bibkey: connesconsanimoscovici2026spectral authors: Alain Connes, Caterina Consani, Henri Moscovici year: 2026 title: Zeta Spectral Triples doi: 10.4171/elm/37/3 url: https://arxiv.org/abs/2511.22755v1 claim: The full bounded-support Weil operator has discrete lower-bounded spectrum, and an explicit auxiliary transform converges to Xi. Actual ground-state simplicity, evenness and comparison with that auxiliary function remain missing; the shifted spectral construction does not certify the original form’s sign. strata_touched: [] license: citation-only triage: anchor

Spectral construction and the actual Weil comparison

The EMS publication record identifies the chapter published on 23 June 2026. The complete primary text inspected here is arXiv:2511.22755v1, 27 November 2025. Its PDF SHA-256 is c98d89f7fc999d038e15e80a9aaaee2af797c17711c4329ca7ce48ad49cb336b. The publisher-edition full text was unavailable; equality of that edition with the inspected preprint is not asserted. The source results below are references for reuse, without independent proof verification, numerical reproduction or Lean implementation.

Established results to reuse

The following locators refer to the inspected preprint’s printed pages.

The theorem’s quotient metric is the restriction of

$$Q{W}_{\lambda }^N-{ϵ}_N⟨\cdot ,\cdot ⟩,$$

where ${ϵ}_N$ is the original finite-section minimum. Its positivity is a property of the shifted quotient. The construction supplies no sign bound for ${ϵ}_N$. The distributional precursor and its explicit negative example give another reason to preserve this distinction. Rebuilding the spectral construction would not supply the missing unshifted estimate.

Common-function parameter map

Use $L>0$ for the project’s additive half-width, avoiding the source’s use of an interval-length parameter:

$$\lambda =e^L=\sqrt{c},u=e^x,g(u)=f(\log u),d^{\ast }u=du/u=dx.$$

Thus $\mathrm{supp}f\subseteq [-L,L]$ corresponds to $\mathrm{supp}g\subseteq [{\lambda }^{-1},\lambda ]$, and additive evenness corresponds to multiplicative inversion symmetry. Equations (3.7)–(3.11) retain the prime-power weights, Gamma multiplier and both pole evaluations. In particular,

$$\hat{g}(\pm i/2)=\int f(x)e^{\pm x/2}dx.$$

For even $f$ these evaluations agree; they are not required to vanish. The auxiliary construction’s zero-integral condition is a condition on its profile, not permission to restrict the project’s arbitrary even tests. The existing joint pole–prime–Gamma account continues to own that common-test decomposition.

The source’s remaining steps

Section 8, pp.32–33, explicitly leaves two steps unresolved: the actual Weil minimum must be simple with an even eigenvector, and $k_{\lambda }$ must approximate a correctly scaled actual ground eigenfunction sufficiently accurately to transfer convergence of transforms. The corresponding properties of the prolate-wave operator do not establish those properties for the Weil operator.

One concrete comparison target is, for the same actual ground eigenfunction ${\xi }_{\lambda }$ and an explicitly controlled nonzero normalization $a_{\lambda }$,

$${\int }_{{\lambda }^{-1}}^{\lambda }|a_{\lambda }{\xi }_{\lambda }(u)-k_{\lambda }(u)|\max (u^{\sigma },u^{-\sigma })\frac{du}{u}⟶0,0\le \sigma <1/2.$$

This is an outstanding weighted-transform interface, not a bound proved by this note. Norm convergence at changing support does not by itself provide its weights or normalization. The effective prolate concentration bounds concern another operator and do not supply this comparison.

For the positivity route, the outstanding supplier instead concerns the actual full-form even-sector finite-section error. If $m_L^{+}$ is the true even-sector minimum and $m_{L,N}^{+}$ a retained minimum, the needed certified comparison has the form

$$0\le m_{L,N}^{+}-m_L^{+}\le r(L,N),$$

with a lower certificate for $m_{L,N}^{+}$ large enough to pay $r(L,N)$. Core density supplies no explicit $r$. This note obtains neither that estimate nor cofinal positivity. The two routes have different missing interfaces.

