Halilsoy-Inspired Residual Extension

• The paper introduces a Halilsoy-inspired residual extension to the Newtonian lunar tidal tensor by incorporating an α-dependent off-diagonal coefficient, χ_H, that rotates the eigenframe.
• The methodology leverages insights from cross-polarized cylindrical gravitational waves to add a distinctive sin(2β) component while preserving the 90° orthogonality of principal axes.
• Magnitude estimates indicate that plausible residuals (|χ_H| ≈ 10⁻³–10⁻²) yield sub-microGal signatures, enabling detection with modern superconducting gravimeters.

Searching arXiv for the specified papers and related Halilsoy context. Halilsoy-Inspired Residual Extension denotes a phenomenological extension of the local Newtonian lunar tidal tensor in which an $\alpha$-dependent off-diagonal residual coefficient, $\chi_H(\alpha,t,\rho)$, is introduced as a testable cross-channel absent from the diagonal Newtonian principal-frame description. In the formulation proposed in "Alpha-Dependent Cross-Tidal Residuals Beyond the Diagonal Newtonian Lunar Tensor: A Halilsoy-Inspired 45° Eigenframe Channel" (Eser et al., 20 May 2026), the dominant lunar tide remains Newtonian, while the added sector is motivated by the off-diagonal tidal structure of Halilsoy’s cross-polarized cylindrical gravitational waves. The extension does not replace standard lunar tidal theory and does not identify the Earth–Moon system with a Halilsoy spacetime; instead, it imports a specific relativistic mechanism as a residual ansatz, preserving the ordinary $90^\circ$ orthogonality of principal axes while rotating the eigenframe and generating a distinct $\sin(2\beta)$ angular signature with extrema at $45^\circ$, $135^\circ$, $225^\circ$, and $315^\circ$ (Eser et al., 20 May 2026).

1. Newtonian baseline and the meaning of the extension

In Newtonian gravity the leading lunar tide at the Earth’s center comes from the Hessian of the lunar potential

$$\Phi_M(\mathbf r)\;=\;-\,\frac{G\,M_M}{|\mathbf D-\mathbf r|}\,,\qquad|\mathbf r|\ll D\,.$$

Expanding to second order in $\mathbf r$ gives the tidal potential

$\chi_H(\alpha,t,\rho)$0

In the local frame with $\chi_H(\alpha,t,\rho)$1-axis along $\chi_H(\alpha,t,\rho)$2 and $\chi_H(\alpha,t,\rho)$3 transverse, the two-dimensional tidal tensor is

$\chi_H(\alpha,t,\rho)$4

This tensor is symmetric, traceless, and diagonal. Its eigenvalues are

$\chi_H(\alpha,t,\rho)$5

with orthogonal eigenvectors along $\chi_H(\alpha,t,\rho)$6 and $\chi_H(\alpha,t,\rho)$7, corresponding respectively to stretching and squeezing (Eser et al., 20 May 2026).

Within this baseline description, no off-diagonal term appears in the principal frame, and the familiar $\chi_H(\alpha,t,\rho)$8 separation of axes follows immediately. A projected acceleration can be evaluated along any direction, including the $\chi_H(\alpha,t,\rho)$9 direction, but in the Newtonian principal frame such a projection is not an independent cross-tidal residual. The Halilsoy-inspired residual extension is therefore defined precisely by the introduction of a new off-diagonal sector beyond the diagonal Newtonian principal-frame tensor, not by a mere re-expression of the standard quadrupolar tide in rotated coordinates (Eser et al., 20 May 2026).

This distinction is central. The proposal is not that standard Newtonian theory secretly contains a separate cross mode, but that one may phenomenologically test for an additional residual structure that would manifest as a rotated eigenframe and a sine-quadrature angular component orthogonal to the ordinary plus-type pattern.

2. Halilsoy motivation: off-diagonal tidal structure in cylindrical waves

The motivating mechanism comes from Halilsoy’s cross-polarized cylindrical gravitational waves. In general relativity, a weak gravitational wave in transverse-traceless gauge produces a tidal tensor

$90^\circ$0

whose plus-polarization is diagonal in some frame and whose cross-polarization appears as equal off-diagonal entries (Eser et al., 20 May 2026).

