# Centered tails and compact dual certificates

This extension concerns finite approximation errors. It does not establish RH or novelty priority. Numerical proposals remain outside the acceptance argument.

## A sharper periodic remainder bound

Let a real period-L sequence take values in [u,v], with mean mu. For a segment of n terms in a single period, its centered sum is at most min(n(v-mu),(L-n)(mu-u)); its negative is at most min(n(mu-u),(L-n)(v-mu)). Maximizing over real n gives L(mu-u)(v-mu)/(v-u), which is at most L(v-u)/4. Whole periods contribute zero, and the argument holds at every starting phase. Constant sequences have zero discrepancy.

Write delta_k(m)=(m mod k)-(k-1)/2 and a=t-sum b_k(k-1)/(2Q), where t is the constant target. Then the residual is a-sum b_k delta_k/Q. The centered sawtooth has total positive mass floor(k^2/4)/2 per period. The product delta_i delta_j lies in [-(i-1)(j-1)/4,(i-1)(j-1)/4] and has period lcm(i,j). Expanding the squared residual, subtracting its exact mean, and applying the triangle inequality therefore gives the phase-uniform bound

    K_centered = |a|/Q * sum_k |b_k| floor(k^2/4)
                 + 1/(8Q^2) * sum_(i,j) |b_i b_j| (i-1)(j-1) lcm(i,j).

Abel summation bounds the weighted remainder beyond M by K_centered/((M+1)(M+2)). The exact mean contributes mean/(M+1). This makes the quadratic part eight times smaller than the earlier range-only bound. It does not assert that a particular residual or its actual error improved: it sharpens the certificate for a fixed candidate.

There is a further moment-sensitive improvement. For a zero-mean periodic sequence, every centered segment is bounded by its total positive mass, equal to half its total absolute mass. With X=delta_i delta_k and C=E[X]=(gcd(i,k)^2-1)/12, E|X-C| <= E|X|+C <= sqrt((i^2-1)(k^2-1))/12+C <= (ik+gcd(i,k)^2-2)/12. Thus replace each quadratic pair factor lcm(i,k)(i-1)(k-1)/8 by

    B_(i,k) = lcm(i,k)/24 * min(3(i-1)(k-1), ik+gcd(i,k)^2-2).

The inequality sqrt((i^2-1)(k^2-1)) <= ik-1 follows by squaring, since the difference of the squared sides is (i-k)^2. No independence assumption is used. The minimum retains the range bound when it is stronger. The quadratic contribution is at most one eighth of the original bound and often substantially smaller.

The checker recomputes the strengthened bound and the exact period mean for saved candidates, reusing their previously verified 128-million-cell rational heads. Direct enumeration tests cover all cyclic phases of small signed coefficient examples and a separate long finite sum with a coarse residual tail.

## Compact exact annihilators

Fix M>N. Choose arbitrary integers T_(N+1),...,T_M and set T_(M+1)=0. Define T_k=-sum_(r>=2,kr<=M) T_(kr) for k=N,...,2, then T_1=-sum_(k=2)^M T_k and alpha_m=T_m-T_(m+1).

Telescoping gives sum_m m alpha_m=sum_k T_k=0 and sum_m alpha_m floor(m/k)=sum_(r>=1,rk<=M) T_(rk)=0 for 2<=k<=N. Thus sum_m alpha_m (m mod k)=0. The vector h_m=m(m+1)alpha_m on the first M cells, zero elsewhere, is exactly orthogonal to every original basis function. Cauchy-Schwarz gives

    d_N^2 >= (sum_m alpha_m)^2 / sum_m m(m+1)alpha_m^2.

The numerator must be nonzero. Neither floating point nor approximate stationarity occurs in this check. Setting the first moment to zero adds a restriction to proposal generation; it is not an extra assumption about RH. A finite weighted least-squares proposal with an auxiliary m column makes this restriction inexpensive in the present numerical examples. Free tails are rationalized on a 2^80 grid; the recurrence produces small integers directly, avoiding an lcm(1,...,N) scale factor.

At M=131072, certificates for N=2048 and N=4096 have maximum coefficient size 73 bits and are about 1 MB each. An N=8192 certificate uses 262144 cells. Increasing the N=4096 support to 1048576 without refining the numerical Gram approximation produced a worse bound, retained as a negative experiment. Rebuilding the numerical head at that support produced a stronger exact certificate. All versions are retained and independently checked; the reported finite lower bound is the largest verified value, not the largest attempted support.

## One more verified induction step

Let L_N be a checked lower bound on d_N^2 and U_(2N) a checked upper bound. For N=2^j, define kappa=(j+1)(1-U_(2N)/L_N). A bound kappa>=1/2 implies

    d_(2N)^2 <= (1-1/(2(j+1))) d_N^2.

The extension checks j=11 and j=12 from exact lower and upper certificates; the resulting constants are in progress-extension-results.json. This extends a finite list. Proving the same inequality for every sufficiently large j remains unresolved; no finite list supplies that quantifier. The earlier residual-line lower bound also allows an exact comparison of the N8192 candidate against every real parameter on the old line.

Run `python3 research/swc-rh/verify_progress_extension.py` from the package root. For direct tail checks run the unittest suite. Large original head sums are reused unless their documented full verification commands are rerun.
