Daniel-Stoker-J-Eidetic-Theory

Eidetic Theory

Daniel James Stoker

 

Published July 3, 2022

Updated May 11, 2025 

 

djs@arizona.edu  

(520) 729-0982

 

Table of Contents

 

Introduction………………………………………………………………………………………………..2

The Formal Basis of Eidetic Theory…………………………………………………………………….3

Eidetic Fundamental Equation.………………………………………………….………………………3

Eidetic Projected Equation………………………………………………………………………………6

Black Hole Thermodynamics – Eidetic Interpretation……………………………………………..…16 

Quantum Eraser Experiment – Eidetic Interpretation………………………………………………..22

Wheeler’s Delayed Choice Experiment – Eidetic Interpretation…………………………………….27

The Holographic Principle – Eidetic Interpretation……………………………………………………31

Entropic Gravity – Eidetic Interpretation……………………………………………………………….33

AdS/CFT Correspondence and Temporal Holography – Eidetic Interpretation……………………35

Holographic Entanglement Entropy – Eidetic Interpretation…………………………………………38

Gravity from Entanglement – Eidetic Interpretation………………………………………………….40

ER = EPR Conjecture – Eidetic Interpretation……………………………………………………..…42

Quantum Extremal Surfaces and the Island Formula – Eidetic Interpretation………………..…..44

Quantum Error Correction in Holography – Eidetic Interpretation…………………………………..47

Reformulating the Stress-Energy Tensor with Entanglement Entropy……………………………..49 Entropic Gravity and Einstein’s Equations from Boundary Entanglement….……………………..52

Boundary Coherence vs. Localization Pressure……………………………………..……………….54

Dark Energy Equation of State in Eidetic Theory…………………………………………………….58

Deriving the Effects of Dark Matter in the Eidetic Theory……………………………………..……60

Calculating the Cosmological Constant in Eidetic Theory…………………………………………..62

The Limitation of Detecting Gravitons (Eide Spheres) in the Bulk…………………………………65

Derivation of General Relativity from the Eidetic Field Equation…………………………………..67

Derivation of Maxwell’s Equations from the Eidetic Field Equation………………………………..69

References……………………………………………………………………………………………….73

 

Eidetic Theory proposes that spacetime, matter, quantum statistics, and gravity all originate from a vast lattice of quantized boundary surfaces called Eide Spheres.

Each sphere is a two-sphere swept along a null time-parameter, so its intrinsic world-volume is (“2 S + T”). Spheres are created only in forward/backward entangled pairs: one component propagates into the future, its partner into the past, both at the speed of light. The lattice is therefore time-symmetric and Lorentz-invariant at the microscopic level. Ordinary quantum waves, classical bodies, and gravitational curvature appear when a very large number of such pairs interfere and then partially decohere.

Eidetic Theory proposes a model to potentially resolve foundational inconsistencies in physics by positing that all physical phenomena—spacetime, mass-energy, quantum behavior, and classical causality—emerge from a single source: Eidetic energy. This energy is not a conventional spacetime field but a quantized, nonlocal coherence encoded in temporally entangled, codimension-one spherical surfaces called Eide Spheres. These surfaces form a fundamental boundary network that propagates bidirectionally through time at speed , with each entangled pair consisting of one expanding and one contracting sphere relative to time’s arrow.

Rather than being embedded in spacetime, this infinite, null-propagating network of Eide Spheres generates spacetime. As standing wave structures formed by temporal interference, they encode coherent information that sources entanglement entropy, which in turn defines the emergent structure and curvature of the bulk.

Crucially, Eidetic Theory reinterprets wavefunction collapse as a topological phase transition: from distributed boundary entanglement (quantum superposition) to localized bulk interaction (measurement). The reverse—recoherence from bulk back to boundary—offers a resolution to the black hole information paradox by preserving unitarity through entanglement recovery.

Unlike models that quantize spacetime or invoke extra dimensions, Eidetic Theory is pre-geometric: spacetime and fields emerge from entangled boundary coherence. This aligns with holographic principles—AdS/CFT, entanglement wedge reconstruction, and quantum error correction—which encode bulk gravity in boundary data. Eidetic Theory advances this by proposing that the Einstein tensor is not fundamental, but a decohered projection of entanglement energy gradients across the Eide Sphere network.

This paper develops the foundational formalism of Eidetic Theory. We begin with its boundary ontology and axioms, attempt a derivation of the emergence of spacetime geometry and Einstein’s field equations, provide possible explanations for the structure of quantum interactions, and address key paradoxes such as black hole evaporation and the measurement problem. Each domain is proposed to arise as a limiting case of the deeper coherent dynamics of Eide Spheres.

Eidetic Fundamental Equation

Four Fundamental Axioms

No.	Axiom	Operational statement
A-1 (Planck Quantization)	All boundary area is quantized in Planck cells.	Creating one Eide Sphere raises the boundary area by exactly ​.
A-2 (Null Propagation)	An Eide Sphere’s generators move exactly on the light-cone.	Intrinsic metric satisfies .
A-3 (Bidirectional Entanglement)	Spheres come only in pairs sharing a common two-surface.	Forward and backward components are coupled in the action.
A-4 (Holographic Projection)	Bulk fields are interference integrals of boundary pairs.	Every spacetime point collects amplitudes from all spheres that pass through .

From Axioms to the Eidetic Fundamental Equation

1. Quantized Operators (A-1)
   

(E-1)

1. Null Constraint (A-2) enforced by a Lagrange multiplier in the action

  (E-2)

1. Third-Quantized Surface Action (A-3)

    (E-3)

1. Surface Euler–Lagrange (“Eidetic Fundamental Equation”)

    (E-4)

 This is the irreducible dynamical law for every bidirectional sphere.

1. Holographic Projection (A-4)

  (E-5)

Fundamental Equation

Symbol	Mathematical definition	Physical / geometric meaning	Units
	A single Eide Sphere world-volume: intrinsic coordinates (λ,θ,ϕ) with topology .	The surface itself is a fundamental degree of freedom (not a field on spacetime). It moves null-like: .	—
		Temporal-entanglement density of the forward and backward components living on the same surface. A complex scalar.	(energy)½
	where is the induced metric .	The intrinsic d’Alembertian—wave operator defined on the surface itself. Encodes how varies as you move across the 2-sphere and along the null time parameter .	1/length²
	Positive, dimensionless normalisation constant (can be set to 1 by rescaling .	Controls the stiffness of entanglement propagation on the sphere; absorbed later into the physical mass .	—
		Planck-cell energy density: the energy cost of one unit of entanglement per Planck area. It supplies the rest-energy that appears as particle mass in the bulk.	energy/area
Equation structure		A light-cone Klein–Gordon equation on each sphere. The first term is kinetic (surface wave curvature); the second is an “entanglement tension” that would shrink the sphere unless balanced by wave motion.	energy/area
Null constraint (implied)	enforced by a Lagrange multiplier (E-2).	Guarantees every point of the sphere moves at speed ; ensures Lorentz invariance and fixes the time-like direction used in .	—

How the Eidetic Fundamental Equation Embodies the Physics of Eide Spheres

Bidirectional pairing 
is non-zero only if both orientations coexist; hence temporal entanglement is built in.

Planck discretization 
derives from the commutator
(E-1).
The “mass term” is the energy of one Planck cell of boundary area.

Null propagation 
Because is null and ​ uses its intrinsic metric, disturbances of move at light speed along the surface.

Dimensionally a Klein–Gordon field 
Compare with :
here but the operator acts on a surface instead of bulk spacetime.

Gateway to bulk physics 
Projecting through

transports (E-4) into the bulk and turns into the particle rest mass via
.

Solving for the Terms in the Eidetic Fundamental Equation

– lets forward and backward waves spread tangentially and along ; responsible for interference patterns that later appear as quantum phases in spacetime.

– tries to confine or “tension” the entanglement; balances the kinetic term to give a stable standing-wave condition.

Net result: a massive, time-symmetric standing wave on every 2-sphere, which—after holographic summation—reproduces both the Klein–Gordon and (in the low-velocity limit) Schrödinger equations for bulk particles, while its stress tensor sources Einstein curvature.

The Eidetic Equation is modeled to act simultaneously as the dynamical law of the boundary lattice and the seed of all familiar field equations in spacetime.

Eidetic Projected Equation

Eidetic Theory posits that spacetime, quantum fields, gravity, and classical observables do not exist independently on a fixed background but emerge from a coherent network of temporally entangled quantum boundaries—called Eide Spheres. These boundary structures encode discrete units of entanglement entropy, and from their gradients, interference, and coherence dynamics, both geometry and gravitational curvature are produced.

Standing Wave Construction of Spacetime

Spacetime emerges from the superposition of forward- and backward-propagating Eide Spheres. Each point in spacetime is defined by a standing wave node:

Where:

: forward-propagating boundary coherence,

: backward-propagating coherence,

 Both propagate at the invariant speed and represent entangled causal boundaries.

These are not classical waves but quantized coherence structures. Their superposition yields standing wave nodes, whose amplitude determines the emergent local spacetime geometry.

Entanglement Energy Density

We define the coherence amplitude field:

From this, the local entanglement energy density is:

Initially, the universe is globally coherent, with a smooth energy distribution. Cosmic decoherence breaks this structure into:

Where:

​: coherent background energy driving large-scale geometry (e.g. cosmic expansion, dark energy),

: decohered, localized energy responsible for mass and classical matter.

The Eidetic Action

We begin with the premise that all physical phenomena — spacetime geometry, energy, and classical structure — arise from a fundamental field:

This field encodes the coherent superposition of forward- and backward-propagating Eide Spheres.

Define the Energy Density:

From the standing wave structure of Eide Spheres:

The entanglement energy density should reflect:

How the field varies across spacetime (i.e., coherence gradients),

Local structure or symmetry-breaking configurations.

So we define the energy density functional as:

Where:

​ is the coherence gradient term, which plays the role of a kinetic term,

is a potential representing:

Preferred configurations (e.g., mass-like localization),

Symmetry-breaking (e.g., phase transitions),

Thresholds for decoherence.

Construct the Action Integral:

The total entanglement energy across the emergent spacetime manifold must be integrated with respect to the covariant volume element:

This ensures general coordinate invariance and captures the proper geometric weight from the metric.

Thus, the full action functional is:

Terms:

measures local coherence gradients; sources entanglement energy and drives curvature.

allows for localized structures (mass-energy), decoherence transitions, and field interactions.

ensures action is a scalar under coordinate transformations; defines the volume of the emergent geometry.

Covariant Derivatives:

Because ​ is not fixed but dynamically emerging, the derivative must respect the curvature of the evolving spacetime. The covariant derivative ​ satisfies:

Ensures compatibility with the Levi-Civita connection.

Therefore, all dynamical and variational quantities must use​ to remain valid in curved and evolving geometry.

The Eidetic Action emerges as the most natural, minimal, and invariant way to capture:

The standing wave coherence structure,

Its variation across an emergent spacetime,

Its potential to undergo decoherence, mass formation, and geometry generation.

Deriving the Covariant Equation of Motion

Let the action be:

Vary the Action with Respect to :

We compute the variation: 

Break it into parts:

Kinetic term:

Potential term:

 

So the variation becomes:

Integrate by Parts on the First Term:

Using the identity for integration by parts in curved space:

Set ​, then:

So the variation becomes:

Stationary Action Principle

Set for arbitrary , which yields the Euler–Lagrange equation:

Rewriting:

​

This is the covariant equation of motion for the entanglement field .

Interpretation:

This equation generalizes well-known wave equations:

If , it becomes the massless Klein-Gordon equation.

If ​, it becomes the massive Klein-Gordon equation with an effective mass ​.

For nonlinear , this allows for self-interacting scalar field dynamics, similar to inflaton or Higgs-type potentials.

Deriving the Gravitational Field Equation

The Eidetic Action is given by:

Variation of the Action:

To derive the gravitational field equation, we vary the action with respect to the metric ​:

We expand the variation explicitly:

Evaluate Variation of :

Recall the known identity:

Thus, we rewrite the second part of the variation as:

Evaluating Variation of the Kinetic Term:

Next, we handle the variation:

Since ​ does not depend explicitly on the metric variation, we get:

Substituting Both Results Into Variation:

Combining the above, the variation of the action is now:

Identifying the Eidetic Energy-Momentum Tensor:

We now define the Eidetic energy-momentum tensor as the functional derivative of the action with respect to the metric:

From our previous equation, we have explicitly:

This is exactly the expression provided.

