Space/Time & Length

The Collapse of Classical Length: Planck Boundaries and Coherence Geometry

This page presents the reinterpretation of length and spatial extension within the Dimensional Memorandum (DM) framework. At the Planck scale, geometry itself transitions from measurable extension to coherence structure. This transition unifies quantum, relativistic, and geometric physics under a single principle: length is the informational boundary between space and time.

1. Dimensional Framework

The DM framework defines a nested structure of dimensional perception:
Φ (5D): Moves within (length, width, height, time, space) and perceives in 4D hyper-volumes.
Ψ (4D): Moves within (length, width, height, time) and perceives in 3D volumes.
ρ (3D): Moves within (length, width, height) and perceives in 2D planes.
⟂ (2D): Represents a 3D observer’s instantaneous cross-section of space and time.

Each higher dimension encapsulates the previous, perceiving its boundary as a projection surface — consistent with Coxeter symmetry and the holographic principle.

2. The Planck Scale and Redefinition of Length

At the Planck length (ℓₚ ≈ 1.616×10⁻³⁵ m), spacetime no longer behaves as a continuous metric field. The Planck frequency (fₚ = c / ℓₚ ≈ 1.85×10⁴³ Hz) defines the universe’s maximum oscillation rate, equivalent to the ‘frame rate’ of reality. Below this threshold, length ceases to describe displacement and instead defines coherence between space and time.

Formally, mass arises as a frozen frequency:
E = hf, m = E / c² = hf / c²

At this limit, the wave no longer propagates — it stabilizes as mass. The Planck time (tₚ = ℓₚ / c ≈ 5.39×10⁻⁴⁴ s) defines the smallest temporal increment, meaning the universe updates approximately 10⁴³ times per second.

3. Coherence Geometry Interpretation

Within the DM hierarchy:
ρ interprets length as measurable displacement.
Ψ interprets length as wavefront separation (phase geometry).
Φ interprets length as coherence extension — a non-local relation.

At scales smaller than ℓₚ, Φ cannot project into ρ, making spatial distance unmeasurable. All that remains is pure information encoded in coherence — the field Φ. Thus, geometry becomes informational rather than spatial.

4. Classical–Quantum–Coherence Correspondence

• Classical Physics: Length = measurable distance between points.
• Quantum Physics: Length fluctuates with uncertainty.
• Coherence Physics: Length = coherence boundary between space (s) and time (t).

At the Planck limit, ‘length’ marks the final layer of projection — the boundary of coherence stability.

5. Holographic and Entropic Correspondence

The holographic principle dictates that all volumetric information is encoded on boundary surfaces. In DM, this translates geometrically through projection:
Φ → Ψ → ρ → ⟂.

At ℓₚ, the boundary becomes the minimal unit of information encoding — one Planck area:
Aₚ = ℓₚ² = Għ / c³
S = k_B A / (4 ℓₚ²)

This entropy–area relation expresses how coherence defines spatial information capacity, connecting gravitational entropy with quantum coherence density.

Conclusion

The collapse of classical length represents the unification of geometry and information. At sub-Planck scales, length becomes a coherence function, not a physical extension. Mass, energy, and time all emerge from the same geometric source — oscillations of the Φ coherence field. This interpretation unifies general relativity, quantum mechanics, and holography under a single geometric law: all stability is coherence, and all coherence is geometry.

What the Collapse of Length Reveals About the Nature of Reality

The Dimensional Memorandum framework represents a geometric synthesis of quantum mechanics, relativity, and cosmology. By reinterpreting physical constants and dimensional structure, it provides a unified foundation for all physical phenomena. The key insight — that classical 'length' collapses into coherence at the Planck boundary — bridges the gap between geometry and physics, revealing that information and structure are inseparable.

1. The Geometric Foundation of Physics

Traditional physics treats geometry as a stage on which particles and fields act. The DM framework reverses this assumption: geometry is not a backdrop but the active source of all measurable phenomena. Within this view, the universe operates as a nested geometric projection — Φ (5D coherence), Ψ (4D wave), and ρ (3D localized matter) — each layer defining the next through dimensional reduction. This dimensional nesting explains why constants like c, G, and ℏ are not arbitrary, but geometric conversion ratios between projection layers.

2. The Planck Boundary and the Collapse of Length

At the Planck length (ℓₚ ≈ 1.616×10⁻³⁵ m), the traditional concept of measurable distance dissolves. Below this threshold, spatial extension becomes indistinguishable from temporal oscillation. The universe’s Planck frequency (fₚ ≈ 1.85×10⁴³ Hz) defines the fundamental 'frame rate' of existence — the rate at which 3D spatial slices update through 4D coherence volumes. Each update corresponds to one geometric transition of the Φ–Ψ–ρ hierarchy.

