Keywords: GW170817, neutron stars, tidal deformability, compactness, structural bounds, envelope methods, Pattern Field Theory, dimensionless constants, fine structure constant, alpha constant.

Problem Statement

Current neutron star inference pipelines produce multi-dimensional posterior distributions over masses, radii, and tidal deformabilities. While such representations are information-rich, they do not directly expose whether the physically admissible region is merely observationally truncated or intrinsically compact.

This work addresses the following question:

Does the GW170817-constrained neutron star state space exhibit a sharp, data-defined structural upper boundary when projected onto a suitable monotone scalar?

We answer this question empirically by defining a scalar envelope functional and extracting its high-quantile boundary across multiple public posterior sets.

Data Sources

We use the public posterior samples from the gw170817-eft-eos dataset in the following categories:

• Mass priors: dns_mass_prior, uniform_mass_prior

• Constraint sets: posterior, posterior_mthresh, posterior_mthresh_maxmass

Each posterior sample contains component masses \(m_1, m_2\), radii \(R_1, R_2\), and tidal deformabilities \(\Lambda_1, \Lambda_2\).

We define per-sample averages: \[C = \frac{1}{2}\left(\frac{G m_1}{R_1 c^2} + \frac{G m_2}{R_2 c^2}\right), \qquad \Lambda = \frac{1}{2}(\Lambda_1 + \Lambda_2).\]

The GW170817 event and its associated inference products are documented extensively in the LIGO/Virgo discovery and follow-up literature, including detection significance, sky localization, mass posteriors, spectrograms, and multi-messenger counterparts. Representative examples include the original detection paper, the neutron-star EOS inference paper, and the public data release documentation. The present work does not reanalyze the raw strain data or detection pipeline outputs, but operates entirely on the released EOS posterior products derived from these analyses.

Definition of the Envelope Functional

Two normalization schemes are used:

• Per-file normalization: \(k = \mathrm{median}(C)\sqrt{\mathrm{median}(\Lambda)}\)

• Fixed normalization: \(k = 2.9802987367\), taken from the dns_mass_prior/posterior_mthresh_maxmass dataset

For each posterior, we define the envelope statistic: \[L_{\text{edge}}(q) = \mathrm{quantile}_q(L), \qquad q \in \{0.99, 0.995, 0.999\}.\]

Results

The full scan across six posterior files yields the following results for the stability-constrained sets under fixed normalization:

\[L_{\text{edge}}(0.995) \in [0.442, 0.450].\]

We therefore define the data-derived structural ceiling:

\[L_{\max}(D_{\mathrm{ref}}) = 0.446 \pm 0.004.\]

Structural Ceiling from GW170817

The scalar envelope statistic \(L = C + k / \sqrt{\Lambda}\) reveals a sharp upper boundary in compactness–tidal deformability space. Using public GW170817 posteriors under threshold-mass and maximum-mass stability constraints, the high-quantile envelope is stable and prior-robust, giving \[L_{\max}(q = 0.995) = 0.446 \pm 0.004.\] This value lies within \(0.6\%\) of the PFT-predicted structural ceiling \(\dfrac{\pi}{7} \approx 0.4488\), derived from the 6-fold projection structure of the Allen Orbital Lattice (6 sectors plus one EQUI axis) normalized over a \(\pi\)-radian span. The empirical bound provides a single-number constraint that any viable equation-of-state model must respect, and the close match to \(\pi/7\) supports the admissibility-based unification framework.

Across quantiles from \(0.99\) to \(0.999\), the envelope shifts by less than \(0.02\), indicating a thin-tailed, sharply bounded distribution.

Unconstrained posteriors produce significantly higher envelope values, demonstrating that the ceiling is not an artifact of the functional form but of the physical stability constraints.

Interpretation

The existence of a sharp upper envelope in a monotone scalar projection implies that the GW170817-constrained neutron star state space occupies a compact admissible region rather than a diffuse or weakly truncated domain.

This statement is purely empirical and independent of any particular microscopic equation of state.

The scalar envelope statistic provides a reproducible method for:

• Tracking structural bounds across events

• Comparing inference pipelines

• Testing convergence toward a universal ceiling

In Pattern Field Theory, such structural bounds are interpreted geometrically via admissibility projections (EQUI) onto stable basins of the Allen Orbital Lattice; a schematic illustration of this mechanism is provided in companion work.