Equating to Einstein’s Tensor:

By analogy with General Relativity (GR), Einstein’s gravitational field equations have the general form:

In Eidetic Theory, gravity is emergent from entanglement coherence, not from traditional classical matter fields. Thus, the Eidetic gravitational equation takes the form:

This explicitly shows how spacetime curvature can arise directly from gradients in entanglement coherence fields, captured by ​.

Eidetic Gravitational Field Equation:

This final equation explicitly encapsulates the foundational concept of Eidetic Theory—gravitational curvature emerges entirely from entanglement coherence gradients rather than classical matter-energy content alone.

Metric Tensor:

: Spacetime Metric

A tensor field describing distances and angles in a given spacetime geometry.

Determines causal structure, curvature, and gravitational interactions.

Interpretation: Encodes the geometry and structure of spacetime itself.

Covariant Derivative of Eidetic Field:

​: Covariant Derivative of the Entanglement Field

Represents the change of the scalar entanglement coherence field as one moves along spacetime directions, accounting for spacetime curvature.

Mathematically:

(Since ​ is scalar, ​ reduces to the partial derivative.)

Interpretation: Measures how the entanglement coherence varies through spacetime. Gradients in this field produce localized curvature.

Contracted Covariant Derivative:

​: Scalar Field Gradient Squared

Represents the scalar quantity obtained by contracting two derivatives with the metric:

Interpretation: Indicates the magnitude squared of the gradient of ​. This scalar quantity is crucial in determining the contribution of entanglement coherence gradients to spacetime curvature.

Potential Function:

: Eidetic Potential

Scalar potential function describing the internal self-interactions and energy density associated with the entanglement coherence field ​.

Typically depends on the field itself, e.g., ​ or more complicated potentials.

Interpretation: Encodes how entanglement coherence inherently contributes to energy density even in the absence of explicit gradients, analogous to potential energy in standard field theories.

Recovering Classical Limits

In low-energy or flat-spacetime limits where:

We recover standard field theory:

For the current treatment, ​ is taken to be a real scalar field encoding the local coherence amplitude of standing wave interference between forward- and backward-propagating Eide Spheres. Extensions to spinor and vector fields—such as those corresponding to Dirac or Maxwell fields—can be incorporated by generalizing the kinetic term to include covariant derivatives appropriate to the field representation (e.g., spinor covariant derivatives with spin connections), and by extending the potential to include appropriate symmetry-breaking, interaction, or gauge-invariant terms. In this way, boundary entanglement dynamics provide a unified formalism from which familiar quantum field equations may emerge as special cases under appropriate symmetry and coherence structures.

Interpretation and Comparison with GR

Element	Eidetic Theory	Classical GR
Metric	Emergent from entanglement structure	Assumed background geometry
Source of Gravity	Coherence gradients:	Matter:
Field equations	Variational principle from boundary coherence	Einstein equations from fixed stress-energy
Time asymmetry	Emerges via decoherence	Imposed via initial conditions

Conclusion

The single unified equation:

is the core of Eidetic Theory. From this principle:

Spacetime is generated,

Quantum dynamics are derived,

Gravitational curvature arises,

Decoherence and classicality emerge.

All physical law is thus understood as a projection of entangled coherence across temporally paired quantum boundaries, giving rise to the classical world from a fundamentally informational structure.

 

Black Hole Thermodynamics (Bekenstein & Hawking, 1973-1975)

The foundation of black hole thermodynamics was laid in the 1970s through the pioneering work of Jacob Bekenstein and Stephen Hawking. Bekenstein proposed that the entropy of a black hole is proportional to the area of its event horizon, not its volume—an insight that fundamentally shifted our understanding of information and gravity. Shortly after, Hawking discovered that black holes emit thermal radiation due to quantum effects near the event horizon, a phenomenon now known as Hawking radiation. These discoveries suggested a deep connection between gravity, quantum mechanics, and thermodynamics, but they also introduced a serious puzzle: if black holes radiate thermally and eventually evaporate, where does the information go?

The black hole information paradox is not a single contradiction, but a clash between three foundational principles of physics: quantum mechanics and its insistence on unitarity, General Relativity and the Equivalence Principle, and the assumption of locality—the idea that information cannot be transmitted faster than the speed of light or across spacelike separations. According to classical General Relativity, information that falls into a black hole is irretrievably lost beyond the event horizon. In contrast, quantum mechanics demands that the evolution of information is unitary and thus cannot be destroyed, and it is typically assumed that this evolution respects locality. The tension between these principles creates a deep paradox, suggesting that at least one of them must be revised. Resolving this paradox remains one of the most profound challenges in theoretical physics.

Eidetic Theory Interpretation

Eidetic Theory offers a resolution to this paradox by redefining the nature of black hole entropy. In this model, black hole entropy is not simply a measure of information stored on the event horizon, but rather a manifestation of deeper quantum boundary structures—Eide Spheres—that encode gravitational energy as entanglement entropy. These lower-dimensional, temporally entangled quantum boundaries propagate at the speed of light, forming a bidirectional network throughout spacetime.

In this framework, when matter collapses to form a black hole, it induces a dense standing wave field composed of forward- and backward-propagating Eide Spheres. These spherical waves interfere to form coherent boundary nodes—standing wave loci—that encode the localized information content of the collapsing mass-energy. These boundaries preserve information in a unitary fashion, with the backward-moving Eide Spheres encoding the infalling data. To an external observer, this information may appear lost—but it is in fact retained within the entanglement entropy of the boundaries network. 

As the black hole evaporates and the event horizon recedes, the entangled backward-time Eide Spheres reemerge in the past, during the epoch preceding the formation of the horizon. Because Eide Spheres carry temporal entanglement, the backward-time Spheres preserve their stored information through their entangled forward-time counterparts—completing the cycle and preserving unitarity.

The Eidetic Field Equation Applied to Black Hole Thermodynamics

This equation links curvature directly to variation of integrated boundary entanglement energy — not stress-energy of matter in the bulk.

Projection of Entanglement Energy by Decoherence

Eide Spheres exist outside of bulk spacetime. They form the quantum boundaries network whose coherence structure gives rise to spacetime geometry. They are not events or particles inside spacetime; rather, they are spherical, temporally entangled surfaces that intersect to form standing wave nodes. These nodes are what we experience as spacetime.

From this, decoherence and observation must be redefined in terms of bulk observers interacting with this standing wave network. Specifically:

Observation = Projective Decoherence of Standing Wave Nodes

When an observer in spacetime makes a measurement (i.e., interacts with mass, energy, or curvature), they do not observe an Eide Sphere directly. Instead, they interact with the local node of the standing wave field formed by forward and backward Eide Spheres. The act of observation is a projective decoherence: it collapses the full entangled state of the Eide Sphere node onto a single observer-accessible state.

We can represent this schematically: 

Where the 4 states come from:

 

Observation acts like a projector:

Where only one state is causally available in the observer’s future light cone. The other 3 amplitudes are coherently preserved but not accessible.

Why Only One State Is Accessible

This projective collapse follows from three principles:

1. Causal Asymmetry: The observer can only access the forward-in-time component (due to light-cone structure).

2. Measurement Decoherence: Decoherence selects a preferred polarization basis, collapsing superpositions into one mode (e.g., vertical or horizontal).

3. No Direct Access to the Boundary: Since Eide Spheres are not in the bulk, their full internal structure cannot be resolved — only what projects onto decohered spacetime can be “seen.”

Resulting Projection Ratio

Therefore:

4 total coherent modes in each Eide Sphere pair (2 temporal × 2 spatial),

1 projected mode accessible per causal observation in the bulk.

So:

Hence, only ¼ of the boundary energy encoded by the Eide Sphere pair contributes to observable quantities like curvature or entropy. The rest remains nonlocal, coherent, and unmeasured, yet still real and essential to the structure of spacetime.

Decoherence in Eidetic Theory is not a process that happens to Eide Spheres — it is what happens when a time-bound observer interacts with the coherent standing wave formed by temporally entangled boundaries. The observer can only access one temporal direction (forward) and one spatial mode (the decohered polarization), collapsing the rich 4-dimensional internal structure of the Eide Sphere pair down to a single, causal, observable state. This projection filters out ¾ of the internal energy, making only ¼ accessible in spacetime. The rest remains in the entangled boundaries network, preserving unitarity without being visible from within the bulk.

Derivation of Observable Entanglement Energy

Let:

: total boundary entanglement energy per unit area,

: observable entanglement energy per unit area in spacetime.

We define the observer-accessible projection as:

Then the integrated observable energy in the field is:

Plugging into the Eidetic Field Equation:

So we see:

The observable curvature is determined by the projected ¼ fraction of total entanglement energy.

The remaining ¾ resides in the non-observable entangled structure, still contributing to coherence and unitarity, but not to causal curvature.

Application to Black Hole Thermodynamic

The Bekenstein-Hawking entropy is:

Eidetic Theory reframes this as:

Where:

Each Planck-area patch holds one Eide Sphere pair,
Each pair contributes Planck energy:

Only ¼ of this energy is causally accessible.

So:

Let:

 ​

Then:

Now plug in:

so then

And simplify to recover:

Hence, the ¼ factor directly emerges from the dimensional projection of the boundary entanglement field due to:

2 inaccessible temporal states,

1 inaccessible spatial polarization.

Final Result: Origin of the 1/4 Factor

In Eidetic Theory, only a quarter of the internal degrees of freedom of each Eide Sphere pair are accessible to a bulk observer. This projection arises from three fundamental constraints: temporal asymmetry restricts causal access to the forward-directed light cone; holographic encoding limits observable information to one bit per Planck area; and decoherence collapses the bidirectional standing wave node into a single observable polarization mode. As a result, only one of the four internal states—arising from two temporal directions and two spatial modes—contributes to the observable curvature or entropy, yielding the Bekenstein-Hawking factor of ¼​.

The Eide Spheres network itself is fundamentally continuous: a coherent field of temporally entangled boundaries encoding gravitational energy. However, during decoherence or horizon formation, this field becomes projectively quantized into discrete Planck-scale tilings—mirroring Fock space quantization in quantum field theory. Entropy and curvature emerge as projections of this continuous entanglement structure, with standing wave interference patterns across Eide Spheres encoding the geometric and informational content of spacetime. Thus, the classical area law reflects a deeper wave-based process where gravity, entropy, and spacetime geometry arise from the same entangled boundary substrate.

Standing Wave Interpretation of Horizon Entropy 

In Eidetic Theory, local spacetime structure emerges from the coherent superposition of forward- and backward-propagating Eide Spheres, forming standing wave nodes. The black hole event horizon is reinterpreted as a surface of maximal boundary coherence, where these nodes constructively interfere at Planck-scale resolution. Each Planck-area patch corresponds to a stationary interference node—an entropic bit encoded by the bidirectional entanglement of Eide Sphere pairs.

This reinterpretation yields three key implications:

1. Entropy arises from interference amplitude—not merely from surface propagation.

2. The event horizon forms a nodal shell where entanglement coherence is maximized.

3. Hawking radiation reflects decoherence-induced node collapse, whereby stored boundary information becomes partially re-emitted into the bulk.

Thus, Eidetic Theory preserves the thermodynamic foundations of Bekenstein and Hawking, while recasting black hole entropy as a manifestation of deeper boundary standing wave dynamics—unifying entropy, curvature, and quantum information through a coherent, wave-based mechanism.

Quantum Eraser Experiment (Scully, M. O., & Drühl, K., 1982)

In the quantum eraser setup, entangled photon pairs are generated. One photon (the signal) travels toward a detection screen where an interference pattern can potentially emerge, while its entangled partner (the idler) carries the which-path information—indicating which slit the signal photon passed through. If this which-path information is retained, the interference pattern vanishes. However, if the which-path information is “erased,” even after the signal photon has already been detected, the interference pattern reappears—but only when the results are correlated (i.e., analyzed in coincidence) with the idler photon. This phenomenon suggests that future manipulation of entangled information can influence the statistical structure of outcomes already recorded, challenging classical notions of causality and time.

