Weil Arithmetic Coupling Second Jet

Abstract

The concrete prime-pole-Gamma divided-difference column has a second exterior jet whose remainder gains an additional N/|m| factor.

The source imports the already bounded arithmetic boundary symbol from WeilArithmeticCouplingJet. It keeps the zeroth and first powers of the interior Fourier index in the exact expansion of 1/(m-n). No replacement of the prime or Gamma terms by an asymptotic model is made.

Theorem 1.1 (Second exterior jet for the actual arithmetic coupling).

$$\mathrm{interiorBand}(N)\wedge \mathrm{outside}(m,N)\Rightarrow \mathrm{norm}(\mathrm{column}(c,v,m)-\mathrm{secondJet}(c,v,m))\le \frac{2\mathrm{BArith}(c)N^2}{\pi \mathrm{abs}(m{)}^2\mathrm{abs}(m)-N}l1(v)$$

Proof. Machine-checked in Lean as D5/S3/Weil/ZetaBridge/WeilArithmeticCouplingSecondJet.arithmetic_coupling_second_jet_error (✓ std3). ∎

Source. Repository-derived.

Commentary.

The exact scalar identity is 1/(m-n)=1/m+n/m^2+n^2/(m^2(m-n)). The previously proved uniform arithmetic boundary budget bounds the last numerator. For |n|<=N<|m| this gives a pointwise remainder proportional to N^2/(|m|^2(|m|-N)). After square summation over |m|>M, the remainder therefore improves by order (N/M)^2 relative to the first jet. The infinite low-rank Gram assembly is deliberately left to the arithmetic certificate rather than asserted as part of this pointwise theorem.

References

• Truth anchor: D5/S3/Weil/ZetaBridge/WeilArithmeticCouplingSecondJet.arithmetic_coupling_second_jet_error
• Dependency: D5/S3/Weil/ZetaBridge/WeilArithmeticCouplingJet