A 0-Dimensional f₀ → Δf → Phase → Planck-Scale Δt in Extended Classical Mechanics (ECM)

DOI: 10.13140/RG.2.2.23191.02729

Soumendra Nath Thakur | ORCiD: 0000-0003-1871-7803 | postmasterenator@gmail.com | December 09, 2025

Abstract

This document presents the 0-dimensional Planck-scale formulation of a photon’s primordial base frequency f₀ and its perturbation by an ultra-small frequency deviation Δf. In Extended Classical Mechanics (ECM), such perturbations convert directly into phase evolution through x° = 360 Δf, while the induced Planck-scale temporal span is determined by Δt = x° / (360° f₀).

Using Δf = 0.16168349753 Hz, the formulation yields the precise Planck-scale event interval Δt = 5.391247 × 10⁻⁴⁴ s and a corresponding phase advance of x° = 58.2060591108°. These values define the boundary between the 0-dimensional origin state and the onset of measurable temporal structure.

The associated energy and mass increments—computed using ΔE = hΔf and ΔM = ΔE / c²—give ΔE = 1.071327 × 10⁻³⁴ J and ΔMᴍ = 1.19165 × 10⁻⁵¹ kg. These represent h-domain intermediate quantities and serve as comparison markers for ECM’s intrinsic pre-Planck energy–mass framework.

This 0-dimensional formulation establishes the foundational boundary condition for ECM’s phase-kernel interpretation: a direct and exact mapping between frequency deviation, phase evolution, and Planck-time distortion. Together, these form the mass–energy–time manifestation kernel that governs the transition from a non-geometric origin to the Planck-surface domain.

1. Substitution and Formulation

Using x° = 360 Δf, the normalized ratio becomes:

(360 Δf) / (360° (2.999 × 10⁴² + Δf)) = 5.391247 × 10⁻⁴⁴

Simplify:

Δf / (2.999 × 10⁴² + Δf) = 5.391247 × 10⁻⁴⁴

2. Brief Derivation (Step-by-Step)

  1. Δf = 5.391247 × 10⁻⁴⁴ (2.999 × 10⁴² + Δf)
         
  2. Δf = (5.391247 × 10⁻⁴⁴)(2.999 × 10⁴²)
    + (5.391247 × 10⁻⁴⁴)Δf
         
  3. Δf (1 - 5.391247 × 10⁻⁴⁴)
= (5.391247 × 10⁻⁴⁴)(2.999 × 10⁴²)
         
  4. (5.391247 × 10⁻⁴⁴)(2.999 × 10⁴²)
= (5.391247 × 2.999) × 10⁻²
≈ 16.168349753 × 10⁻²
= 0.16168349753
         
  5. Δf = 0.16168349753 Hz

3. Tie-Together (Derived Quantities)

x° = 360 Δf
   = 360 × 0.16168349753
   = 58.2060591108°

Δt = 5.391247 × 10⁻⁴⁴ s

4. Concise Data Summary (Full Precision)

QuantitySymbolValueNotes
Original photon frequency f₀ 2.999 × 10⁴² + 0.16168349753 Hz Planck-scale carrier + Δf
Frequency difference Δf 0.16168349753 Hz Exact value for Δt match
Phase shift x° 58.2060591108° = 360 Δf
Planck-scale time interval Δt 5.391247 × 10⁻⁴⁴ s ECM event interval
Energy increment ΔE 1.071327 × 10⁻³⁴ J = h Δf
Effective mass increment ΔMᴍ 1.19165 × 10⁻⁵¹ kg = ΔE / c²

5. Short Descriptive Paragraph

Using the ECM relation x° = 360 Δf inside the normalized Planck-scale ratio yields an exact linear expression for Δf. Solving produces Δf = 0.16168349753 Hz, corresponding to a phase evolution of 58.2060591108° accumulated across the Planck-scale interval Δt = 5.391247 × 10⁻⁴⁴ s. The associated energy and mass follow directly from ECM frequency-governed relations, forming a consistent manifestation kernel linking Δf → phase → Δt in the 0-dimensional limit.

