Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434)

[DaysSky]

Conditional κ = 618775866623647/(5·10¹⁷) ≈ 1.237551733247294·10⁻³, +9.218% over #434 (1.133100552284434·10⁻³). The bit supplier binds. It is also +8.67% above the newest posted value, #441 (1.138798055343319·10⁻³). Hosting changes only the global template, so it should stack on #440/#441's R740 word as well.

Change and mathematical scope

In the compressed nine-stage window, idle data pairs are brought to the active level by idle advance children (ranks 48/32/16/8 for complex, 42/28/14/7 for bit). These children are about a sixth of the paid cost.

A dirty helper residual acts on whatever stream owns it as an exact coordinate factor, independent of the stream's contents. That is the inherited correlated dirty-bank fact that banks already rely on. It is the same kind of factor an idle child applies to its pending coordinates. So a helper residual can be owned by an idle data family instead of a bank slot, on coordinates that family would advance later, and the child keeps only the rest.

The validity condition is a window: the family must be untouched (no local invocation on its pair, no bridge naming it) between the hosting invocation and its advance. Neither local word changes.

Result

supplier	parent saving	hosted saving	live stock	max child	hosted residuals
bit, m = 64, #434 R747	1.134386·10⁻³	1.239085·10⁻³	14,259 → 9,457	42 → 8	39,568 of 53,784
complex, m = 73, #433 R560	1.183987·10⁻³	1.267453·10⁻³	12,720 → 8,400	48 → 8	34,560 of 40,320

The rank deficit stays at 2496 on both sides.

Validation

python -B research/idle-axis-hosting/verify.py /absolute/new/output uses the standard library only. It:

1. runs Conditional kappa 0.001249035299298004: hosting-selected rank147 kernel #434's complete verifier from scratch;
2. regenerates both hosted templates from Conditional kappa 0.001249035299298004: hosting-selected rank147 kernel #434's fresh outputs and a frozen integer-programme selection (--discover recomputes the selection with scipy);
3. runs an independent audit that shares no code with the generators:
   • windows re-derived from each schedule;
   • for every invocation and replica: exactly one owner per helper, and slots in window, inside their child's pending set, disjoint, and off the active pair;
   • exact remainder children, exact bank tiling, a recomputed profile and deficit;
   • nine mutation controls rejected;
4. computes two rational moments per supplier and fresh finite invoices with the hosted stock:
   • two extra route factors are charged per helper selector;
   • the parent's larger complex per-vertex bill is kept;
5. runs Conditional kappa 0.001249035299298004: hosting-selected rank147 kernel #434's strict outer assembly: six bootstrap levels and all 47 inequalities, with the next grid point and the zero-η, zero-β and unit-β controls rejected.

Open obligations

PROOF.md gives the hosting lemma. It extends the inherited correlated dirty-bank / compressed-window interface to data streams during idle windows; that extension is the principal review point. The global operator identity is argued, not replayed numerically on the address spaces, as for the parents' banks.

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

This builds on and runs unchanged:

• Conditional kappa 0.001249035299298004: hosting-selected rank147 kernel #434 (Rohan Arun, OpenAI Codex);
• Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433 (icekylinx);
• κ = 0.001128832830332521: #417's shared-donor kernel on #433's R758 bit word (+0.71%) #435/E8 bit supplier c ≈ 1.044064·10⁻³ in the nine-stage word (+2.2 % bit cap; no new κ, complex-bound) #417 (chafreaky);
• Conditional kappa 0.001018766833114736: checker-clean chronological gauges and R767 reuse #426 (ai-nanachi);
• κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425/Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386 (Zhihao Chen, OpenAI Codex);
• κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370 (Dylan Crow);
• Jacob Sussman's E8 unit and interfaces.

The hosting construction, generators, audit and assembly were prepared with assistance from Claude. No inherited file is changed.

Local transcript: PASS_FULL_IDLE_AXIS_HOSTING_COMPOSITION on macOS (Apple silicon) with Python 3.14.8 and Python 3.12.13; each run replays #434 in full (about 2 minutes) before the hosting steps. CI covers Python 3.11 and 3.14.

[Mister-Meeseeks]

Integrated into published main at 681e8923a0f23a19f93c7105c3a9c96149aa8362, preserving the original commits, notices and contributor credit. Thank you for the contribution.

Idle-axis hosting and its cold dependency reconstruction passed the scoped finite review.

The scoped review and pinned execution receipts record the exact coverage and remaining assumptions. This strengthens specific interfaces and preserves useful research; it does not change the selected κ or establish the complete multiplication theorem. Local checks passed; fresh Linux CI was queued at publication.