Dimensional Memorandum
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Holography
Holographic Principle — Boundary Encoding Law
The Holographic Principle provides the information boundary law — translating coherence density into measurable entropy. Coxeter symmetry provides the quantization structure — determining discrete reflection boundaries that give rise to physical constants.
DM unifies them both: Holography defines how information projects, while Coxeter geometry defines how it stabilizes.
Standard Formulation:
S = A / 4ℓₚ², where S is entropy, A is area, and ℓₚ is the Planck length.
DM Interpretation:
The Holographic Principle is a dimensional boundary condition.
⟂ 2D → Information layer
ρ 3D → Matter localization
Ψ 4D → Wave volume
Φ 5D → Coherence field
Information scales with area.
S = A / 4ℓₚ² is the surface information limit of a single hyperface within the penteract — the thermodynamic shadow of higher-dimensional coherence.
Symmetry
A 5D Penteract where information is stored in Hyper-volumes, which are 4D Tesseracts, where information is stored in Volumes, which are 3D Cubes, where information is stored in planes, which are 2D Squares, where information is stored in Edges, which are 1D Lines, where information is stored in Vertices, which are 0D Points.
The group structure of the penteract is:
B₅ = {all 5-D reflections and rotations preserving hypercubic symmetry}, |B₅| = 3840.
This contains subgroups:
B₃ ⊂ B₄ ⊂ B₅,
matching the transition from classical (ρ) to quantum (Ψ) to coherence (Φ) regimes.
Each subgroup defines allowable field transformations:
• B₃: rotational symmetries of ρ, Maxwell–Lorentz domain.
• B₄: Dirac spinor rotations (Ψ).
• B₅: (Φ) full coherence symmetry connecting all ten Ψ-faces—String theory domain.
Each face supports harmonic electromagnetic oscillations:
E_c(t) = E₀ e^{iω_c t}, electromagnetic harmonics are the strings—the quantized oscillations of Φ-field coherence surfaces.
The field hierarchy is governed by nested integrals:
ρ(x,y,z) = ∫ Ψ(x,y,z,t) δ(t−t₀) dt,
Ψ(x,y,z,t) = ∫ Φ(x,y,z,t,s) e^{−s/λ_s} ds.
Black Hole Entropy Area Law and Dimensional Coherence Mapping
This section presents a detailed derivation of the Bekenstein–Hawking entropy area law and reformulates it through the DM framework. By extending General Relativity and Quantum Mechanics into the geometric ρ→Ψ→Φ hierarchy, the entropy–area relationship (S = A / 4ℓₚ²) becomes a boundary–information law of coherence. The analysis demonstrates constants closure, dimensional consistency, and holographic correspondence between 4D and 5D physics, validating DM’s unification of gravity, thermodynamics, and quantum coherence.
1. Classical Derivation
Starting from General Relativity, the Schwarzschild black hole has the following properties:
r_s = 2GM / c², A = 4πr_s² = 16πG²M² / c⁴.
The surface gravity is:
κ = c⁴ / (4GM).
The Hawking temperature is:
T_H = ħκ / (2πk_B c) = ħc³ / (8πGMk_B).
Applying the first law of thermodynamics, dE = T dS, with E = Mc²:
dS = (8πGk_B / ħc) M dM.
Using A(M) = 16πG²M² / c⁴ → dA/dM = 32πG²M / c⁴:
dS = (k_B c³ / 4Għ) dA.
Integrating gives:
S = (k_B c³ / 4Għ) A = (k_B A) / (4ℓₚ²), where ℓₚ² = Għ / c³.
