Gravitational Composition and Relational Invariance

From Galactic Surface-Density Regularities to a Falsifiable TSTOEAO Gravitational Research Program

John Swygert

September 24, 2026

Abstract

Recent work on Verlinde-inspired emergent gravity has renewed attention to a long-standing galactic regularity: the characteristic central surface density inferred for dark-matter halos. Youngsub Yoon reported in 2026 that a modified emergent-gravity construction yields a characteristic phantom-halo surface-density scale consistent with the observational scale previously reported for galaxies. The calculation builds on Yoon’s 2024 proposal that baryonic and apparent-dark gravitational accelerations combine quadratically rather than additively. This paper asks a prior question: does the existing mathematical architecture of TSTOEAO independently select any gravitational composition law? The answer is presently no. Encoded Equilibrium, EC-1/EC-2, the Universal Coordinate Principle, relational form invariance, dynamic container encoding, and the recently constructed relational inverse operator provide machinery for declaring states, extracting histories, testing observability, transporting structural criteria, and preserving failure conditions. They do not yet supply a gravitational field equation, relativistic metric, matter-coupling prescription, or unique composition operator. Rather than hiding that boundary, the present paper converts it into a falsifiable research program. Additive, quadratic, generalized inner-product, and nonlinear composition are separated as competing hypotheses. The relational inverse operator T_rel is incorporated as a possible observational and identifiability layer—not as a substitute for gravitational dynamics. The paper establishes tests involving rotation curves, surface-density scaling, gravitational lensing, environment, wide binaries, clusters, and cross-domain observability. A TSTOEAO gravitational domain module is therefore mathematically formulable but remains exploratory until an independently motivated response law produces locked quantitative predictions that survive comparison with established gravitational models.

Keywords: TSTOEAO; Encoded Equilibrium; gravitational composition; relational invariance; relational inverse operator; emergent gravity; dark matter; surface density; observability; gravitational lensing; falsifiability

1. Introduction

Galactic dynamics presents a persistent interpretive problem. Observed orbital accelerations in galaxies do not, under ordinary Newtonian inference from visible baryonic matter alone, generally match the accelerations required by measured rotation curves. The standard cosmological interpretation introduces non-baryonic dark matter whose gravitational contribution depends on halo structure and galaxy-formation history. Modified-gravity and emergent-gravity approaches instead ask whether at least part of the discrepancy reflects a change in the effective gravitational description.

A particularly interesting empirical regularity concerns the product of a fitted halo core radius and central density, commonly written μ₀D = r₀ρ₀. Donato and collaborators reported an approximately constant characteristic central dark-matter surface density across a broad range of galaxies. The exact universality, population dependence, profile dependence, and interpretation remain empirical questions; nevertheless, the existence of a characteristic scale is sufficiently persistent to motivate theoretical explanation.

In 2026, Youngsub Yoon reported that a Verlinde-inspired emergent-gravity construction predicts a characteristic phantom-halo surface-density scale near the observed galactic scale. The published report gives approximately 170 M☉ pc⁻², while the cited observational range is approximately 90–220 M☉ pc⁻². The calculation uses a gravitational relation developed in Yoon’s 2024 work in which total acceleration is related to baryonic and apparent-dark components quadratically rather than additively.

The purpose of the present paper is not to adopt that result. It is to use it as a stringent external test of a separately developed theoretical program. TSTOEAO has recently progressed from gravitational-well and nested-container intuitions toward relational form invariance, dynamic container encoding, relational observability, and an explicit candidate relational inverse operator. Those developments sharpen the question: can the existing framework independently determine how gravitational contributions compose, or can it at least determine which hidden gravitational decompositions are identifiable from observed geometric and dynamical histories?

The distinction is decisive. A theory that selects a quadratic rule only after learning that the rule reproduces a desired astronomical scale is accommodating the observation, not predicting it. A theory that independently derives a composition operator and then survives withheld tests is doing something substantially stronger. This paper therefore begins from a null position: no gravitational composition law is granted in advance.

2. Empirical Motivation: Galactic Surface Density

The characteristic halo surface-density quantity is commonly defined as

μ₀D ≡ r₀ ρ₀.

It has dimensions of mass per area. Multiplication by Newton’s constant therefore produces an acceleration scale:

G μ₀D ~ acceleration.