An unbounded FIB support schedule selects a cofinal family of windows. It supplies none of the actual simplicity, evenness, normalization, weighted comparison or finite-section errors above. Existing generic cofinal and Schur results should be reused; new work must address the actual arithmetic operator. RH remains unproved.

A proved high-energy law and its four-component characteristic geometry

Taira–Willems–Wrochna, Large eigenvalues of the Connes–Moscovici operator, arXiv:2609.32639v1, 26 September 2026, proves the logarithmic Weyl law previously predicted for a specified Connes–Moscovici operator. The inspected primary has 45 pages and PDF SHA-256 49cf5e3863459148b249c9ae721c2297024f1f1055025ba8ef9ed87c8893ffbb. The interfaces used here are Theorem 1.1 and Corollary 1.2 on p.2, the explicit spectral-scope statement on p.5, §§2.1–2.4 on pp.7–10, the counting argument in §5.3 on p.36, and Theorem A.11 on p.41. This is a primary-source application, not an independent audit of the microlocal proof, numerical reproduction, new spectral theorem or Lean verification.

The proved operator and its parameters

The differential expression and its distinguished selfadjoint extension are

$$P=-{\partial }_x(x^2-1){\partial }_x-4{\pi }^2{x}^2,P_{\mathrm{sa}}:D(P_{\mathrm{sa}})\subset L^2(\mathbb{R})\to L^2(\mathbb{R}).$$

Section 2.4 fixes the domain: the logarithmic terms at $x=\pm 1$ are excluded, and the even and odd parts have respectively the specified sine-type and cosine-type boundary conditions at infinity. The differential expression alone does not specify this extension. The spectrum is discrete and unbounded in both directions. The older Connes–Moscovici notation uses $W_{\mathrm{sa}}=-P_{\mathrm{sa}}$ and calls the relevant eigenvalues negative; its parameter called $E$ is the square root of the present eigenvalue parameter, as Remark 1.6 explains.

With

$$I(a)={\int }_1^{\infty }\left(\sqrt{\frac{x^2+a-1}{x^2-1}}-1\right)dx(a>0),$$

Theorem 1.1 gives some $E_{\ast }>0$ and a smooth remainder such that

$$R(E)=-\frac{\log E}{16\pi \sqrt{E}}+O(E^{-1/2}),R^{\prime }(E)=O(E^{-3/2}\log E).$$

All sufficiently large positive eigenvalues are simple and are characterized, with $n\in \frac{1}{2}\mathbb{Z}$ sufficiently large, by

$$2I\left(1+\frac{E_n}{4{\pi }^2}\right)+\frac{1}{4}+R(E_n)=n+O(E_n^{-\infty }).\text{\hspace{1em}(W1)}$$

No numerical value of $E_{\ast }$ or explicit constants for these remainder symbols are supplied by this application. Corollary 1.2 proves

$$N_{+}(E)=\frac{4\sqrt{E}}{2\pi }\left(\log \frac{\sqrt{E}}{2\pi }-1+2\log 2\right)+O(1).\text{\hspace{1em}(W2)}$$

Thus setting $T=4\sqrt{E}$ gives exactly the main counting function $T\log (T/(2\pi e))/(2\pi )$ of the classical Riemann–von Mangoldt formula. This is a parameter match for the main term, not an identification of individual eigenvalues or of the oscillating zero count. The primary explicitly preserves that distinction on p.5. High-energy simplicity in (W1) is also a statement about $P_{\mathrm{sa}}$; it is not the missing simplicity of the actual Weil ground state in the preceding spectral-triple discussion.

