q3 design

Claim boundary

The computation certifies one conditional exponent improvement. It reuses the q1 exact certificate as the previous bound and does not rerun or change q1's finite Oppermann certificates.

The only external inputs are the RH short-interval theorem quoted by Chamberland--Straub and the Sorenson--Webster endpoint $N=70{,}500{,}000{,}000{,}000$. The new arithmetic is exact and network-free.

Independent paths

• make_certificate.py constructs rational logarithm enclosures by a power-of-two reduction and a bounded atanh series.
• verify_certificate.py reconstructs every field, canonicalizes every rational, and checks all signs without trusting decimal fields.
• verify_float.c separately evaluates the splice with long double and checks wide diagnostic margins. The rational Python verification is authoritative.
• run_all.sh first invokes the q1 exact verifier on q1's committed rh_delta.json, then regenerates, byte-compares, and checks the q3 certificate.

Invariants

Let $d=0.22524401991935$, $\alpha=2+d$, $X=N^{2/\alpha}$, and $A=X^{d/2}=N^{d/\alpha}$. The certificate proves

It verifies that the logarithmic margin for Chamberland--Straub condition (5) is negative. It then retains the quadratic Taylor term from their inequality (6), bounds its relative contribution from below, and proves the resulting logarithmic overlap margin positive.

Finally, an exact positive lower bound for the derivative shows that the strengthened sufficient inequality continues to hold for every $x\geq X$. The finite proposition covers $x<X$.