This dimensional correspondence is mathematically genuine but physically incomplete. It does not determine the coefficient, the applicable galaxy population, the halo-profile dependence, the role of baryonic geometry, or the behavior in environments outside ordinary rotationally supported galaxies.

Yoon’s 2026 analysis reports an analytic emergent-gravity estimate μ₀D ≈ 10²·²⁴ M☉ pc⁻², consistent with the earlier observational estimate near 10²·¹⁵ M☉ pc⁻² within the quoted uncertainties. This numerical agreement motivates investigation, but agreement with one characteristic scale cannot determine the underlying gravitational ontology.

3. Yoon’s Quadratic Composition Proposal

Yoon’s 2024 analysis proposed that total acceleration g, baryonic Newtonian acceleration g_B, and an apparent-dark acceleration g_D satisfy

g² = g_B² + g_D²,

rather than the additive expression

g = g_B + g_D.

The quadratic relation was derived within a specific treatment of Verlinde-inspired emergent gravity and subsequently used in the 2026 surface-density calculation. It is therefore a model-dependent physical result, not a generic identity obtained merely by naming two gravitational contributions.

The TSTOEAO question is consequently stricter: does any independently developed TSTOEAO principle require quadratic composition, additive composition, or another rule? At present, no.

4. Existing TSTOEAO Architecture

4.1 Encoded Equilibrium and EC-1/EC-2

TSTOEAO summarizes Encoded Equilibrium through the structural grammar

V = E × Y.

EC-1 and EC-2 impose discipline on declared states, routes, receivers, boundaries, and observable outcomes. This architecture is valuable for preregistration and for preventing definitions from changing after results are known. It does not, without additional physical identifications, specify a gravitational potential, metric, stress-energy tensor, acceleration field, action, or matter coupling.

4.2 Universal Coordinate Principle and relational form invariance

The Universal Coordinate Principle motivates the search for relational quantities preserved across admissible descriptions. In the gravitational branch, this can be represented schematically by a common domain G_T, model overlays M_D, and a candidate transported invariant I_R:

M_D₁ → G_T ← M_D₂

I_R(M_D₁) = I_R(M_D₂), when a valid transport map exists.

The principle supplies a reason to search for invariant relationships between dark-matter, emergent-gravity, modified-gravity, and other descriptions. It does not itself supply the missing field equation or composition operator. Recent TSTOEAO work therefore shifted away from coordinate-dependent ‘relational compression’ toward relational form invariance: the target must be a structure whose definition survives the permitted change of description.

4.3 Dynamic Container Encoding

Dynamic Container Encoding introduced a temporal extension. Instead of asking whether a final geometry resembles the environment that produced it, the framework asks whether the history of an observed system retains information about hidden outer conditions acting through nested transformations. The generic chain is

c₂(t) → c₁(t) → x(t) → Γ_G,

where Γ_G is the observed geometric history and c₂ is a hidden outer drive. This is directly relevant to gravitational inference because astronomical observations frequently provide effects—motions, lensing, spatial distributions, histories—from which hidden causal structure is inferred.

4.4 Candidate relational inverse operator

The recent paper From Dynamic Container Encoding to a Candidate Relational Inverse Operator converted the container intuition into an explicit computational object. Its candidate map is

T_rel = W* ∘ Π_out ∘ 𝒟_L ∘ Φ,

where Φ converts observed geometry into dimensionless relational features, 𝒟_L forms a delay embedding, Π_out isolates a preregistered persistent-mode subspace, and W* maps the retained relational coordinate to a selected descriptor of the hidden outer drive. Thus

Γ_G → T_rel → θ̂₂ → Q̂_held-out.

The same work defines a relational observability matrix and conditioning criterion. In reduced coordinates,

O_R(q) = [C_r; C_r A_r; C_r A_r²; … ; C_r A_r^(q−1)],

ρ_R = σ_min(O_R|S_out) / σ_max(O_R|S_out).

This construction is explicitly provisional and deliberately close to established delay-coordinate, Koopman, inverse, and observability methods. Its scientific value is not that it automatically creates a new invariant, but that it makes the hidden-cause inference problem executable and exposes it to baseline comparison.

4.5 Relational information accounting: a necessary correction

Subsequent TSTOEAO work reported an important negative result: the first singular-value-ratio observability quantity was not stable under changes of sampling, delay depth, noise, and representation, and the first inverse operator did not consistently outperform conventional delay-based reconstruction. This correction is central to the present gravitational program. T_rel and ρ_R cannot be treated as established universal invariants. They are candidate tools whose conditioning, identifiability, and transfer properties must be tested again in any gravitational application.

