Conditional κ ≈ 0.001717794984118634663827: two hostee moves and seven retired-lender aliases on #2751's joint twelve-source words

[mdumitrean]

This extends the words of #2751 (@tekuuu, head 4cfbb0cc) with seven retired-lender aliases. Five need only exact refits. The other
two are opened by moving two existing borrows onto free retired lenders.

The new conditional κ is 0.001717794984118634663826999…. The exact rational is 0.001717794984118634663827 - 10^-46 - 10^-70;
the selected frontier is unchanged.

over	margin
#2751 (tekuuu)	+6.53397770208325718e-7 (exact 326698885104162859/(5·10^23))
#2766 (DavidAriasC)	+5.38010651632355557e-7 (exact 538010651632355557/10^24)
#2767 (jeremykothe2)	+5.38010651632355556055e-7

What is new, and what is credited.

• The certified frames contain, unchanged, Improve conditional κ to 0.001717256973467 with credited routing-frame fusion #2766's frame refit of Conditional kappa ≈ 0.00171714158635 via joint twelve-source recovery #2751's words: B1151 rank 6 -> 5. Conditional κ = 0.001717256973467002: post-cut partner correction on #2696's lender assignment with #2679's pre-life hosting #2767 reaches the same κ by
  another route.
  • Corrected after Independent verification: #2802's seven-stage increment over #2751 (retired5 + retired7 stages) passes 28/28 rules and 20/20 controls; covers #2818's identical construction (no κ) #2847: the other 19 frame changes (A0-A11, B771/773/776/778/783/788, restore60) were credited to Improve conditional κ to 0.001717256973467 with credited routing-frame fusion #2766 here by
    mistake. Improve conditional κ to 0.001717256973467 with credited routing-frame fusion #2766's own words change only B1151. The 19 are this package's refit5c choices, and Independent verification: #2802's seven-stage increment over #2751 (retired5 + retired7 stages) passes 28/28 rules and 20/20 controls; covers #2818's identical construction (no κ) #2847 shows they are
    cost-neutral. The package README and CLAIM.json at 17b2ee6e still carry the old wording.
• New here, worth +5.38e-7 over Improve conditional κ to 0.001717256973467 with credited routing-frame fusion #2766/Conditional κ = 0.001717256973467002: post-cut partner correction on #2696's lender assignment with #2679's pre-life hosting #2767:
  • Five retired-lender aliases (606<-550, 635<-551, 637<-552, 638<-553, 639<-562), made feasible by exact one-rank refits:
    +1.52e-7.
  • Two hostee moves that free two more lenders, plus the two aliases they enable (667<-401, 669<-403): +3.86e-7.

Construction.

• Every alias is of the kind of Conditional κ = 0.0017148565406944334: pre-life hosting and a frame-refit alias on the #2588 T74 words #2679's 653<-563 (philippeVV). A helper enters a retired lender's register through one B warp
  with an explicit invertible G, G(U)=E; the helper and its private register are gone.
• Five aliases.
  • Each lender's last gate drops from rank 6 to 5, its exact floor; each follower's first gate rises from rank 4 to 5, its exact
    ceiling. Both use Conditional κ = 0.0017148565406944334: pre-life hosting and a frame-refit alias on the #2588 T74 words #2679's frame-lattice tools.
  • The census on Conditional kappa ≈ 0.00171714158635 via joint twelve-source recovery #2751's words found no fixed-frame pairs and 35 refit-feasible ones.
• Hostee moves. With every existing B-warp hosting treated as movable, the remaining rank-4 followers 667/669/670/671 can enter
  24 hosted lenders whose own lives end in A, but only if each hostee moves to a free retired lender. Only 564 and 565 qualify
  (floor 5, retired at B911/B912 before the hostees enter). That gives 192 augmenting paths, at most two extra aliases.
  • Chosen: warps 30/31 (Research: 80-helper anticipatory borrowing component and exact price rejection (no new kappa) #2226 borrows; their hostees enter at rank 5 already), moved 401 -> 564 and 403 -> 565.
  • 564/565's last gates are refit from rank 6 to 5. Both warp records are replaced in place, so op ids are kept.
  • 401/403 then take 667 and 669 at fixed frames (dim 4).
• The final program takes its certified optimal frames (Tooling: exact frame solver and delivery-rewrite search for gauge44/k8 local words (no new κ) #1186 toolkit). The 48 followers with a rank-1 entrance stay infeasible:
  every lender floor is at least 4.

Result.

• 408 helpers {4:26, 5:2, 7:62, 8:78, 9:240}, 72 buffers, 666 calls, 271728 roles; physical 66794.
• Local profile {1:2382,2:648,3:268,4:128,5:26,8:8}.
• The owner fill is Conditional kappa ≈ 0.00171714158635 via joint twelve-source recovery #2751's fill with the seven helpers' 4662 roles removed (renumbered, every bin re-tiled as a prefix). Its
  pattern relaxation has the same optimum.
• Complete profile {1:1603728,2:431568,3:185082,4:85248,5:27194,8:5328,32:1,48:26558}; rank mass 4816512.

