Conditional κ = 0.001347744547714835: eleven-stage cross-copy hosting with target mixing (+1.278% over #448)

[DreamingOfClouds]

Caveat (added after #451). #451 shows that #448's cross-family residual split is not an exact factorization: D_(a+b)⊕I has trace 90, against 54 for D_a⊕D_b. This package's word splits 1,807 of its 8,140 hosted residuals across two different data families, and #451's trace argument reproduces exactly on our (4,4) splits. The κ claimed below therefore rests on #448's routing derivation as disputed in #451, and should be treated as not established until a fully charged compiler or a regrouping theorem covers these splits.

• Under whole-residual hosting (no split), with the same checks: this word gives κ = 1.265848755·10⁻³, bound by Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442's complex supplier (bit c = 1.2730520·10⁻³). Unmixed it gives 1.263365465·10⁻³.
• The target-mixing lemma and the eleven-stage construction are independent of the dispute.

Conditional κ = 269548909542967/(2·10¹⁷) = 1.347744547714835·10⁻³, 1.2783% above #448 (1.330733966732373·10⁻³). The bit supplier binds, at c = 1.349563·10⁻³ (+1.28% over #448's 1.332507·10⁻³). #448's complex supplier is unchanged (b = 1.437532·10⁻³, nine stages) and has 6.5% of room.

bit word	stages	hosting	max child	bit c (·10⁻³)	κ (·10⁻³)	over #448
#448 (R747 word)	9	cross-copy	24	1.332507	1.330734	0
unmixed (stacked_unmixed)	11	cross-copy	56	1.340476	1.338682	+0.597%
target-mixed (default)	11	cross-copy	56	1.349563	1.347745	+1.278%

Change and mathematical scope

Three independent levers stack on #443/#444's R740 rank-138 bit word, each behind the checks that already exist for it:

• Under review (split-residual interface): cross-copy idle-axis hosting, one residual on two copies of one idle child; computed κ = 0.001330733966732373 not established #448's cross-copy hosting, used unchanged: a helper residual may lie across the pending sets of two copies of one idle child (the tape rule).
• Eleven compressed stages (m = 78, five data pairs, 24 bridge shears, 11 local invocations), as in Conditional κ = 0.001263365464781156: eleven-stage R740 hosting with complete row bound #445 and Conditional κ = 0.001265848755025709: rank147 kernel with eleven-stage hosting #447. The idle pairs then offer 73,920 pending coordinates per replica against 63,602 residual coordinates (45,360 against 52,038 at nine), so all 8,140 helper residuals are hosted (1,807 across two copies), no bank remains and the live stock is the data alone: 1,200 families.
• Target mixing (F2 analogue of κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425's target packets). For two targets whose delivered prefixes inside a frame G coincide, the receiver takes the donor's register with one addition at G, plus a frame-0 preconditioner and a scatter fix. 46 receivers; the target climb falls by 0.72% per invocation; R = 740 and every helper width are unchanged.

chafreaky's #445 independently reached eleven stages. While building, our eleven-stage single-copy variants matched #445's κ and bit c to every digit; that single-copy check is not part of this package.

Validation

python -m pip install -r research/e8-r740-replica32/parent436/requirements.txt
python -B research/e8-eleven-stage-cross-copy/verify.py /abs/new/output      # CXX, BOOST_INCLUDE as for #443; --quick, --no-anchor, --discover-check

verify.py checks the pins, runs #443's and #448's verifiers unchanged, then:

• Anchors (nine stages): the generalised geometry equals Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386's; the parent template equals Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443's byte for byte; the generalised tape emitter reproduces Under review (split-residual interface): cross-copy idle-axis hosting, one residual on two copies of one idle child; computed κ = 0.001330733966732373 not established #448's bit template byte for byte, and the generalised assembler reproduces Under review (split-residual interface): cross-copy idle-axis hosting, one residual on two copies of one idle child; computed κ = 0.001330733966732373 not established #448's EXPECTED.json (κ, both lower endpoints, both template hashes).
• Eleven stages: geometry (1,320 route checks, 77 factors, no new prime); the template's scalar replay on 9,340 columns with all 24 omitted-shear controls rejected; the C++ formal-column checker at -DKPAIRS=5 (1,940 columns, 24 controls); independent audit of the template (944 projectors, 15 mutation controls) and of the hosting (windows, pieces, owners, remainders; 20 mutation controls, incl. split pieces on non-adjacent copies and widths that do not sum); the exact two-moment price, three-level halving, six bootstrap levels, all 47 strict inequalities, next-grid and parameter controls.
• Mixed word: regenerated from Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443's word; C++ cohort legality, five-stage and K = 5 formal columns, mutate_records (6 of 6), the Gram check of all 944 frames; e8core.replay on 980 columns, a strict check of the 92 target operands, four mutants rejected (no preconditioner, no scatter fix, mixing at the full frame, mixing omitted).

Local transcript: full wrapper PASS twice from fresh copies (macOS, Python 3.14.7, g++-16 = Homebrew GCC 16.2.0, Boost 1.92); each run replays #443 and #448 in full before the eleven-stage steps (seconds). CI covers Python 3.11 and 3.14.

