresh run this prints SAME for all three files.

Full artifact check. RESULT.json lists 824 artifacts. Comparing the fresh run with the run that produced the committed receipts, 616 are byte-identical. The other 208 are equal once timings, machine paths, and hashes of files that differ only in timings are factored out. No artifact differs in substance.

Prepared by G. V. Chudnovsky with Anthropic Claude assistance.

[gvconly]

Sixth package (head 007a7c64): two providers mixed call by call — conditional κ = 0.001675867534384272.

research/gauge44-slots-mixed-arcflow. Even replicas run the deepen word (16 restored entrances, 10 deepened ends; provider gauge44r16, word 6ace902a…). Odd replicas run a sibling with 12 restored entrances (provider gauge44r12, word 69c93a05…). Each provider serves 36 calls, so the average is 14 restorations per call, which the balance package showed to be the right supply, with no broken entrance group in either word.

• The whole-owner map packs both inventories with idle {48: 2463} and no small remainder.
• The scalar program, 72-call schedule, idle stock, windows and PR589's bit supplier are unchanged.
• SPEC dispatches per call. 114 admission controls and 23 controller mutations are rejected.
• code/ is research/gauge44-layered's two-provider derivative, unchanged.

Gain. +0.0050 % over the balance package and +0.1918 % over the selected #1094. On the selected supplier, it is +0.0035 % above the highest claims in the October 13 queue census (#1239/#1218/#1193, 0.001675809683500501). The T74 family (#2530 onward) claims more; those claims are conditional on the unresolved T74/#1319 supplier admission and the PR1673 outer.

Whole-head check. The command in the execution notes above replays all six packages from the cold chain. CLAIM.json (schema pr1175-claims/3) lists all six packages in the census field names, with packing caps and realised fractions.

Balance package change. The balance package's committed wholehead/RESULT.json now records the PR635 supplier path relative to the repository instead of as an absolute path. verify_chain.py writes it that way from now on, a one-line change.

Prepared by G. V. Chudnovsky with Anthropic Claude assistance.