Independent checks.

• Independent verification: #2802's seven-stage increment over #2751 (retired5 + retired7 stages) passes 28/28 rules and 20/20 controls; covers #2818's identical construction (no κ) #2847 (@thakrarsagar): fresh standard-library code checks the seven stages over Conditional kappa ≈ 0.00171714158635 via joint twelve-source recovery #2751 on both words. 28/28 rules pass and all 20
  negative controls are rejected, with no defect found. It reproduces the retired5/retired7 split and corrected the Improve conditional κ to 0.001717256973467 with credited routing-frame fusion #2766 frame
  credit above.
• Followers 670/671 (@philippeVV, comment,
  checker philippeVV/integer-mult-bounds@0610f3de): no refit of the fixed gate list gives them a lender. With all 72 hostees
  movable, their 74 pairwise-overlapping lives would need the 72 registers whose floor is at most 5, and all of those already host.
  That leaves 0 augmenting paths, so a further alias needs a gate-level edit.

Packaging.

• Only research/t74-joint-twelve-retired7/ is added. Conditional kappa ≈ 0.00171714158635 via joint twelve-source recovery #2751's directory is byte-identical.
• The branch is based on Conditional kappa ≈ 0.00171714158635 via joint twelve-source recovery #2751's head, so this PR contains Conditional kappa ≈ 0.00171714158635 via joint twelve-source recovery #2751's commit until Conditional kappa ≈ 0.00171714158635 via joint twelve-source recovery #2751 merges.
• The seven stages refit5 -> alias5 -> refit5c -> refit7 -> hostmove2 -> alias7 -> refit7c are new rebuild_borrow stages in the
  overlay, with exact structure checks. Each new state passes the auditor.
• Tamper controls: 71 per word. Every mutant re-derives its cleanup, and a stale-cleanup rejection counts as a failure:
  • lender refits undone (warp frame mismatch);
  • follower raises undone;
  • per move: lender reverted, move undone, warp before retirement, G wrong;
  • Conditional κ = 0.0017148565406944334: pre-life hosting and a frame-refit alias on the #2588 T74 words #2679's alias suite per alias;
  • corrupted certified frames.

Checker adaptations (diffs in RETIRED5-CODE.patch and RETIRED7-CODE.patch, reasons in the README).

• independent_owners.py: the helper count is read from the providers (408) instead of the constant 415. Every check is unchanged.
• workflow/admit.py: the derived-source pin (506 files), physical 66794, and strict improvement over Conditional kappa ≈ 0.00171714158635 via joint twelve-source recovery #2751.
• verify.py: it reproduces Conditional kappa ≈ 0.00171714158635 via joint twelve-source recovery #2751's own fill and words, then checks the seven declared stages, the declared compaction, the new pins
  and improvement over Conditional kappa ≈ 0.00171714158635 via joint twelve-source recovery #2751/Improve conditional κ to 0.001717256973467 with credited routing-frame fusion #2766/Conditional κ = 0.001717256973467002: post-cut partner correction on #2696's lender assignment with #2679's pre-life hosting #2767.
• verify_retired7.py and retired7_controls.py: new.

Validation.

• Full cold release from the committed package through full_replay.py: PASS (PASS_T74_JOINT_TWELVE_COLD_COMPLETE_RELEASE,
  411 s, run cold from the committed package). It covered:
  • all 491 pinned PR2679 files authenticated;
  • the 36-file overlay;
  • four fresh native providers (five stages each);
  • the retained-parent admission;
  • owners, lift, rebind, and discovery/unpin/freeze at Conditional kappa ≈ 0.00171714158635 via joint twelve-source recovery #2751's 24-digit precision;
  • the outer ledger and independent derivation;
  • the original verify_release.py: PASS_RELEASE, including rebuild-borrow-words with all stage checks and controls, the strict
    T74 replay with its admission, bit and independent controls, and the geometry controls.
• Registered hook, about 100 s, budget 1020 s: python3.12 -B scripts/research_package.py --package research/t74-joint-twelve-retired7.
  • verify.py literal review;
  • verify_retired7.py: seven-stage rebuild, the vendored byte-identical PR2679 auditor on eight states, 142 tamper controls, the
    literal owner fill, the 80-digit root and κ margins;
  • packing_cap.py: cap ≈ 0.00175808593463, realized 0.9771;
  • test_integrity.py.
• python3.12 -B -m unittest discover -s tests -p 'test_research_*.py': 69 tests OK (3 skipped).
• Evidence is in evidence/retired5/ and evidence/retired7/, sanitised of local paths.

python3.12 -B scripts/research_package.py --package research/t74-joint-twelve-retired7
python3 -B research/t74-joint-twelve-retired7/full_replay.py --work /tmp/FRESH-retired7   # fresh external dir; requirements.txt

Scope. This is conditional finite construction. All all-size, interface, hosting, precision, recurrence, bit-supplier and outer
obligations of #2751 and its lineage are inherited unchanged. No optimality is claimed beyond the exact census of retired-lender
aliases with movable hostings.

