MATH, 10 PRIMO, 10 - Vol. 2: The Universal Ladder - One Shift Identity Unifies All Heegner Prime-Run Families; and the Independently Verified Prime-Gap Record g=354 at p=4,302,407,359

Abstract

Volume 2 establishes and certifies the Universal Ladder: the shift identity n^2+an+b = m(m+1)+C_D with m=n+(a-1)/2 and C_D=(1-D)/4 shows that every member of every prime-rich quadratic family at fixed discriminant D=a^2-4b evaluates one universal sequence U_D(m)=m(m+1)+C_D. From it we derive the ladder theorem (rungs a_j=a_0+2j have exact prime runs R_0-j) and verify 145/145 rungs across all six odd class-number-one discriminants with triple confirmation, resolving the single apparent exception D=-7 (mirror degeneracy U(-1)=U(0)=2). The same verification cycle certifies, by four independent paths, the new maximal prime gap g=354 after p=4,302,407,359 (merit 15.9586), discovered live by the PRIMES-LOOP engine at cycle 3417, superseding g=336 (merit 15.2247); and the exact anchor pi(3*10^9)=144,449,537 by dual independent computation. Every quantitative claim was verified and re-verified in two complete rounds (16/16 checks) by an independent master script before submission. Includes PDF, LaTeX source and all verification scripts.