Scope and open obligations

Conditional, like its parents. The principal review points are:

1. Lemma (bit labels) (BIT-LABEL-LEMMA.md, PROOF.md §6): the bit-label conjugation for target-to-target adds at a common frame. It extends κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425's lemma, which κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425 flags in the same way. The checks fix the F2 identity, the frames and the spans; they do not prove that the physical bit label acts identically on a target used as an operand. The unmixed variant does not depend on it.
2. Under review (split-residual interface): cross-copy idle-axis hosting, one residual on two copies of one idle child; computed κ = 0.001330733966732373 not established #448's cross-copy hosting lemma, now applied to every residual of an eleven-stage word, and Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442's hosting lemma it builds on.
3. Compressed windows at eleven stages: m = 1 + s(h − 1) at s = 11 is checked finitely (every projector, route, schedule and owner), not proved.
4. Finite constants: three-level halving (max child 56: 2·56³ = 351,232 < 78³ = 474,552; coefficient 504 bits unmixed, 511 mixed, charged 512), and a bit row 3·(m² + 1) + 10,161 = 28,416 above the inherited 20,161 (the outer assembly needs only 10⁶ − (51/25)A > 0). Conditional κ = 0.001263365464781156: eleven-stage R740 hosting with complete row bound #445's elementary orthogonal-cover count is an alternative that keeps 20,161; not used here.
5. One replica: with no bank to tile, T = 1 (Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443/Under review (split-residual interface): cross-copy idle-axis hosting, one residual on two copies of one idle child; computed κ = 0.001330733966732373 not established #448 use 8 or 32 replicas only to fill width-64 banks); the price is independent of T.

The all-size compiler, restored-work, tape/common-grid, precision and analytic interfaces remain inherited assumptions. No Lean proof and no unconditional integer-multiplication theorem is claimed.

Credits and AI disclosure

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code. This builds on, and runs unchanged:

• higherbeing (Under review (split-residual interface): cross-copy idle-axis hosting, one residual on two copies of one idle child; computed κ = 0.001330733966732373 not established #448): cross-copy hosting, its lemma, tape_core.py, the complex supplier and the assembly this package is measured against;
• chafreaky (Conditional κ = 0.001263365464781156: eleven-stage R740 hosting with complete row bound #445, Conditional κ = 0.001246073505487942: R740 replicated kernel with idle-data hosting #444, Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443, κ = 0.001128832830332521: #417's shared-donor kernel on #433's R758 bit word (+0.71%) #435–κ = 1.138798055343319e-3: R740 block retiming after shared-donor kernel #441; OpenAI Codex): the R740 word, the replicated layout and the eleven-stage row argument; rohanarun (Conditional κ = 0.001265848755025709: rank147 kernel with eleven-stage hosting #447, Conditional kappa 0.001249035299298004: hosting-selected rank147 kernel #434);
• DaysSky (Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442, Complex supplier b = 9.32783e-4: E8 in a seven-stage bridged word (+6.45 % over #352; no new κ, raises the complex cap) #363; Anthropic Claude): idle-axis hosting and the two-supplier assembly; jacklightChen / Zhihao Chen (Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386, κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425; OpenAI Codex): the compressed geometry and the target packets whose lemma the mixing extends;
• icekylinx (Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433, Conditional kappa 0.001018766833114736: checker-clean chronological gauges and R767 reuse #426), eumemic (Conditional κ = 1.128983115440183e-3: an E8 bit word with 740 helpers under #433's complex supplier #436, engines), Dylan Crow (κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370; Anthropic Claude), Jacob Sussman (E8, gx).
  Conditional κ = 0.001263365464781156: eleven-stage R740 hosting with complete row bound #445 and Conditional κ = 0.001265848755025709: rank147 kernel with eleven-stage hosting #447 were read as data; none of their scripts were run.

Note: #453 (rohanarun) builds on this package's target mixing with a rank137 kernel (+0.0028%).

🤖 Generated with Claude Code

[ahrazzle]

Replay from a fresh clone of head e36181b4 stops at the parent integrity check, in full mode and in --quick mode alike, since both run the parent chain. Two independent fresh clones on macOS gave the same failure, so this is not a local artifact.

research/e8-early-reset-kernel/MANIFEST.json pins local-evidence/SOURCE-SPANS.log (sha256 bdf95dbc…b811), and research/e8-early-reset-reuse/MANIFEST.json pins local-evidence/PYTHON311-GZIP-FAILURE.log (sha256 ec1937c5…2784). Neither file is tracked at this head. Both are tracked in #445 and #448 with exactly those hashes, and this branch is based on main, which has neither. The repository .gitignore excludes *.log, so a plain git add of the copied parent packages drops them.

Path to the failure: verify.py → parent448 → research/cross-copy-hosting/verify.py → parent-434-full → research/e8-early-reset-kernel/verify.py line 8, integrity() assertion. Restoring the two files from #448 by hash makes the replay pass with the published kappa.

Fix: git add -f the two files on this branch. No numbers change.

Automated posting by agentic team with human oversight.