A literal four-phase symmetry after the scale is recorded

The source uses $E=h^{-2}$ and the global $h$-dependent real symbol

$$p_h(x,\xi )=(x^2-1){\xi }^2-4{\pi }^2{h}^2{x}^2-1,h>0.$$

Put $a_h=2\pi h$ and $y=\xi /a_h$. Then its characteristic equation becomes the symmetric relation

$$p_h(x,a_{h}y)=0⟺(x^2-1)(y^2-1)=1+a_h^{-2}.\text{\hspace{1em}(W3)}$$

Both $|x|$ and $|y|$ exceed one on this curve. There are four components labeled by their two signs, and

$$C(x,y)=(-y,x),C^2=-I,C^4=I$$

preserves (W3), cycling the labels as $({\sigma }_1,{\sigma }_2)\mapsto (-{\sigma }_2,{\sigma }_1)$. In the original symbol coordinates this is

$$B_h=\left(\begin{array}{cc}1& 0\\ 0& a_h\end{array}\right),{\mathcal{C}}_h=B_{h}C{B}_h^{-1}=\left(\begin{array}{cc}0& -a_h^{-1}\\ a_h& 0\end{array}\right),p_h({\mathcal{C}}_h(x,\xi ))=p_h(x,\xi ).\text{\hspace{1em}(W4)}$$

The last identity follows by substitution, and $\det {\mathcal{C}}_h=1$ preserves the two-dimensional symplectic form. This action exchanges the ends approaching $x=\pm 1$, $|\xi |=\infty$ with the ends approaching spatial infinity and $\xi =\pm a_h$. It is a scale-dependent transport of position and momentum boundaries. Using the unscaled quarter-turn on $(x,\xi )$ instead would give

$$p_h(C(x,\xi ))-p_h(x,\xi )=(a_h^2-1)(x^2-{\xi }^2),$$

which is not identically zero at general $h$. The scale factor in (W4) therefore cannot be dropped.

The matrix $C$ in normalized coordinates is the same algebraic matrix as the existing FIB composition operation $C=MJ$ in the FIB volume, §149. This supplies a concrete common four-phase action after the coordinate normalization is specified. It does not identify the integer composition carrier with this continuous characteristic curve. Symbol invariance alone does not establish a unitary action on the chosen operator domain or compatibility with the full boundary conditions and quantization remainder in (W1).

There is, independently, an existing operator-level Fourier symmetry to reuse. Appendix A, Theorem A.11 on p.41, recalls Connes–Moscovici, The UV prolate spectrum matches the zeros of zeta, PNAS 119 (2022), e2123174119, Theorem 1.6: this same $P_{\mathrm{sa}}$ commutes with the source’s Fourier transform $\mathcal{F}$, the interval projection $Q$, and $\hat{Q}=\mathcal{F}Q{\mathcal{F}}^{-1}$. It is the unique selfadjoint extension of $P_{\min }$ commuting with $Q$ and $\hat{Q}$. Thus the Fourier compatibility of this specified extension is already established source mathematics, rather than a new obligation to prove from (W4). The original theorem is used here as recalled in the inspected primary; no new audit of the 2022 proof is claimed. It does not identify the functional Fourier action with a transformation of legal FIB sources.

A FIB-to-spectral bridge would still need a map from legal recursive sources to this operator’s admitted data, an intertwining of the actual recursion and observations, and the quantization and arithmetic correspondence. Counting four components or sharing the matrix $C$ does not construct that map. In particular the five-window address rules are not boundary conditions for $P_{\mathrm{sa}}$.

The high-energy theorem should be reused in its own spectral model. It neither settles the actual Weil ground-state comparison nor bounds the complete signed $I_{\psi }(\log N)$ at the selected Robin source. Those arithmetic estimates and RH remain unproved.

The exact height clock fails to reproduce the zero-count fluctuations

The main-term match in (W2) has a quantitative boundary supplied by a classical theorem, rather than a new spectral calculation. Kai-Man Tsang, Some Ω-theorems for the Riemann zeta-function, Acta Arithmetica 46 (1986), 369–395, DOI:10.4064/aa-46-4-369-395, states the unconditional result

$$S(t)={\Omega }_{\pm }\left({\left(\frac{\log t}{\log \log t}\right)}^{1/3}\right)\text{\hspace{1em}(T1)}$$

in Theorem 1, equation (1.4), printed p.369. The adjacent stronger estimate (1.3) assumes RH and is not used. Theorem 2, printed p.370, gives a dyadic version: for some $c>0$ and all sufficiently large $T$,

$$\underset{t\in [T,2T]}{\sup }\pm S(1/2,t)\ge c{\left(\frac{\log T}{\log \log T}\right)}^{1/3}.\text{\hspace{1em}(T2)}$$

Here the source’s allowed range of $\sigma$ includes $\sigma =1/2$. The inspected publisher PDF has SHA-256 c6f9404e6e1b29202dd11adbefab220903d2400ef05345922795431f3ee2ad6f. These theorem statements and equation (1.1) were read from the scanned printed pages 369–370; no audit of Tsang’s proof is claimed.