This negative result strengthens rather than weakens the methodological requirement: a gravitational module must preserve failure boundaries instead of renaming unstable quantities as invariants.

5. The Gravitational Composition Problem

Let g_B denote acceleration associated with measured baryonic matter and let g_X denote an additional effective contribution. The symbol g_X is deliberately neutral: it could represent dark matter, apparent-dark gravity, a boundary response, a new field, or another effective term.

The most general starting point is

g_obs = 𝒞(g_B, g_X; Θ),

where 𝒞 is a composition operator and Θ contains any required constants, geometry, environmental variables, or state parameters. Existing TSTOEAO principles do not uniquely determine 𝒞.

Candidate	Representative form	Independent justification required
Additive	g = g_B + g_X	Linear field equation or equivalent superposition principle
Quadratic	g² = g_B² + g_X²	Derived physical orthogonality, finite-boundary law, or equivalent mechanism
General inner-product	g² = g_B² + g_X² + 2g_Bg_X cos θ_R	Defined relational space, metric/inner product, and physical meaning of θ_R
Nonlinear	F(g,g_B,g_X;Θ)=0	Action, constitutive law, field equation, or independently motivated response law

6. Mathematical Identity Versus Physical Law

For vectors A and B in an inner-product space,

||A + B||² = ||A||² + ||B||² + 2 A·B.

If A·B = 0, the magnitudes combine quadratically. That is a mathematical identity under an orthogonality condition. It does not establish that baryonic gravity and an apparent-dark contribution are orthogonal physical channels.

An exploratory generalized form is

g² = g_B² + g_X² + 2 g_B g_X cos θ_R.

This equation is not proposed here as TSTOEAO gravity. It exposes what a derivation would require. A physical theory would need to define the relational space, define θ_R, derive its value or evolution, and explain when the cross term vanishes. Selecting θ_R = π/2 solely because it reproduces Yoon’s quadratic relation would reverse the required direction of inference.

Statement	Classification
Vector-sum norm contains a dot-product cross term	Mathematical identity
Orthogonal components combine quadratically in norm	Mathematical identity under an orthogonality condition
Newtonian gravitational potentials superpose	Physical consequence of a linear field equation
Baryonic and apparent-dark accelerations combine quadratically	Model-dependent physical claim
A nonlinear gravitational interpolation equation applies	Physical postulate or consequence of a specified action
Different gravitational descriptions preserve a common observable	Possible correspondence requiring a defined mapping

7. What the Relational Inverse Operator Adds

The inverse-operator work does not solve the composition problem, but it changes how the problem can be attacked. A gravitational theory must distinguish two questions that are often conflated:

Dynamics: what equations determine the gravitational response?
Identifiability: given the observables, can the hidden contributions or boundary conditions be uniquely or stably reconstructed?

T_rel belongs to the second category. It may help test whether a hidden gravitational descriptor leaves a recoverable trace in a time-resolved or population-resolved observable history. It cannot substitute for the first category.

This distinction suggests a new gravitational inference chain:

hidden gravitational/boundary state → physical response law → observable history Γ_G → relational inference → hidden-state estimate.

If two competing physical decompositions generate observational histories that are indistinguishable within measurement precision, no inverse operator can recover information that the forward dynamics have erased. This yields a useful information-theoretic constraint:

recoverable hidden gravitational information ≤ information retained in Γ_G.

Consequently, non-observability is itself a scientific result. A viable TSTOEAO gravitational program should predict where reconstruction becomes rank-deficient, poorly conditioned, or degenerate.

8. From Inverse Observability to Composition-Law Testing

A practical program can now be stated. Rather than fitting one composition law to all galaxies and declaring success, construct competing forward models with frozen composition rules, generate their predicted observable histories, and ask whether relational observables distinguish them out of sample.

For each model H_k,

H_k : {baryons, boundary state, environment} → Γ_G^(k).

A relational feature map then produces

I_R^(k) = Ψ[Γ_G^(k)],

where Ψ may use normalized curvature, acceleration-profile shape, tidal structure, lensing profile, temporal evolution, or other preregistered observables. The scientific question is not whether a classifier can be trained after seeing all outcomes. It is whether a structural distinction derived or calibrated in one regime predicts discrimination in a withheld regime.

