Relational Compression in Gravitational Domains
A Scale-Invariant TSTOEAO Framework for Investigating Quantum–Gravitational Unification
John Swygert
Ivory Tower Publishing
September 12, 2026
Research Update — September 12, 2026: This paper records the original formulation of the Relational Compression hypothesis and is retained unchanged for provenance. A subsequent investigation mathematically tests and refines the hypothesis, including coordinate-independent formulations of gravitational depth and relational transformation.
Abstract
This paper proposes a formal research direction within TSTOEAO for investigating whether gravitation and quantum behavior may represent different manifestations of a deeper relational structure rather than fundamentally incompatible descriptions of reality.
The central hypothesis is that increasing gravitational depth may be representable as scale-dependent relational compression within one overarching relational-coordinate domain \(G_T\). Observable physical descriptions are treated as domain overlays \(M_D\), while physically meaningful invariants are sought across transformations between those overlays.
The investigation arose partly from consideration of the golden ratio,
\[ \phi=\frac{1+\sqrt5}{2}\approx1.6180339887, \]
whose defining self-similar relationship,
\[ \phi=1+\frac1\phi, \]
provides a simple mathematical example in which the relationship between whole and part reproduces itself across scale.
However, this paper explicitly does not assume that \(\phi\) governs gravitational structure, nor is derivation of the golden ratio established as an objective of the theory. Its appearance in the development of the hypothesis is recorded as intellectual provenance. Any future emergence of \(\phi\), or any other privileged scaling constant, must result independently from physically motivated equations rather than being inserted as an assumption.
Likewise, failure to recover \(\phi\) would not by itself falsify the relational-compression hypothesis. The underlying invariant may manifest differently under different physical conditions, may depend upon multiple coupled variables, or may produce no privileged constant at all.
The principal research question is therefore:
Does gravitational depth produce a scale-dependent relational transformation under which a physically meaningful invariant survives, and can that invariant provide a common mathematical structure connecting gravitational and quantum descriptions?
The framework is intentionally falsifiable. Failure to identify such an invariant, failure of the proposed transformations to reproduce established physics, or failure of derived predictions constitutes evidence against the proposed formulation.
01 Introduction
Modern physics possesses two extraordinarily successful frameworks whose mathematical structures remain difficult to reconcile.
General relativity describes gravitation through dynamical spacetime geometry. Matter and energy influence geometry, while geometry constrains the motion of matter and energy.
Quantum theory describes physical systems through states, amplitudes, probabilities, operators, correlations, and measurement relationships.
Each succeeds spectacularly within its domain.
Yet their conceptual architectures differ profoundly.
The conventional problem of quantum gravity is therefore often posed as an attempt to quantize gravity, geometrize quantum phenomena, or construct a more fundamental framework from which both emerge.
TSTOEAO suggests another possibility.
Instead of requiring one domain description to become fundamental and absorbing the other into it, both may be treated as overlays of one deeper relational domain.
Let the overarching relational-coordinate domain be
\[ G_T. \]
Let domain-specific representations be
\[ M_D. \]
General relativity and quantum theory may then represent different \(M_D\) overlays mapped into the same \(G_T\).
The primary task is not initially to force the mathematics of one overlay onto the other.
It is to identify the relational structures that survive translation between them.
02 The Universal Coordinate Principle
TSTOEAO assumes one overarching relational-coordinate domain:
\[ G_T. \]
Physical, informational, geometric, biological, cognitive, or other domain descriptions are not necessarily independent universes of structure. They may instead be representations projected from or mapped into portions of the overarching relational domain.
Thus,
\[ M_D \rightarrow G_T. \]
Multiple domain descriptions may coexist,
\[ M_{D_1},M_{D_2},M_{D_3},\ldots \]
without requiring their native coordinates to be identical.
The objective is to determine whether some relational property
\[ I_R \]
remains invariant under translation:
\[ I_R(M_{D_1}) = I_R(M_{D_2}) = \cdots. \]
If such an invariant exists, it may permit transport of information or law between representations that otherwise appear mathematically incompatible.
03 Points as Relational Realms
Ordinary geometry commonly treats a point as dimensionless.
The present framework introduces a different interpretation.
A plotted point may function as a coordinate within one domain while simultaneously providing access to an arbitrarily rich internal relational domain.
