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Quantum Prime Spectral Theory IV: Rigidity of the Zero–Type Spectrum via the Mixed Spectral Channel Charles Gon¸calves Abstract We establish a rigidity theorem for the zero–type spectrum of the canonical Hamiltonians arising in Quantum Prime Spectral Theory (QPST). After fixing the canonical Archimedean normalization and removing the rigid leading terms of order Tlog Tand Tin the zero–type Weyl law, we show that the remaining fine spectral structure is completely controlled by an intrinsic off–diagonal trace invariant. This invariant is defined through a mixed arithmetic–Archimedean spectral channel, encoded by a tempered spectral measure naturally associated with the global delocalizing unitary of the QPST framework. We prove that equality of this mixed spectral channel implies equality of the zero–type spectral functional, up to a fixed regular gauge. Under a minimal discreteness assumption on the zero–type spectrum, this yields rigidity at the level of the underlying spectral multiset. The resulting rigidity mechanism is purely operator–theoretic and does not rely on probabilistic assumptions, random matrix models, or conjectural statistical input. 1
1 Introduction Quantum Prime Spectral Theory (QPST) provides a canonical operator–theoretic framework for encoding the arithmetic and Archimedean structures underlying the Riemann zeta function within a single self–adjoint Hamiltonian. In the first three parts of this program, we established the existence and canonicity of the QPST Hamiltonian, proved a Weyl law with the correct universal coefficient for the associated zero–type spectrum, and identified the structural origin of the fine error term as arising from an intrinsic arithmetic–Archimedean interference channel. The purpose of the present work is to show that, once this canonical architecture is fixed, the remaining freedom in the zero–type spectrum is illusory. We prove that the fine spectral data is rigidly determined by a single off–diagonal trace invariant associated with the mixed spectral channel, leading to a uniqueness result for the zero–type spectrum under minimal spectral assumptions. 2 Setup and Canonical Invariants In this section we fix notation and recall the canonical operator–theoretic objects underlying the Quantum Prime Spectral Theory (QPST) framework. All constructions below are intrinsic and do not depend on probabilistic or statistical assumptions. 2.1 Canonical Hilbert space decomposition The QPST framework is based on a canonical orthogonal decomposition of the Hilbert space H=Harit ⊕ Harq, encoding, respectively, the arithmetic and Archimedean degrees of freedom. We denote by Parit, Parq the orthogonal projections onto the corresponding subspaces. This decomposition is fixed once and for all and is independent of any delocalization or masking procedure applied at the level of the Hamiltonian. 2.2 The QPST Hamiltonian and the physical class C Let Kfull =Karit ⊕Karq denote the canonical undeformed operator, where Karit has pure point spectrum given by the arithmetic frequencies klog p, and Karq acts by multiplication by the real spectral parameter on the Archimedean sector. A global delocalizing unitary operator Uacting on His used to mix the arithmetic and Archimedean components, producing the conjugated operator K0:= UKfullU†. An admissible bounded Schur–Hadamard multiplier SMis then introduced, and the physical QPST Hamiltonian is defined by the symmetrized expression H:= 1 2SMK0+K0SM. We denote by Cthe class of all Hamiltonians obtained by this procedure. Every H∈ C is self–adjoint and canonically normalized. 2
2.3 Trace functionals and zero–type decomposition For an even Schwartz test function φ∈ S(R), the trace functional TH(φ) := Trφ(H) admits a canonical decomposition of the form TH(φ) = Tzero,H(φ)+A(φ) + Carch bφ(0) + Rsmooth(φ). Here Adenotes the purely arithmetic contribution supported on the frequencies klog p,Carch bφ(0) is the Archimedean term fixed by normalization, and Rsmooth is a regular remainder. The zero–type functional Tzero,H is defined by subtraction and encodes the discrete spectral data of interest. 2.4 Gaussian probes and Weyl normalization To probe the mean spectral density and the fine structure of the zero–type spectrum, we employ the Gaussian family ψT(λ):=e−λ2/(2T2), T > 0. The associated zero–type counting functional is ZH(T) := Tzero,H(ψT). For every H∈ C, the zero–type spectrum satisfies a Weyl law of the form ZH(T) = 1 2πTlog T+c T +EH(T), where the coefficient 1/(2π) is universal and fixed by the Archimedean sector, cis a constant depending on normalization, and EH(T) denotes the zero–type error term. Throughout this work, the terms of order Tlog Tand Tare regarded as rigid and are removed by normalization. 2.5 The mixed spectral channel The central object of the present work is the off–diagonal or mixed spectral channel, defined by the distribution ΞH(u) := TrPariteiuHParq +ParqeiuHParit, u ∈R. This distribution captures the interference between the arithmetic and Archimedean sectors induced by the global delocalization. It defines a tempered distribution on R, and there exists a unique tempered Radon measure µmix,H such that ΞH(u) = ZR eiux dµmix,H(x) in the sense of distributions. For ψ∈ S(R) even, we define the mixed trace functional Tmix,H(ψ) := 1 2πZRb ψ(u) ΞH(u)du. In particular, for the Gaussian family ψT, we set Emix,H(T) := Tmix,H (ψT) = ZRc ψT(x)dµmix,H(x). 3