This subsection introduces the Extended Classical Mechanics (ECM) framework describing how a strictly 0-dimensional origin state—defined only by the primordial base frequency f₀—transforms into a measurable Planck-scale temporal interval Δt through an incremental frequency modulation Δf. This marks the earliest definable moment and initiates the emergence of a 3-dimensional micro-gravitational state.

At the 0-dimensional origin, ECM employs the phase–time relation:

Tₓ° = x° / (360° f₀)

Within the 0-dimensional domain, ECM excludes spatial constructs—no wavelength λ, no propagation v = c, no NAM transformation, and no acceleration mapping aᵉᶠᶠ. Under these restrictions, classical energy consistency applies:

ΔEₕ = h Δf
ΔMₕ = ΔEₕ / c²

The modulation Δf is interpreted through the ECM phase increment:

x° = 360 Δf

applied to the primordial base frequency: f₀ = 2.999 × 10⁴² Hz.

This displaces the primordial state from symmetry, producing a measurable Δt that marks the boundary between 0D origin and the Planck threshold. Beyond this threshold, ECM predicts the wavelength compresses below the Planck length (λ < ℓₚ), the acceleration aᵉᶠᶠ diverges, and the effective propagation speed vᵉᶠᶠ becomes ultra-superluminal.

Stabilization occurs at the Planck boundary (λ = ℓₚ, v = c) where the ECM constant k transitions into the Planck constant h. This forms the foundation for deriving k, computing ΔEₕ, ΔMₕ, and determining the ultra-superluminal vᵉᶠᶠ characteristic of ECM inflation.

6. ECM Reinterpretation of Inflation

In ECM, the ultra-superluminal velocities vᵉᶠᶠ ≫ c found in the pre-Planck state do not violate physical limits, because c has not yet emerged as a propagation boundary. Prior to the Planck surface, no spatial metric exists—no λ, no v = c constraint, and no geometric propagation rules.

Pre-Planck Regime: λ ≪ ℓₚ, Extreme aᵉᶠᶠ, and Emergent v > c

As the system descends toward the pre-Planck domain, λ ≪ ℓₚ. ECM predicts:

ECM replacement dispersion (illustrative):

fᵉᶠᶠ = f₀ (ℓₚ / λ)ᵅ
vᵉᶠᶠ = fᵉᶠᶠ λ = f₀ λ^(1−ᵅ) ℓₚᵅ

For α > 1 and λ ≪ ℓₚ, both quantities diverge, producing the natural ECM mechanism for inflation.

7. Mechanism for ECM Cosmic Inflation

  1. 0D: Δt and Δf from phase alone. No λ, no c.
  2. Pre-Planck: λ ≪ ℓₚ ⇒ aᵉᶠᶠ, fᵉᶠᶠ, vᵉᶠᶠ blow up.
  3. Mass–energy projection: NAM → −ΔPEᴇᴄᴍ → ΔMᴍ.
  4. Inflationary burst: ultra-superluminal expansion.
  5. Planck surface: λ = ℓₚ, v = c, k → h.

8. Algebraic Summary

ΔEₕ = h Δf
ΔMₕ = ΔEₕ / c²

k = h (fₚ / f₀)
ΔEₖ = k Δf
ΔMₖ = ΔEₖ / c²

9. Conclusion

ECM inflation arises from the pre-Planck descent where λ ≪ ℓₚ forces extreme aᵉᶠᶠ, enormous vᵉᶠᶠ, and rapid conversion of NAM into mass–energy. Stabilization at the Planck surface restores λ = ℓₚ, v = c, and k → h.

10. Pre-Planckian 0-Dimensional Data Summary (Full Precision)

This subsection presents the exact numerical constants governing the 0-dimensional ECM seed transition f₀ → Δf → phase → Δt prior to the emergence of wavelength λ and the post-Planck constraint c. All quantities listed here belong strictly to the pre-Planckian, non-geometric regime where ECM relations are governed by phase-based temporal definition and frequency-driven increments of energy and effective mass.