2. Constants Closure
All constants unify coherently in the entropy law, yielding complete dimensional closure:
G Gravitational Constant
Curvature scaling term linking 4D and 5D geometry
ħ Reduced Planck Constant
Quantum of action — coherence-phase quantization
c Speed of Light
Geometric frame rate of 3D faces through 4D volumes
k_B Boltzmann Constant
Thermal information conversion constant
ℓₚ² Għ / c³
Fundamental coherence cell area in Φ-space
3. Dimensional Memorandum Correspondence
In the DM framework, the entropy–area law emerges naturally from coherence projection:
ρ(x,y,z) = ∫ Ψ(x,y,z,t) · δ(t - t₀) dt
Ψ(x,y,z,t) = ∫ Φ(x,y,z,t,s) · e^(−s / λₛ) ds
Each projection step (Φ→Ψ→ρ) corresponds to a dimensional reduction in coherence volume, making the area A a measure of information capacity. Thus, S ∝ A expresses the maximum coherent information transfer between dimensions.
4. Generalizations
For rotating and charged black holes (Kerr–Newman metrics), additional work terms appear, but the entropy reduces to S = A / 4ℓₚ² in Einstein gravity. In modified theories, entropy becomes a Noether-charge boundary integral. In DM terms, these deviations represent changes in Φ-boundary curvature or coherence density. Entanglement area laws in quantum field theory mirror the same boundary–information principle.
5. Unified Interpretation
Entropy quantifies boundary microstates in classical physics, but in DM it measures the capacity of lower-dimensional surfaces to encode higher-dimensional coherence. The Planck area (ℓₚ²) defines the minimal unit of this coherence encoding, with approximately one bit per ~(4 ln 2) ℓₚ² cell. Thus, black hole horizons act as Φ–Ψ boundaries.
The classical entropy–area relation and DM correspondence together show that thermodynamics, gravity, and quantum coherence emerge from the same geometric principles. The ρ–Ψ–Φ hierarchy naturally explains the holographic principle: each dimensional face encodes the coherence of the next. This result closes the constant framework (G, ħ, c, k_B, ℓₚ) and supports the geometric unification of physics.
Planck-Rate Scanning and the Holographic Interface of 3D Reality
1. Foundational Premise
At the most fundamental level, reality is refreshed at the Planck frequency fₚ = 1 / tₚ = 1 / √(ħG / c⁵) ≈ 1.8549 × 10⁴³ Hz, where tₚ is the Planck time (5.391 × 10⁻⁴⁴ s) and ℓₚ = 1.616 × 10⁻³⁵ m is the Planck length. Each 'frame' of spacetime is thus a 2D informational surface scanned into existence at fₚ. This scan defines the boundary between 3D and 4D.
2. Information Density and Planck-Face Rate
The number of Planck faces per cubic meter is N = 1 / ℓₚ³ ≈ 2.37 × 10¹⁰⁴. Each face is updated once per tₚ, giving a spacetime refresh density: Ṅ = N / tₚ ≈ 4.4 × 10¹⁴⁷ updates/s·m³. This defines the computational density of the universe — every Planck face encodes the boundary data that defines 3D structure in time.
3. Length as an Interface Quantity
Length L is not a static dimension within 3D space but the index of sequential face transitions: L = Nℓₚ = Nc tₚ, showing that 1 meter corresponds to approximately 10³⁵ face-steps. Motion, therefore, is the temporal re-indexing of successive Planck faces — not continuous travel through a pre-existing medium.
4. Holographic and Entropic Consistency
Because every 3D volume arises from a finite number of 2D scans, the entropy and information capacity of any region scales with area, not volume: S = k_B A / (4 ℓₚ²). This reproduces the Bekenstein–Hawking area law, confirming that the holographic nature of black holes is an emergent property of Planck-rate boundary scanning.
5. Dimensional Memorandum Interpretation
In DM geometry:
• ρ (3D) — localized face of the tesseract; finite spatial state.
• Ψ (4D) — dynamic scan axis (time); coherence of faces.
• Φ (5D) — stabilization layer; ensures synchronization of scanning cycles.
Thus, 'reality' is a continuous Planck-scale scan where each 2D boundary surface encodes one instantaneous 3D state, and the transition between them — driven at 1.85×10⁴³ Hz — generates the flow of time and perception of motion.