Thus a point
\[ P\in G_T \]
need not represent ontological emptiness.
It may represent a coordinate whose internal relational structure contains further coordinates,
\[ P_1,P_2,P_3,\ldots \]
each of which may themselves contain additional relational structures.
Likewise, \(P\) may itself exist within progressively larger containing structures.
Consequently, relational organization may be represented recursively across scale.
This produces a nested architecture,
\[ \cdots \supset G_{n-1} \supset G_n \supset G_{n+1} \supset \cdots, \]
without requiring any individual observational scale to constitute the ultimate boundary of reality.
A coordinate may therefore be externally small while internally unbounded in relational complexity.
04 Fractals and the Container
Fractal structures are relevant not because the universe must literally be a mathematical fractal at every scale, but because fractal behavior demonstrates an important principle:
Recurring structure can reveal properties of the rules and constraints under which the structure is generated.
A pattern is therefore potentially informative about its containing domain.
Different generating constraints produce different families of structures.
Accordingly, recurring relational patterns observed within physical reality may provide information about the constraints imposed by the physical container in which those patterns arise.
The relevant scientific progression is therefore
\[ \text{observed pattern} \rightarrow \text{generating relationship} \rightarrow \text{domain constraint} \rightarrow \text{allowable transformation} \rightarrow \text{prediction}. \]
The theory must not stop at identifying similarity.
It must derive the rule producing it.
05 The Golden Ratio as Intellectual Provenance
One important conceptual step in developing the present hypothesis arose from considering the golden ratio.
The golden ratio satisfies
\[ \phi=\frac{1+\sqrt5}{2} \]
and
\[ \phi^2=\phi+1. \]
Equivalently,
\[ \phi=1+\frac1\phi. \]
Geometrically,
\[ \frac{a+b}{a} = \frac{a}{b}. \]
In words:
The relationship of the whole to the larger part equals the relationship of the larger part to the smaller part.
This is noteworthy because the relationship reproduces itself despite a change in scale.
Within the relational framework developed here, that mathematical property suggested a broader question:
Can a relationship remain invariant while the scale at which that relationship manifests changes?
This observation contributed directly to the development of the gravitational-compression hypothesis.
It is therefore recorded explicitly.
However, the golden ratio is not assumed to be a gravitational constant, universal scaling law, or expected experimental result.
Any subsequent derivation must proceed without requiring
\[ \phi. \]
Should \(\phi\) later arise independently from physically justified conditions, that would constitute a result requiring investigation.
Should another value arise, that value must be accepted instead.
Should no privileged value arise, the hypothesis must accommodate that result as well.
This separation between origin of hypothesis and target of derivation is essential.
06 Gravity as Relational Transformation
Gravity provides an unusually valuable testing ground because general relativity already supplies precise quantitative relationships between observers occupying different gravitational environments.
For a static observer outside a nonrotating spherical mass, the Schwarzschild metric contains the factor
\[ \alpha(r) = \sqrt{1-\frac{r_s}{r}}, \]
where
\[ r_s=\frac{2GM}{c^2} \]
is the Schwarzschild radius.
For appropriate static observers,
\[ d\tau = dt \sqrt{1-\frac{r_s}{r}}. \]
Thus quantities measured relationally between observers change systematically with gravitational depth.
As
\[ r\rightarrow\infty, \] \[ \alpha(r)\rightarrow1. \]
As
\[ r\rightarrow r_s^+, \] \[ \alpha(r)\rightarrow0. \]
The meaning of these relationships depends upon the observer and coordinate description.
This provides precisely the kind of environment in which relational-coordinate theory should be tested.
07 The Relational Compression Hypothesis
Define a gravitational relational-compression transformation
\[ C_g(r). \]
The term compression does not initially claim that physical space is literally squeezed into a fractal object.
Rather, it refers to systematic changes in relational accessibility, temporal rate, frequency, causal structure, energetic description, or informational representation as gravitational depth changes.
The proposed sequence is
\[ M(r_1) \xrightarrow{C_g} M(r_2) \xrightarrow{C_g} M(r_3) \rightarrow\cdots. \]
The hypothesis asks whether there exists some physically meaningful invariant
\[ I_R \]
such that
\[ I_R[M(r_n)] = I_R[M(r_{n+1})]. \]
If no meaningful invariant exists, the hypothesis fails in its present form.
