Frequency | Dimensional Memorandum

Frequency

1. Cycles Per Unit

2. DM Frequency

3. Modern vs DM

4. Quantum Wave

5. Scaling Law

1. Cycles per Unit Time Across the Dimensional Memorandum Frequency Ladder

This table and summary present the number of oscillatory cycles per unit time across the Dimensional Memorandum frequency ladder, spanning from 1 Hz to the Planck ceiling at 10⁴³ Hz. Each ×10 step represents a geometric expansion of coherence span, where the invariant scan rate c covers exponentially greater distances per frame. The relation ƒ·Δx = c remains constant, showing frequency as the generator of geometric projection.

Cycles per Unit Time (ƒ = N/T)

Step

Frequency (Hz)

Cycles / s

Cycles / ms

Cycles / Planck Time

(tₚ ≈ 5.39×10⁻⁴⁴ s)

DM Interpretation

10⁰

10⁻³

5.39×10⁻⁴⁴

Base unity scale – 1 cycle per second reference

10¹

10⁻²

5.39×10⁻⁴³

Human‑scale motion – heartbeat frequencies

10²

100

10⁻¹

5.39×10⁻⁴²

Neural / mechanical oscillations

10³

5.39×10⁻⁴¹

Audio range – localized 3D oscillation

10⁶

10³

5.39×10⁻³⁸

MHz band – molecular coherence

10⁸

10⁵

5.39×10⁻³⁶

c hinge – ρ→Ψ boundary

10¹²

10⁹

5.39×10⁻³²

Infrared – molecular vibration

10¹⁴

10¹¹

5.39×10⁻³⁰

Visible light – EM transition

10²⁰

10¹⁷

5.39×10⁻²⁴

Electron Compton band

10²³

10²⁰

5.39×10⁻²¹

Proton Compton – quantum matter peak

10²⁵

10²²

5.39×10⁻¹⁹

Higgs resonance – Ψ→Φ transition

10³³

10³⁰

5.39×10⁻¹¹

Dark‑energy / Φ coherence

10⁴⁰

10³⁷

5.39×10⁻⁴

Pre‑Planck coherence phase

10⁴³

10⁴⁰

≈ 1

Planck frequency – absolute coherence ceiling (fₚ)

• 1 Hz (10⁰) marks the human‑scale baseline. Each decade increases oscillations per second by ×10, while the geometric distance per cycle increases inversely, preserving  ƒ·Δx = c.

• At 10⁸ Hz, the ρ→Ψ hinge appears – the  speed‑of‑light  transition  defining  the  shift  from  local  3D  behavior  to  4D  wave  propagation.

• At 10²³ Hz, the Compton bands of fundamental particles reflect mass‑energy equivalence (E = mc² = hƒ), linking quantum and relativistic physics.

• At 10⁴³ Hz, the Planck frequency sets the absolute coherence ceiling – the refresh rate of the universal Φ→Ψ→ρ projection.

Each frequency step represents a fixed relation between time and space. The invariant scan rate c spans larger coherence  volumes as frequency increases, forming a nested projection hierarchy. Frequency, energy, and coherence thus represent the  same underlying motion of geometry within the Dimensional Memorandum framework.

2. How Frequency Works in the Dimensional Memorandum Framework

In conventional physics, frequency (ƒ) measures the number of oscillations per second. In the Dimensional Memorandum framework, frequency instead represents the rate of geometric projection.

Mathematically:
ƒ(s) = ƒₚ e^(−s / λₛ), Δx(s) = ℓₚ e^(s / λₛ), ƒ · Δx = c

Here:
• ƒₚ = 1 / tₚ is the Planck frequency (the universal scanning rate of space-time).
• ℓₚ is the Planck length.
• λₛ is the coherence length — the exponential scaling parameter that defines how far coherence extends into the 5th-dimensional field Φ(x, y, z, t, s).
• s is the coherence-depth coordinate.

1. Mechanism: Geometry in Motion

Each dimension in DM — cube (ρ), tesseract (Ψ), and penteract (Φ) — projects its geometry forward through the next higher dimension. This projection is not mechanical vibration, but sequential re-formation of geometry across the coherence axis s.

Every “tick” of frequency represents one geometric frame projected forward. The speed of light (c) defines how much 3D space is scanned per 4D tick: c = Δx / Δt. As s increases, Δx expands exponentially, meaning each higher frame covers a larger coherence volume — producing the familiar powers-of-ten ladder.