2.6 Canonical decomposition of the error term After removal of the rigid Weyl contributions, the zero–type error term admits an exact canonical decomposition EH(T) = Earit,H(T)+Emix,H (T)+Ereg,H(T), where Earit,H denotes the purely arithmetic contribution, Emix,H is governed by the mixed spectral channel defined above, and Ereg,H is a bounded regular term. The analysis of Quantum Prime Spectral Theory III shows that the purely arithmetic contribution is subdominant and that all nontrivial fine spectral structure is carried by the mixed term. This observation motivates the rigidity results established in the following section. 3 Rigidity via the Mixed Spectral Channel In this section we establish the main rigidity results of the present work. We show that the mixed spectral channel introduced in Section 2 provides a complete invariant for the fine structure of the zero–type spectrum, once the rigid Weyl contributions have been removed. 3.1 Equivalence of mixed spectral invariants We begin by recording the equivalence between the various representations of the mixed spectral channel. Theorem 3.1 (Equivalence of mixed invariants).Let H∈ C be a QPST Hamiltonian. The following data are equivalent: 1. the mixed correlation distribution ΞH(u); 2. the associated tempered Radon measure µmix,H; 3. the mixed trace functional Tmix,H(ψ)for all even ψ∈ S(R); 4. the mixed error term Emix,H(T)for all T > 0. Proof. By definition, ΞHis the Fourier transform of the tempered measure µmix,H, hence (1) and (2) are equivalent. Given ΞH, the mixed trace functional is obtained by Tmix,H(ψ) = 1 2πZRb ψ(u) ΞH(u)du, which proves the equivalence with (3). Taking ψ=ψTyields (4). Conversely, the family of Gaussian test functions {ψT}T >0forms an approximate identity in the Schwartz class. Equality of Emix,H (T) for all Ttherefore implies equality of the underlying tempered measure, and hence of ΞH. 3.2 Rigidity of the zero–type error We now show that the mixed spectral channel rigidly determines the nontrivial part of the zero–type error term. 4
Theorem 3.2 (Rigidity of the fine error term).Let H1, H2∈ C be QPST Hamiltonians with the same canonical Archimedean normalization. Assume that ΞH1= ΞH2as tempered distributions. Then, for all T > 0, EH1(T)−EH2(T) = Earit,H1(T)−Earit,H2(T). In particular, the mixed contribution to the zero–type error term is identical for H1and H2. Proof. By Theorem 3.1, equality of ΞH1and ΞH2implies Emix,H1(T) = Emix,H2(T) for all T > 0. Since the Archimedean normalization is fixed, the regular terms Ereg,H1and Ereg,H2coincide. Subtracting the canonical decompositions EH=Earit,H +Emix,H +Ereg,H for H=H1, H2yields the stated identity. 3.3 Rigidity of the zero–type spectral functional We now pass from rigidity of the error term to rigidity of the zero–type spectral functional itself. Theorem 3.3 (Rigidity of the zero–type functional).Let H1, H2∈ C satisfy ΞH1= ΞH2and have the same canonical normalization. Assume moreover that the purely arithmetic contribution to the zero–type error is negligible in the sense that Earit,H1(T)−Earit,H2(T) = o(1) as T→ ∞. Then Tzero,H1(ψ) = Tzero,H2(ψ)for all even ψ∈ S(R). Proof. By Theorem 3.2, the difference between the zero–type counting functionals for H1and H2 is controlled by the arithmetic term alone. Under the stated hypothesis, this difference vanishes in the Gaussian family ψT. Since linear combinations of Gaussian functions and their derivatives are dense in Seven(R), the equality extends to all even Schwartz test functions. 3.4 Spectral rigidity under a discreteness assumption Finally, we translate rigidity of the zero–type functional into rigidity at the level of the underlying spectrum. Theorem 3.4 (Spectral rigidity of the zero–type spectrum).Assume, in addition to the hypotheses of Theorem 3.3, that the zero–type functional admits a discrete spectral representation of the form Tzero,H(ψ) = X n mnψ(λn), where {λn}⊂Ris a discrete set with |λn| → ∞ and mn∈N. Then equality of the mixed spectral channels implies equality of the zero–type spectra as multisets: {λ1,n with multiplicities}={λ2,n with multiplicities}. Proof. Under the discreteness assumption, Tzero,H is represented by a purely atomic measure. Equality of the corresponding distributions on Seven(R) implies equality of the underlying atomic measures, and hence of the spectral multisets. 5