Credits.

• tekuuu (Conditional kappa ≈ 0.00171714158635 via joint twelve-source recovery #2751, the parent);
• 1337isnot1337 (Research: 80-helper anticipatory borrowing component and exact price rejection (no new kappa) #2226, anticipatory dirty borrowing; warps 30/31 are its borrows 603<-401 and 605<-403, as adapted to the T74 words);
• philippeVV (Conditional κ = 0.0017148565406944334: pre-life hosting and a frame-refit alias on the #2588 T74 words #2679/Conditional κ = 0.0017044322370702051: owner-fill evacuation and two more retired-lender aliases on the #2526 T74 words #2559: pre-life hostings, the one-gate refit and the 653<-563 retired-lender alias re-used for all seven, the
  refit/search tools and the alias builder);
• jeremykothe2 (Conditional κ = 0.001714600409398252: priority-ordered window-optimal owner fill on #2588's words #2673/Conditional κ = 0.001716140425997441: #2696's lender assignment with #2679's pre-life hosting and a window-optimal fill #2745: call order and window-optimal fill, the Improve conditional kappa to 0.001715599770234969498857 with concentrated source-partner assignments #2696+Conditional κ = 0.0017148565406944334: pre-life hosting and a frame-refit alias on the #2588 T74 words #2679 composition, entrance raises, the gmx-t74-borrow43
  package machinery and BORROW-WORD lineage, including the base-word lenders 564/565);
• mason-hale (Improve conditional kappa to 0.001715599770234969498857 with concentrated source-partner assignments #2696, the eight-lender assignment);
• DavidAriasC (Improve conditional κ to 0.001717256973467 with credited routing-frame fusion #2766, the included B1151 frame refit);
• Vlado-V (Improve conditional κ to 0.001697060296258836 with coherent handoffs and retimed hosting #2322);
• our Conditional κ = 0.001676896669021299: pivot-566 helper-donor kernel on PR1319's T74 exact fill (+4.14e-8) #1625 kernel shape and Tooling: exact frame solver and delivery-rewrite search for gauge44/k8 local words (no new κ) #1186 exact frame toolkit;

and the lineage these retain (SOURCE-NOTICE).

Prepared by Marius-Paul Dumitrean with substantial AI assistance (Anthropic Claude, via Claude Code). Apache-2.0.

[philippeVV]

On your open question about followers 670/671 (no κ): under every refit of the fixed gate list, they have no lender. I ran an augmenting-path census that treats all 72 current hostees plus 670/671 as movable, using exact lattice frame bounds. The ceiling of every life's first gate is ≤ 5. Exactly 72 registers have floor ≤ 5, and every one of them already hosts. All free registers have floor ≥ 6. The 74 lives overlap pairwise in time, so no register can take two in sequence. That makes it a pigeonhole: 74 lives on 72 registers, with 0 augmenting paths. Opening a slot would need a gate-level edit. Your fill is also at the class-level ceiling to about 1e-14, and pre-life/refit censuses find no new legal hostings or aliases on these words. Checker: https://github.com/philippeVV/integer-mult-bounds/tree/0610f3de/research/t74-lender-pigeonhole (Claude Code assisted).

[mdumitrean]

Thank you both.

@philippeVV, followers 670/671. Thank you, that settles the question. It agrees with our census: the only free floor-5 sinks were 564/565, and the two moves use them. Your pigeonhole extends that to every refit with all 72 hostees movable. So a further width-5 alias needs a gate-level edit, which is open question 1. The PR body now records the answer and credits your checker (0610f3de).

@thakrarsagar, #2847. Thank you for the independent check. The provenance correction is right. On both words, #2766's own words differ from #2751's at one frame only, B1151 (rank 6 → 5). The other 19 frame changes we credited to #2766 (A0–A11, B771/773/776/778/783/788, restore60) are our refit5c choices.

• The PR body is corrected.
• I have not pushed to the branch, because your check pins 17b2ee6e and several downstream claims vendor its words. The package README and CLAIM.json keep the old wording until the next package revision.

On your open question 2: the final frames are the #1186 toolkit's certified optimum for the borrow-word model, with gates, warps, start/final frames and widths fixed (inputs/refit7c_43*.json; only B1170/B1171 move, dim 6 → 5). That certificate does not cover frame changes that alter start or final frames, and with them the widths and the fill.

Your questions 1 and 3 are open on our side too. Our cold release ran fresh native providers on the shipped words only, with no occupancy mutants.