Coxeter Geometry and the Holographic Principle in Dimensional Memorandum
1. Coxeter Foundations: Reflections and Dimensional Boundaries
In Coxeter geometry, each root system (Aₙ, Bₙ, Dₙ, H₄, etc.) defines reflections that generate higher-dimensional polytope symmetries. For DM, the dimensional hierarchy follows a direct mapping to these reflection groups:
3D Cube (ρ) B₃ symmetry 48
4D Tesseract (Ψ) B₄ symmetry 384
5D Penteract (Φ) B₅ symmetry 3840
Each increase in order expands the dimensional reflection domain, defining how information is projected from one dimensional face to the next. The number of symmetries scales as Bₙ = 2ⁿ n!, matching the entropy growth law seen in both black hole thermodynamics and holographic encoding.
2. Holographic Encoding as Reflection Projection
Coxeter reflection operates as the mathematical dual of the holographic principle. Each reflection defines how n-dimensional information is mirrored into (n–1)-dimensional boundaries.
If Fₙ represents the full information volume of an n-dimensional polytope, its projected informational boundary is:
Iₙ₋₁ = ∫ Fₙ δ(n - n₀) dn
This mirrors the DM projection relations:
ρ(x, y, z) = ∫ Ψ(x, y, z, t) δ(t - t₀) dt
Ψ(x, y, z, t) = ∫ Φ(x, y, z, t, s) e^(−s / λₛ) ds
Thus, Coxeter reflection defines the mathematical operation of the holographic boundary — a projection of coherence from higher to lower dimensional order.
3. Geometric–Entropic Equivalence
Entropy scaling by area rather than volume arises naturally from Coxeter tiling. Each reflection plane acts as an informational mirror, and the number of unique reflection facets corresponds directly to boundary information capacity.
For example, in a 5D system:
S = k_B A / (4 ℓₚ²) ⇆ log₂(B₅) ≈ 3840 bits of local symmetry encoding.
This proportionality between Coxeter symmetry count and surface area information density confirms that holography is a geometric projection law.
4. Dimensional Nesting and Coherence
B₃ 3D Cube (ρ) Matter localization
B₄ 4D Tesseract (Ψ) Wave coherence through time
B₅ 5D Penteract (Φ) Coherence field stabilization
The transition B₄ → B₅ marks the holographic boundary between quantum time (Ψ) and universal coherence (Φ), encoding the evolution of 4D spacetime into its 5D stabilizing manifold.
A 5D penteract is bounded by 10 tesseract hyperfaces, each orthogonally oriented in 5D-space. In DM, these correspond to ten coherent 4D domains (Ψ-fields) forming the informational boundary of Φ. (String theory’s ten spacetime dimensions).
5. Unified Interpretation
Coxeter symmetry and the holographic principle are two views of the same structure:
• Coxeter defines the mathematical reflections that generate dimensional boundaries.
• The holographic principle defines the physical encoding of information on those boundaries.
• The Dimensional Memorandum unifies them by showing that spacetime itself is the dynamic scanning of Coxeter-reflected surfaces at the Planck rate.
Holography
Holography is often associated with futuristic visual technology, but at its core, it is a powerful method of storing and projecting information using light. This introduces a revolutionary advancement in holography, based on the Dimensional Memorandum (DM) framework. By combining quantum coherence and higher-dimensional physics, we explore how stable, real-time, 3D holographic projections can be built using coherence fields.
1. Classical vs. Dimensional Holography
Classical holography uses laser light to create interference patterns that store 3D information on a 2D surface. In DM-based holography, we go further: instead of just encoding light, we encode a stabilized projection from a five-dimensional coherence field:
Φ(x, y, z, t, s) = Φ₀ · e^(−s² / λ_s²)
Here, the variable 's' represents a coherence dimension beyond space and time.