If one exists, its mathematical character becomes the principal object of investigation.
08 Gravitational Depth as Scale Transformation
The central proposal is:
Increasing gravitational depth may correspond to increasing relational compression while preserving one or more deeper relational invariants.
This does not mean that every observable quantity remains unchanged.
Indeed, observable quantities may transform substantially.
What remains invariant would instead belong to the relational architecture connecting those quantities.
Symbolically,
\[ X_n \neq X_{n+1}, \]
while
\[ R(X_n)=R(X_{n+1}), \]
where \(R\) represents a deeper relational property.
This distinction is critical.
A theory of relational invariance does not require identical measurements across different frames.
It requires preservation of a relationship beneath those frame-dependent measurements.
09 Self-Similarity Under Gravitational Compression
Suppose successive gravitational domains exhibit transformations
\[ C_1,C_2,C_3,\ldots. \]
A self-similar condition could take a general form such as
\[ F(C_n,C_{n+1}) = F(C_{n+1},C_{n+2}). \]
A simple ratio condition might be
\[ \frac{C_{n+1}}{C_n} = \frac{C_{n+2}}{C_{n+1}}, \]
although no particular functional form is assumed here.
A more sophisticated transformation could involve curvature, proper time, redshift, causal accessibility, entropy, area, action, phase, or another dimensionless relational quantity.
The correct transformation must emerge from physics rather than aesthetic preference.
10 Why the Golden Ratio Must Not Be Inserted
If the analysis begins by demanding
\[ x=1+\frac1x, \]
then obtaining
\[ x=\phi \]
would prove nothing beyond elementary algebra.
Therefore the theory must instead derive its governing equations from independent physical requirements.
Only afterward may the resulting constants be evaluated.
The scientifically meaningful sequence is
\[ \text{physical assumptions} \rightarrow \text{relational condition} \rightarrow \text{equation} \rightarrow \text{solution}. \]
Not
\[ \phi \rightarrow \text{equation designed to recover }\phi. \]
The distinction prevents confirmation bias and numerological pattern matching.
10.1 The Golden Ratio Is Not a Pass–Fail Criterion
Failure to recover
\[ \phi\approx1.6180339887 \]
does not by itself falsify relational compression.
The central hypothesis concerns the existence of a physically meaningful relational scaling law or invariant under changes in gravitational depth.
The value through which that invariant appears may depend upon additional physical variables, boundary conditions, or domain-specific projections.
Accordingly, an observed scaling relationship may take a more general form,
\[ R = F(r,J,Q,S,\kappa,\ldots), \]
where gravitational depth, angular momentum, charge, entropy, curvature, or other relevant variables influence the manifestation of the underlying relationship.
The golden ratio may therefore prove to be:
a special-case value;
a limiting value;
one component of a multivariable relationship;
a projected manifestation of a deeper invariant;
or unrelated to the final gravitational law.
The theory must distinguish between the fundamental relational invariant and the numerical value through which that invariant appears under particular physical conditions.
Thus,
\[ I_R\neq\text{necessarily }\phi. \]
Even where
\[ \phi \]
appears observationally, it may itself be a projection,
\[ \phi_{\mathrm{obs}} = \Pi(I_1,I_2,I_3,\ldots), \]
of a more fundamental relational structure.
A result different from \(\phi\) may therefore indicate that multiple physical influences are altering, modulating, or projecting the underlying relational structure.
The correct result is whatever follows from independently justified physics.
11 General Relativity as a Container Description
One possible interpretation suggested by the present framework is that general relativity predominantly describes the relational geometry of the physical container.
This statement must be qualified.
General relativity applies locally as well as globally and is not merely an external-observer theory.
Nevertheless, its fundamental mathematical object is spacetime geometry.
The Einstein field equations,
\[ G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}, \]
relate spacetime geometry to stress-energy.
In this sense, general relativity provides a theory of the relational structure of the containing spacetime domain.
12 Quantum Theory as Internal Relational Structure
Quantum mechanics describes systems differently.
Its formalism centers upon states, observables, amplitudes, superposition, correlations, operators, and measurement outcomes.
One possible TSTOEAO interpretation is therefore:
General relativity and quantum mechanics may encode different levels or projections of one deeper relational architecture.
