# Complete real-query response to clock-rate distortion

This develops Conjecture A from `concepts_2026_09_10/CONCEPTS.md`. It proves
the tenfold target for both degree-eleven weightings, with a stronger bound.
The source remains the full integer envelope, not a multiplicative subclass.

## Model and distinction from older certificates

Use exactly 8,900 times t_j=(2j+1-8900)/10, the prior positive even mass-one
weights W, and X=[Phi_1,...,Phi_50,Psi_1,...,Psi_11]. Here
Phi_m(t)=exp(-it log m), and the Psi are the positive-leading W-orthonormal
polynomials in x=t/890. Let G=X*WX, G>=fI, 0<f<=1, and

    ell_n=e_n* G^-1 X*W,   L_n=n² ell_n,   n=2,...,50.

The real query is Re L_n. All source coefficients are independent integers
0<=a(m)<=d_14(m), with a(1)=1. Constant nuisance is absorbed by Phi_1 and its
fitted coordinate discarded. The same unrestricted degree-eleven complex
polynomial nuisance and arbitrary-phase sinusoidal drift remain admissible.

The arithmetic bias A_n, digital centering bound Xi, and Gram lower floor f
are inherited complete premises, with their previous replay routes. The
new derivative channels, actual inverse, variations, moments and complete
derivative-series bounds below are freshly reconstructed with Arb.

## 1. The derivative and a complete quadratic remainder

Put F(t)=sum_m a(m)m^-2 exp(-it log m). Absolute convergence of every needed
derivative follows from the positive Dirichlet identity sum d_14(m)m^-z=
zeta(z)^14 for real z>1. In particular define complete positive sums

    D_r=sum_m d_14(m)(log m)^r/m²  (r=1,2).

For any real slope error s, the scalar exponential remainder, valid on the
whole real axis, gives

    F((1+s)t)-F(t)
      = s sum_m a(m)m^-2 (-it log m) exp(-it log m) + R(t),
    |R(t)| <= s² t² D_2/2.

Since ||ell_n||_(W->C)<=1/sqrt(f), any bound H_n for the absolute real
decoded first derivative proves, simultaneously for every allowed source,

    |Re ell_n [F((1+s)t)-F(t)]|
      <= |s| H_n + s² D_2 ||t²||_W/(2 sqrt(f)).                 (1)

This is already divided by n² as in Conjecture A. We now bound H_n without
discarding the oscillatory acquisition geometry.

## 2. A direct finite prefix

For each 2<=m<=100 reconstruct the exact physical column
v_m(t)=(-it log m)exp(-it log m) and all 49 real channels

    h_nm >= |Re (G^-1 X*W v_m)_n|.

Their complete prefix contribution is bounded by

    P_n=sum_(m=2)^100 d_14(m)h_nm/m².                         (2)

Every prefix response passes through the actual full augmented inverse
before a magnitude is taken. No assumption that the independent coefficient
envelope is attained simultaneously is required; triangle inequality gives
a valid uniform bound. The m=1 derivative is exactly zero.

## 3. Every intermediate frequency, including polynomial channels

For a finitely supported sequence r on Z, let

    V_2(r)=sum_j |r_j-2r_(j-1)+r_(j-2)|.

Both ends are extended by zero. Exact finite reindexing yields
|(sum_j r_j z^j)| <= V_2(r)/|1-z|² for |z|=1, z!=1.

For the ith arithmetic normal channel, take r_j=w_j t_j and
z=exp[-i log(m/i)/5]. For a polynomial normal channel k take
r_j=w_j t_j Psi_k(x_j) and z=exp[-i log(m)/5]. The initial phase and the
factor -i have modulus one; the remaining scalar multiplier is log m.

For EVERY integer 101<=m<=N=10^12 and 1<=i<=50,

    .07 < log(m/i)/10 < 2.8 < pi.

The same interval contains log(m)/10. The endpoints satisfy
sin(.07)>1/16 and sin(2.8)>1/16. Concavity of sine on [0,pi] therefore proves
|1-z|>1/8 throughout both bands. These inequalities are reconstructed with
outward elementary functions; there is no finite frequency scan.