Quantity Symbol Value Notes
Original photon frequency f₀ ≈ 2.999 × 10⁴² Hz with Δf correction 0.16168349753 Hz Planck-scale seed frequency with Δf correction
Frequency difference Δf 0.16168349753 Hz Computed to match Δt exactly
Phase shift x° 58.2060591108° Using the ECM relation x° = 360 Δf
Planck-scale temporal span Δt 5.391247 × 10⁻⁴⁴ s 0-D event interval (exact)
Energy increment ΔEₕ 1.071327 × 10⁻³⁴ J ΔEₕ = h Δf — h-domain intermediate energy, not intrinsic 0-D ECM energy
Effective mass increment ΔMₕ 1.19165 × 10⁻⁵¹ kg ΔMₕ = ΔEₕ / c² — h-domain intermediate mass, not intrinsic 0-D ECM mass

Note: The quantities ΔEₕ and ΔMₕ are derived using the Planck constant h for comparison purposes. They represent **h-domain intermediate constructs** and are distinct from the intrinsic 0-D ECM energy Eₖ,ᴇᴄᴍ and mass Mₖ,ᴇᴄᴍ.

Descriptive Summary

Applying the ECM phase law x° = 360 Δf to the Planck-normalized temporal ratio yields an exact linear solution for Δf. This produces the precise frequency increment Δf = 0.16168349753 Hz, corresponding to a phase advance of 58.2060591108° across the pre-geometric event interval Δt = 5.391247 × 10⁻⁴⁴ s. The matching energy and mass increments follow directly from ECM’s frequency-governed relations, forming a consistent manifestation kernel that connects frequency, phase evolution, and the Planck-scale temporal structure at the 0-dimensional limit before spatial extension and the emergence of c.

11. ECM Pre-Planckian Dynamics and Inflation-Equivalent Amplification

In Extended Classical Mechanics (ECM), the emergence of propagation, energy, and mass is governed by existence transformations (NAM ↔ ΔPEᴇᴄᴍ ↔ ΔMᴍ) rather than geometric curvature or relativistic postulates. The following three-phase sequence (A → B → C) describes how a photon-like excitation evolves from a 0-dimensional non-propagating state into the Planck-surface regime where vᵉᶠᶠ → c and familiar physics begins.

12. A. 0-Dimensional Pre-Emergence (vᵉᶠᶠ,₀ = 0, no λ)

In the primordial 0-dimensional stage, no spatial metric exists; therefore, the excitation has:

The ECM effective acceleration at this stage is defined by the first displacement Δd₁:

Δd₁ = vᵉᶠᶠ,₀ Δt + ½ aᵉᶠᶠ (Δt)², vᵉᶠᶠ,₀ = 0

Substituting vᵉᶠᶠ,₀ = 0 and Δd₁ = 3 × 10⁸ m over Δt = 1 s gives:

aᵉᶠᶠ = 6 × 10⁸ m/s²

This marks the germination of motion from a non-geometric state and initiates Phase B.

B. Pre-Planck Amplification (Superluminal vᵉᶠᶠ ≫ c)

Once a finite characteristic length appears (λ > 0) but remains far below the Planck length (λ ≪ ℓₚ), the excitation undergoes ECM amplification. Effective frequency and velocity follow:

fᵉᶠᶠ = f₀ · (ℓₚ / λ)ᵅ,  vᵉᶠᶠ = fᵉᶠᶠ · λ = f₀ · λ^(1−ᵅ) · ℓₚᵅ

For α > 1 and λ ≪ ℓₚ, both fᵉᶠᶠ and vᵉᶠᶠ increase explosively, producing:

vᵉᶠᶠ ≫ c

This is a pre-metric effective velocity; ECM interprets it as the driver of inflation-equivalent behaviour prior to the emergence of the Planck surface.