A concrete exact solution carrying both polarizations is given by Halilsoy’s cross-polarized cylindrical wave. In a local orthonormal cylindrical frame $90^\circ$1, the transverse $90^\circ$2 tidal block reads

$90^\circ$3

where

$90^\circ$4

The off-diagonal entry is

$90^\circ$5

which is nonzero whenever $90^\circ$6 and $90^\circ$7 (Eser et al., 20 May 2026).

Because this is a symmetric $90^\circ$8 block, its principal-axis rotation angle $90^\circ$9 satisfies

$\sin(2\beta)$0

As $\sin(2\beta)$1 grows large compared to $\sin(2\beta)$2, one drives $\sin(2\beta)$3 while preserving the $\sin(2\beta)$4 separation of the two principal axes (Eser et al., 20 May 2026).

The import of this construction is algebraic rather than ontological. The Earth–Moon system is not modeled as a cylindrical-wave spacetime. Instead, Halilsoy’s wave provides an example in which an off-diagonal tidal entry does not destroy orthogonality but rotates the local eigenframe. That is the mechanism adopted as a guide.

3. Definition of the $\sin(2\beta)$5-dependent residual coefficient $\sin(2\beta)$6

The simplest phenomenological two-dimensional ansatz proposed for the lunar tidal tensor is

$\sin(2\beta)$7

For a nonzero off-diagonal coefficient $\sin(2\beta)$8, the general relation

$\sin(2\beta)$9

implies that the entire eigenframe is rotated by

$45^\circ$0

(Eser et al., 20 May 2026).

Rather than leaving $45^\circ$1 as an arbitrary constant, the construction matches the Newtonian-side rotation ratio to the Halilsoy wave ratio: $45^\circ$2 This defines the effective Halilsoy-induced residual

$45^\circ$3

The full extended tensor is then

$45^\circ$4

with

$45^\circ$5

In this sense, the extension is a residual addition to the Newtonian tensor rather than a reformulation of it (Eser et al., 20 May 2026).

A plausible implication is that the formal role of $45^\circ$6 is to parameterize deviations from the purely plus-aligned Newtonian principal frame in a way that is directly tied to an explicit off-diagonal tidal mechanism rather than to an unconstrained phenomenological fit.

4. Eigenframe rotation and spectral structure

For a symmetric matrix

$45^\circ$7

the eigenvalues are

$45^\circ$8

and the principal-axis rotation obeys $45^\circ$9. Applying this to the extended lunar tensor with $135^\circ$0, $135^\circ$1, and $135^\circ$2 yields

$135^\circ$3

and

$135^\circ$4

(Eser et al., 20 May 2026).

Because the tensor remains symmetric, the two eigenvectors remain orthogonal. The extension therefore does not alter the $135^\circ$5 separation of principal axes; it rotates the entire eigenframe away from the original plus-aligned frame. In the cross-dominant regime $135^\circ$6, one has $135^\circ$7 (Eser et al., 20 May 2026).

This feature resolves a potential misconception. The residual channel is not introduced by replacing the ordinary lunar geometry with a fundamentally different non-orthogonal structure. The proposal preserves the symmetric-tensor geometry of principal directions and modifies only their common orientation and the directional decomposition of the projected acceleration.

5. Projected acceleration and the $135^\circ$8 residual channel

Let

$135^\circ$9

be a unit horizontal direction making angle $225^\circ$0 with the Earth–Moon axis. At the surface, with $225^\circ$1, the tidal acceleration along $225^\circ$2 is

$225^\circ$3

Defining the lunar-tide scale

$225^\circ$4

one obtains

$225^\circ$5

The first two terms are the standard plus-type projection, with peak-to-peak amplitude $225^\circ$6, while the last term is a pure $225^\circ$7 residual (Eser et al., 20 May 2026).

Writing

$225^\circ$8

the extrema of the residual occur where $225^\circ$9, namely at

$315^\circ$0

In particular, at $315^\circ$1,

$315^\circ$2

(Eser et al., 20 May 2026).

The significance of this decomposition lies in the orthogonality of the sine-quadrature channel to the usual plus channel. The proposal therefore singles out a directional fingerprint that is not equivalent to a rescaling, phase shift, or coordinate rotation of the standard Newtonian projection.

6. Magnitude estimates, observational strategy, and relation to other Halilsoy extensions

Using

$315^\circ$3

the paper states that a residual coefficient $315^\circ$4 corresponds to

$315^\circ$5

and that for $315^\circ$6 one gets

$315^\circ$7

(Eser et al., 20 May 2026).