The inverse-operator paper’s strongest proposed test—zero-shot structural transfer—therefore has a direct gravitational analogue. Calibrate a normalized observability or distinguishability boundary in one gravitational regime, freeze it, and test whether it prospectively identifies distinguishable versus degenerate composition laws in another regime without redesign.

9. Requirements for an Independent TSTOEAO Gravitational Derivation

A genuine derivation must proceed in the following direction:

Independently justified physical postulates.
A closed set of equations with explicit source, matter coupling, and boundary conditions.
A derived composition operation or field response.
Solutions for independently specified galactic mass distributions.
A predicted acceleration profile and characteristic surface-density scale without using that scale as calibration.
Relativistic completion sufficient to predict photon trajectories and gravitational lensing.
Locked predictions for observations not used in construction.
Declared failure conditions that remain failures after the data are revealed.

If the theory predicts

μ_char = C a_*/G,

both C and a_* must be derived or independently fixed. If either is selected using the target galactic surface-density relation, the calculation is a fit rather than an independent prediction.

10. Candidate Physical Extensions

10.1 Boundary-dependent response

g = 𝒞(g_B, g_X; B).

The boundary state B must be independently measurable and specified without using the unexplained acceleration itself. Candidate observables could include external gravitational field, baryonic truncation scale, tidal environment, or another preregistered structural descriptor. A valid model must predict the sign and magnitude of the boundary effect before confirmatory data are inspected.

10.2 Cosmologically determined acceleration scale

a_*(z) = F[H(z), c, G, …].

If a characteristic acceleration is cosmologically determined, the functional relation must be fixed and should imply redshift evolution. That immediately produces a falsifiable prediction.

10.3 Additional dynamical field

A new field requires a field equation, source coupling, energy accounting, and relativistic embedding. This is the most demanding route but also the route capable of jointly predicting massive-particle dynamics and lensing with the same parameters.

10.4 Recursive boundary evolution

A temporally evolving Encoded Equilibrium could be investigated where past outflows, star-formation history, tidal encounters, or other state changes alter subsequent gravitational structure. Because conventional galaxy formation already contains feedback and halo response, distinctness requires a quantitative prediction not already entailed by those models.

11. Observational Program

11.1 Galactic rotation curves

Rotation-curve tests should split training and validation by galaxy, not by radial point, because measurements within a galaxy share distance, inclination, mass-to-light, and structural uncertainties. Candidate composition laws must be frozen before evaluation on held-out galaxies. Hierarchical errors-in-variables analysis should propagate correlated uncertainties and intrinsic scatter.

11.2 Surface-density scaling

The characteristic μ₀D relation should be reconstructed under multiple halo-profile assumptions. Covariance between fitted core radius and central density must be quantified. If the apparent constancy changes substantially with profile choice, its status as a model-independent invariant is weakened.

11.3 Gravitational lensing

A gravitational composition rule that explains rotation curves is incomplete unless embedded in a framework that predicts how photons respond. Rotation and lensing should therefore be jointly tested with the same physical parameters. A model that requires unrelated parameters for the two observables has not yet supplied a unified gravitational mechanism.

11.4 Environment

Isolated galaxies, satellites, groups, and cluster environments provide a boundary test. If a proposed relational or external-field variable matters, its effect must be specified before comparison. Conventional formation history, tides, and selection effects must be modeled rather than attributed automatically to TSTOEAO.

11.5 Beyond galaxies

Wide binaries provide a low-acceleration regime at scales where a galactic dark-matter contribution is expected to be negligible. Galaxy clusters provide the opposite extreme, with large mass discrepancies and complex environments. A composition law intended as a gravitational principle should state whether it applies to these regimes and why.

12. Relational Observability Experiment for Gravity

The gravitational branch can adapt the nested-drive architecture without claiming that gravity itself is a container in a literal mechanical sense. Define a controlled simulation or observational ensemble in which an outer descriptor c₂ represents an independently known environmental or boundary quantity, c₁ represents an intermediate gravitational or baryonic state, and Γ_G represents a measured dynamical/lensing history.

c₂ → c₁ → gravitational response → Γ_G.

