RELATIONAL ACCESSIBILITY AND PHASE CLOSURE
A Prospective Boundary-Coupling Test for Gravitational-Wave Ringdown
John Swygert | October 16, 2026
Abstract
A useful test of a relational theory must predict a discriminating observable before the relevant data are examined. We construct a prospective gravitational-wave ringdown test motivated by the TSTOEAO emphasis on encoded equilibrium, boundaries, and accessible pathways. Rather than claiming that interference or quadratic quasinormal-mode coupling is new, we define a dimensionless, phase-sensitive coupling ratio, an explicit general-relativistic reference model, and a cross-event closure residual. We derive the invariance of the ratio under a common time-origin shift, identify its failure under independent phase-convention changes, and formulate a conditional prediction: after controlling for remnant parameters and excitation geometry, any additional boundary-mediated contribution must yield a reproducible residual structure across events that cannot be absorbed into conventional nonlinear ringdown physics. We specify model comparisons, noise controls, injection tests, selection criteria, and falsification conditions. The result is a mathematically explicit hypothesis-testing protocol, not a demonstrated deviation from general relativity or a unique derivation from TSTOEAO.
Keywords: TSTOEAO; gravitational waves; black-hole ringdown; quasinormal modes; nonlinear coupling; relational boundaries; phase closure; prospective prediction.
1. Research problem and physical context
The Swygert Theory of Everything AO (TSTOEAO) employs V = E × Y as a relational grammar: realized expression depends not solely on an available input but also on the encoded conditions governing its accessibility. Applied to gravitational-wave science, the productive question is whether the transition from a strongly interacting merger to a settling remnant contains a reproducible, measurable relational signature beyond those predicted by general relativity (GR). A metaphor about hidden pathways is not sufficient. It must be converted into a model with units, nuisance parameters, null predictions, and failure conditions.
Black-hole ringdown is already described by damped quasinormal modes (QNMs); nonlinear GR predicts second-order combination modes and couplings. Thus, neither a new spectral line at a sum frequency nor a nonzero quadratic coupling constitutes evidence for TSTOEAO. Recent LIGO–Virgo–KAGRA tests, including GW250114, remain broadly consistent with Kerr ringdown and GR. The candidate contribution here is a cross-event, phase-sensitive *residual test* against a GR-calibrated coupling prediction, not the rediscovery of mode mixing [1–5].
2. Observable definition
For a fixed mode basis and waveform convention, write the complex analytic ringdown signal as a sum of linear modes and a quadratic contribution:
h(t) = Σₐ cₐ exp(−i ωₐ t) + Σₐᵦ cₐᵦ exp[−i(ωₐ + ωᵦ)t] + n(t). (1)
Here ω = Ω − i/τ denotes a complex QNM frequency, c denotes a complex excitation coefficient, and n includes instrumental noise and model discrepancy. Mode labels include angular indices and overtone labels. Equation (1) is a local fitting representation; it does not assume that every fitted term is separately identifiable in actual data.
For a selected parent pair a,b and daughter ab, define the normalized complex coupling:
Kₐᵦ = cₐᵦ / (cₐ cᵦ), provided |cₐ cᵦ| > 0. (2)
The normalization removes the leading product of parent amplitudes in a quadratic-response approximation. It is not intrinsically detector-independent: angular projection, extraction conventions, mode mixing, source inclination, and fitting choices must be held fixed or modeled. Because strain amplitudes and fitted coefficients depend on normalization conventions, K must be compared only within a consistently calibrated analysis.
Let Kᴳᴿₐᵦ(χ, q, ι, …) be the corresponding prediction or calibrated distribution from second-order perturbation theory and numerical-relativity (NR) injections, conditioned on remnant spin χ, mass ratio q, viewing geometry ι, and relevant excitation parameters. Define the complex residual
Rₐᵦ = Kₐᵦ − Kᴳᴿₐᵦ(χ, q, ι, …). (3)
The null hypothesis is that R is statistically consistent with GR modeling uncertainty and detector noise. A TSTOEAO-motivated alternative permits a structured boundary-coupling term, without asserting its physical existence:
Kₐᵦ = Kᴳᴿₐᵦ + λ F(χ, q, ι, B; θ), (4)
where λ is a dimensionless coupling parameter, B is a specified boundary descriptor and θ is a set of shared parameters. Unless B and F are defined independently of the tested waveform, (4) is only a flexible fit and not a genuine prediction. The minimum viable version therefore uses a pre-registered F, fixed before the holdout events are analyzed.
3. A phase-closure result
Consider changing the common reference time by Δt. In a consistent analytic-mode convention, the coefficients transform as cₐ → cₐ exp(−iωₐΔt), cᵦ → cᵦ exp(−iωᵦΔt), and cₐᵦ → cₐᵦ exp[−i(ωₐ+ωᵦ)Δt]. Substitution into (2) gives
K′ₐᵦ = [cₐᵦ e^(−i(ωₐ+ωᵦ)Δt)] / [cₐcᵦ e^(−i(ωₐ+ωᵦ)Δt)] = Kₐᵦ. (5)
Thus K is invariant to a *common* time-origin shift if the daughter frequency equals the sum of its parent complex frequencies. This is a direct algebraic consequence of the assumed quadratic frequency relation, not a new law of nature. It provides a useful internal consistency check. An independently excited daughter with a shifted frequency will not in general share this cancellation.