The bridge formula

$315^\circ$8

shows three enhancement channels: large $315^\circ$9, spatial “resonances” where $$\Phi_M(\mathbf r)\;=\;-\,\frac{G\,M_M}{|\mathbf D-\mathbf r|}\,,\qquad|\mathbf r|\ll D\,.$$0 is big, and temporal phases near $$\Phi_M(\mathbf r)\;=\;-\,\frac{G\,M_M}{|\mathbf D-\mathbf r|}\,,\qquad|\mathbf r|\ll D\,.$$1. In the weak-cross limit $$\Phi_M(\mathbf r)\;=\;-\,\frac{G\,M_M}{|\mathbf D-\mathbf r|}\,,\qquad|\mathbf r|\ll D\,.$$2, since $$\Phi_M(\mathbf r)\;=\;-\,\frac{G\,M_M}{|\mathbf D-\mathbf r|}\,,\qquad|\mathbf r|\ll D\,.$$3,

$$\Phi_M(\mathbf r)\;=\;-\,\frac{G\,M_M}{|\mathbf D-\mathbf r|}\,,\qquad|\mathbf r|\ll D\,.$$4

so the residual is linear in $$\Phi_M(\mathbf r)\;=\;-\,\frac{G\,M_M}{|\mathbf D-\mathbf r|}\,,\qquad|\mathbf r|\ll D\,.$$5 (Eser et al., 20 May 2026).

For plausible magnitudes, the paper states that a conservative bound might take $$\Phi_M(\mathbf r)\;=\;-\,\frac{G\,M_M}{|\mathbf D-\mathbf r|}\,,\qquad|\mathbf r|\ll D\,.$$6, Bessel-factor $$\Phi_M(\mathbf r)\;=\;-\,\frac{G\,M_M}{|\mathbf D-\mathbf r|}\,,\qquad|\mathbf r|\ll D\,.$$7, and phase $$\Phi_M(\mathbf r)\;=\;-\,\frac{G\,M_M}{|\mathbf D-\mathbf r|}\,,\qquad|\mathbf r|\ll D\,.$$8, giving $$\Phi_M(\mathbf r)\;=\;-\,\frac{G\,M_M}{|\mathbf D-\mathbf r|}\,,\qquad|\mathbf r|\ll D\,.$$9. This would imply

$\mathbf r$0

or a few $\mathbf r$1Gal. More realistically, one might expect $\mathbf r$2–$\mathbf r$3, corresponding to sub-$\mathbf r$4Gal residuals (Eser et al., 20 May 2026).

The proposed observational strategy is likewise explicit. Modern superconducting gravimeters routinely achieve sub-$\mathbf r$5Gal precision. The procedure consists of: modeling and subtracting the standard Newtonian lunar and solar tides plus ocean-loading, atmospheric, hydrological, solid-Earth response, instrumental drifts, and related effects; expressing the residual as

$\mathbf r$6

and checking whether $\mathbf r$7 and mapping $\mathbf r$8 (Eser et al., 20 May 2026). Because the sine-quadrature channel is orthogonal to the usual plus channel, it cannot be absorbed into errors in the plus-type tidal model. A statistically significant nonzero $\mathbf r$9 with the predicted $\chi_H(\alpha,t,\rho)$00 extremal directions would directly test the ansatz (Eser et al., 20 May 2026).

In a broader Halilsoy context, the phrase “Halilsoy-inspired” should not be conflated with other Halilsoy constructions. "On the properties of a deformed extension of the NUT space-time" (Narzilloev et al., 2020) discusses a stationary extension of the Zipoy–Voorhees metric linked to NUT spacetime, where the parameter $\chi_H(\alpha,t,\rho)$01 is interpreted as a gravitomagnetic or NUT-type charge rather than an ordinary rotation parameter. That work concerns a two-parameter deformation of NUT with altered geodesic structure and closed time-like curves (Narzilloev et al., 2020). The lunar residual model, by contrast, uses Halilsoy inspiration specifically at the level of off-diagonal tidal structure and eigenframe rotation, not at the level of identifying the Earth–Moon system with a Halilsoy or NUT spacetime. This suggests that the common thread across these distinct usages is the appearance of nontrivial off-diagonal sectors with geometric consequences, although the physical settings and intended applications are different.