Let V_k enclose the complete variation of the appropriate sequence and J_k
its L1 norm. Using the COMPLETE positive derivative mass after the prefix,

    D_1,>100 = D_1 - sum_(m=2)^100 d_14(m)log(m)/m²,

the contribution from every 101<=m<=N to normal channel k is bounded by
64 V_k D_1,>100. Using this full mass includes some remote terms twice when
the following bound is added, which is conservative.

## 4. All remote coefficients and sampling aliases

For every m>N, m^-2<=N^-1/2 m^-3/2. Hence

    sum_(m>N) d_14(m)log(m)/m²
       <= N^-1/2 [-d/dz zeta(z)^14]_(z=3/2) =: M_remote.       (3)

Use the always valid Fourier bound |sum r_j z^j|<=J_k, including frequencies
arbitrarily near every later sampling alias. The complete normal-channel
bound for all m>100 is consequently

    C_k=64 V_k D_1,>100 + J_k M_remote,     k=1,...,61.        (4)

Let U_nk >= |(G^-1)_nk| be freshly reconstructed. The full decoded derivative
bound, now including every integer, is

    H_n=P_n+sum_(k=1)^61 U_nk C_k.                           (5)

The source coefficients need not be multiplicative; multiplicativity is used
only to identify the positive d_14 envelope and its convergent total series.

## 5. Conjecture A follows by an exact strict comparison

With h_s=10^-14, (1) and (5) give a rational upward C_n. The program uses a
positive rational lower approximation to sqrt(f) only in upper error bounds.
It verifies max C_n < tau_W(1)/10. Since f<=1, this is stronger than the
conjectured max C_n <= tau_W(1)/(10 sqrt(f)). Thus no reversed square-root
rounding direction enters the comparison.

The finite prefix cutoff and remote cutoff are proof computation choices;
neither truncates the admissible source. Full source coverage includes every
large-frequency alias. The bound is uniform for the entire slope interval,
with a complete quadratic remainder, not merely its derivative at zero.

## 6. Offset symmetry and valid joint budgets

Define (Jy)_j=conjugate(y_(M-1-j)). This is an antilinear W-isometry. Arithmetic
columns are fixed by J, while J Psi_k=(-1)^k Psi_k. Thus the augmented span
is invariant. Uniqueness of weighted least squares gives
ell_n(Jy)=conjugate(ell_n(y)) for arithmetic n. In particular ell_n exp(-iat
log m) is real for every real a. Absolute convergence then gives

    Re ell_n F(at+b)
      =sum_m a(m)m^-2 cos(b log m) ell_n exp(-iat log m).

The real query is exactly even in offset b at each fixed slope a. By
|1-cos u|<=u²/2 and ||ell_n||<=1/sqrt(f), uniformly in a,

    |Re ell_n [F(at+b)-F(at)]| <= D_2 b²/(2 sqrt(f)).           (6)

There is no missing slope-offset cross term: first decompose the full change
as [F(at+b)-F(at)]+[F(at)-F(t)] and use (6) uniformly in a, then (1).
Complex query error need not be even and is not claimed to obey (6).

For slope radius h_s and offset radius h_b the complete real coefficient
error is therefore

 E_n=A_n+n²[(eta+kappa+Xi)/sqrt(f)+rho/f
           +h_s H_n+h_s² D_2 ||t²||_W/(2 sqrt(f))
           +h_b² D_2/(2 sqrt(f))] + B K_n(3(1+h_s)).          (7)

Here K_n is the prior complete direct query-response bound for sinusoidal
drift, reevaluated at the NEW effective frequency. It already includes n².
The affine clock preserves the entire polynomial nuisance space and shifts
the already arbitrary sinusoidal phase, so no polynomial leakage is omitted.
The query spectrum replaces the old drift term; (1) replaces the old clock
allowance. Both must not be charged twice. Strict E_n<1/2 certifies rounding
the real part of every recovered coefficient.

The actual-data normal residual rho must still be independently certified.
It cannot establish sensor, clock, drift or mismatch model membership.
The expanded clock boxes are mathematical contracts in model time units;
they assert no empirical apparatus calibration.