The preregistered target is a descriptor θ₂ = P[c₂]. The candidate relational pipeline may use

T_rel[Γ_G] = W* Π_out 𝒟_L Φ[Γ_G],

but it must be compared against endpoint-only, standard regression, native-model inverse, and shuffled-history/null baselines. Because the earlier TSTOEAO observability ratio was representation-sensitive, no fixed ρ_R threshold should be imported uncritically. Conditioning measures must be revalidated and their dependence on sampling, noise, and feature representation reported.

The strongest version removes the fitted output map from cross-regime transfer. A structural observability boundary is calibrated in Regime A, frozen, and then asked to predict in Regime B which hidden gravitational descriptors are recoverable. If this reduces to standard observability theory, it should be classified accordingly rather than claimed as excess TSTOEAO content.

13. Falsification Matrix

Observation	Interpretation
μ₀D varies systematically with independently measured baryonic or environmental variables	Weakens strict universality; may support conditional scaling
μ₀D changes strongly with halo-profile choice	Weakens model-independent invariant status
Quadratic composition fails on held-out rotation curves while another locked rule succeeds	Rejects the tested quadratic model in that domain
Rotation and lensing cannot be reproduced with the same physical parameters	Weakens the proposed gravitational mechanism
A boundary variable adds no held-out predictive information	Weakens that boundary-dependent extension
Relational inverse performance does not exceed endpoint/null baselines	Weakens dynamic-history encoding for that target
A proposed cross-domain observability boundary fails after freezing	Rejects that transport claim
A successful TSTOEAO operator is mathematically equivalent to an established method with no excess prediction	Classify as established-method correspondence, not new physical law

14. Levels of Claim

To prevent conceptual correspondence from being mistaken for physical discovery, the gravitational program should maintain explicit claim levels.

Level	Meaning
Level 1	Conceptual analogy or descriptive correspondence
Level 2	Formal reconstruction using mathematics already established elsewhere
Level 3	Independently specified TSTOEAO structure yields an excess, prospective, falsifiable prediction
Level 4	Repeated cross-regime success plus a coherent physical field theory and relativistic completion

The present gravitational composition work is Level 1–2. The relational inverse operator is an explicit computational construction, but its components have close relatives in established observability, delay-coordinate, and Koopman methods, and subsequent tests identified instability in the first proposed observability quantity. A Level 3 gravitational claim has not yet been earned.

15. Discussion

The most important result of this investigation is negative but constructive: existing TSTOEAO doctrine does not force Yoon’s quadratic acceleration relation. It also does not force ordinary additive composition. This prevents an attractive external result from being retrofitted into the theory as though it had been predicted.

At the same time, the recent relational-invariance and inverse-observability work materially advances the research program. It supplies a language for separating representation from invariant structure, a computational route for testing whether hidden outer conditions remain inferable from observed histories, explicit conditioning and failure concepts, and a cross-domain transfer standard. These tools address identifiability. They do not yet address gravitational dynamics.

This separation suggests a disciplined division of labor. A future gravitational domain module must first propose independently motivated physics that determines 𝒞. Relational methods can then test whether the resulting hidden states, boundary variables, or competing decompositions are observationally distinguishable and whether a normalized structural criterion transfers between regimes.

The generalized inner-product expression is useful in this context not as a proposed law but as a diagnostic. If future TSTOEAO mathematics defines a relational metric under which physically distinct gravitational channels possess a calculable inner product, then additive, quadratic, or intermediate composition could become derived consequences of geometry in that relational space. Until such a metric is independently justified, θ_R remains an illustrative placeholder.

The program must also resist a second temptation: treating a recoverable hidden cause as proof that the inferred cause is ontologically unique. Inverse problems can be non-identifiable. Different physical models may generate the same observables. A successful T_rel reconstruction under one model therefore does not prove that nature uses that model unless competing forward descriptions are excluded by additional measurements.

16. Research Determination

A TSTOEAO gravitational domain module is mathematically formulable, but it is not yet independently derived.

The existing framework supplies architecture for declaring Encoded Equilibrium, defining relational observables, preserving coordinate discipline, constructing candidate inverse maps, testing observability, and preregistering failure. It does not select a gravitational composition law or provide a relativistic gravitational field theory.

Accordingly, additive, quadratic, generalized inner-product, and nonlinear composition should remain separate competing hypotheses. Yoon’s quadratic relation is a valuable external target because it makes a specific quantitative claim and has been used to obtain a galactic surface-density scale consistent with observation. It should not be adopted merely because of that agreement.