Eidetic Theory Interpretation:

In the Eidetic Theory interpretation, the signal and idler photons are entangled not only with each other, but also with a standing wave field of temporally entangled Eide Spheres encoding Eidetic energy—quantum boundary structures composed of temporally entangled forward- and backward-propagating components. These standing wave nodes encode spatial and temporal information, forming the substrate from which bulk events emerge. 

The delayed erasure of the idler’s which-path information influences the entangled configuration of these Eide Spheres, thereby modifying the quantum boundary conditions that define the bulk geometry into which the signal photon’s behavior is ultimately resolved. Although the signal photon has already been detected, its observed behavior was not fully classical until the boundary configuration—including the idler’s outcome—became complete. In this view, the apparent retrocausality arises not from information traveling backward in time, but from the fact that the bulk event only emerges after coherence across the full network of temporally entangled boundaries is achieved.

We start with a typical entangled photon-pair state (Signal S and Idler I):

Where, 

This is before any which-path information is measured. 

In Eidetic Theory, this state is not just a wavefunction floating in space – it is encoded across a network of time forward and time backward Eide Spheres, forming temporal quantum boundaries.

We can represent this embedding as:

Where,

The Eide Spheres carry the nonlocal, temporally-bidirectional information about both photons. 

In quantum mechanics, detection of S results in wavefunction collapse to

In Eidetic Theory, detection corresponds to the localization of a standing wave node within the Eide Sphere field—where coherent interference between temporally bidirectional components collapses into a bulk-resolved event, however, the geometry of this event is still entangled with the unresolved structure – due to the Idler not being measured yet. 

 

Thus,

 

Bulk localization of S occurs, but is still conditioned on .

 

Now when the Idler is measured, say in a basis that erases which-path information:

 

Transform,

 

Where = Erased basis (interference basis)

 

Now the total state is, 

 

Where, 

 

Now, conditioned on coincidence counts with , the signal photon’s bulk state regains interference patterns:

 

Probability of S detected at position x:

 

Where, 

 

In Eidetic Theory, wavefunction collapse is not instantaneous or absolute – it is conditional on total boundaries resolution:

 

Where:

 

And the final bulk outcome is governed by the resolved configuration of Eide Spheres, which can be modified by later actions (idler measurement). 

 

In the Eidetic Theory interpretation of the quantum eraser experiment, prior to any detection, the wavefunction of the entangled photon pair is fully distributed across the network of Eide Spheres—lower-dimensional quantum boundaries that encode both spatial and temporal information. When the signal photon is detected, this initiates a partial boundaries-to-bulk transition, localizing the signal photon within emergent bulk reality. However, the configuration of the boundaries—the entanglement structure of the Eide Spheres—still retains the complete which-path information of the system. If the idler photon’s which-path information is later erased, this action reconfigures the Eide Spheres network, thereby altering the boundary conditions that define the signal photon’s bulk event. As a result, even though the signal photon has already been detected, the interference pattern can re-emerge in the coincidence counts, because the bulk geometry of the signal photon’s event was not fully resolved until the entire entangled boundary configuration—including the idler’s outcome—was completed.

 

So this is not delayed choice, it’s delayed decoherence, it’s delayed standing wave node resolution—bulk reality emerges only after the full interference structure of temporally entangled Eide Spheres is finalized.

 

In Eidetic Theory, causality is not violated because bulk events emerge only after all entangled boundary configurations are finalized. Time symmetry in the boundary field ensures consistency, while local observers perceive asymmetry due to directional entanglement access.

 

To fully understand how the delayed decoherence of the signal photon arises and how the final outcome is selected, we must now examine the deeper structure of probability and localization within Eidetic Theory. Specifically, we must explore how the surface entanglement energy density across the Eide Spheres network naturally leads to the emergence of the Born Rule and defines the coherence window within which free will operates.

 

Looking at this from a more quantum fundamental perspective, in Eidetic Theory, a quantum system is fundamentally a distributed entanglement across an infinite network of forward- and backward-propagating Eide Spheres. The wavefunction does not merely represent an abstract probability amplitude but encodes the coherent boundary structure:
specifically, the distribution of entanglement energy density across the Eide Spheres network.

Each possible future outcome corresponds to a region within the network where Eide Spheres are coherently aligned toward that outcome. The amplitude (outcome) describes the local strength of boundary entanglement corresponding to that possibility.

Importantly, because Eide Spheres are temporally entangled pairs — with forward- and backward-moving components — the physical energy associated with a coherent outcome involves contributions from both the amplitude and its complex conjugate.

Thus, the surface entanglement energy density ​ associated with a particular outcome is proportional to:

where is the complex conjugate of .

This quadratic scaling arises from the physical interference energy of the standing wave field—where both the forward and backward Eide Sphere components contribute equally to the local amplitude. The squared modulus represents the coherent standing wave energy density at a given nodal point, which governs the likelihood of localization.

This immediately introduces a quadratic dependence on the wavefunction amplitude, matching the structure of the Born Rule.

Localization into the classical bulk occurs when the quantum system decoheres — that is, when its distributed entanglement across the Eide Spheres network collapses into a finite, classical configuration.

In Eidetic Theory, decoherence is not external or arbitrary but occurs when the local entanglement tension across the Eide Spheres exceeds a critical threshold, denoted .

At the moment of decoherence, the system transitions from a coherent superposition of possibilities into a localized outcome within the bulk.
However, this transition is weighted by the surface entanglement energy density at each potential outcome:

regions of the Eide Spheres network with higher ​  — meaning stronger coherent alignment — are more stable against decoherence.

when decoherence occurs, outcomes with higher ​ have higher probability of being realized in the bulk.

Thus, the probability of a particular outcome emerging from the boundary-to-bulk transition is proportional to the local surface entanglement energy density, which, as shown, is proportional to .

Formally:

which is the Born Rule.

The Born Rule, in the context of Eidetic Theory, is not an imposed statistical postulate but a direct consequence of the geometric structure of temporally entangled Eide Spheres, the energy density scaling of coherent entanglement configurations, and the dynamical boundary-to-bulk transition governed by entanglement tension thresholds.

This interpretation provides a physically grounded and inevitable explanation for why quantum probabilities are determined by the square modulus of the wavefunction, eliminating the mystery traditionally associated with the Born Rule in standard quantum mechanics.

Eidetic Theory derives the Born Rule as a natural consequence of the physics of quantum boundaries as the wavefunction amplitude encodes the density of boundary entanglement associated with each possible outcome and the surface entanglement energy scales quadratically with the amplitude, . During decoherence, the probability of localization into a particular outcome is proportional to the local surface entanglement energy. Therefore, the Born Rule emerges without needing to be postulated.

In this view, probability in quantum mechanics reflects a physical competition among entangled boundary structures, rather than an abstract statistical rule.

Thus, in the Eidetic Theory interpretation of the quantum eraser experiment, phenomena such as delayed choice, interference recovery, and probabilistic outcome selection are not mysterious or paradoxical. They are natural consequences of the boundary-to-bulk transition process, governed by the entanglement structure of Eide Spheres across both temporal directions. The experiment does not imply retrocausality or abstract statistical interference. Instead, it reveals the fundamentally nonlocal and bidirectional structure of entangled boundary energy — where bulk outcomes only emerge once the full standing wave configuration of forward- and backward-temporal Eide Spheres is finalized. Decoherence is not an instantaneous collapse, but a dynamically resolved transition from continuous boundary coherence to localized classical geometry. Free will corresponds to the finite window during which coherent influence over branching outcomes remains possible. This reinterpretation grounds the Born Rule in physical energy scaling and recasts quantum measurement as a geometric resolution of temporally distributed entanglement.

The Born Rule, as derived in Eidetic Theory, arises from the surface entanglement energy density scaling with the square of the wavefunction amplitude, . This scaling reflects not an arbitrary statistical axiom, but a physical competition among coherent boundary structures — regions of the Eide Sphere network with varying degrees of temporal entanglement alignment. Outcomes with stronger local coherence density are more likely to decohere into the classical bulk, making probability a geometric consequence of boundary energy configuration. In this framework, causality remains globally consistent, time symmetry is preserved at the level of the boundary field, and quantum measurement is reinterpreted as a process of boundary entanglement resolution rather than an abrupt or mysterious collapse.

The quantum eraser experiment, in this light, is not merely a curiosity of quantum strangeness but a direct empirical signature of the deeper boundary physics that underlies quantum reality. It reveals a standing wave ontology in which time, causality, and probability all emerge from the coherent interference geometry of bidirectionally entangled Eide Spheres. Decoherence is the resolution of this geometry into a classical event — not a loss of information, but a localization of it — showing that what we perceive as quantum measurement is, fundamentally, a projection from an entangled boundary field into a causally consistent spacetime.

Wheeler’s Delayed Choice Experiment (Wheeler, J. A., 1984)

In Wheeler’s delayed choice experiment, a photon passes through a double-slit apparatus, where it can exhibit either wave-like interference or particle-like which-path behavior. The twist lies in the timing: the decision to insert or remove the which-path detector—thereby determining whether the photon behaves as a wave or a particle—is made only after the photon has already passed through the slits. Astonishingly, the photon appears to “know” what kind of measurement will be performed in the future, displaying interference when no which-path detector is present, and particle-like behavior when one is.

Eidetic Theory Interpretation:

In Eidetic Theory, the photon does not follow a classical trajectory through space, but instead exists as a standing wave node formed by the interference of bidirectionally propagating Eide Spheres—quantized boundary structures carrying Eidetic energy that give rise to all fields, particles, and classical geometry—across the slits. These temporally entangled boundaries encode both past and future measurement configurations in their interference pattern. Prior to detection, the photon exists as a distributed boundary resonance—its final position in the bulk emerging only after the standing wave configuration is resolved.

Because Eide Spheres are entangled across time, the future measurement choice—such as whether to insert or remove the which-path detector—is already encoded within the entangled configuration of Eidetic energy across the Eide Spheres network. Thus, the photon does not retroactively alter its past; rather, the emergent event reflects a coherent geometry constructed from boundary conditions that already encompassed both past and future configurations.

This is not a violation of causality but a manifestation of delayed decoherence: the bulk event of the photon is not fully defined until the interference pattern stored in the Eide Sphere boundaries decoheres into a classical configuration. The future measurement updates the standing wave configuration across the boundaries, finalizing the projection into the bulk.

The photon state passing through the double slit (paths A and B) is:

Where, 

In Eidetic Theory, this superposition is not floating abstractly, it is physically encoded across temporally entangled Eide Spheres:

Where, 

The Eide Spheres network — a temporally bidirectional lattice of quantum boundaries — spans both the past (before the slit) and the future (after detection), coherently binding both possibilities. 

A key Eidetic Theory theory process is that:

Events only emerge in the bulk after the entire configuration of the boundaries – including future measurement choices – is resolved. 

It is important to note that later, we will explore how the reverse of this process describes the interior of a black hole, where events in the bulk are shifted back to the boundaries—becoming unresolved, distributed, and re-entangled across the Eide Spheres network. In this reversal, the causal structure flips: rather than boundary information collapsing into localized events, localized events dissolve into boundary entanglement—effectively de-localizing both position and time. This absorption of events by the boundary is a defining feature of black hole interiors in Eidetic Theory and offers a resolution to the black hole information paradox.

Let be the Boundaries-to-Bulk Projection Operator, which maps coherent standing wave configurations of the Eidetic energy field — encoded across the Eide Spheres — into localized classical events within emergent spacetime

Thus, the emergent bulk state of the photon is:

But crucially,

If not which-path detector:

If which-path detector inserted:

If which-path detection occurs after the photon has passed the slits, Eidetic Theory says:

The boundary state updates:

depending on whether the detector distinguishes paths. 

If no which-path detection:

remains coherent. 

In the final bulk state of the photon, we find, 

No which-path detector (interference pattern):

Where, 

Which-path detector inserted (no interference pattern):

Interference disappears because the configuration of the boundaries decoheres the two paths. 