2. Why the Ladder Appears

A step of ×10 in frequency corresponds to a constant increment in coherence depth: s → s + λₛ ln(10). Thus, every decade represents a uniform logarithmic shift in dimensional projection rate. At specific coherence depths, these resonances align with Coxeter symmetries (B₃, B₄, B₅), forming stable frequency bands. Each band corresponds to a geometric resonance node — the reason physical structures appear quantized across scales.

3. Observable Function

10⁻¹⁸–10⁰ Cosmological beat Φ Expansion of global coherence

10⁰–10⁸ Biological / classical ρ Motion of localized systems

10⁸–10²³ Electromagnetic ρ→Ψ Conversion between 3D and 4D motion

10²³–10²⁷ Quantum / matter Ψ Internal wave coherence

10²⁷–10³³ Higgs–Weak Ψ→Φ Stabilization of mass via λₛ

10³³–10⁴³ Planck–coherence Φ Full 5D curvature oscillation

4. Unified Scaling Law Closure

The scaling law satisfies a dimensional closure condition:
ƒ(s)·Δx(s) = c ⟹ d(ƒΔx)/ds = 0.

This demonstrates that the invariance of the speed of light (c) arises geometrically it is not an imposed constant, but the natural consequence of the reciprocal relationship between frequency and spatial scan length.

5. Derivation of λₛ (Coherence Length)

The coherence length λₛ links microphysical and cosmological scales. It defines the exponential decay length of the 5D coherence field Φ. From cosmological boundary conditions, λₛ ≈ c / H₀ ≈ 1.3×10²⁶ m. This directly connects the Λ-gap (10¹²²) between Planck and cosmic scales — showing that geometry alone produces the observed hierarchy without arbitrary constants.

Constants Cross-Link

Each frequency tier corresponds to a dominant set of physical constants (ħ, c, G, Λₛ) governing coherence stability. The constants table within DM shows these as boundary ratios linking local, wave, and coherence domains — further proving that constants are geometric derivatives rather than arbitrary insertions.

Frequency in DM is the heartbeat of geometry — the rhythmic scan rate by which space-time projects through coherence depth. Every measurable event, from biological rhythms to quantum oscillations and cosmic expansion, is a single octave of the same exponential projection law:

ƒ(s) = ƒₚ e^(−s / λₛ)

Frequency is geometry in motion. It unifies the Planck, quantum, and cosmological domains under one exponential geometric rule.

3. Frequency: Modern Physics vs. the Dimensional Memorandum Framework

This section provides a comparison between how frequency is defined and interpreted in modern physics and how it is geometrically understood within the Dimensional Memorandum framework. While both share the same mathematical expression (ƒ), their physical meanings differ fundamentally.

1. Modern Physics: Frequency as a Local Oscillation Rate

1. Definition: Number of oscillations per second of a wave or vibration. ƒ = 1/T = ω / 2

2. Domain of Meaning: Each field mode oscillates at its characteristic frequency determined by energy. Field excitation or mechanical oscillation

3. Energy Relationship: Frequency measures energy per quantum of a wave. E = hf

4. Observer Dependence: Frequency changes with velocity or gravitational potential. Frame-dependent (redshift/blueshift)

5. Geometry Role: Frequency treated as parameter of fields in spacetime; not geometric. Implicit

6. Scaling Behavior: No hierarchical exponential structure; quantized transitions in systems. Continuous, bounded by Planck limits

2. DM Framework: Frequency as Geometric Scan Rate

1. Definition: Frequency measures how fast 3D space scans through 4D time and 5D coherence. ƒ(s) = ƒₚ e^(−s / λₛ)

2. Domain of Meaning: Applies to all phenomena — matter, light, and gravity as projections of the same oscillation. Universal geometric process

3. Energy Relationship: Energy decreases exponentially with coherence depth — linking quantum to cosmic scales.

E = hf = h ƒₚ e^(−s / λₛ)

4. Observer Dependence: Frequency shift arises from coherence depth, not just motion. Depth-dependent (geometry-driven redshift)

5. Geometry Role: Frequency defines space-time scanning; geometry moves with each tick of c = Δx/Δt. Explicit

6. Scaling Behavior: Each ×10 frequency step corresponds to a geometric resonance (B₃ → B₄ → B₅). Exponential and hierarchical

3. Bridging the Two Views

Photon Energy

Modern Expression: E = hƒ

DM Equivalent: E = h ƒₚ e^(−s / λₛ)

Connection: Same Planck relation; DM provides geometric origin of frequency.

