Written conditional result, independently audited (../../audits/prime-gap-multistrip/AUDIT.md): assuming every nontrivial zeta zero has real part <=7/8, every integer n>=1 has a prime strictly between n^7 and (n+1)^7. Neither a novelty claim nor kernel formalization. All inherited explicit-formula and verified-height assumptions are unchanged from routes/prime-gap/RESULTS.md; this is a refinement of its density bound.
Exact additional source rows
Kadiri-Lumley-Ng, arXiv:2101.12263v1, Lemma4.14 (4.71) and section5 Table1: https://arxiv.org/pdf/2101.12263v1 . For T>=3.0610046e10, N(s,T)<=A_s T^[8(1-s)/3](log T)^[5-2s]+C_s(log T)^2. The rows (s,A_s,C_s) actually printed are (.75,5.277,4.403),(.8,6.918,3.997),(.85,8.975,3.588), (.86,9.441,3.514),(.87,9.926,3.430). The implementation rounds upward to (5.28,4.41),(6.92,4),(8.98,3.59), (9.45,3.52),(9.93,3.44). Counts have beta>s, positive ordinate, multiplicity. The Lemma4.14 versus Theorem1.1 delta-condition issue was already resolved in the previous route. Source constant calculations are cited, not reproduced.
Layer-cake reduction
Let s0=.75,s1=.8,s2=.85,s3=.86,s4=.87,s5=.875 and x>1. For every beta<=s5,
x^(beta-1) <= x^(s0-1)
- sum[j=0..4] (x^(s[j+1]-1)-x^(s[j]-1))*1[beta>s[j]].
The right side telescopes to the right endpoint of beta's strip, or to x^(s0-1) below the first strip. It is therefore a pointwise upper bound. All coefficients are positive, so substituting upper bounds for the strip counts is valid. No subtraction of independently bounded counts occurs. After multiplying by G(gamma)=g(gamma/a) and summing positive ordinates >H, the new high-zero error is
2x^(-.25) S_high(a) + 2 sum[j] Delta_j(x) R_j(a),
where R_j integrates the jth density bound against -G', from max(H,4.5a) to infinity, exactly as in the audited eighth-power route. That route's endpoint repair is inherited: the integration endpoint remains an Arb ball, never rounded upward to discard a positive interval. The low-zero, trivial-zero, and prime-power terms are unchanged.
For logx>=8 each Delta decreases in x. Indeed, writing y=logx, Delta=y*integral[sj,sj+1] exp[-(1-t)y]dt, every integrand times y decreases when y>=1/(1-t); t<=7/8 makes y>=8 sufficient. Thus on a logn interval [La,Lb], evaluate every Delta at x=exp(7La), and the nonnegative zero-sum majorants at a(Lb). This produces a valid interval bound.
Complete cover
764 closed intervals of width1/16 cover logn in [2.25,50]. Arb at256bits proves every margin>.13045; the weakest cell begins at29.125. Since2.25<log10, this handles n>=10 to exp50. Trial division through isqrt verifies explicit prime witnesses and strict membership for n=1,...,9.
For L=logn>=50, the inherited tail routine at k7,theta=.75 bounds the baseline, low zeros, trivial zeros and prime-power subtraction. It is only a bound on these quantities, not an assumption of QRH(.75). Use a<=1+n/7<=n, replacing a by n in each positive zero sum. Then B=4.5n>H. For row j put r=8(1-sj)/3,q=5-2sj,t=sj+1,d=7(1-t)-r. We have r+q/logB<.8, q<4, 4.5^r<3. The power moment at exponent.8 is<3. Therefore, discarding the negative part of Delta for an upper bound,
2 Delta_j R_j <=18 A_j n^(-d)(L+2)^4 +6 C_j n^-7(1-t) (L+2)^2.
Both monomials decrease on L>=50, verified from4/(L+2)<d and 2/(L+2)<7(1-t). Adding their values at50 to the baseline tail gives total error<.012961<1. All stated decimal inequalities are Arb assertions. This completes the candidate conditional implication for all integer n.
Boundaries and falsification
The previous single-.85-split bound failed at k7 nearL30; replacing it with single-.8 fails much worse. The success comes from resolving beta strips, not from ignoring those negative tests. Same layered implementation for k6 fails at L29 with negative margin approximately -56.04, while eventual positivity returns. This is a failure of the bound, not of sixth-power primes. The external density estimate remains a cited theorem, and the inherited explicit-formula source check is complete in ../../audits/prime-gap/SOURCE-CHECK.md. The critical-line source addendum is in ../../audits/kln-restoration/SOURCE-CHECK.md.
The independent audit reconstructed the pointwise layer identity and the monotonicity of Delta, checked moment and tail constants, and replayed the cover at higher precision. Its scope and shared inherited code are disclosed in ../../audits/prime-gap-multistrip/AUDIT.md.
Prior-art scope: the earlier route checked KLN and Cully-Hugill-Johnston arXiv:2402.04272v3 section5. Density arguments and asymptotic prime-in-power results are classical. Worldwide priority for the explicit all-n conclusion under precisely QRH(7/8) remains unresolved; no first-result claim is made.
The doors
Active k6 loss: finite-width beta bins and the absolute zero-sum majorant near the verified-height transition. Frozen choices: these five source rows, quadratic spline, verified H, and simple log-power majorants. Finer bins or continuous density integration retain the new density information class. The measured k6 failure is not an optimality certificate. No additional weight optimization or brute-force prime search was needed for k7.