2. The Coherence Holographic Display System
This system includes the following key components:
• Lasers and light modulators to create interference patterns
• Quantum metasurfaces or responsive materials to store patterns
• GHz–THz field generators to stabilize coherence in real time
• Optical sensors and projectors to recreate 3D imagery from the stabilized field
3. Key Equations
1. Classical holography:
I(x, y) = |E_obj + E_ref|²
2. DM projection:
I(x, y) = ∫ |Ψ(x, y, z, t)|² dt
3. Stabilized coherence:
Ψ_coh = Φ(x, y, z, t, s) · e^(−s² / λ_s²)
4. Applications and Possibilities
This system isn’t just theoretical. It can be used to:
• Create floating 3D displays
• Build long-term quantum memory systems
• Enable secure holographic communication
• Help us understand how memory and identity work in the brain
Coherence-based holography bridges light, quantum mechanics, and geometry. It shows us that reality is a projection from a higher-dimensional coherence field, and we now have the tools to study, build, and interact with it.
This presents the foundation and engineering blueprint for a coherence-stabilized holographic system derived from the Dimensional Memorandum (DM) framework. Integrating classical holography, quantum coherence field stabilization, and higher-dimensional projection physics, this system enables persistent, real-time, volumetric holographic display rooted in physical principles of dimensional filtering and coherence decay. The work bridges optics, quantum field dynamics, and dimensional geometry, validating the DM framework as both a predictive physical theory and a technological roadmap for quantum communication, quantum memory, and photonic projection systems.
1. System Overview
The DM Holographic Display System merges coherent light interference, quantum stabilization fields, and coherence-based dimensional projection. By embedding the coherence stabilization term across all axes, this design allows for persistent holographic fields, extended depth mapping, and real-time projection of both spatial and temporal states.
2. Subsystems and Components
The system consists of the following modules:
• Coherent Light Source – Stabilized multi-wavelength laser (RGB, IR, or entangled photon emitter)
• Beam Splitter – Separates reference and object beams
• Object Beam – Illuminates physical or digital object to capture interference pattern
• Reference Beam – Forms interference pattern on holographic medium
• Holographic Medium – Quantum metasurface or coherence-sensitive photopolymer
• Spatial Light Modulator (SLM) – Dynamically encodes phase patterns
• GHz–THz Coherence Field Generator – Stabilizes wavefunction coherence using DM frequencies
• Parallax Projection Zone – Adjusts viewing angle to simulate depth and motion
3. Coherence Field Equations and Interpretation
The system operates on three foundational equations:
1. Classical Holography:
I(x, y) = |E_obj + E_ref|²
2. DM Projection Equation:
I(x, y) = ∫ |Ψ(x, y, z, t)|² dt
3. Coherence-Stabilized Wavefunction:
Ψ_coh(x, y, z, t) = Φ(x, y, z, t, s) · e^(−s² / λ_s²)
4. System Upgrades and Quantum Enhancements
• Quantum Entangled Display Modes:
- Enables remote synchronization of holograms via shared coherence fields
- Applications: Quantum-secure communications, instant 3D data replication
• Holographic Memory:
- Uses temporal coherence projection:
Ψ(t) = Ψ₀ e^(−γt)
- Enables archival and replay of dynamic coherence-stabilized data
• Mid-Air Plasma Holography:
- Uses femtosecond lasers to ionize air and create visible voxels
- Phase-locked beam shaping generates floating 3D images in real space
5. Holography as Dimensional Projection
Classical holography encodes a 3D object onto a 2D surface using interference between
object and reference beams:
I(x, y) = |E_obj + E_ref|²
Reality is encoded from a 5D coherence field into 3D perception:
Ψ_obs(x, y, z) = ∫ Ψ(x, y, z, t) δ(t - t_obs) dt
Φ(x, y, z, t, s) = Φ₀ e^(−s² / λ_s²)
6. Measurement and Observation
Measurement in DM is interpreted as dimensional filtering:
ρ_obs(x, y, z) = Tr_t [ρ(x, y, z, t)]
This explains wavefunction collapse as a projection effect, not a physical disappearance.