Define the phase closure angle
Φₐᵦ = arg(cₐᵦ) − arg(cₐ) − arg(cᵦ) = arg(Kₐᵦ) mod 2π. (6)
A crucial limitation: Φ is not invariant under arbitrary independent rephasings of the mode basis. Physical comparisons require a specified spherical/spheroidal harmonic convention, polarization frame, waveform extraction scheme, and detector response. A purported universal phase without these controls is not meaningful.
4. A concrete, prospective alternative
For an exploratory benchmark only, choose a preselected dimensionless remnant variable u (e.g., remnant spin χ) and a localized complex residual. This shape is not a consequence of TSTOEAO and should not be represented as one:
F(u) = exp{−(u − u₀)² / (2w²)} exp(iφ₀). (7)
The Gaussian profile is an explicitly phenomenological search template. Its center u₀, width w, and phase φ₀ are not predicted by V = E × Y. They must be frozen using an independent physical calculation or a separate training sample. An observed peak would not identify its cause: instrumental, waveform, or astrophysical alternatives must be evaluated.
For a predeclared near-window N and far-window F, define a contrast in the GR-subtracted, inverse-variance-weighted complex residual:
T = mean_w(R | N) − mean_w(R | F). (8)
A nonzero T is necessary but insufficient: detector calibration, waveform systematics, correlated parameter estimation, and selection effects can all produce apparent structure. A physical claim requires posterior predictive checks and independent replications with different detectors and waveform families.
5. Illustrative calculation and sensitivity
Take a hypothetical parent pair with cₐ = 0.20 e^(i0.4), cᵦ = 0.10 e^(−i0.2), and a GR quadratic coefficient Kᴳᴿ = 0.30 e^(i0.5). Their product amplitude is |cₐcᵦ| = 0.020, and the GR daughter amplitude is 0.006 in the same arbitrary, consistently normalized units. If a boundary term of magnitude λ = 0.05 is aligned with the GR coupling, the extra daughter amplitude would be 0.020 × 0.05 = 0.001; the total daughter amplitude would be 0.007. If instead its phase opposed the GR term, the total would be 0.005. These numbers are synthetic examples, not astrophysical forecasts, observed strains, or values derived from TSTOEAO.
More generally, a residual amplitude |δcₐᵦ| = |cₐcᵦ| |λF| can be below present sensitivity even if the corresponding dimensionless coupling is finite. Future detector sensitivity can improve the constraint on λ, but the actual detectability depends on signal-to-noise, spectral overlap, event population, calibration, and parameter degeneracies. No numerical forecast in strain units is justified without a specified detector noise spectrum and source population.
6. Locked prediction protocol
Training stage: freeze the parent–daughter mode triplet using a prespecified selection score computed only on simulated GR injections: projected daughter signal-to-noise, linear-mode leakage, mode-frequency separation relative to damping widths, angular-mode mixing, and stability under start-time variation. Publish the complete candidate list, ranking rule, excluded configurations, and analysis conventions before holdout selection.
Physical specification: define u and a physical boundary descriptor B independently of the tested ringdown; freeze u₀, w, φ₀, λ prior or predicted amplitude range, and complex-residual direction before opening the holdout. Without an independently motivated relation for at least one of these, classify the Gaussian as exploratory, not a TSTOEAO prediction.
Identifiability: conduct GR-only injections across masses, spins, inclinations, mass ratios, sky positions, detector networks, calibration realizations, and ringdown start times. Include linear-mode leakage, quadratic-mode overlap, spherical–spheroidal mixing, extraction uncertainty, and selection bias. Report recovery coverage and the smallest residual that the complete pipeline can reliably distinguish.
Blind holdout: freeze waveform conventions (spin-weighted harmonic normalization and phase, polarization basis, reference time, extraction prescription, detector response, and complex coefficient definition). Fit the preselected events without altering the template; evaluate T with a covariance-aware likelihood, posterior predictive diagnostics, and multiplicity corrections.
Replication: require consistency across detectors, waveform models, independent event groups, and a separately frozen second holdout set.
Falsification: distinguish (i) exclusion of a fixed, preregistered nonzero amplitude/phase alternative at demonstrated sensitivity from (ii) failure to detect an unconstrained λ. A null result constrains a specified model; it cannot disprove all relational boundary hypotheses. Reject apparent discoveries that disappear in GR injections, calibration perturbations, or independent waveform analyses.
7. What would count as a new scientific result?
A new result would require more than statistically unusual residuals. First, an explicit TSTOEAO boundary equation must determine at least one nontrivial quantitative feature of F before the measurement. Second, a GR-plus-known-astrophysics baseline must fail on independent data where the pre-registered TSTOEAO prediction succeeds. Third, the feature must persist under realistic modeling uncertainties and independent replication. A null result is also useful: it rules out the specific coupling model over the tested parameter range, rather than proving that no unknown relational physics exists.
