# A real-query response spectrum and a certified amplitude–frequency frontier This develops the first proposed concept using the unchanged degree-eleven arithmetic experiment. Its new frequency bounds concern **real amplitudes B and real phase phi** in B sin(omega x+phi). Polynomial nuisance remains arbitrary complex degree <=11; sensor and mismatch errors may be complex. For a complex drift amplitude, the previous complex-modulus response bound remains available; the real-query reduction below cannot be substituted. ## 1. The exact query response Let X=[Phi_1,...,Phi_50,Psi_1,...,Psi_11], G=X*WX>=fI, and L_n=n² e_n* G^-1 X*W, ell_n=Re L_n, n=2,...,50. The times and weights are exactly those reconstructed in the degree-eleven physical model: 8,900 symmetric times and positive, even, mass-one weights. The arithmetic source coefficients are real integers, independently bounded by 0<=a(m)<=d14(m), a(1)=1, for every integer m. No multiplicativity of the actual source or a number-field realization is asserted in this branch. Let (Jy)_j=conjugate(y_(8899-j)). The augmented column space is invariant under this antilinear isometry, and its arithmetic coordinates satisfy L_n(Jy)=conjugate(L_n y). The proof by uniqueness of weighted least squares is in the [framework](../framework/PROOFS.md). An odd real reading vector therefore has purely imaginary L_n response; an even real vector has real response. Consequently ell_n sin(omega x)=0, r_n(omega)=ell_n cos(omega x)=L_n cos(omega x) is real, sup_phi |ell_n sin(omega x+phi)|=|r_n(omega)|. (1) This is an analytic symmetry statement. Numerical checks that an interval contains zero are consistency checks, not the proof of exact parity. The full real-query response spectrum is K_n(Omega)=sup_(|omega|<=Omega) |r_n(omega)|. (2) It is even in frequency and nondecreasing as a function of the bandwidth Omega. We do not assume that r_n itself is monotone or that its largest absolute value occurs at the interval endpoint. ## 2. Signed polynomial channels with a complete remainder Exact polynomial nuisance removal implies L_n x^k=0 for 0<=k<=11. Reconstruct signed real intervals for h_nk=L_n x^k at even k=12,...,64. Taylor's theorem **through degree 65**, whose odd cosine coefficient vanishes, gives r_n(omega)=P_n(omega²)+R_n(omega), P_n(v)=sum_(k=12,14,...,64) (-1)^(k/2) h_nk v^(k/2)/k!, |R_n(omega)| <= n² m66 |omega|^66/(sqrt(f) 66!), (3) where m66>=||x^66||_W is reconstructed over all 8,900 readings. The weighted decoder norm is <=n²/sqrt(f). Both the finite coefficients and the remainder retain the exact physical inverse; no saved floating Gram is substituted. The auxiliary Taylor degree 64 is a proof computation, not a larger nuisance fit: the decoder still has 61 columns and uses the same readings. For each design there are 49*27=1,323 signed moment channels. All are rebuilt with Arb at 320 bits. Their signed rational enclosures are then used in interval polynomial evaluation. A separate 448-bit replay reconstructs every signed moment, the complete norm, and additional actual cosine and sine/phase responses. All inherited full arithmetic-tail premises remain. ## 3. Complete bandwidth coverage and legitimate lower witnesses For bandwidths Omega in {1,3,5,6,7,8,9,10}, the producer covers [0,Omega(1+10^-8)] by adjacent closed cells of width (1+10^-8)/128. On every cell it evaluates the signed polynomial with interval arithmetic and adds the order-66 remainder at the cell's upper endpoint. The maximum is a certified upper bound for every real phase and every frequency in the declared band, including every clock slope of magnitude at most 10^-8. The independent consumer checks all 1,280 cells per design, not only the saved maxima or a grid of values. Lower response witnesses use **nominal** frequencies j/128<=Omega. These belong to every admitted clock family by choosing slope zero. At each such point the polynomial's absolute-value lower endpoint minus the complete remainder is a lower bound for the true response. Keeping lower witnesses inside the nominal band is essential: a witness allowed only by the larger frequency guard could not obstruct a budget with a smaller slope family. The checker rejects out-of-family witnesses explicitly. Thus for any allowed slope family within the guard we obtain k_n^- <= sup_(|omega|<=Omega_eff) |r_n(omega)| <= k_n^+, where the upper guard may be a conservative superset of that family's actual effective frequencies. The endpoints and all rounding gates are rationally checked. A small outward cushion in stored bounds permits higher-precision interval replay without assuming identical rounding paths. ## 4. The amplitude frontier belongs to a stated error budget Let b_n be the complete coefficient error bound excluding sinusoidal drift. It includes the infinite centered arithmetic tail, source-clock distortion, separate sensor/mismatch bounds, digital centering, and the required actual normal residual. Assume m_n=1/2-b_n>0 for all n. The response bounds give B_safe = min_n m_n/k_n^+, B_budget_upper = min_(n:k_n^->0) m_n/k_n^-. (4) Every nonnegative B