The Type-I Seesaw as a Relativistic Fermionic Proximity Effect: Spectral Moments, EFT Consistency, and Sterile-Sector Reconstruction

Short version: this is a solid theory paper, and I agree with the reader’s accept. It introduces no new physics, but it does introduce a new organizing structure for the tree-level type-I seesaw: the paired lepton-number-violating and lepton-number-conserving moment towers, the mixed Nambu-Hankel positivity lift, the common annihilating recurrence, the matrix-pencil reconstruction, and the residue compatibility identity. I checked the main factorizations. They are direct finite-rank Gram/Hankel decompositions and they work. The one-generation benchmark is honest and correct; the paper carefully distinguishes the exact Takagi pole mass from the Schur-complement derivative expansion, including the r⁶ mismatch, rather than sweeping it under the rug.

The strongest part is also the part that limits immediate application. All the consistency tests and reconstruction theorems are stated for the gauge-invariant source/Green’s-basis moments before equations-of-motion reduction. The paper states this limitation explicitly in Section IV, which I appreciate, but it does not supply the pullback map to a conventional nonredundant SMEFT coefficient set. Until that map is built, the results cannot be imposed on global-fit Wilson coefficients as they are normally presented. That is a practical gap, not a logical flaw. The tree-level scope and the finite visible mass set are also clearly flagged.

The citation pattern is fine. The author cites the relevant seesaw-matching, positivity-bound, and Hankel-reconstruction literature, and the self-citations are to related previous work, not padding. No code is shipped and the numerical section is synthetic, but the formal proofs are explicit enough that the math can be checked by hand.

Who is this for? Neutrino EFT and seesaw-model readers who want a rigorous way to test whether a set of high-order Wilson coefficients could have come from a finite tree-level type-I completion, and who are willing to work in the source basis. I would send it to a serious referee and expect a publishable paper after a revision that either constructs the operator-basis pullback or makes the absence of that map even more prominent in the abstract. The stress-test note’s no-significant-objection matches my reading.

The paper is mathematically self-contained and its central theorems are proven. The weakest link is the practical connection to conventional SMEFT coefficients, but the paper transparently states this limitation. There is no internal inconsistency or unstated assumption that threatens the core argument.

The reader correctly identifies the mapping between source-basis moments and standard SMEFT coefficients as the least secure assumption. However, this is a task left for future work, not a gap in the presented proofs. The mathematical results stand on their own; the verdict of ACCEPT is appropriate.

If the results hold, this work provides a mathematically rigorous and falsifiable framework for extracting sterile-sector information from arbitrarily high-order EFT coefficients in the type-I seesaw. The moment constraints—particularly the mixed Nambu-Hankel positivity and the common annihilating polynomial—are new and would allow consistency tests even if only a few moments are known. The paper is notable for its transparency: the limitations of the source-basis formulation and the need for pullback to standard SMEFT coefficients are clearly stated. The algebraic proofs rely only on the explicit discrete measure, requiring no dispersion relations or additional assumptions. This elevates the seesaw inverse problem to a well-posed spectral reconstruction.

This paper claims that the canonical type-I seesaw—the standard explanation for tiny neutrino masses via heavy sterile partners—has an exact low-energy fingerprint that reaches far beyond the usual dimension-five mass term. When the sterile fields are integrated out, the active neutrino's two-point function is controlled by two response functions, one lepton-number-violating and one lepton-number-conserving, whose Taylor coefficients form paired moment towers generated by one common set of sterile masses. The paper proves that these towers obey strict positivity and recurrence conditions, and that a matrix-pencil calculation can reconstruct the visible sterile masses and their aggregate couplings from low-energy coefficients alone. If correct, this turns arbitrarily high-order effective-field-theory coefficients into consistency tests and reconstruction probes for the heavy neutrino sector, though the tests are formulated in a redundant source basis and must be mapped to standard operator bases before use.