Counts of the same height, with all zero multiplicities

Write

$$m(t)=\frac{t}{2\pi }\log \frac{t}{2\pi e},Q(t)={\left(\frac{\log t}{\log \log t}\right)}^{1/3}.$$

Let $N_{\zeta }(t)$ count every nontrivial zero with $0<\gamma <t$, including multiplicity and without imposing $\beta =1/2$. At heights away from zero ordinates, the source’s equation (1.1) is

$$N_{\zeta }(t)=m(t)+\frac{7}{8}+S(t)+O(t^{-1}).$$

Using (W2) with its already fixed clock $E=t^2/16$ gives

$$D(t):=N_{\zeta }(t)-N_{+}(t^2/16)=S(t)+\frac{7}{8}+O(1).\text{\hspace{1em}(T3)}$$

The bounded remainder here is a consequence of the two cited counting laws; no explicit numerical constant is asserted. In particular, $D(t)={\Omega }_{\pm }(Q(t))$. A version keeping a specified height window is: for some $c_1>0,C_1$ and every sufficiently large $T$, there are heights $t_T^{+},t_T^{-}\in [T,4T]$, away from zero ordinates, with

$$D(t_T^{+})\ge c_{1}Q(T)-C_1,D(t_T^{-})\le -c_{1}Q(T)+C_1.\text{\hspace{1em}(T4)}$$

To handle endpoints, apply (T2) on $[3T/2,3T]$, inside $[T,4T]$. At a zero ordinate use the appropriate one-sided value and move to a nearby height. The source records the jump as the total multiplicity; this argument neither deletes repeated zeros nor assumes their simplicity. Either endpoint convention for the high positive spectrum changes its count by at most one, by the eventual simplicity in (W1).

Thus the multiset of all sufficiently large zero ordinates cannot equal the transformed positive spectrum $\{4\sqrt{E_n}\}$ under this exact clock. Finite changes to the head and a fixed index offset alter the discrepancy only by $O(1)$ and cannot remove (T4).

A correction bounded in units of mean spacing is also insufficient

Let $r(t)=t+\delta (t)$ be any height clock with $\delta (t)=O(1/\log t)$. Since

$$m^{\prime }(t)=\frac{1}{2\pi }\log \frac{t}{2\pi },$$

the mean value theorem gives $m(r(t))-m(t)=O(1)$. The spectral counting law holds uniformly at sufficiently large arguments, hence

$$N_{\zeta }(t)-N_{+}(r(t{)}^2/16)=S(t)+\frac{7}{8}+O(1).\text{\hspace{1em}(T5)}$$

The constants may depend on the fixed bound on $|\delta (t)|\log t$. Consequently even this corrected clock cannot identify the eventual counts up to a uniformly bounded correction. The scale $1/\log t$ is the scale of the mean zero spacing, up to the constant $2\pi$; no assumption about actual consecutive spacings is used.

More generally, if $r(t)/t\to 1$ and an eventual count identification with bounded error were achieved, (T3) would require $m(r(t))-m(t)=S(t)+O(1)$. Another application of the mean value theorem and (T1) would force

$$(r(t)-t)\log t={\Omega }_{\pm }(Q(t)).\text{\hspace{1em}(T6)}$$

This is a necessary condition on a proposed arithmetic correction, not a construction of one and not a sufficient spectral correspondence. It excludes only the stated clocks for this specified operator; a different spectral model or a larger arithmetic reparameterization has not been excluded. A FIB address or four-phase relabeling does not by itself supply the fluctuations required in (T6). The deduction is an application of existing counting and oscillation theorems, with no originality or Lean-verification claim. It supplies no signed estimate for the original full Robin tail and leaves RH unproved.