Bulk reality of the photon is projected from the resolved configuration of the boundaries, which inherently includes both the past conditions and future measurement choices. 

Bulk outcome arises only from the global resolution of the boundaries entanglement network:

Where, 

In Eidetic Theory, the photon does not “know” the future in a classical causal sense. Instead, the future measurement choice is already encoded within the entanglement structure of its Eide Sphere boundaries, which span both temporal directions—past and future. The observed bulk event is a projection from this fully resolved boundary configuration, creating the appearance of retrocausality while in fact preserving unitarity and coherence within the nonlocal, temporally entangled boundaries network.

This perspective reframes quantum indeterminacy not as randomness born of ignorance, but as the result of incomplete boundary resolution. Once the full entangled structure—including future interactions—is defined, the bulk outcome becomes fixed. Thus, the so-called “delayed choice” is not a violation of causality but a manifestation of how events in the bulk are emergent from deeper, temporally bidirectional quantum boundaries. In this way, Eidetic Theory preserves the integrity of both quantum mechanics and relativistic causality, offering a unified framework in which time, information, and geometry arise coherently from the entanglement fabric of the universe.

Wheeler’s delayed choice experiment does not imply retrocausality in the classical sense. Instead, it demonstrates that the apparent “choices” made after a quantum system passes through an apparatus are already embedded in the nonlocal, temporally symmetric structure of Eidetic energy encoded in the Eide Spheres. The observed behavior arises not from an event at the slit, but from a finalized standing wave configuration that coherently includes both past and future measurement conditions. Decoherence, in this view, is not a sudden collapse but a geometrically governed transition: the projection of a resolved boundary entanglement into the bulk. In Eidetic Theory, what appears paradoxical is clarified — the delayed “choice” is simply the finalization of the boundary structure from which the bulk event is projected. Measurement is not the collapse of possibility, but the decoherence of boundaries into a single projection.

The Holographic Principle (’t Hooft, 1993; Susskind, 1995)

The Holographic Principle, originally proposed by Gerard ’t Hooft and developed by Leonard Susskind, suggests that all of the information contained within a volume of space can be described by degrees of freedom encoded on its boundary. This principle was inspired by the thermodynamic properties of black holes—particularly the discovery that black hole entropy is proportional not to volume, but to the area of the event horizon.

Susskind’s formulation generalized this insight beyond black holes, proposing that the entire universe may be a hologram, with its fundamental physics encoded on a two-dimensional surface at the boundary of spacetime. This laid the conceptual foundation for later developments such as the AdS/CFT correspondence and quantum gravity dualities.

Eidetic Theory Interpretation:

Eidetic Theory builds directly on the logic of the Holographic Principle but extends it in a transformative direction. While the traditional Holographic Principle treats the boundary as a passive record—a mathematical encoding of bulk dynamics—Eidetic Theory posits that the boundary is the active generator of physics: the bulk emerges as a projection from the coherent entanglement geometry of these lower-dimensional boundaries.

In this framework, the boundaries are not idealized limits or abstractions, but physically real, quantized surfaces of entanglement energy—Eide Spheres—that generate both spacetime geometry and its connectivity. Rather than a single global boundary at infinity, Eidetic Theory envisions an infinite, distributed network of locally entangled quantum boundaries that form the scaffolding from which all bulk events arise.

These Eide Spheres propagate bidirectionally in time, forming coherent standing wave interference patterns across the boundary network. Each localized bulk event corresponds to a resolved node within this interference field. Geometry, causality, and even measurement outcomes emerge only when this entangled configuration decoheres into a finite, localized result—explaining why spacetime appears classical while arising from non-classical coherence.

By identifying Eidetic energy with boundary entanglement entropy, Eidetic Theory reframes the holographic surface not as a descriptor of information, but as the dynamic source of bulk geometry. This recasts the boundary-bulk relationship as generative and causal, rather than merely dual. Gradients in boundary entanglement energy density produce what we observe as curvature. In this view, Einstein’s field equations emerge as the large-scale limit of microscopic entanglement gradients across the Eide Spheres.

This perspective transforms the Holographic Principle into a constructive mechanism. The Eide Spheres are real, null-propagating, temporally entangled quantum surfaces that encode, transmit, and shape the information and structure of the universe. Their form obeys area laws and satisfies holographic entropy bounds, reinforcing the idea that the true degrees of freedom of reality lie not in the bulk, but on the boundaries that define it.

 

Because the Eide Spheres network is temporally bidirectional, the boundary structure encodes both forward- and backward-propagating entangled information. This ensures time-symmetric unitarity at the boundary level, even as decoherence leads to apparent time asymmetry in the emergent bulk.

The area law emerges naturally because Eidetic energy exists only on the boundaries—never in the bulk. In this way, Eidetic Theory not only honors the core insight of the Holographic Principle but operationalizes it: boundaries are not merely where gravity is encoded—they are where gravity originates.

Moreover, because information in Eidetic Theory is encoded nonlocally across overlapping Eide Spheres, the structure naturally exhibits features analogous to quantum error correction. A single bulk event can be redundantly reconstructed from multiple, independent boundary subsets. This redundancy reinforces the stability of classical geometry and explains the resilience of spacetime to local disturbances: it is a redundancy-protected projection from a coherent entanglement substrate.

Entropic Gravity (Jacobson, 1995)

In 1995, Ted Jacobson made a groundbreaking discovery: Einstein’s field equations could be derived from the first law of thermodynamics. Rather than treating gravity as a fundamental force, Jacobson proposed that it emerges as an entropic effect—a macroscopic response to information flow across causal horizons. By applying the Clausius relation

to local Rindler horizons, he demonstrated that spacetime curvature arises from energy flux and entropy change. This energy flux corresponds to matter crossing a local causal horizon; the temperature is the Unruh temperature perceived by an accelerated observer; and the entropy is proportional to the area of the horizon. When applied to every spacetime point with local Rindler observers, this thermodynamic principle yields the Einstein field equations:

This insight reframed General Relativity as a thermodynamic theory of spacetime, where the geometry we perceive is a coarse-grained result of microscopic quantum degrees of freedom. Yet Jacobson’s derivation, while elegant, left open a fundamental question: what is the microscopic origin of this entropy?

Eidetic Theory Interpretation

Eidetic Theory answers this question by identifying a concrete, physical origin for gravitational entropy: Eidetic energy, the entanglement energy encoded in the structure of Eide Spheres. These are bidirectionally time-entangled, quantized 2S+T surfaces that propagate at light speed in both temporal directions. The universe is threaded with a dynamic lattice of these null-like boundaries, and it is this lattice that gives rise to emergent bulk geometry.

Where Jacobson treated energy flux abstractly, Eidetic Theory reinterprets this flux as a physically quantized field of Eidetic energy, manifesting as observable curvature gradients in the bulk geometry. Each Eide Sphere carries a differential of coherence between forward- and backward-propagating wavefronts, forming an entropic standing wave field. As the network evolves, local decoherence corresponds to energy flow across these boundaries, and gradients in Eidetic energy density deform the bulk spacetime—i.e., they create curvature.

In this view, the entropy is not a statistical approximation, but a measurable quantum energy quantity:

where:

​ is the change in entanglement energy across the Eide Sphere field,

​ is the boundary quantum gravitational temperature (an Unruh-like temperature defined on the null surfaces).

Thus, the Clausius relation becomes:

The flux across an infinitesimal patch of an Eide Sphere is:

where:

is the entanglement energy density of the Eide Sphere field,

is a local timelike vector field,

is the hypersurface element on the local null patch .

In Jacobson’s formulation, combining this relation with the Raychaudhuri equation leads to:

In Eidetic Theory, however, curvature arises not from matter but from boundary entanglement:

That is:

or more generally,

In this formulation, curvature is sourced by the structure and gradient of the entanglement network, not by a classical stress-energy tensor. Since Eide Spheres propagate bidirectionally in time, forming null-like coherence fields, the spatial and temporal density gradients of their configuration determine entropy flow:

Thus, gravity is not a geometric reaction to matter—it is a geometric consequence of maintaining coherence in a temporally entangled quantum boundary network. Spacetime curvature arises from the entropic tension between forward- and backward-moving Eide Spheres. Local curvature is the visible trace of coherence regulation.

This gives Jacobson’s thermodynamic insight a physical foundation: the entropic force we perceive as gravity emerges from the dynamical behavior of Eide Spheres as they propagate, entangle, and decohere across spacetime. Entropy is not an abstract statistical construct—it is a quantized energy field encoded in the geometry of entangled surfaces.

In short: Eidetic Theory provides the missing microscopic mechanism behind entropic gravity, transforming entropy from a statistical inference to a structural property of spacetime’s quantum boundary architecture. The Einstein field equations are no longer fundamental—they are emergent constraints imposed by the need to preserve coherence across a dynamically evolving, entangled boundary network.

AdS/CFT Correspondence and Temporal Holography (Maldacena, 1997; Susskind, 2001–present)

The AdS/CFT correspondence proposes a duality between a gravitational theory in a (d+1)-dimensional Anti-de Sitter (AdS) bulk and a d-dimensional conformal field theory (CFT) on its boundary. In this framework, all bulk gravitational dynamics—including black hole formation and evaporation—are fully encoded in boundary quantum correlations. The extra radial coordinate in AdS maps to the renormalization group (RG) scale of the CFT, associating spatial depth with energy resolution.

AdS/CFT remains the most concrete realization of the Holographic Principle and provides a path to resolving the black hole information paradox: since the boundary theory is unitary, information is preserved even if it appears lost behind a classical event horizon.

However, applying this framework to our own universe—de Sitter (dS) space with positive cosmological constant—remains an open problem. Unlike AdS, dS has no spatial boundary; instead, it has cosmological horizons and future/past spacelike boundaries. Causal disconnection increases with cosmic expansion, limiting access to a global boundary. Any holographic dual for dS must therefore be formulated with respect to temporal boundaries.

Temporal Boundaries and the Need for a New Holographic Framework

Physicists like Leonard Susskind have proposed that holography in dS space may require encoding information not on spatial boundaries, but on temporal surfaces—particularly the Big Bang and the future boundary at infinity. These concepts align with time-symmetric approaches to quantum mechanics, such as the two-state vector formalism, in which both initial and final conditions contribute to quantum outcomes.

However, these ideas lie beyond the reach of AdS/CFT, which is rooted in the spatial and causal structure of AdS space. No complete dS/CFT correspondence has yet been established.

Eidetic Theory Interpretation:

Eidetic Theory generalizes the Holographic Principle beyond AdS by introducing a physical mechanism: Eide Spheres—quantized 2S+T boundary surfaces that propagate bidirectionally in time from all mass-energy events. These structures do not reside at fixed spatial infinities but radiate dynamically through time, forming a network of entangled quantum boundaries that span all of spacetime.

In this view:

Holography is not a static duality between separate boundary and bulk regions.

Instead, it is a generative mechanism: bulk geometry emerges from the evolving entanglement structure of Eide Spheres.

This mechanism is local, dynamic, and applicable to any spacetime geometry—including de Sitter.

Each Eide Sphere pair consists of one forward- and one backward-propagating boundary. Their bidirectional entanglement encodes the full causal, energetic, and geometric information of the bulk. Thus, unlike AdS/CFT, which encodes information at spatial infinity, Eidetic Theory encodes it across spacetime itself—at every point where mass-energy exists.

We write the emergent bulk metric as:

Where:

 

: entanglement entropy stored in Eide Spheres at nearby boundary points.

 

: projection operator from boundary entanglement to bulk geometry.

In AdS, this reduces to the standard boundary projection at spatial infinity. In dS, the same mechanism operates from the temporal boundaries—the Big Bang and the asymptotic future—because Eide Spheres radiate along null-like temporal surfaces.

This generalization leads to the unified principle:

Where:

​: emergent bulk geometry

​: local entanglement energy density of the Eide Sphere network

​: spatial or temporal gradient operator across boundary surfaces

: curvature-generating functional

Summary: Generalized Holography from Eide Spheres

Framework	Boundary Type	Mechanism	Bulk Emergence
AdS/CFT	Spatial at infinity	CFT encodes bulk physics	RG flow = radial depth in AdS
Eidetic Theory	Temporal everywhere	Eide Spheres entangle forward/backward time	Bulk emerges from bidirectional entanglement

In AdS, the conformal boundary arises from spatial overlap of Eide Spheres in negatively curved geometry.