The next mathematical task is to seek an independently motivated physical response law and determine what it predicts before the target galactic scale is used for calibration. In parallel, the relational inverse machinery can be used to ask whether the competing hidden decompositions are distinguishable from rotation, lensing, environment, and time- or population-resolved structure.

17. Conclusion

The central question is no longer whether TSTOEAO can be made compatible with a successful gravitational relation. Almost any sufficiently flexible framework can be made compatible after the fact. The scientific question is whether its independently developed structure can constrain gravitational composition before the relevant observation is revealed.

Current evidence does not justify that claim. Encoded Equilibrium and the Universal Coordinate Principle do not select an exponent, cross term, or interpolation law. Relational form invariance identifies what kind of structure should be sought. Dynamic Container Encoding asks whether hidden conditions leave recoverable histories. The candidate relational inverse operator makes that question executable. Its subsequent limitations demonstrate why conditioning, representation dependence, and comparison with established methods must remain explicit.

The resulting research program is therefore deliberately vulnerable. Derive a physical response law independently. Freeze it. Predict rotation and lensing together. Test surface-density scaling, environment, wide binaries, and clusters. Use relational inference only where the forward model preserves recoverable information. Publish null results and degeneracies. Do not redefine Encoded Equilibrium after failure.

If one composition rule survives those tests and produces an additional quantitative prediction unavailable to its comparators, a TSTOEAO gravitational domain module will have earned further development. If none survives, the failure boundary is itself the result.

Physical postulates → composition law → observable predictions → relational identifiability → withheld tests → independent reveal.

References

1. Donato, F., et al. (2009). A constant dark matter halo surface density in galaxies. Monthly Notices of the Royal Astronomical Society, 397, 1169–1176.

2. Lelli, F., McGaugh, S. S., & Schombert, J. M. (2016). SPARC: Mass models for 175 disk galaxies with Spitzer photometry and accurate rotation curves. The Astronomical Journal, 152, 157.

3. McGaugh, S. S., Lelli, F., & Schombert, J. M. (2016). Radial acceleration relation in rotationally supported galaxies. Physical Review Letters, 117, 201101.

4. Verlinde, E. (2017). Emergent gravity and the dark universe. SciPost Physics, 2, 016.

5. Yoon, Y. (2024). Testing Verlinde’s emergent gravity with the low acceleration gravitational anomaly observed in wide binary stars. Physics of the Dark Universe, 45, 101551. https://doi.org/10.1016/j.dark.2024.101551.

6. Yoon, Y. (2026). A characteristic phantom-halo surface-density scale in Verlinde-inspired emergent gravity. Physics of the Dark Universe, 102462. https://doi.org/10.1016/j.dark.2026.102462.

7. Swygert, J. (2026). From Static Fractal Resemblance to Dynamic Container Encoding: A TSTOEAO Framework for Testing Whether Geometric Histories Reveal the Forces, Constraints, and Motions of Nested Dynamical Environments. Ivory Tower Publishing.

8. Swygert, J. (2026). From Dynamic Container Encoding to a Candidate Relational Inverse Operator: A TSTOEAO Mathematical Construction for Nested-Drive Observability Without Direct Use of the Native Governing Equation. Ivory Tower Publishing / The Journal of TSTOEAO.

9. Swygert, J. (2026). From Relational Observability to Relational Information Accounting: Dynamic Container Encoding, Relational Closure, Synergistic Information, Causal Identifiability, and Intervention-Based Information Flow. Ivory Tower Publishing / Ivory Tower Journal.

10. Kamb, M., Kaiser, E., Brunton, S. L., & Kutz, J. N. (2020). Time-Delay Observables for Koopman: Theory and Applications. SIAM Journal on Applied Dynamical Systems, 19.

11. Bejarano, F. J., & Zheng, G. (2014). Observability of linear systems with commensurate delays and unknown inputs. Automatica, 50(8), 2077–2083.

12. Sundaram, S., & Hadjicostis, C. N. (2007). Delayed Observers for Linear Systems With Unknown Inputs. IEEE Transactions on Automatic Control, 52(2), 334–339.

13. Givoli, D. (2021). A tutorial on the adjoint method for inverse problems. Computer Methods in Applied Mechanics and Engineering, 380, 113810.

Gravitational Composition and Relational Invariance: From Galactic Surface-Density Regularities to a Falsifiable TSTOEAO Gravitational Research Program