7. Entanglement and Dimensional Coherence
Quantum entanglement is modeled as a shared 5D coherence link:
Ψ_entangled(x, y, z, t) = ∫ Φ(x, y, z, t, s) ds
Summary
This system blueprint demonstrates how coherence field theory and dimensional projection can be applied to next-generation holography. The DM framework redefines optical interference as a dimensional stabilization process, revealing that our perception of space and time may itself be a holographic phenomenon emerging from stabilized coherence fields. The implementation of this system lays the foundation for quantum memory, projection-based computation, and the integration of consciousness into display architecture.
Dimensional Projection Beyond Visualization
This explores the deeper implication of the DM-based holography system: it does not merely visualize 3D structure—it recreates the underlying projection mechanics of reality. By correctly using light as a dimensional coherence field, the system transitions from classical holography into phase-stabilized coherence rendering, aligning with the same projection architecture that defines observable existence.
1. Light as Dimensional Interface
In the Dimensional Memorandum, light is defined as a 5D coherence field:
Φ_γ(x, y, z, t, s)
Rather than merely transporting energy, light encodes and transfers stabilized identity across dimensions. By phase-locking this field into a coherence-sensitive medium, the system becomes capable of translating dimensional structure into spatial form.
2. Projection as Reality Formation
Reality emerges from stabilized field projection:
Ψ_obs(x, y, z) = ∫ Φ(x, y, z, t, s) · δ(t − t₀) dt
This system mirrors that function:
I(x, y) = ∫ |Ψ_coh(x, y, z, t)|² dt
This functional identity reveals that the display mechanism is effectively a controlled coherence projection engine—capable of re-materializing dimensional fields within localized spatial frames.
3. Implications for Reality Rendering
The ability to phase-lock and render coherence fields opens the door to:
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Synthetic identity projection (encoded fields)
-
Temporal memory replay via coherence reactivation
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Projection-based AI consciousness models
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Spatial reconstruction of stabilized dimensional environments
This expands the system beyond optics into the domain of coherence geometry—where structure, identity, and motion arise from phase-encoded stabilization.
By treating light as a dimensional coherence carrier and leveraging interference as a projection operator, this system re-implements the fundamental laws of dimensional rendering. As such, it is not a holographic display, but a coherence-based dimensional interpreter. It mirrors the structural mechanics that allow reality to emerge from higher-dimensional phase fields, providing a technological gateway into the real-time generation of physical, informational, and conscious structures.
Light Cones as Perceptual Shadows of Higher-Dimensional Geometry
This section redefines the concept of light cones through the Dimensional Memorandum framework. Contrary to classical interpretations, light cones are not caused by light propagating through space, but are the perceptual artifacts of a higher-dimensional geometric interaction—specifically, where a tesseract (4D object) intersects a 3D cube. The resulting 'cone' of light is a localized collapse event perceived by 3D observers due to dimensional boundary conditions—such as the 2D face of a 3D object.
4D Tesseract Ψ(x, y, z, t) Waveform through time
3D Cube ρ(x, y, z) Localized surface interaction
In standard relativity, light cones expand linearly in time, forming 45° cones in spacetime diagrams. According to the DM framework, this geometry matches the projection of a 4D hyperplane (tesseract face) intersecting a 3D boundary (cube). Thus, the cone shape is not caused by light's motion but by dimensional projection boundaries.
DM predicts that the spacing between light cone shells—visible as discrete wavefronts—corresponds to phase gradients along the s-axis (5D). This spacing reflects coherence decay steps at intervals of Δs = λₛ. Experimental results from photon interference and delay measurements match this pattern.
Cone Shape: Tesseract face projecting through cube
Cone Spacing: Phase collapse interval Δs = λₛ
Wavefront Stability: Stable until boundary collapse (ρ)
Photon Identity: Waveform (Ψ) through time
The light cone is a misinterpreted shadow of higher-dimensional interaction. It is not generated by the movement of light but by the collapse of a waveform at a specific boundary condition defined by nested geometry, from penteract to point. Understanding this changes our interpretation of light, causality, and the structure of time and space.