In dS, the temporal boundary is a future/past attractor of Eide Spheres, dynamically encoding cosmic evolution.

In both cases, spacetime emerges as a projection from quantum boundary entanglement, not as a fixed background.

This framework of temporal holography offers a powerful resolution to the black hole information paradox—not merely by appealing to unitarity in a dual boundary theory, but by embedding information directly in the bidirectional entanglement lattice of Eide Spheres. Because Eide Spheres propagate forward and backward in time from all mass-energy events, they naturally encode a complete, time-symmetric record of causal and quantum information across the entire history and future of the universe. In this view, no information is ever truly lost; it is coherently distributed across temporal boundaries and preserved within the structure of spacetime itself.

To summarize: in AdS spacetime, holography operates via spatial boundaries at infinity, where bulk geometry reflects energy scales and renormalization group flow through the AdS/CFT duality. In contrast, de Sitter space lacks such spatial boundaries, but Eidetic Theory enables holography through temporal boundaries—cosmological horizons at the beginning and end of time. Here, bulk geometry emerges not from fixed conformal surfaces, but from evolving temporal entanglement encoded in Eide Spheres. Eidetic Theory thus transforms the Holographic Principle from a mathematical constraint into a dynamical, physical process, wherein gravity and geometry actively arise from the networked propagation of entangled quantum boundaries—providing a complete, universal mechanism of spacetime emergence that applies to both AdS and our own de Sitter universe.

Holographic Entanglement Entropy (Ryu & Takayanagi, 2006)

In 2006, Shinsei Ryu and Tadashi Takayanagi introduced a groundbreaking result that helped crystallize the deep connection between quantum information and spacetime geometry. Known as the Ryu-Takayanagi (RT) formula, their proposal equates the entanglement entropy of a region in a conformal field theory (CFT) with the area of a minimal surface in the corresponding anti-de Sitter (AdS) bulk spacetime. Specifically, the entanglement entropy ​ of a boundary region is given by:

where ​ is the minimal surface in the AdS bulk homologous to , and is Newton’s constant. This formula not only provided a powerful computational tool in holographic theories, but also lent deep support to the idea that spacetime geometry is emergent from quantum entanglement.

The RT formula implies that the very structure of spacetime—its shape, curvature, and causal connectivity—is encoded in the entanglement patterns of a lower-dimensional boundary theory. What had once appeared as a thermodynamic anomaly in black hole physics—the area scaling of entropy—was now understood as a general feature of holography. The black hole area law became part of a broader principle linking geometry to quantum information theory.

Eidetic Theory Interpretation:

Eidetic Theory naturally incorporates—and extends—the logic of the Ryu-Takayanagi framework by proposing that entanglement entropy is not merely a boundary condition or computational tool, but the physical source of gravitational effects in the bulk. In this model, Eide Spheres are quantized 2S+T surfaces of entangled Eidetic energy, propagating at the speed of light in both forward and backward time directions. These temporally entangled quantum boundaries encode entropy not as an abstract statistical quantity, but as a measurable, dynamic energy field—the fundamental driver of spacetime geometry.

If Eidetic energy is identical to boundary entanglement entropy, as Eidetic Theory proposes, then the Ryu-Takayanagi formula acquires a deeper physical meaning: the area of a minimal surface is not merely a proxy for entropy—it is a projection of the quantized Eidetic energy stored in Eide Spheres intersecting that region. These surfaces obey area laws because Eide Spheres themselves are quantized null boundaries, and their collective configuration defines the emergent structure of the bulk.

In standard holography, the area is computed geometrically and only indirectly linked to quantum states. In Eidetic Theory, the surface arises from the physical propagation of quantized Eidetic energy, such that geometry itself is an emergent expression of boundary entanglement dynamics.

Thus, Ryu-Takayanagi surfaces are not just mathematical tools for calculating entropy—they correspond directly to coherent configurations of Eide Spheres. Each such surface reflects a specific entanglement geometry that gives rise to the curvature, topology, and connectivity of spacetime itself.

Restating the Entropy-Energy Relationship in Eidetic Terms

Where:

is the total Eidetic energy encoded in Eide Spheres intersecting ,

​ is the effective quantum gravitational boundary temperature (e.g., Unruh-like).

From Eidetic Theory, we also obtain a reformulation of the area-based expression:

​

Due to polarization and causal projection constraints, each Eide Spheres pair contains four internal states (two temporal, two polarization), but an observer interacts with only one temporal and one polarization state per interaction. This causally limits the accessible information to ¼​​, aligning with the Bekenstein-Hawking entropy factor:

This implies:

So that:

An Eidetic entanglement projection law could thus be stated:

The gravitational entropy of a bulk region is the quantized projection of Eide Sphere boundary entanglement energy across its minimal surface.

Eidetic Theory transforms the Ryu-Takayanagi formula from a tool of holographic computation into a direct law of nature. In this view, entanglement is not an auxiliary boundary statistic—it is the physical substance of gravity. The area of a minimal surface reflects the quantized Eidetic energy encoded in Eide Spheres intersecting that region. Spacetime emerges not from smooth manifolds, but from coherence patterns in a null-propagating entanglement field. RT becomes an operational law: a geometrical fingerprint of the quantum coherence structure that gives rise to gravity itself.

Gravity from Entanglement (Van Raamsdonk, 2010)

In 2010, Mark Van Raamsdonk proposed a bold and elegant insight that significantly advanced the modern understanding of spacetime in the context of quantum gravity. Through thought experiments involving quantum fields and the AdS/CFT correspondence, he demonstrated that the geometric structure of spacetime emerges from patterns of quantum entanglement. Specifically, increasing entanglement between regions in a boundary quantum field theory causes the corresponding bulk geometry to become more connected, while decreasing entanglement leads to disconnection—and ultimately, to the disintegration of the bulk spacetime itself.

This revolutionary perspective suggested that spacetime is not fundamental, but emergent—a geometric manifestation woven from the entangled states of an underlying quantum system. Van Raamsdonk’s work helped solidify the idea that quantum entanglement is not merely a feature of quantum systems, but the essential fabric from which space and time arise. However, his approach remained largely conceptual: while it established that entanglement governs spacetime connectivity, it did not specify a physical medium or mechanism through which this entanglement generates geometry.

Eidetic Theory Interpretation:

Eidetic Theory fully embraces—and extends—Van Raamsdonk’s insight by proposing that entanglement is not merely responsible for the emergence of spacetime; it is quantized gravitational energy. In this model, Eide Spheres are the fundamental units of that energy: quantized, temporally entangled boundaries that propagate at the speed of light and define the geometric structure of the bulk. These surfaces are not emergent from deeper degrees of freedom—they are the quantized Eidetic field itself, encoding and transmitting the informational content of mass-energy throughout the universe and generating spacetime in the process.

Where Van Raamsdonk demonstrated that entanglement governs the connectivity of spacetime, Eidetic Theory provides a concrete mechanism: the entangled network of Eide Spheres constructs and connects spacetime from the bottom up. Their bidirectional temporal propagation encodes both the causal structure and geometry of the universe, while their density and configuration determine curvature and gravitational interactions. As Eide Spheres radiate outward from mass-energy in time-symmetric pairs, they weave a nonlocal, entropic network whose topology gives rise to the observable fabric of spacetime.

In this picture, Van Raamsdonk’s principle—that more entanglement leads to more connected spacetime—finds a direct physical expression: the density of temporally entangled Eide Spheres is proportional to the local connectivity of spacetime.

Define the local connectivity function at point bulk as:

Where:

is the local Eidetic energy density from overlapping Eide Spheres at

Then the bulk geometry is governed by:

This implies that spacetime curvature at point arises solely from the local configuration of Eide Sphere entanglement—the curvature is a geometric projection of quantum boundary density.

If entanglement is removed between regions, Eide Spheres no longer span between them. The entropic flux across the shared boundary collapses. The Eide Sphere network fragments, and the bulk region loses its causal coherence. Spacetime disconnects—it rips.

This realizes Van Raamsdonk’s thought experiment as a physical medium: spacetime is built by the causal stitching of entangled Eide Spheres.

Discrete Formulation of Bulk Curvature from Eidetic Energy

Modeling spacetime geometry as a bulk projection of quantized Eidetic energy, we express: 

Where are discrete Eide Sphere interaction points. 

Each delta function term represents a localized spike in Eidetic energy at , and the sum of these projections yields smooth classical curvature in the large-scale limit.

Recovering General Relativity

Treat the classical stress-energy tensor as the large-scale limit of this projection:

As quantum coherence across Eide Spheres decoheres, the discrete entanglement structure averages into an effective field. The classical stress-energy tensor of General Relativity arises—not as a fundamental input, but as an emergent projection of Eidetic energy geometry.

Einstein’s theory becomes the decohered macroscopic limit of a fundamentally entangled quantum boundary network.

What Van Raamsdonk proposed as a powerful heuristic—that entanglement creates spacetime—becomes, in Eidetic Theory, a concrete physical model: quantized Eidetic energy and boundary entanglement entropy are the same phenomenon.

Eide Spheres are the physical structures that generate spacetime and its curvature. Rather than merely mapping entanglement onto geometry, Eidetic Theory identifies entanglement as the physical origin of gravity itself. It offers a fully quantized framework in which the universe’s structure and connectivity emerge not from pre-existing spacetime, but from the coherence structure of quantum information distributed across dynamically entangled boundaries.

Eidetic Theory thus completes Van Raamsdonk’s vision:not only does entanglement construct spacetime—it is gravity.

ER = EPR Conjecture (Maldacena & Susskind, 2013)

In 2013, Juan Maldacena and Leonard Susskind proposed a bold unification of quantum entanglement and spacetime geometry in the form of the ER = EPR conjecture. According to this idea, each pair of entangled particles—such as those involved in the Einstein-Podolsky-Rosen paradox—is connected by a non-traversable wormhole, or Einstein-Rosen bridge. In this framework, entanglement and geometry are not merely related, but two manifestations of the same fundamental phenomenon. The conjecture built a striking bridge between General Relativity and quantum mechanics, suggesting that the microscopic fabric of spacetime might consist of countless entangled connections, each with its own geometric signature.

Eidetic Theory Interpretation:

Eidetic Theory honors the deep intuition behind ER = EPR—that the structure of spacetime arises from quantum entanglement—but it does not adopt the wormhole metaphor as a literal mechanism. Instead, it offers a more fundamental and fully quantized reformulation. In this framework, Eide Spheres—lower-dimensional, null-like, temporally entangled quantum boundaries—are the fundamental carriers of Eidetic energy, not existing in spacetime, but generating it.

Each bidirectionally entangled pair of Eide Spheres does not connect two spatial points via a tunnel. Rather, it projects coherence across time, encoding geometric and causal relations through entropic alignments of boundary information. The wormhole of ER = EPR is replaced by a nonlocal, temporally extended entanglement network. This structure preserves the spirit of ER = EPR, but reframes it in a more physically grounded and dynamic architecture.

Geometry Without Wormholes

Where the ER = EPR conjecture implies hidden spatial geometry (non-traversable wormholes) underlying entanglement, Eidetic Theory asserts that entanglement is geometry. There are no latent bridges stitched behind the scenes. Instead, the connectivity, curvature, and causal flow of spacetime emerge from the density, orientation, and temporal coherence of Eide Spheres. The entangled structure of the universe is encoded in this boundary network—not through tunnels, but through entropic projections of quantized coherence across null-like surfaces.

Eide Spheres carry Eidetic energy, and their intersecting propagation patterns define the shape and topology of the bulk. Geometry is no longer inferred from entanglement—it is constructed by it.

Temporal Entanglement and Causal Stitching

A key generalization offered by Eidetic Theory is that Eide Spheres are temporally entangled: each pair propagates both forward and backward in time. This enables the encoding of causal coherence across the temporal dimension, allowing information to be preserved not just nonlocally in space, but bidirectionally in time.