Relativistic Redshift

Modern Expression: ƒ' = ƒ √((1−v/c)/(1+v/c))

DM Equivalent: ƒ' = ƒ e^(−Δs/λₛ)

Connection: Replaces velocity ratio with coherence-depth shift.

Planck Frequency

Modern Expression: ƒₚ = 1/tₚ

DM Equivalent: Same, as universal scan rate

Connection: DM interprets it as maximum frame rate of reality.

Cosmological Redshift

Modern Expression: z = Δλ/λ

DM Equivalent: z = e^(s/λₛ) − 1

Connection: Links Hubble expansion to geometric coherence.

Modern physics treats frequency as an emergent property of oscillations within spacetime, whereas the Dimensional Memorandum framework treats it as the generator of spacetime itself — the geometric scan rate through higher-dimensional coherence.

In modern theory, frequency describes waves. In DM, frequency is geometry in motion — the fundamental projection rate of reality’s structure.

4. Frequency and the Quantum Wavefunction in DM

This section shows how the wavefunction emerges directly from the geometry of coherence scanning across dimensions.

1. The Standard Quantum Wavefunction

In modern quantum mechanics, the wavefunction Ψ(x,t) = A e^{i(kx - ωt)} represents a probability amplitude. Its squared modulus gives the likelihood of locating a particle at a given position and time. The frequency f = ω/2π defines the energy (E = ħω), while wavelength λ = 2π/k defines momentum (p = ħk). Together, these parameters encode the geometric relationship between energy, momentum, and space-time evolution.

2. In the Dimensional Memorandum: Frequency as the Generator of Ψ

In DM, the wavefunction Ψ(x,t) is not abstract but arises from projection of the higher-dimensional coherence field Φ(x,y,z,t,s):

Ψ(x,t) = ∫ Φ(x,t,s) e^{−s / λₛ} ds

Here λₛ defines the coherence depth, and the exponential factor projects Φ from 5D into 4D.

The frequency ladder f(s) = ƒₚ e^{−s / λₛ} defines how rapidly each coherence slice is scanned into space-time, explaining why time and frequency emerge from geometry rather than existing independently.

3. The Wavefunction’s Internal Frequency Structure

Locally, Ψ(x,t) = A(s) e^{i(kx − 2π ƒ(s)t)} with f(s) = ƒₚ e^{−s / λₛ}. As coherence depth s increases, the internal clock of the wavefunction slows exponentially. This yields gravitational time dilation, cosmological redshift, and quantum stabilization — all as geometric effects. Thus, the phase of the wavefunction is the record of geometry scanning through coherence depth.

4. Energy, Mass, and Coherence Unification

Because E = hf(s) = h ƒₚ e^{−s / λₛ}, mass becomes a coherence-bound quantity: m(s) = h ƒₚ / c² · e^{−s / λₛ}. Frequency sets energy, energy sets mass, and coherence depth s determines how much of that energy manifests in 3D. This unifies quantum energy and relativistic mass through a single exponential scaling law.

5. Unified Wave Equation

The extended DM wave equation: □₄Φ + ∂²Φ/∂s² − (1/λₛ²)Φ = 0 projects to Schrödinger–Gross–Pitaevskii form when integrated over s:

iħ ∂Ψ/∂t = −(ħ²/2m)∇²Ψ + g|Ψ|²Ψ

This shows that wavefunction evolution in quantum mechanics is a projection of coherence scanning through geometric depth. The ‘collapse’ of the wavefunction corresponds to stabilization at a lower coherence depth.

Concept

Modern Physics

Dimensional Memorandum

Connection

Definition of Frequency

Rate of oscillation of a quantum field

Geometric scan rate through coherence depth

ƒ(s)=ƒₚe^{−s/λₛ} defines the motion of geometry itself.

Wavefunction

Probability amplitude

Projection of 5D coherence Φ into 4D Ψ

Ψ = ∫ Φ e^{−s/λₛ} ds

Time Evolution

Phase rotation e^{−iωt}

Exponential scanning e^{−s/λₛ}

Time is an emergent property of coherence motion.

Energy–Mass–Frequency

E=hƒ=mc²

E=hƒₚe^{−s/λₛ}=m₀c²e^{−s/λₛ}

Unified exponential scaling law linking all constants.