This does not mean that quantum mechanics applies only inside gravitational wells or that general relativity applies only outside them.
Such a literal distinction would be incorrect.
The proposed distinction is structural rather than spatial.
General relativity emphasizes the geometry of the containing physical domain.
Quantum theory emphasizes relational possibilities and states manifested within physical domains.
13 The Quantum–Gravity Bridge
If both theories are overlays of \(G_T\),
\[ M_{\mathrm{GR}}\rightarrow G_T \]
and
\[ M_{\mathrm{QM}}\rightarrow G_T, \]
then unification may not require replacing one theory with the other.
Instead, the problem becomes
\[ \boxed{ \text{Find }I_R \text{ preserved across } M_{\mathrm{GR}} \leftrightarrow M_{\mathrm{QM}} } \]
The invariant must possess sufficient mathematical content to recover known behavior within both domains.
A valid candidate would therefore have to satisfy at least three conditions:
reproduce an established gravitational relationship;
reproduce an established quantum relationship;
generate at least one nontrivial consequence not inserted independently into either derivation.
Without the third condition, the framework may represent an elegant reformulation rather than a new physical theory.
14 The Horizon as a Relational Test Case
The Schwarzschild horizon is particularly interesting because different coordinate descriptions can make the same physical region appear profoundly different.
A distant Schwarzschild observer associates extreme redshift and coordinate-time behavior with approach toward the horizon.
A freely falling observer crossing the horizon of a sufficiently large black hole may encounter no locally singular event at the horizon itself.
Thus,
\[ \text{same physical region} + \text{different relational coordinates} \rightarrow \text{very different descriptions}. \]
A theory emphasizing relational-coordinate overlays should be especially capable of explaining which features are coordinate dependent and which are invariant.
The horizon therefore constitutes an important early test environment.
15 Relational Compression and Information
A potentially stronger formulation of the hypothesis concerns information rather than literal spatial compression.
Increasing gravitational depth may alter:
temporal relationships;
observable frequencies;
accessible causal regions;
information exchange;
entropy descriptions;
horizon accessibility;
energetic relationships;
measurement relationships between observers.
Accordingly, relational compression may ultimately describe compression of accessible relational information rather than contraction of ordinary geometric distance.
This interpretation is more readily connected to established physics and therefore provides a preferable initial test.
16 A Candidate Research Program
The investigation should proceed without assuming its desired result.
Stage I — Schwarzschild Geometry
Begin with the simplest gravitational field.
Construct dimensionless relational quantities from combinations of
\[ r,\quad r_s,\quad d\tau,\quad dt, \]
redshift, acceleration, curvature quantities, and other invariant or observer-specific measures.
Determine whether physically motivated recursive transformations exist.
Stage II — Search for Relational Invariants
Identify quantities preserved under successive changes in gravitational depth.
Reject quantities that are coordinate artifacts.
Stage III — Test for Self-Similarity
Determine whether any transformation possesses a recurring scale relation.
Do not assume a constant beforehand.
Stage IV — Introduce Additional Physical Factors
Test whether any candidate invariant changes under the introduction of additional variables such as
\[ J,\quad Q,\quad S,\quad \kappa,\ldots \]
or other physically relevant quantities.
Determine whether deviations from simple scaling arise from genuine modulation of the invariant, from projection effects, or from failure of the proposed structure.
Stage V — Extend to Kerr Geometry
Repeat the analysis for rotating masses.
If the proposed invariant disappears immediately under rotation, its claimed universality must be reconsidered.
Stage VI — Quantum Overlay
Determine whether the same invariant has a natural representation in quantum states, phases, correlations, amplitudes, information measures, or operator relationships.
Stage VII — Cross-Domain Transport
Attempt transport
\[ M_{\mathrm{GR}} \rightarrow I_R \rightarrow M_{\mathrm{QM}} \]
and
\[ M_{\mathrm{QM}} \rightarrow I_R \rightarrow M_{\mathrm{GR}}. \]
Stage VIII — Prediction
Derive something not inserted as an assumption.
Only here does the framework become genuinely predictive.
17 What Would Count as a Prediction?
A valid prediction might take one of several forms.