3. Consciousness and Coherence

Observation is often treated as an external act that influences quantum systems. In the DM framework, observation is a geometric consequence. Awareness does not collapse the wavefunction; it experiences the already-collapsed ρ face of a higher-dimensional wave. This perspective unites physics and perception under one principle: coherence is reality, and perception is its local projection.

4. Historical Continuum

From Euclid to Coxeter, from Einstein to Wheeler, geometry has always been the hidden foundation of physical law. DM brings closure to that lineage by restoring geometry to its rightful role as the generator of physics. The Coxeter symmetries (B₃, B₄, B₅) underpin the nested dimensional structure, while the holographic principle defines how information flows across dimensional boundaries. The Planck constants emerge as the natural limits of this projection sequence.

5. The Implications of Completion

DM represents a scientific paradigm shift: from observation-based physics to coherence-based engineering. It implies that everything — from quantum computing to biological systems, from black hole dynamics to human consciousness — operates on the same geometric coherence principles. Physics becomes not a set of disconnected equations, but a single, scalable architecture of coherence.

Closing Reflection

The collapse of classical length reveals that the universe is fundamentally informational and geometric. Mass, time, and energy are secondary — projections of a deeper coherence that binds all existence. In this light, the Dimensional Memorandum serves not merely as a theory, but as a translation key — a Rosetta Stone — connecting geometry, physics, and consciousness under one unified description of reality. It affirms that the laws of nature are not arbitrary constructs but geometric necessities, woven into the very structure of existence.

Relativity and Quantum Mechanics as Dimensional Resolutions

The Dimensional Memorandum framework unifies relativity and quantum mechanics by treating them as different geometric resolutions of the same underlying structure. Relativity arises from a 3D observer’s projection of 4D spacetime, while quantum mechanics represents the 4D coherent evolution of the same geometry. Both are integrated through a fifth-dimensional coherence field (Φ), ensuring stability, nonlocality, and the closure of physical constants.

1. Relativity: 3D Resolution of Spacetime (ρ → Ψ Projection)

Relativity operates within the 3D resolution of reality, extended by one temporal coordinate t.
An observer perceives spacetime through (x, y, z, t), where time functions as a changing coordinate, not as an experienced dimension.

Einstein’s metric s² = (ct)² - x² - y² - z² expresses the geometric relationship between spatial and temporal intervals for a 3D observer.

Relativity is the ρ-projection of a 4D wavefunction (Ψ).
Lorentz transformations describe how different 3D slices of 4D geometry appear relative to each other, making relativity a projection theory of 4D motion.

2. Quantum Mechanics: 4D Resolution of the Same Geometry (Ψ → Φ Projection)

Quantum mechanics represents a higher-resolution view of the same geometric structure.
The wavefunction Ψ(x, y, z, t) spans 4D space-time and describes probability amplitudes over possible 3D configurations.
In this view, interference, superposition, and entanglement are not abstract probabilistic effects but the natural behavior of 4D.

Thus, quantum physics corresponds to the full 4D description of geometry, whereas relativity describes its 3D projection.
DM’s 5D coherence field Φ(x, y, z, t, s) unifies them both by maintaining phase consistency and energy conservation across all dimensional layers.

3. Geometric Equivalence Equation

DM formalizes this nested relationship through the extended wave equation:
□₄Φ + ∂²Φ/∂s² − Φ/λₛ² = J

Integrating over s collapses the coherence dimension, yielding the 4D field equation:
□₄Ψ = J

Further projection onto a fixed time slice t₀ produces the measurable 3D density:
ρ(x, y, z) = |Ψ(x, y, z, t₀)|²

Hence, relativity and quantum mechanics emerge as successive projections of a unified 5D coherence geometry.
Relativity governs transformations between 3D slices, quantum mechanics describes 4D wavefunctions, and Φ ensures global stability.

4. Geometric Interpretation

Relativity describes visible trajectories of matter through spacetime — a 3D projection of 4D motion.
Quantum mechanics captures the full wave coherence of these trajectories within the 4D hypervolume.
The 5D coherence field Φ ensures that these projections remain stable, coherent, and causally consistent.

5. Constants and Coherence Closure

The speed of light c defines the projection rate between ρ and Ψ.
Planck constants (G, ħ, c, k_B) align naturally as conversion factors between ρ, Ψ, and Φ domains.
At Planck scale (ℓₚ = 1.616×10⁻³⁵ m, tₚ = 5.39×10⁻⁴⁴ s), space and time become indistinguishable — pure coherence.