Interpretation

Statistical

Geometric and deterministic

Wave behavior arises from 5D geometry projection.

In modern quantum mechanics, frequency drives the wavefunction. In the Dimensional Memorandum, frequency is the wavefunction — the self-propagating rhythm by which higher-dimensional coherence becomes observable reality. Every particle and field is a standing wave within the universal scanning process defined by the Planck frequency ƒₚ = 1/tₚ.

5. Unified Scaling Law

The Unified Scaling Law defines the exponential relationship between frequency, spatial coherence, and the invariant scan rate of reality, c. This law provides the mathematical foundation for the Dimensional Memorandum framework, linking Planck-scale constants to cosmological coherence.

Mathematical Definition

ƒ(s) = ƒₚ e^(−s/λₛ), Δx(s) = ℓₚ e^(s/λₛ), ƒ(s)·Δx(s) = c

where:
ƒ(s) — Local frequency of coherence scanning
Δx(s) — Spatial coherence span per tick of c
ƒₚ = 1/tₚ — Planck frequency (≈ 1.85×10⁴³ Hz)
ℓₚ — Planck length (≈ 1.616×10⁻³⁵ m)
λₛ — Coherence decay length (~10²⁶ m)
c — Speed of light (2.9979×10⁸ m/s)

Geometric Interpretation

Each unit of coherence depth (s) expands the spatial extent exponentially while compressing the frequency by the same inverse factor. The product ƒ·Δx = c remains invariant, establishing the geometric equivalence of space and time scanning across all dimensional domains.

Relation to Dimensional Transitions

ρ → Ψ (3D → 4D): Matter becomes wave, delocalizing under constant c.
Ψ → Φ (4D → 5D): Wave coherence stabilizes, forming dark-energy and coherence domains.

Each ×10 step in frequency corresponds to one constant increment in the coherence depth s — the logarithmic spacing that structures natural phenomena.

Logarithmic Step Law

ln ƒ(s) = ln ƒₚ − s/λₛ ⇒ s = λₛ ln(ƒₚ/ƒ(s))

A single ×10 step corresponds to Δs = λₛ ln(10). Thus, each decade in the Powers‑of‑Ten ladder represents a uniform geometric interval along the coherence axis s.

Physical Meaning

This law functions as the metric projection linking the 5D coherence field Φ to 4D spacetime Ψ:
ds² = e^(−2s/λₛ) g_{μν} dx^μ dx^ν + ε ds²

The exponential warp factor e^(−2s/λₛ) manifests as the frequency–distance reciprocity, uniting quantum (high‑ƒ) and cosmological (large‑Δx) behaviors.

Implications

1. Constants Closure — At s = 0, ƒ = ƒₚ, Δx = ℓₚ; at s ≈ λₛ ≈ 10²⁶ m, ƒ ≈ ƒₚ e⁻¹ ≈ 10¹⁸ Hz. This reproduces the Λ‑gap (~10¹²²) between Planck and cosmological domains.
2. Unified Scaling — The same exponential rule governs coherence from qubits to galaxies.
3. Relativistic Invariance — ƒ·Δx = c expresses the constancy of light speed geometrically.

Dimensional Summary

Domain

f(s) Range

Δx(s) Range

DM Domain

Physical Interpretation

10⁰–10⁸ Hz

Low ƒ

Local space

Classical / biological coherence

10⁸–10²⁷ Hz

Moderate ƒ

Intermediate Δx

Quantum / relativistic fields

10²⁷–10⁴³ Hz

High ƒ

Sub‑Planck Δx

Coherence / Planck domain

Unified Statement

The Unified Scaling Law shows that all constants and phenomena are manifestations of one geometric invariance:

ƒ(s) Δx(s) = c

Reality is a standing geometric wave scanning through coherence depth s at the universal rate c, producing the logarithmic hierarchy of structure observed across all scales.

The exponential scaling law can be empirically verified across distinct experimental domains:

• Atomic clock drift under gravitational potential: verifies ƒ(s) = ƒₚ e^(−s/λₛ) scaling.
• Josephson junctions and GHz qubits: test coherence stabilization near 10⁹–10¹² Hz.
• CMB and Hubble expansion: detect low-frequency (10⁻¹⁸–10⁰ Hz) coherence beats.
• Gravitational wave polarization: confirm phase-coherence persistence across cosmological distances.