PFT Interpretation: Structural Link to the Fine Structure Constant

The empirical ceiling \(L_{\max} \approx 0.446 \pm 0.004\) aligns with Pattern Field Theory (PFT), in which compactness–tidal space is interpreted as a projection of the 6-fold Allen Orbital Lattice (AOL). In this framework, the structural ceiling arises as the purely geometric bound \(\dfrac{\pi}{7} \approx 0.4488\) from AOL projection symmetry.

Independently, PFT proposes a candidate geometric origin for the fine structure constant \(\alpha \approx 1/137\) from 6-fold QuantaHex quantization, schematically expressed as \[\alpha \approx \frac{\ln 6}{RT \cdot 6},\] where \(RT \ln 6\) represents a fundamental partition scale in the AOL construction. Although no direct physical identification is claimed in the present work, the appearance of the same 6-fold structural arity in both bounds suggests that the neutron-star ceiling and dimensionless constants may reflect a common underlying geometric constraint in PFT. A full treatment of the fine structure constant lies outside the scope of this paper and is deferred to dedicated work.

Conclusion

We have demonstrated the existence of a data-defined structural ceiling in compactness–tidal space using GW170817 EOS-inference posteriors. The ceiling is sharp, stable, and prior-robust under fixed normalization, with \[L_{\max}(q=0.995) \approx 0.446 \pm 0.004.\] This establishes a falsifiable, event-testable structural bound derived directly from data.

In the broader Pattern Field Theory framework, such structural ceilings are interpreted as manifestations of underlying geometric admissibility constraints. Similar constraints appear in independent candidate treatments of dimensionless constants, including the fine structure constant, although no such derivation is claimed or required for the present result. Future multi-messenger observations will determine whether the neutron-star ceiling reported here is a universal feature of compact-object structure and whether it participates in a deeper unifying geometric pattern.

Supplementary Material

The following supplementary figure is provided to demonstrate the robustness of the envelope ceiling across individual posterior datasets and prior variants used in this study.

• Figure S1. Per-dataset envelope statistics \(L_{\mathrm{edge}}(q=0.995)\) across all posterior files used in the analysis. The narrow spread relative to the global ceiling demonstrates that the observed structural bound is not dominated by any single dataset or prior choice.

Glossary

• \(C\) (Compactness). The dimensionless relativistic compactness parameter, defined as \[C = \frac{G M}{R c^2},\] where \(M\) is the gravitational mass, \(R\) the circumferential radius, \(G\) the gravitational constant, and \(c\) the speed of light. It measures how close a star is to relativistic gravitational collapse.

• \(\Lambda\) (Dimensionless tidal deformability). The dimensionless quadrupolar tidal deformability parameter that characterizes how easily a neutron star is deformed by an external tidal field during inspiral. It is related to the Love number and scales approximately as \((R/M)^5\).

• \(L\) (Scalar load functional). The monotone scalar envelope functional defined in this work by \[L = C + \frac{k}{\sqrt{\Lambda}},\] which compresses the two-dimensional compactness–tidal deformability space into a single dimensionless load coordinate.

• \(k\) (Normalization constant). A positive scalar chosen to fix the relative scaling between the compactness and tidal terms in the load functional. In this work both per-file normalization and fixed normalization schemes are used.

• \(L_{\mathrm{edge}}(q)\) (Envelope statistic). The high-quantile envelope statistic defined as the \(q\)-quantile of the empirical distribution of \(L\) over a posterior sample. It estimates the location of the upper structural boundary in load space.

• \(L_{\max}\) (Data-derived structural ceiling). The inferred global upper bound of the envelope statistic under physically constrained posteriors, reported in this work as \[L_{\max}(q=0.995) \approx 0.446 \pm 0.004.\] This value represents an empirically observed structural ceiling in compactness–tidal space.

• Envelope (or envelope boundary). The sharp upper boundary of the distribution of the scalar load functional induced by physical stability constraints, as revealed by high-quantile statistics.

• EOS (Equation of State). The microscopic relation between pressure, density, and composition of neutron star matter, which determines the mass–radius and tidal response of neutron stars.

• GW170817. The first observed binary neutron star merger detected by LIGO and Virgo, providing joint gravitational-wave constraints on neutron star masses, radii, and tidal deformabilities.

• Admissible region. The subset of the neutron star state space (in compactness–tidal coordinates) that is consistent with both observational data and physical stability constraints.

References

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Document Timestamp and Provenance

This document is part of Pattern Field Theory (PFT) and the Allen Orbital Lattice (AOL). It defines a data-derived structural envelope statistic and specifies methods and replication procedures used by subsequent papers in the series. Pattern Field Theory (PFT) and related marks are claimed trademarks. Any research, derivative work, or commercial use requires an explicit license from the author.