What carries the argument. The load-bearing object is the exact active-sector embedding kernel written in Nambu–Gorkov form, a two-component doubling that packages normal and pairing correlators into one kernel; it is encoded by the paired response functions $A(Q^2)=\sum_i M_i y_i y_i^T/(Q^2+M_i^2)$ and $B(Q^2)=\sum_i y_i y_i^\dagger/(Q^2+M_i^2)$. Its power series defines the moment towers $A_n=\sum_i y_i y_i^T/M_i^{2n+1}$ and $B_n=\sum_i y_i y_i^\dagger/M_i^{2n+2}$, which are the objects the theorems study. The machinery has four pieces: the shifted block-Hankel matrices $H_r^{(s)}=[B_{a+b+s}]_{a,b=0}^r$, proven positive semidefinite; the mixed Nambu–Hankel blocks $C_n(\mu)=\begin{pmatrix}\mu^2 B_n & \mu A_n\\ \mu A_n^\dagger & B_{n-1}^*\end{pmatrix}$, which supply cross-family Cauchy–Schwarz constraints; the annihilating polynomial $P(t)=\prod_j(t-M_j^{-2})$, which gives a common linear recurrence for both towers; and the generalized eigenproblem $H_1 v=xH_0 v$, which converts the moments back into masses $M_j=x_j^{-1/2}$ and, through Vandermonde fits, into aggregate residues.

Core claim. The central discovery is that the seesaw is a relativistic proximity effect at the level of quadratic response functions: integrating out the sterile Majorana reservoir induces both an anomalous self-energy—the light Majorana mass—and a normal residue correction in the active sector. In the sterile Takagi basis, the basis that diagonalizes the complex symmetric Majorana matrix to nonnegative masses, the exact embedding kernels are $A(Q^2)=\sum_i M_i y_i y_i^T/(Q^2+M_i^2)$ and $B(Q^2)=\sum_i y_i y_i^\dagger/(Q^2+M_i^2)$, and their low-energy expansions generate the two towers $A_n=\sum_i y_i y_i^T/M_i^{2n+1}$ and $B_n=\sum_i y_i y_i^\dagger/M_i^{2n+2}$. The paper proves that the normal tower is a positive matrix Stieltjes sequence, so all shifted block-Hankel matrices are positive semidefinite; that a dimensionally rescaled mixed Nambu–Hankel sequence is also positive and yields matrix Cauchy–Schwarz inequalities linking the two towers; and that both towers obey one common annihilating polynomial whose roots are the inverse squared sterile masses. A matrix pencil built from the normal moments therefore recovers the visible masses and aggregate residues, and isolated nondegenerate poles satisfy the additional identity $S_jS_j^\dagger=M_j^2\,\mathrm{Tr}(R_j)\,R_j$. Nambu doubling is shown to introduce no new physical degrees of freedom.

The premise it rests on. The theorems are stated for a redundant source-basis set of low-energy operators before equations of motion are used to simplify them; if data come as ordinary on-shell effective-field-theory coefficients and the mapping back to this source basis is not applied, every positivity, recurrence, and reconstruction statement can fail.

The leading moments reproduce the Weinberg coefficient and the dimension-six kinetic operator, so the standard dimension-five plus dimension-six seesaw relations are just the first two terms of one exact nonlocal kernel.

Every finite canonical type-I completion must satisfy the common annihilating polynomial on both towers, making deviations in measured source-basis coefficients a direct test for a single tree-level seesaw origin.

Scalar trace moments count the distinct visible sterile mass supports, while the stabilized block-Hankel rank gives the minimal realization order of the normal response.

The mixed Nambu–Hankel conditions can expose paired sterile directions that a degenerate normal residue hides, including cases where the aggregate anomalous residue vanishes.

A matrix pencil from the normal moments reconstructs the visible sterile masses and aggregate residues; unique recovery of individual sterile fields requires additional multiplicity information.

Editorial inference: at loop level these tree-level constraints should stop being exact, and the natural observable is how recurrence residuals scale, which could quantify radiative leakage from the sterile sector without reconstructing any poles.

Editorial inference: the matrix Cauchy–Schwarz bounds imply a concrete cross-check—lepton-number-conserving data at one derivative order can bound lepton-number-violating data at an adjacent order, so the flavor structure of high-dimension operators is not independent.

Editorial inference: the same paired-kernel machinery should transfer to inverse and linear seesaws by encoding pseudo-Dirac poles in the reservoir kernel, a direction the paper mentions without working out its moment formulation.

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