It could identify:
a previously unknown dimensionless relationship;
a privileged gravitational depth arising independently from the equations;
a specific scaling law;
a measurable relation between gravitational and quantum observables;
a new correction term;
a boundary at which one relational regime transforms into another;
an independently derived numerical constant;
a variable-dependent scaling law;
a phenomenon predicted before its observation.
The prediction must distinguish TSTOEAO from a framework that merely redescribes known results.
18 What Would Falsify the Proposal?
The framework should be considered weakened or falsified if:
no meaningful relational invariant can be found;
proposed invariants depend entirely upon arbitrary coordinates;
the compression formalism fails to reproduce established gravitational physics;
the proposed bridge contradicts experimentally confirmed quantum results;
different gravitational geometries destroy the claimed invariant without a principled explanation;
constants appear only after equations are selected specifically to produce them;
deviations are repeatedly explained away by introducing unconstrained new variables after the fact;
predicted phenomena fail experimental or observational testing.
Failure is scientifically valuable.
A theory that cannot lose cannot meaningfully win.
19 The Fractal Interpretation
The word fractal is used cautiously.
The hypothesis does not require literal exact fractality throughout spacetime.
Instead, it proposes investigation of whether relational self-similarity occurs across transformations of scale.
The distinction is
\[ \text{geometric fractality} \neq \text{necessarily relational self-similarity}. \]
A system may preserve relational structure without reproducing identical geometric shapes.
This broader concept may be more useful for quantum gravity than ordinary visual fractality.
20 Compression Without Destruction
Compression ordinarily implies loss.
But mathematical and informational compression can preserve essential structure.
The relevant question is therefore whether gravitational compression could transform observable manifestation while preserving deeper relational content.
That would suggest
\[ \text{manifestation changes} \]
while
\[ \text{relational identity persists}. \]
Such a principle would provide a natural candidate for transport between apparently incompatible domain descriptions.
21 Observer, Observation, and Participation
Physics distinguishes among observers and reference frames without requiring consciousness as a fundamental ingredient of every measurement.
TSTOEAO nevertheless emphasizes that every description exists relationally.
An observer never possesses the totality of \(G_T\).
The observer occupies a coordinate within it.
Measurements therefore represent constrained relational access to the larger domain.
Different observers may legitimately generate different descriptions while participating in the same underlying reality.
This provides a natural interpretation of apparently contradictory descriptions that are actually coordinate dependent.
22 Nested Domains and Apparent Infinity
If a coordinate can provide access to an arbitrarily rich internal relational structure, then observational scale need not correspond to ontological complexity.
A point viewed externally may contain a domain whose internal structure cannot be inferred merely from its apparent external size.
Thus,
\[ \text{coordinate size} \not\Rightarrow \text{relational complexity}. \]
This principle may prove relevant wherever physical systems exhibit enormous information density, recursive structure, or scale-dependent accessibility.
23 The Container Hypothesis
The strongest provisional statement developed here is therefore:
The physical universe may be a relational container whose observable structures reveal the constraints governing allowable transformations within that container.
Different containers, if they exist, need not possess identical relational laws.
Their characteristic patterns could therefore differ.
Observable structure would then provide indirect information about the properties of the containing domain.
This hypothesis remains speculative until mathematical constraints and discriminating predictions are derived.
24 The Greater Projection Problem
A domain description may be understood as a projection.
A projection preserves some relationships while obscuring others.
Consequently, two apparently incompatible models could represent different projections of a common underlying structure.
The word project is therefore unintentionally appropriate.
A scientific project constructs a representation.
A mathematical projection maps structure between spaces.
TSTOEAO asks whether physical theories themselves may be projections whose shared relational invariants reveal the deeper domain from which they arise.
25 Theological Interpretation Is Separate From the Physical Derivation
A metaphysical interpretation naturally arises from this framework.
One may ask whether an ultimate substrate underlying all relational domains corresponds to what theological traditions call God.
Under such an interpretation, omnipresence would not require a distant entity receiving information from separate locations.
Events would instead occur within the encompassing substrate itself.
This could be expressed metaphorically as:
The universe does not report its experiences to God; its experiences occur within that which God encompasses.
This interpretation may possess philosophical and theological value.
It is not required by the physical hypothesis, however.
The gravitational and quantum analysis must remain valid or invalid independently of theological belief.
Keeping these domains distinguishable protects both the science and the philosophy from being used improperly to prove one another.