This framework directly addresses the black hole information paradox. Information entering a black hole becomes entangled with Eide Spheres that propagate both inward (toward singularity) and outward (to future infinity). During evaporation, this temporal entanglement allows information to be recovered—not by traversing a wormhole, but by maintaining entanglement across temporal boundaries.

This causal stitching across time preserves non-local quantum coherence without invoking traversable or hidden geometries. The same functional role attributed to ER bridges is fulfilled by temporally entangled Eide Sphere networks.

The Eidetic Equivalence Principle

Eidetic Theory therefore generalizes ER = EPR. It replaces the duality between entanglement and wormhole geometry with a unified identity: entanglement is geometry. This leads naturally to what might be called the Eidetic Equivalence Principle:

Quantized entanglement and emergent geometry are not dual descriptions—they are the same physical phenomenon, viewed from boundary and bulk perspectives.

This principle removes the need for dual constructs such as “wormhole = entanglement” and instead reveals that spacetime is the unfolding expression of temporally entangled quantum boundaries.

In summary, the ER = EPR conjecture proposed that quantum entanglement implies the existence of hidden wormhole geometries linking entangled systems. Eidetic Theory advances this idea by asserting that quantum entanglement is the geometry—specifically, that Eidetic energy, encoded in temporally entangled Eide Spheres, is the fundamental source of spacetime curvature and connectivity. In this view, wormholes are not required; they are metaphors for a deeper mechanism—entropic coherence across quantum boundaries. Information preservation during black hole evaporation and the maintenance of causal structure arise naturally from the bidirectional temporal entanglement within the Eide Sphere network. ER = EPR is not contradicted but rather encompassed—revealed as a special case of a more universal principle:

All spacetime structure emerges from the quantized entanglement of boundaries.

Quantum Extremal Surfaces and the Island Formula (Engelhardt & Wall, 2019; Almheiri et al., 2020)

In 2019, Netta Engelhardt and Aron Wall introduced the concept of Quantum Extremal Surfaces (QES)—surfaces that extremize the generalized entropy, defined as the sum of a geometric area term and the bulk entanglement entropy. This refinement extended the Ryu-Takayanagi framework to dynamical and quantum-corrected spacetimes, laying the foundation for the Island Rule: in evaporating black holes, the entropy of Hawking radiation is computed not just from exterior regions but from “islands” inside the black hole defined by these QES surfaces.

This formalism achieved a significant breakthrough: it showed that the entanglement entropy of Hawking radiation does not grow endlessly but instead follows a Page curve, rising and then falling in a way consistent with unitary quantum evolution. By including interior islands within the entanglement wedge, the QES framework demonstrated how information from behind the horizon can be recovered from exterior radiation, offering a viable resolution to the black hole information paradox.

Eidetic Theory Interpretation:

Eidetic Theory shares the conceptual core of the QES framework: entanglement entropy is the engine of gravitational dynamics. But while the QES formulation selects entropy-extremizing surfaces computationally, Eidetic Theory provides a physical mechanism for their existence: bidirectionally entangled Eide Spheres.

In this view, Eide Spheres are temporally entangled quantum boundaries propagating at the speed of light. These structures encode quantized gravitational energy and entropic tension, and their intersections define real surfaces of geometric and informational balance. QES surfaces, in Eidetic Theory, are the physical loci where the entropic tension from intersecting Eide Spheres reaches local equilibrium.

The islands that appear in QES entropy formulas correspond, in Eidetic Theory, to regions enclosed by entangled forward and backward Eide Spheres. These islands are not mathematical artifacts—they are emergent spacetime domains arising from the causal and entropic structure of the Eide Sphere network.

This model also provides a unitary mechanism for black hole information recovery. As a black hole forms and radiates, Eide Spheres propagate from the collapsing matter in both time directions: forward-directed spheres become correlated with Hawking radiation, while backward-directed spheres encode infalling information into earlier boundary slices. Over time, the entanglement network reconnects, re-coheres, and preserves information across time, forming a nonlocal causal web that enforces the Page curve without invoking non-unitary evolution.

Eidetic Reinterpretation of the Generalized Entropy

The QES condition that extremizes generalized entropy,

,

is reinterpreted in Eidetic Theory as the entropic equilibrium condition across Eide Spheres. The generalized entropy functional becomes:

,

where:

​​ is the observable entropy per Planck area due to causal projection limits,

is the area of each Eide Sphere patch intersecting the QES surface,

is the boundary entanglement energy contributed by overlapping Eide Spheres through the region .

These terms physically represent the quantized entropic flux and energy density of the Eide Sphere lattice at a given point.

Thus, a Quantum Extremal Surface in Eidetic Theory is defined as the physical surface where bidirectional Eide Sphere entanglement reaches a state of net equilibrium. The QES is not a saddle point in an entropy functional—it is a quantum gravitational boundary surface where coherence stabilizes into emergent geometry.

Island Rule as Entropic Geometry

In Engelhardt’s model, islands are added to the entanglement wedge to preserve unitarity. In Eidetic Theory, they arise naturally as regions bounded by bidirectionally entangled Eide Spheres. The island is the region where backward-time Eide Spheres from the interior reconnect with forward-time spheres from the exterior—forming a closed temporal entanglement loop across spacetime.

As the black hole evaporates, this entropic structure defines the Page curve dynamically: entropy rises early, but as backward-propagating Eide Spheres begin to influence earlier slices, information encoded in their coherence begins to project back toward the boundary, reversing the entropy growth. This yields the Page curve as a direct consequence of temporally symmetric entanglement propagation.

Eidetic Extremal Surface Principle

A Quantum Extremal Surface in Eidetic Theory is the locus of local equilibrium in the entropic tension of intersecting, temporally entangled Eide Spheres. It is where coherence becomes curvature, and where bidirectional quantum information stabilizes into the structure of classical spacetime.

The QES condition thus becomes a physical equation of balance across the Eide Sphere network. The island formula becomes not a computational tool, but a projection law for the causal geometry of information-preserving spacetimes. What appears in semiclassical theory as entropy extremization, emerges in Eidetic Theory as a coherence stabilization surface—the true physical scaffolding of the universe.

 

Quantum Error Correction in Holography (Harlow, 2016)

In 2016, Daniel Harlow proposed a profound reinterpretation of the AdS/CFT correspondence: that holographic spacetimes exhibit the structure of quantum error-correcting codes. In his paper “The Ryu-Takayanagi Formula from Quantum Error Correction”, Harlow showed that:

Bulk operators in AdS can be redundantly reconstructed on multiple boundary subregions—a defining property of error correction.

Entanglement wedge reconstruction implies that the geometry associated with a boundary region is recoverable from the boundary data.

Boundary entanglement nonlocally encodes bulk information, shielding it from localized erasures and perturbations.

This perspective reframes holography not simply as geometric duality, but as an information-theoretic protocol where the boundary entanglement pattern encodes, protects, and reconstructs the bulk.

Eidetic Theory Interpretation:

Eidetic Theory naturally manifests this quantum error correction structure—not as an analogy, but as a physical mechanism. The emergent bulk geometry is constructed from a redundant, temporally entangled network of boundaries called Eide Spheres. These are lower-dimensional, 2-spherical surfaces that encode gravitational information and project it into the bulk.

Let:

​​: quantum state associated with Eide Sphere ​,

: logical operator acting in the emergent bulk,

​: local encoding unitary on boundary patch ​.

Then bulk observables are redundantly encoded as:

​

Each represents a partial encoding of the observable into the Eide Sphere ​, and the sum reconstructs the full bulk operator from distributed boundaries entanglement.

Likewise, the emergent bulk energy arises from the coherent sum of local boundaries contributions:

This redundancy gives rise to resilience: local perturbations in the Eide Spheres network do not destroy the encoded geometry, because the entanglement is nonlocal and distributed—hallmarks of quantum error correction.

Boundaries Curvature and Bulk Decoherence

To model how decoherence propagates into the bulk, we define:

​: boundaries entanglement energy density tensor, measuring curvature in the entanglement field across Eide Spheres

: local decoherence tensor in the bulk.

Then decoherence is driven by curvature in the boundaries entanglement network:

Here, is the scalar boundaries entanglement energy density, and its second covariant derivatives express entropic tension. Coherent, smooth entanglement across Eide Spheres yields stable spacetime; curvature in this field signals decoherence, manifesting as geometric instability in the bulk.

Eidetic Theory reframes quantum error correction as the stabilizing dynamics of spacetime itself. Specifically:

Logical encoding of bulk observables occurs nonlocally across temporally entangled Eide Spheres.

Decoherence in the bulk arises from curvature in the boundaries entanglement network.

Correction occurs via entropic coherence restoration, smoothing the boundaries curvature and restoring stable bulk geometry.

This extends Harlow’s interpretation beyond AdS/CFT: any emergent spacetime supported by Eide Spheres inherits these error correction properties. The universe is not stabilized by background spacetime—it is actively maintained by a self-correcting, entangled quantum boundaries network.

Reformulating the Stress-Energy Tensor with Entanglement Entropy 

In classical General Relativity (GR), Einstein’s field equations relate spacetime curvature to the distribution of mass-energy:

where is the classical stress-energy tensor representing matter and radiation.

In Eidetic Theory, mass-energy is not fundamental. Instead, curvature arises from the entanglement entropy encoded across temporally entangled quantum boundaries, called Eide Spheres. The classical tensor ​ is replaced by an entanglement-energy tensor ​, which captures the flow of Eidetic energy—quantized energy sourced by entropic correlations across the Eide Sphere network:

Here:

: forward-time Eidetic energy flux

: backward-time Eidetic energy flux

These are null-propagating and temporally entangled, and satisfy a unitarity-preserving conservation law:

This expresses the divergence-free propagation of Eidetic energy across boundary layers, ensuring that information and coherence are conserved throughout spacetime evolution. Eide Spheres move at the speed of light and project geometry through their bidirectional entanglement structure.

Variational Definition from Generalized Entropy

We now express ​ as a functional derivative of the generalized entropy:

Then:

This ties variations in geometry directly to variations in boundary entanglement entropy, anchoring spacetime curvature in the dynamics of Eide Sphere coherence.

Entanglement Curvature Tensor and Entropy Propagation

Curvature arises from local gradients in Eide Sphere entanglement. We define an entanglement-induced curvature tensor:

where encodes the local density and directional flux of temporally entangled Eide Spheres. Entropy itself evolves according to:

This equation implies that curvature is not simply sourced by entropy—but that entropy gradients are actively generating spacetime curvature. Geometry responds to how entanglement propagates and redistributes.

Modified Einstein Field Equations in Eidetic Theory

In this framework, Einstein’s equations become:

Here, Eidetic energy replaces classical mass-energy, and is a consequence of bidirectionally entangled null-like structures encoded in the Eide Spheres network.

Schwarzschild-like Metric from Entanglement

To analyze static, spherically symmetric spacetimes, consider a Schwarzschild-like line element:

In classical GR, the Schwarzschild function is:

In Eidetic Theory, the mass term is replaced by an entropic energy density:

Imposing gauge condition , the full metric becomes:

This replaces the curvature source with entanglement entropy density, yielding regular behavior near the origin. If small , if with , then:

The event horizon is located approximately at:

As near the core, spacetime flattens smoothly, eliminating singular behavior.

From Classical GR to Entanglement Geometry

This derivation replaces the stress-energy source of GR with an entanglement-encoded tensor and shows how spacetime curvature emerges from boundary information flow. In classical GR, geometry reacts to pre-existing energy in spacetime. In Eidetic Theory, spacetime itself is emergent—a response to the entanglement structure of temporally coherent quantum boundaries.

Thus, Einstein’s equations appear in Eidetic Theory as the large-scale, decohered limit of a deeper quantum boundary theory. The classical field equations are not fundamental—they are thermodynamic approximations of a unitary entanglement propagation law across Eide Spheres.

Toward an Entropic Interpretation of GR

In the next section, we begin with the Clausius relation and Jacobson’s thermodynamic derivation of Einstein’s equations, reinterpreting them through entanglement entropy saturation across temporally entangled Eide Spheres. This will reveal that Einstein’s equations correspond to an entropic equilibrium condition, not a gravitational field law in the classical sense.