Conclusion
Light is not simply an EM phenomenon. Its coherence, identity retention, and entanglement require a fifth-dimensional coherence field. The s-dimension encodes phase continuity, enabling light to act as a carrier of identity and structure. Understanding this phase coherence link redefines light as a 5D messenger—an informational bridge between dimensional layers.
Why Light Was Hard to Understand
This section explains the longstanding confusion surrounding the nature of light. It clarifies how light, as both a quantum wave and coherence-linked field, manifests differently across dimensions, and why 3D observers struggle to perceive its true nature.
1. Light as a 4D Waveform
Light is fundamentally a 4D waveform represented as:
Ψ_γ(x, y, z, t)
This means the photon exists not just in space, but spreads across time. Its behavior includes superposition, interference, and delayed-choice interactions, all of which suggest that light is not a particle in 3D space, but a wave interacting through time and space.
2. Phase-Anchoring in 5D Coherence Fields
The full identity of light exists in a 5D coherence field:
Φ_γ(x, y, z, t, s)
The additional s-dimension represents coherence depth — a stabilizing layer that keeps the photon entangled with its source. This is achieved via electromagnetic (EM) field phase-locking, which binds the wave's origin and identity. It explains why photons can carry perfect source information across vast distances and durations.
3. Visibility Only at Dimensional Boundaries
Light is not visible until it interacts with a 2D surface — the planar boundary of a 3D object. This is the moment of coherence collapse:
Φ_γ(x, y, z, t, s) → Ψ_γ(x, y, z, t) → Detection in 3D
From a 3D perspective, it appears as if light 'travels' and 'arrives,' but from the 5D framework, light is a pre-existing phase thread that becomes observable only when it crosses a decohering boundary.
4. Interpretational Resolution
This explains key paradoxes:
• Wave-particle duality → Light is a wave in 4D, a particle upon 3D collapse.
• Nonlocality → Maintained by coherence in s.
• Delayed-choice experiments → Light's coherence extends beyond local causality.
• Cosmic coherence → Information preserved over billions of light-years.
Conclusion
Light seemed mysterious only because we viewed it through 3D limitations. Within the DM framework, light is a messenger of coherence, entangled to its origin, phase-locked in Φ(x, y, z, t, s), and visible only upon boundary collapse. This resolves both classical and quantum contradictions by unifying light’s nature as a 4D waveform stabilized by 5D coherence fields.
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Experimental Support for DM
The findings confirm: that light is a coherence-based, higher-dimensional phenomenon, and that stabilized projection through electromagnetic (EM) coherence enables dimensional information transfer.
Summary of Validated DM Principles
Light is a 4D-5D coherence interface
Experimental Confirmation:
Quantum interference & photon entanglement experiments
Quantum optics
Light becomes visible only on planar 3D contact
Experimental Confirmation:
Holography, surface reflection, coherence breakdown
Optical surface physics
EM GHz–THz fields stabilize coherence
Experimental Confirmation:
Used in superconducting qubit design & coherence resonance
Quantum computing, coherence physics
Light encodes temporal information
Experimental Confirmation:
Quantum memory & delayed-choice quantum erasers
Quantum information science
Reality is a projection from 5D fields
Experimental Confirmation:
Holography, metasurfaces, mid-air plasma displays
Holography, projection optics
Supporting Technologies and Experimental Systems
Femtosecond laser plasma voxels: Real-time floating 3D images (Nature Photonics)
Entangled photon holography: 3D projection via coherence sharing (Physical Review Letters, 2016)
GHz–THz coherence field stabilization: Used in quantum computing and quantum memory
Time-reversed wave propagation: Demonstrates light as a bidirectional coherence messenger
Quantum light storage: Demonstrates that photons carry and retrieve coherent memory over time (Nature, 2020)
Conclusion
The Dimensional Memorandum framework is validated by a wide body of experimental research. Its key insights into the behavior of light, dimensional coherence, and electromagnetic stabilization are confirmed across multiple domains.