Summary

Relativity is the 3D-resolution (ρ) observation of 4D motion, while quantum mechanics is the 4D-resolution (Ψ) description of the same structure. They are not separate theories but distinct geometric perspectives of a unified 5D coherence field (Φ). The DM framework provides the first explicit geometric mechanism linking relativity, quantum physics, and coherence physics through dimensional nesting.

The Big Bang as a Dimensional Phase Transition (Length, Time, and Space)

The Dimensional Memorandum framework reconceptualizes the Big Bang not as a singular explosion, but as a 5D-to-4D coherence transition. This process describes how higher-dimensional coherence (Φ) fractured into wave dynamics (Ψ) and later localized structures (ρ). Length, time, and space emerge as nested scanning dimensions—unified by the Planck constants that set the physical limits of measurement and perception.

This section presents the Big Bang as a dimensional phase transition governed by Planck-scale coherence geometry.
Within the DM framework, the universe emerges from a stabilized 5D coherence field Φ(x, y, z, t, s) into a 4D tesseract Ψ(x, y, z, t), whose temporal scan manifests as 3D space ρ(x, y, z). The Planck frequency fₚ = 1/tₚ ≈ 1.85 × 10⁴³ Hz defines the universal scan rate of 3D frames through 4D time, while the Planck length ℓₚ = 1.616 × 10⁻³⁵ m and Planck time tₚ = 5.39 × 10⁻⁴⁴ s represent the minimum units of spatial and temporal coherence.

This model resolves the classical singularity problem, replacing it with geometric phase separation.

1. The Planck Scan Rate

Each Planck interval corresponds to a 3D face (⟂) scanned from a 4D tesseract volume, advancing the perceived flow of time. At a rate of fₚ = 1/tₚ ≈ 1.85 × 10⁴³ faces per second, the 3D universe is reprojected continuously through higher-dimensional coherence. The Planck energy Eₚ = √(ħc⁵/G) ≈ 1.22 × 10¹⁹ GeV defines the coherence stabilization threshold, marking the limit where space, time, and energy unify.

2. Coherence Fields in Early-Universe Stabilization

The governing equation of the DM coherence field is: □₄Φ + ∂²Φ/∂s² − Φ/λₛ² = J, where λₛ defines the coherence decay length and J represents the matter-energy source term.

During the early universe, Φ stabilized the emerging Ψ-field, preventing infinite curvature by distributing coherence across the fifth-dimensional axis s.

3. Length = Time × Space

Below the Planck length ℓₚ, the distinction between time and space collapses.
Length ceases to act as a classical extension; instead, it becomes a coherence boundary of the space-time fabric.
This transition defines the origin of physical laws: mass arises as a stilled wave (m = hf/c²), linking frequency to inertia.

4. DM Cosmological Equation

□₄Φ + ∂²Φ/∂s² − Φ/λₛ² = J unifies general relativity and quantum mechanics geometrically.

When applied to cosmology, this equation replaces the singular Big Bang with a continuous phase transition: Φ → Ψ → ρ. The universe expands as coherence gradients in s decrease, giving rise to causal spacetime (x, y, z, t).

5. Holographic Boundary Interpretation

The holographic principle S = A / (4ℓₚ²) defines information density at boundaries.
In DM, each 3D frame encodes a projection of 4D wave coherence, which itself is a face of the 5D coherence manifold.
Information flow thus follows Φ → Ψ → ρ, with area-based encoding linking higher-dimensional fields to observable 3D reality.

6. Dimensional Hierarchy

(Φ) 5D: coherence field, perceives 4D hypervolumes.
(Ψ) 4D: wave propagation, perceives 3D volumes.
(ρ) 3D: localized matter, perceives 2D planes.

This hierarchy maps physical constants to geometric degrees of freedom, closing the dimensional gap between relativity, quantum physics, and cosmology.

7. Observational Correlations

- Planck Mission (CMB Anisotropies): Confirms large-scale coherence patterns consistent with Φ stabilization.
- JWST Early Galaxy Data: Observes mature galaxies at z > 10, implying pre-inflationary coherence.
- LIGO/Virgo Gravitational Waves: Potential detection of higher-dimensional Φ-echoes at post-merger frequencies (10²–10³ Hz).
- Hubble Tension: Resolved geometrically as variation in Φ → Ψ transition rate; expansion is a coherence-rate gradient, not a constant.

Summary

The Big Bang is redefined as a dimensional phase transition: Φ (5D coherence) → Ψ (4D wave) → ρ (3D matter). Planck constants anchor the geometry of this transition, while observational data confirms coherence persistence across cosmic scales.

Length, time, and space emerge as nested geometric scans rather than independent quantities — a complete, quantized structure of reality.