26 Why This Distinction Matters
A theory of everything should not succeed because spiritual language makes the physics beautiful.
Nor should metaphysical possibilities be dismissed simply because they are not themselves physical measurements.
The proper relationship is
\[ \text{physics tests measurable structure} \]
while
\[ \text{philosophy examines its broader interpretation}. \]
TSTOEAO may provide a relational grammar capable of allowing both conversations without confusing their evidentiary standards.
27 From Pattern Recognition to Prediction
The present work is not yet a theory of quantum gravity.
It is a proposed route toward one.
The transition required is
\[ \text{intuition} \rightarrow \text{formal definition} \rightarrow \text{derivation} \rightarrow \text{constraint} \rightarrow \text{prediction} \rightarrow \text{test}. \]
The framework becomes scientifically consequential only when the later stages are reached.
The next decisive task is therefore mathematical.
28 Immediate Mathematical Question
The first question should be stated without reference to the golden ratio:
Does a physically motivated transformation describing increasing gravitational depth admit a nontrivial self-similar relational invariant?
If yes:
What equation determines that invariant?
Only after solving it should the resulting value or functional relationship be examined.
If the answer is
\[ 1.6180339887\ldots, \]
that result would be remarkable precisely because it was not requested.
If the answer is another constant, that is equally important.
If the answer is a variable-dependent expression rather than a constant, that may be more informative still.
If no privileged scaling relationship appears, that too constrains the theory.
29 Stronger Quantum–Gravity Question
The deeper question is:
Can the same relational invariant be represented in both gravitational geometry and quantum structure while preserving its identity under transformation between the two domain overlays?
Symbolically,
\[ I_R(M_{\mathrm{GR}}) = I_R(M_{\mathrm{QM}}). \]
If such an equality can be derived rather than asserted, it may provide a bridge between the two theories.
A still stronger result would show that the apparent forms of the invariant differ between domains while being generated by the same deeper structure:
\[ I_R \xrightarrow{\Pi_{\mathrm{GR}}} R_{\mathrm{GR}}, \] \[ I_R \xrightarrow{\Pi_{\mathrm{QM}}} R_{\mathrm{QM}}. \]
This would permit domain-specific manifestations without requiring identical surface mathematics.
30 The Standard Required for a Theory of Everything
TSTOEAO should not claim success merely because it provides a language capable of describing many systems.
To qualify as a physical theory of everything, it would need to demonstrate substantially more.
At minimum it would have to:
recover established physics where established physics is successful;
explain why apparently different theories emerge in their respective domains;
derive their relationship from one coherent structure;
reduce arbitrary assumptions rather than multiplying them;
generate new testable consequences;
survive serious attempts at falsification.
The present paper proposes one possible path toward that standard.
It does not claim that the standard has yet been achieved.
Conclusion
The central insight developed here is that scale-dependent transformation need not destroy relational identity.
A physical domain may change radically in its observable manifestation while preserving a deeper relational structure.
Gravity provides a natural environment in which to investigate this possibility because gravitational depth already produces systematic transformations among time, frequency, geometry, energy, causality, and observer-dependent description.
TSTOEAO therefore proposes treating increasing gravitational depth as a candidate form of relational compression.
The objective is not to discover the golden ratio.
The golden ratio served as an intellectual catalyst because it demonstrates, with unusual mathematical clarity, that a relationship between whole and part can reproduce itself across scale.
That historical origin is deliberately preserved.
But the mathematics must now proceed without it.
Nor would a result different from the golden ratio invalidate the central hypothesis. The observable scaling may depend upon additional physical variables, may represent a projection of a deeper invariant, or may take the form of a function rather than a universal constant.
The central scientific question is
\[ \boxed{ \text{What relationship survives gravitational compression?} } \]
The deeper question is
\[ \boxed{ \text{Does that same relationship survive translation into the quantum domain?} } \]
And the decisive question is
\[ \boxed{ \text{What does it predict that we did not already know?} } \]
If no such invariant exists, the hypothesis must be revised or rejected.
If one does exist, and if it independently reproduces both gravitational and quantum behavior while generating a novel verified prediction, then a relational route toward quantum–gravitational unification may have been found.
The container must ultimately answer for itself.
Copyright © John Swygert 2026
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