3.5 Interpretation The results above show that the mixed spectral channel ΞHprovides a complete invariant for the fine structure of the zero–type spectrum within the QPST framework. Once the canonical architecture and normalization are fixed, no additional spectral freedom remains beyond that encoded in the off–diagonal arithmetic–Archimedean interference. 4 Consequences and Outlook The rigidity results established in Section 3 have several immediate conceptual and structural consequences. We summarize them below and briefly outline the natural directions that follow from the present analysis. 4.1 Completeness of the mixed spectral channel Theorems 3.2–3.4 show that the mixed spectral channel ΞHprovides a complete invariant for the fine structure of the zero–type spectrum within the QPST framework. Once the canonical Archimedean normalization and the rigid Weyl terms are fixed, the off–diagonal arithmetic–Archimedean interference encoded by ΞHexhausts the remaining spectral freedom. In particular, any two Hamiltonians in the physical class Cthat share the same mixed spectral channel necessarily share the same zero–type spectral functional, and, under a minimal discreteness assumption, the same zero–type spectrum as a multiset. This establishes a genuine rigidity phenomenon at the operator–theoretic level. 4.2 Structural exclusion of spectral arbitrariness The rigidity mechanism identified here shows that the apparent freedom in the fine structure of the zero–type spectrum is illusory. While the arithmetic and Archimedean sectors independently contribute rigid and regular components to the trace, all nontrivial spectral fluctuations are forced to pass through the mixed channel. As a consequence, no additional parameters, statistical assumptions, or external models can be introduced to modify the fine spectral structure once the mixed spectral invariant is fixed. In particular, the zero–type spectrum cannot be tuned or perturbed without altering the underlying off–diagonal trace invariant. 4.3 Relation to non–Poissonianity In Quantum Prime Spectral Theory III, it was shown that, under minimal non–degeneracy assumptions on the delocalizing unitary, the mixed spectral measure µmix,H cannot be purely atomic. The present rigidity result strengthens this observation by showing that such non–Poissonian behavior is not merely a statistical feature, but a structural necessity. Indeed, since the mixed channel is a complete invariant, any admissible zero–type spectrum must reflect the same intrinsic correlations encoded in µmix,H . Absence of correlations is therefore excluded at the level of operator architecture, independently of any probabilistic interpretation. 6
4.4 Implications for the Hilbert–P´olya paradigm Although the present work is formulated entirely in operator–theoretic terms, it has direct implications for the Hilbert–P´olya program. Any spectral realization of the nontrivial zeros of the Riemann zeta function within the QPST framework must arise from a Hamiltonian whose mixed spectral channel coincides with the canonical one dictated by the arithmetic–Archimedean architecture. In this sense, the rigidity theorem reduces the problem of spectral identification to the determination of a single intrinsic invariant. Once this invariant is fixed, the associated zero–type spectrum is forced and admits no further freedom. 4.5 Outlook Several natural directions emerge from the present analysis. First, the rigidity theorem suggests a strategy for spectral identification based on the explicit computation or characterization of the canonical mixed spectral channel associated with the Riemann zeta function. Second, the operator– theoretic nature of the invariant opens the possibility of studying fine spectral statistics, such as pair correlation, as consequences rather than assumptions. More broadly, the results indicate that any successful spectral approach to the Riemann zeros must involve a nontrivial and rigid interference between arithmetic and Archimedean degrees of freedom. The QPST framework provides a natural setting in which this principle can be made fully explicit and mathematically precise. 7
A Density of Gaussian Test Functions in the Even Schwartz Class In this appendix we justify the density statement used in Theorem 3.3, namely that equality of spectral functionals tested against Gaussian functions implies equality on the entire even Schwartz class. A.1 The Schwartz space and even subspace Let S(R) denote the Schwartz space of rapidly decreasing smooth functions on R, and let Seven(R) denote the closed subspace of even Schwartz functions. It is well known that S(R) is a nuclear Fr´echet space, stable under differentiation, multiplication by polynomials, and Fourier transform. A.2 Hermite–Gaussian basis Let g(λ) := e−λ2/2 denote the standard Gaussian. The Hermite functions hn(λ):=Hn(λ)g(λ), where Hndenotes the n–th Hermite polynomial, form a complete orthogonal basis of L2(R) and belong to S(R). Moreover, the even Hermite functions {h2n}n≥0form a basis of the even subspace. It follows that finite linear combinations of derivatives of Gaussian functions are dense in Seven(R) with respect to the Schwartz topology. A.3 Density of Gaussian families Consider the one–parameter family of Gaussian functions ψT(λ):=e−λ2/(2T2), T > 0. By differentiation with respect to λand T, and by taking finite linear combinations, one generates functions of the form P(λ)e−λ2/(2T2), where Pis an even polynomial. After rescaling, these functions span the even Hermite–Gaussian subspace. Since the even Hermite functions form a dense subspace of Seven(R), it follows that the linear span of the family {ψT}T>0and its derivatives is dense in Seven(R). A.4 Consequence for spectral functionals Let F1, F2be continuous linear functionals on Seven(R). If F1(ψT)=F2(ψT) for all T > 0, then F1and F2coincide on a dense subset of Seven(R). By continuity, this implies F1(ψ) = F2(ψ) for all ψ∈ Seven(R). This justifies the extension step used in the proof of Theorem 3.3. 8
