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Consistent n-scale Cipolla Repacked Probe

This is a supporting exact probe, not the official top-level benchmark suite. The canonical repository benchmark now lives in power_of_ten_anchor_suite.

This probe tests one narrow question: whether a fully derived k*(n) can beat li_inverse_seed in the exact large regimes without introducing chosen coefficients.

The construction keeps the existing backbone form, but derives both the bend and the lift from the same n-scale Cipolla expansion.

Let

$$ L = \ln n $$

$$ \ell = \ln\ln n $$

$$ P(n) = n\left(L + \ell - 1 + \frac{\ell - 2}{L}\right) $$

The bend term is the exact 1/log^2 n Cipolla correction routed through the existing bend slot:

$$ d(n) = -n\frac{\ell^2 - 6\ell + 11}{2L^2} $$

The order-5 candidate puts the next three Cipolla terms into the lift slot:

$$ P_3(\ell) = \frac{2\ell^3 - 21\ell^2 + 84\ell - 131}{6} $$

$$ P_4(\ell) = \frac{6\ell^4 - 92\ell^3 + 588\ell^2 - 1908\ell + 2666}{24} $$

$$ P_5(\ell) = \frac{24\ell^5 - 490\ell^4 + 4380\ell^3 - 22020\ell^2 + 62860\ell - 81534}{120} $$

$$ k_5^*(n) = \frac{n(P_3(\ell)/L^3 - P_4(\ell)/L^4 + P_5(\ell)/L^5)}{P(n)^{2/3}} $$

$$ \widehat{p}_n = round(P(n) + d(n) + k_5^*(n)P(n)^{2/3}) $$

This is algebraically the Cipolla asymptotic expansion through 1/log^5 n, repacked into the existing backbone + bend + lift structure.

Strongest Finding

The order-5 repacked seed is the first fully derived candidate in this line that beats li_inverse_seed on the exact large-regime stage families already committed in the repository.

  • stage_a max seed ppm: cipolla_log5_repacked = 0.070933, li_inverse_seed = 0.089328
  • stage_b max seed ppm: cipolla_log5_repacked = 0.005033, li_inverse_seed = 0.010553
  • published exact grid max seed ppm: cipolla_log5_repacked = 4163.125782, li_inverse_seed = 4382.740215
  • reproducible exact baseline max seed ppm: cipolla_log5_repacked = 5050.456465, li_inverse_seed = 5271.714558

Pointwise on the published power-of-ten grid, the order-5 repacked seed wins from 10^4 through 10^16 and then loses from 10^17 upward.

Large-Regime Family Split

stage_a

family order-5 max ppm li max ppm order-5 mean ppm li mean ppm order-5 median ppm li median ppm
boundary_window 0.033778 0.048934 0.021563 0.033257 0.021562 0.033256
dense_local_window 0.070933 0.089328 0.029186 0.041800 0.023242 0.035048
off_lattice_decimal 0.038038 0.050585 0.009875 0.017588 0.007580 0.015647

stage_b

family order-5 max ppm li max ppm order-5 mean ppm li mean ppm order-5 median ppm li median ppm
boundary_window 0.000811 0.005202 0.000669 0.003575 0.000669 0.003575
dense_local_window 0.005033 0.010553 0.001724 0.004808 0.001053 0.003943
off_lattice_decimal 0.001466 0.003699 0.000991 0.001616 0.001089 0.001452

Artifacts