Entropic Gravity and Einstein’s Equations from Boundary Entanglement (Eidetic Theory Formulation) 

Ted Jacobson famously derived Einstein’s field equations by assuming that the Clausius relation holds for all local Rindler horizons through each spacetime point, with entropy proportional to the horizon area. He found that requiring this thermodynamic identity to hold implied the Einstein equation:

In Eidetic Theory, spacetime curvature is not sourced by stress-energy in the bulk, but by gradients in boundary entanglement entropy across the Eide Sphere network. The Einstein field equations emerge as a macroscopic limit of a more fundamental entropic equilibrium condition between localized decoherence in the bulk and entanglement energy density on temporally bidirectional boundary structures.

Thermodynamic Foundation

Following Jacobson’s insight, consider the Clausius relation:

 

We reinterpret each term:

: energy flux through a local boundary patch of Eide Spheres.

: Unruh temperature experienced by an accelerated observer near the boundary.

: change in entanglement entropy across the boundary surface area.

Let the entropy be proportional to the area of the local boundary:

  with  

A boost Killing vector defines the local Rindler horizon, and energy flux is:

where ​ is the local energy-momentum tensor of the matter fields, and is the cross-sectional area element of the horizon.

Variational Principle from Entropic Functional

We define the Eidetic Entropy Functional across boundary surfaces :

where is the entanglement energy density of Eide Spheres at point x, defined as:

Demanding stationarity of this entropy functional under variation of the spacetime geometry:

This yields a geometric equation for the curvature of spacetime as a response to entanglement energy:

In this equation:

: the bulk decoherence energy density, a projection from residual Eide Sphere standing waves.

: an effective entanglement energy-momentum tensor, arising from net gradients in coherence and decoherence across Eide Sphere intersections.

We identify the cosmological constant from the vacuum entanglement field:

Thus, the Einstein tensor arises not as a classical geometric quantity, but as a response to the local entropic configuration of Eide Sphere entanglement patterns:

Interpretation

In this framework, Einstein’s Field Equations are an emergent condition of entropic equilibrium:

The curvature ​ tells how boundary entanglement is distributed.

The matter energy tensor represents the loss of boundary coherence into the bulk (decoherence).

The cosmological constant arises from persistent residual coherence across spacetime (dark energy).

Summary Equation

We summarize the entropic derivation of spacetime geometry as:

This equation replaces the role of Einstein’s field equations in standard GR, while grounding curvature in quantum entanglement entropy across dynamically evolving Eide Spheres boundaries.

Boundary Coherence vs. Localization Pressure

Having established that the geometry of spacetime emerges from gradients in standing-wave entanglement energy—encoded by temporally entangled Eide Spheres—we now ask: what determines whether a system remains delocalized in this entangled boundary network or collapses into a localized event in the bulk?

Eidetic Theory answers this through a fundamental dynamical principle: the competition between boundary entanglement coherence and local interaction pressure. This tension governs whether a quantum system remains nonlocally extended across Eide Spheres, or transitions into emergent spacetime as a localized, classical object.

This coherence-localization duality provides a unifying framework not only for quantum measurement and wavefunction collapse, but also for black hole interiors, Hawking radiation, and the emergence of causal structure itself. It is the phase dynamic underlying all physical events.

Entanglement–Interaction Competition

We define:

Boundary Entanglement Density:

The surface density of Eide Spheres across a local quantum boundary patch. This quantifies the system’s nonlocal coherence—its capacity to resist collapse.

Interaction Energy Density:

The effective localization pressure exerted by interacting energy. It scales with the square of the characteristic frequency , or inversely with the square of its wavelength.

The phase behavior of a quantum system is determined by the balance between these two quantities:

If

the system remains delocalized across Eide Spheres.

If

the system undergoes wavefunction collapse into the bulk.

Here, is a proportionality constant related to Planck-scale geometry.

Phase Dynamics: Measurement and Black Holes

At low frequencies (long wavelengths), ​ is small, and boundary coherence ​ can resist localization. The system persists in the entangled boundary state.

At high frequencies (short wavelengths), the localization pressure overcomes coherence, and the system transitions into bulk spacetime—resulting in classical localization and wavefunction collapse.

This same mechanism governs black hole interiors.

Outside the horizon: interaction energy density ​ dominates over entanglement energy density ​, enforcing localization.

At the horizon: the system reaches a critical threshold where .

Inside the black hole: ​, and recoherence begins.

As mass-energy falls further inward:

The surrounding Eide Sphere density ​ increases.

Localized degrees of freedom dissolve back into the boundary entanglement network.

Spacetime itself fades; what remains is pure boundary coherence.

Thus, black hole interiors are not singularities, but regions of high-density recoherence, where the bulk dissolves and localization vanishes.

Collapse and Recoherence as Mirror Phases

Wavefunction collapse and black hole interiors are two sides of the same phase process:

Collapse: A system under high interaction pressure collapses out of the boundary into the bulk.

Recoherence: A system under high entanglement density recoheres back into the boundary from the bulk.

These are inverse transitions across the coherence–localization threshold.

The critical frequency for localization is given by:

In high entanglement regions (e.g., near black holes), is high—ordinary localization is suppressed.

In low entanglement regions (e.g., flat spacetime), is low—systems localize more readily.

Universal Phase Engine

Eidetic Theory thus frames reality as a dynamic tension between:

Boundary coherence, sustained by nonlocal entanglement of the Eide Sphere network;

Localization pressure, driven by local energetic interactions.

From this entanglement-interaction dialectic arises the entire physical world:

Phenomenon	Eidetic Interpretation
Wavefunction Spread	Boundary entanglement dominates; coherence is maintained.
Measurement / Collapse	Local interaction pressure exceeds entanglement; localization into bulk spacetime occurs.
Decoherence	Gradual accumulation of interactions erodes boundary coherence.
Recoherence (e.g., BH)	Entanglement overtakes interactions; system returns to boundary state.
Black Hole Evaporation	Slow leakage of recohered information back into emergent bulk.

The causal structure of the universe—space, time, and geometry—emerges from this underlying contest between nonlocal coherence and local interaction.

Where entanglement dominates, systems remain delocalized, temporal directionality fades, and spatial relationships dissolve. Where interactions dominate, coherence collapses, and classical spacetime with causal order is born.

This boundary-to-bulk tension is the phase engine of the cosmos—governing particles, fields, time’s arrow, gravitational wells, and the fate of information. Every quantum measurement, every moment of decoherence, every black hole evaporation event is a ripple on the deeper entanglement sea.

From this perspective, spacetime is not a fundamental arena, but a contingent projection: a visible phase outcome of when and where boundary coherence fails to resist localization.

 

Dark Energy Equation of State in Eidetic Theory

In Eidetic Theory, the residual coherent standing wave energy density survives cosmic decoherence and acts as a smooth, large-scale energy field. This field sources gentle curvature on cosmological scales and manifests observationally as dark energy.

We begin with the Eidetic Field Equation in its covariant form:

 

We aim to determine the effective equation of state parameter:

for the energy component sourced by ​.

Although ​ is a standing wave coherence field, on cosmological scales it behaves like a perfect fluid with:

Energy density:  ​

Pressure:     ​ (to be derived)

In the large-scale limit, where General Relativity is recovered, the stress-energy tensor of a perfect fluid is:

where:

is the energy density,

is the pressure,

is the 4-velocity of the fluid elements.

For a spatially homogeneous and temporally stable field, ​ exhibits:

,

.

Meaning its energy density ​ is nearly constant in space and time.

Since ​ is nearly constant, the primary variation in the action:

comes from varying the volume element ​. Standard variation yields:

Thus:

And the gravitational field equation becomes:

This is formally equivalent to the Einstein equation with a cosmological constant , where:

Since a term proportional to implies: 

,

we immediately recover:

The residual standing wave coherence field ​, which permeates the emergent spacetime in Eidetic Theory, naturally acts as a cosmological constant:

It is not a fundamental scalar field,

It is not vacuum energy in the conventional sense,

It is a persistent quantum boundary structure, whose smooth standing wave coherence couples gravitationally through variation of the emergent geometry.

Thus:

Dark energy is not a mystery fluid—it is the signature of residual coherence in the quantum boundaries network that gives rise to spacetime itself.

This yields:

as a direct result of the entanglement geometry of the Eide Sphere field.

 

Deriving the Effects of Dark Matter in the Eidetic Theory

 

The emergent gravitational field equation in Eidetic Theory is:

where:

​ = smooth, coherent bulk entanglement energy (cosmic-scale dark energy),

= decohered localized energy structures (mass-energy).

Under cosmic decoherence, most localized structures fully collapse into matter—forming stars, gas, and dust. These contribute to the gravitational field via:

However, not all boundary regions decohere completely. Some regions retain partial quantum coherence with the standing wave network. These regions:

Maintain residual boundary entanglement,

Continue to couple to spacetime curvature,

Lack localized, radiating mass-energy.

We therefore write:

where:

​: the gravitational effect of residual Eide Sphere coherence — interpreted observationally as dark matter.

The full field equation now reads:

Here, both baryonic and coherent localized energy contribute to curvature, but only the baryonic term emits light. The coherent residuals act like invisible mass-energy that sources gravity—but not photons.

In a region like a galaxy:

Observed baryonic mass: ​,

Effective gravitational mass (curvature source):

where

Thus:

Observed rotation curves and gravitational lensing are determined by ​,

But electromagnetic radiation only traces .

This mismatch creates the observed dark matter phenomena.

In cases like the Bullet Cluster, where gravitational lensing reveals mass displaced from the visible gas, Eidetic Theory provides a natural explanation: the residual coherent standing wave fields remain spatially distinct from the baryonic matter that has been stripped away by collision. The entanglement energy of these nonlocal coherent structures continues to curve spacetime, even though no ordinary mass resides there.

In Eidetic Theory, then, the gravitational effects attributed to dark matter arise from partially decohered boundary structures that retain quantum coherence with the standing wave network. These are not composed of localized mass-energy and do not emit or absorb light, but they still source curvature due to their residual entanglement energy density.

To summarize, the residual coherence of the Eide Spheres network contributes to curvature without requiring dark matter particles. In classical GR, this appears as “missing mass.”  In Eidetic Theory, it is the gravitational memory of incomplete decoherence.

Formally, this decomposition can also be embedded into the covariant action-based formulation:

Here:

corresponds to the smooth background energy ​ (dark energy),

​ captures fully decohered matter,

​ represents the dark matter effect as partially decohered but still gravitationally active standing wave structures.

This two-level approach—starting with the intuitive heuristic and culminating in the covariant formulation—reveals how Eidetic Theory geometrizes dark matter as residual quantum coherence in the gravitational field equations. Dark matter is not particulate, exotic, or decoupled from known physics. It is the geometric imprint of incomplete decoherence in the quantum boundaries that source spacetime. These residual coherent standing wave regions remain gravitationally active despite lacking localized mass. The classical “missing mass” problem is resolved as a mismatch between luminous matter and entanglement energy density—not as evidence for unseen particles, but for unseen coherence.

Calculating the Cosmological Constant in Eidetic Theory

At the origin of the universe, the energy density of the standing wave network formed by Eide Spheres is at the Planck scale:

Numerically:

Thus, if coherence were perfect, the energy density driving spacetime would be Planck-scale.

Over cosmic time, expansion, quantum fluctuations, and the growth of gravitational structures drive widespread cosmic decoherence.

As a result, almost all standing wave modes lose coherence and only a tiny fraction of the original standing wave modes remain coherent across the observable universe.

Thus, the effective residual energy density today is:

where:

is the fraction of coherent modes remaining.

The amount of residual coherence scales with the area-based ratio of the Planck length to the Hubble radius:

where:

(Planck length),

(current Hubble radius),

is determined by the dimensional scaling of decoherence.

The coherent standing wave structures are 2-spherical (surfaces) and cosmic decoherence occurs primarily across surfaces, not volumes. This is consistent with the holographic principle: degrees of freedom scale with area.

Thus:

Ratio:

Thus:

Thus:

Observed value (from supernovae, CMB, large scale structure):

There is no need for fine-tuning or cancellation mechanisms—only the natural suppression arising from cosmic decoherence of standing wave structures across causal surfaces.

