Prove2Me

Rank sum of Jain.bitR8 equals 55 395 128 48155\,395\,128\,481

Proved
IntMul.Kappa.R8.bitR8_rankSum

by Ethan Cheung Wang · Oct 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationround-eight

Rank sum of the round-eight bit histogram.

Let hbit=Jain.bitR8h^{\mathrm{bit}}=\mathrm{Jain.bitR8} be the child-width histogram of Jain's round-eight bit interchange (opposite bank orders on round-seven witness 2). Its weighted rank sum equals

histRankSum(hbit)=∑ww nw=55 395 128 481,\mathrm{histRankSum}(h^{\mathrm{bit}})=\sum_w w\,n_w=55\,395\,128\,481,

the same value as for Jain.bitR7plain. This matches bit_rank_sum in lean/Round8.lean / the bit-phase rank certificate of round8_certificates, and is consistent with the opp-bank recompile preserving total child width while reshaping the histogram bins.

Formalization Note. histRankSum is from IntMul_KappaFramework; bitR8 is from IntMul_KappaJainData.

Preamble
import Definitions.Def_IntMul_KappaFramework
import Definitions.Def_IntMul_KappaJainData

set_option autoImplicit false
set_option relaxedAutoImplicit false
Formal statement
namespace IntMul.Kappa.R8

open IntMul.Kappa IntMul.Kappa.Jain

/-- Rank sum of the round-eight bit histogram (opposite bank orders); equals the plain rank sum. -/
theorem bitR8_rankSum : histRankSum bitR8 = 55395128481 := by
  sorry

end IntMul.Kappa.R8
Source
S. Jain, integer-mult-kappa, lean/Round8.lean theorem bit_rank_sum; Prove2me IntMul.Kappa.round8_certificates (bit-phase rank sum); lean/round7-histograms.json bitplain.s and round8 bit hist share s=55395128481

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