The present paper establishes a possible route from relational language to a constrained gravitational-wave experiment. It does not yet supply a unique field equation, new stress-energy coupling, boundary evolution law, or quantitatively derived non-GR prediction. Those remain necessary to turn this method into a novel physical theory. The distinction matters: an imaginative test design can be worthwhile without being misrepresented as an established discovery.
8. Broader transfer: the relational-closure search
The same construction generalizes to systems in which two independently characterized excitations interact at a boundary. Measure the joint response, subtract the independently validated interaction model, normalize by the parent excitations, and test a preregistered cross-condition residual. Candidate domains include nonlinear optics, coupled mechanical resonators, and biological signaling networks. Their governing equations differ; transferring the *measurement logic* does not establish a universal new force or universal coupling constant. The transferable hypothesis is methodological: unrecognized pathways are most effectively tested through interaction-specific, controlled residuals rather than through unqualified appeals to emergence.
9. Minimal boundary dynamics: a derived toy response
A narrow phenomenological Gaussian is not dictated by relational language. To demonstrate what an actual dynamical derivation would require, introduce a hypothetical damped boundary degree of freedom x driven by the product of two parent-mode excitations: ẋ + (γ + iΩB)x = g a(t)b(t), with γ > 0. Let a(t)=Aa exp(−iωa t), b(t)=Ab exp(−iωb t), and assume a steady forced component x(t)=X exp[−i(ωa+ωb)t]. Then X/(Aa Ab)=g/[γ + i(ΩB−ωa−ωb)]. This follows by substitution, not by invoking V = E × Y. The symbols ωa,b may be complex; the expression is valid for the forced particular solution where the denominator is nonzero and initial transients are separately modeled.
If the observed daughter coefficient receives δcab = ηX, the predicted toy residual is Rtoy = ηg/[γ + i(ΩB−ωa−ωb)] in a fixed extraction convention. For real drive frequency Ωd, this produces |Rtoy| = |ηg|/[γ²+(ΩB−Ωd)²]½ and arg(Rtoy)=arg(ηg)−atan2(ΩB−Ωd,γ). Thus amplitude and phase vary together: a resonance-centered enhancement is accompanied by a dispersive phase rotation. This correlated signature is a stricter target than an arbitrary Gaussian with constant phase.
Crucial limitations: this is a generic driven-oscillator model, not a unique TSTOEAO field equation or evidence of new physics. Conventional nonlinear GR, detector response, and waveform-model errors may produce analogous features. A physical theory must independently identify x, ΩB, γ, g, η, their units and covariance, and their relation to measurable remnant properties. Only then can the response be advertised as a parameter-constrained physical prediction.
10. Convention and identifiability audit
Time-origin invariance of K is conditional on the exact complex-frequency sum relation and consistent coefficient transport. It does not make the phase closure Φ invariant under mode-basis rephasing. Freeze the spin-weighted spherical or spheroidal harmonic basis, azimuthal origin, polarization frame, extraction radius or null-infinity convention, complex strain sign, detector projection, and normalization. Apply the same transformations to measured K and GR reference KGR; test the covariance of R under allowed coordinate/basis changes.
Preselect modes only after simulation-based ranking of expected daughter recoverability against contamination. Quantify the condition number of the mode-fit design matrix and leakage of parent fundamentals and overtones into each daughter template. If the daughter cannot be separately identified at the preregistered threshold, report a non-informative event rather than interpreting an unstable phase.
11. Prospective discrimination and power
The target is not a nominal residual magnitude but a forecast for a complete experiment. For complex residual vector r and covariance C estimated from injections, an illustrative matched-template squared signal-to-noise is ρ² = f† C⁻¹ f |λ|², where f is a fixed candidate pattern over events and † denotes conjugate transpose. This Gaussian-noise expression is only a planning approximation; correlated systematics, uncertainty in KGR, event selection, and non-Gaussian posteriors require full end-to-end injection tests. No detector-era detection threshold is claimed here.
A decisive study should pre-register: selected modes and event inclusion rules; the GR reference ensemble and its uncertainty; all phase conventions; boundary descriptor and any toy-model parameter priors; injection-derived minimum detectable effect; one primary complex-valued test statistic; trial-factor treatment; held-out event set; replication criteria; and explicit null-result interpretation. The Gaussian template and oscillator response must be evaluated as separate alternatives rather than retroactively interchanged.
12. Conclusion
A normalized complex coupling offers a disciplined route to search for departures from a GR-calibrated nonlinear ringdown baseline. Its common-time-origin invariance is algebraic, and its phase requires stringent convention control. We now distinguish a freely specified Gaussian search template from a minimally derived, but nonunique, damped-boundary toy response whose amplitude and phase are linked. Neither is yet a prediction derived from TSTOEAO. The decisive next advance is an independently justified boundary dynamics with frozen parameters, demonstrated identifiability, and a blinded, replicable test against GR and instrumental alternatives.
References
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