In Eidetic Theory, the observed small but nonzero cosmological constant arises naturally from the residual coherent standing wave energy density of Eide Sphere structures that survive cosmic decoherence. The suppression factor scales with the square of the ratio of the Planck length to the cosmological horizon length, yielding and correctly predicting the dark energy density without requiring unnatural cancellations or fine-tuning.

The cosmological constant is thus not a mysterious vacuum energy but the entropic shadow of decoherence in the boundary network that gave rise to spacetime.

 

The Limitation of Detecting Gravitons (Eide Spheres) in the Bulk

While conventional quantum field theory treats gravitons as hypothetical quantized excitations of the gravitational field, Eidetic Theory reinterprets them as manifestations of a deeper boundary coherence structure that gives rise to spacetime itself.

In the framework of Eidetic Theory, the fundamental quantum structures termed Eide Spheres are not particles or fields within spacetime—they are the pre-spacetime quantum structures from which the geometry of spacetime itself emerges. Eide Spheres are quantized, temporally entangled boundary surfaces that propagate at the speed of light, continuously encoding the coherent structure of bulk spacetime geometry. From this fundamental perspective, an essential consequence follows: Eide Spheres cannot decohere within the bulk, and hence, cannot be detected as localized quanta.

This result is in agreement with the impossibility arguments for graviton detection first articulated by Freeman Dyson and other subsequent analyses in quantum gravity. Dyson (2004) argued that individual gravitons would be impossible to detect not merely due to technological limitations, but because the very act of attempting to detect a graviton would require gravitational effects far too small to resolve against background noise — rendering detection operationally meaningless even in principle. 

Dyson’s argument shows that detecting individual gravitons is not just technologically unfeasible—it is conceptually incoherent within semiclassical frameworks. Eidetic Theory sharpens and deepens this insight: it is not merely that gravitons elude detection, but that they cannot even exist as measurable entities inside the bulk. They are not part of the observable inventory of spacetime—they are its invisible scaffolding.

In other words, in Eidetic Theory, this limitation becomes more fundamental: it is not simply a matter of measurement precision, but a consequence of the ontological role of Eide Spheres.

The reasoning proceeds as follows:

Eide Spheres Are the Substrate of Bulk Reality
Eide Spheres are not quanta within spacetime—they are temporally entangled quantum boundaries that generate spacetime’s coherence. They form the standing wave lattice whose gradients give rise to curvature, localization, and causality.

Detection Requires Decoherence
All detectable events in the bulk arise through decoherence. To “observe” a quantum system, it must transfer entangled information into classical degrees of freedom within the bulk. Detection is decoherence.

Eide Spheres Cannot Be Decohered from the Bulk
Because Eide Spheres are the source of coherence, they cannot decohere relative to the structure they define. This would require collapsing the very entanglement that makes localization possible—an ontological contradiction.

Attempting Detection Would Disrupt Spacetime
Any attempt to measure an Eide Sphere directly would be equivalent to perturbing the boundary entanglement fabric itself. Rather than yielding a graviton detection, it would result in the dissolution of the coherent structure of spacetime.

Dyson’s Argument Reinforces the Principle
Dyson showed that any apparatus capable of detecting a graviton would gravitationally collapse before registering a signal. Eidetic Theory explains this not just as practical limitation, but as a reflection of the impossibility of isolating a single Eide Sphere from the infinite entanglement network it forms.

Thus, gravitons (Eide Spheres) cannot be detected as individual quanta in the bulk because:

They do not exist inside spacetime as localized objects.

Detection requires decoherence, but Eide Spheres cannot decohere relative to the bulk—they generate the decoherence structure itself.

Attempting detection would amount to collapsing the bulk spacetime coherence that defines the classical world.

Energy and observational limits derived from Dyson’s analysis are consistent with this, but the Eidetic explanation is deeper — it shows that even in principle, a graviton cannot be isolated or detected.

In this view, Eide Spheres are fundamentally pre-bulk phenomena: they define the boundary entanglement from which the bulk emerges. The bulk can measure decoherence among localized energy distributions, but it cannot resolve the foundational coherence of its own structure.

Let the quantum entanglement structure defining the bulk be denoted by ​, sourced by the network of Eide Spheres .
Detection requires an interaction Hamiltonian ​ satisfying:

However, because ​ is generated by the themselves, applying ​ to an individual ​ implies:

i.e., any attempt to isolate and measure ​ leads outside the defined coherent structure of bulk spacetime — a physically undefined operation. Thus, detection is impossible.

In this view, asking to detect a graviton is like trying to observe the mirror by looking only at the image inside its reflection. The mirror—the Eide Sphere network—cannot be seen from within the reflection it produces.. Eide Spheres are not gravitational signals in spacetime—they are the very quantum scaffolding upon which spacetime is built.

Derivation of General Relativity from the Eidetic Field Equation

In Eidetic Theory, spacetime is not a preexisting arena but an emergent structure arising from gradients in a fundamental quantity: the bulk entanglement energy density, denoted . This energy density is sourced by the standing wave coherence of temporally entangled Eide Spheres—quantized, bidirectional boundary surfaces that form the informational substrate of spacetime.

We begin with the Eidetic action:

Where:

is the bulk entanglement energy density, emerging from standing wave coherence of temporally entangled Eide Spheres,

is the emergent volume element of the inverse spacetime metric ​.

Classical Limit: Decohe rence and Matter Coupling

In the fully decohered (classical) limit, reduces to the matter Lagrangian ​, a function of classical fields , their covariant derivatives , and the metric :

The action becomes:

Metric Variation and the Emergent Stress-Energy Tensor

To derive the field equations, we vary the action with respect to :

Using the standard identity:

We write the full variation:

This yields the entanglement energy-momentum tensor:

In the decohered limit where , this becomes:

which is the standard stress-energy tensor in General Relativity.

The Emergent Gravitational Field Equation

From the variation of the entanglement-based action, the Eidetic Field Equation is:

In the classical limit, this reduces to:

This is Einstein’s field equation—not postulated, but derived from a deeper entanglement-based action.

Interpretation in Eidetic Theory

In General Relativity, curvature is sourced by the stress-energy of matter.

In Eidetic Theory, curvature is more fundamentally sourced by entanglement gradients in the network of Eide Spheres.

Matter corresponds to localized decoherence events—regions where boundary coherence collapses into emergent bulk structure.

The Einstein tensor satisfies the Bianchi identity:

Because the action is diffeomorphism-invariant, we obtain:

This reflects conservation of entanglement energy flow—not just classical energy-momentum—across the emergent spacetime.

This derivation shows that General Relativity is a low-energy, decohered limit of a more fundamental quantum-coherent framework rooted in boundary entanglement. The stress-energy tensor of classical physics emerges from decohered gradients in entanglement energy, and the curvature of spacetime reflects the geometric response of the bulk to these gradients.

In this view, gravity is not a force between masses—it is the manifestation of changes in the coherence structure of the Eide Sphere network. Both spacetime and matter arise from the same substrate: the quantum standing wave coherence of temporally entangled boundary structures.

This framing offers a powerful bridge between quantum mechanics and General Relativity, positioning gravity as a consequence of the geometry of entanglement rather than a fundamental interaction in its own right.

Derivation of Maxwell’s Equations from the Eidetic Field Equation

Let be the directional boundary coherence flux at spacetime point , sourced by the alignment of Eide Sphere entanglement phases. It  encodes non-scalar coherence information — directional correlations in boundary entanglement.

This vector potential is emergent — it does not exist on a fixed manifold, but within a decohered region of the Eide Spheres network.

We define the entropic phase field strength tensor:

This structure is antisymmetric by definition: , and captures the rate of change of coherence alignment across spacetime directions.

We apply the exterior derivative (wedge product) to the 2-form field , yielding:

This is the Bianchi identity and implies the two homogeneous Maxwell equations:

These follow from the geometry of entanglement flux and require no sources — they are topological consequences of the antisymmetric coherence field.

Let represent the local entropic decoherence current, sourced by an imbalance in the number or phase weight of forward- and backward-temporally entangled Eide Spheres.

​: a net coherence imbalance gives rise to charge,

: a net directional transport of coherence yields current.

This current is locally conserved:

This conservation is a geometric consequence of entanglement structure — not of particle number.

We define a Lagrangian density that encodes the energy of directional entanglement structure:

The first term is the internal energy of the phase field,

The second term is the coherence–decoherence coupling, i.e., how entanglement flux interacts with decoherence events (i.e., “charge”).

The total action is:

We vary this with respect to :

Demanding for arbitrary ​, we obtain:

This gives:

These are the inhomogeneous Maxwell equations.

In Eidetic Theory, does not represent particle flow. It arises from:

Temporal asymmetries in entanglement coherence,

Differences in forward vs backward Eide Sphere emission,

The presence of localized decoherence, which breaks temporal symmetry.

Thus, charge is not fundamental — it is an emergent residue of entanglement asymmetry.

The potential can be redefined without changing the physical field:

This reflects the redundancy in phase labeling of entanglement across the Eide Spheres network. The bulk is sensitive only to phase differences — not absolute labels.

Gauge symmetry is therefore a boundary information redundancy, not a fundamental symmetry of nature.

Define components in the observer’s rest frame:

These fields arise from the time-space and space-space entanglement gradients respectively.

They obey:

All four Maxwell equations are thus recovered.

Maxwell’s equations are not postulated but emerge from:

The antisymmetric structure of directional coherence ,

The coupling between entanglement phase flux and decoherence currents ,

Gauge symmetry as entanglement phase relabeling,

Conservation laws from coherence continuity,

And all observed electromagnetic dynamics arising from bulk responses to structured boundary entanglement.

The derivation of Maxwell’s equations within the Eidetic framework reveals that electromagnetism — like gravitation — is not a fundamental interaction in the traditional sense, but an emergent feature of the structured entanglement of boundary coherence. The electromagnetic field arises from directional asymmetries in the phase-aligned network of Eide Spheres, and charge itself is recast as a localized temporal imbalance in boundary entanglement. Just as curvature in General Relativity reflects spatial gradients in decohered entanglement energy, electromagnetic phenomena reflect directional and temporal gradients in coherence structure. Gauge symmetry, rather than being a foundational principle, emerges here as a redundancy in labeling entangled boundary phases — a freedom of representation within the deeper topology of the entanglement network. Together, these insights unify gravity and electromagnetism as dual aspects of a single geometric substrate: the coherent, temporally bidirectional structure of Eide Spheres. This reinforces the core claim of Eidetic Theory — that all classical fields and interactions are the bulk manifestations of more fundamental patterns of entanglement on quantum gravitational boundaries.

Eidetic Theory does not merely reinterpret existing physics—it integrates them into a single causal framework. It provides a concrete mechanism by which coherence becomes curvature and information becomes spacetime. The theory retains the empirical successes of General Relativity and quantum mechanics, while resolving their foundational paradoxes through a shared substrate: the bidirectional, temporally entangled geometry of the Eide Spheres.

According to Eidetic Theory, phenomena as diverse as gravitational attraction, black hole interiors, cosmic acceleration, quantum entanglement, and wavefunction collapse all emerge from a single underlying dynamic—the competition between nonlocal boundary coherence and local interaction density within an infinite quantum entanglement network. Spacetime, mass, and gravity are not fundamental entities, but projected effects of this hidden network. The arrow of time and the transition from superposition to classicality are byproducts of coherence loss under the strain of localization pressure.

In this view, the universe is not built from spacetime and particles—but from coherence itself, encoded across a vast network of temporally entangled quantum boundaries. It is from these boundaries that all geometry, causality, and physical reality emerge.

As the mathematical structure of Eidetic Theory continues to be developed, it opens the door to new testable predictions—offering a path toward unifying quantum gravity, resolving the information paradox, and reimagining the very fabric from which the universe is made.

Daniel James Stoker holds degrees of B.S. in Physics, B.S.H.S. in Physiological Sciences, a M.S. in Computer Sciences Database Technologies, and has completed a PGP in Artificial Intelligence and Machine Learning: Business Applications.

djs@arizona.edu 

(520) 729-0982

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With assistance from ChatGPT by OpenAI in refining language, structure, and mathematical framing.