C. Planck Surface Stabilisation (vᵉᶠᶠ → c)

When the characteristic length reaches the Planck scale:

λ → ℓₚ, t → tᴘ

ECM amplification saturates and physical quantities transition into post-Planck values:

Post-Planck dynamics follow, where c becomes the operational limit.

Summary of A → B → C Evolution

  1. A: 0-D pre-emergence, vᵉᶠᶠ,₀ = 0, λ = 0, infinite aᵉᶠᶠ potential.
  2. B: λ ≪ ℓₚ, explosive fᵉᶠᶠ and vᵉᶠᶠ ≫ c, ECM inflation-equivalent amplification.
  3. C: λ → ℓₚ, vᵉᶠᶠ → c, k → h, post-Planck physics begins.

13. Pre-Planckian 0-Dimensional ECM Summary (Energy–Mass Formulation)

This subsection presents the ECM-defined energy and mass content of the 0-dimensional photon using the ECM proportionality constant k instead of Planck’s constant. The fundamental relations are: Eₖ,ᴇᴄᴍ = k f₀ and Mₖ,ᴇᴄᴍ = Eₖ,ᴇᴄᴍ / c², representing the intrinsic energetic and manifestational properties of the photon before wavelength λ or geometric propagation emerge. The initial 0-D propagation velocity is vᵉᶠᶠ,₀ = 0, with effective acceleration aᵉᶠᶠ = 6 × 10⁸ m/s².

Quantity Symbol Value Notes
Original photon frequency f₀ 2.999 × 10⁴² Hz Seed frequency at 0-dimensional origin
ECM proportionality constant k (value defined by ECM scaling) Energy constant governing 0-D manifestation; replaces h
ECM energy content (0-D photon) Eₖ,ᴇᴄᴍ k × 2.999 × 10⁴² J Using Eₖ,ᴇᴄᴍ = k f₀, intrinsic ECM energy
ECM effective mass (0-D photon) Mₖ,ᴇᴄᴍ (k × 2.999 × 10⁴²) / c² kg Using Mₖ,ᴇᴄᴍ = Eₖ,ᴇᴄᴍ / c², pre-Planck mass
Propagation constraint vᵉᶠᶠ,₀ 0 m/s Initial effective propagation velocity in 0-D regime
Effective acceleration aᵉᶠᶠ 6 × 10⁸ m/s² Initial germination acceleration from non-geometric state

14. Descriptive Summary

The h-domain quantities (Δf, x°, Δt, ΔEₕ, ΔMₕ) are intermediate constructs for Planck-normalized transitions and are excluded from the intrinsic 0-D ECM formulation.

15. Pre-Planck Proportionality Constant (ECM constant) ECM defines the 0-dimensional photon through its proportionality-based energy Eₖ,ᴇᴄᴍ = k f₀ and corresponding mass Mₖ,ᴇᴄᴍ, independent of wavelength, phase propagation, or Planck’s constant. Together, these form the fundamental pre-Planck ECM state prior to the emergence of space, distance, or the speed limit c.

15. Original Photon Frequency & Pre-Planck (ECM) Constant

Original Photon Frequency

Presented in two complementary formats: a compact scientific form and a high-precision expanded form.

f₀ = 2.999 × 10⁴² + 0.16168349753 Hz
f₀ = 2.99900000000000000000000000000000000000000016168349753 × 10⁴² Hz

16. Pre-Planck Proportionality Constant (ECM constant)

Shown as a leading term (matching Planck's constant digits) plus a tiny ECM correction. The correction is many orders of magnitude smaller than the leading term and encodes the pre-Planck scaling factor (fᴘ / f₀). 

k = 6.62606868 × 10⁻³⁴ − 3.57227729 × 10⁻⁷⁷ J·s
k = 6.62606868000000000000000000000000000000000000357256 × 10⁻³⁴ J·s

17. Planck Frequency & Constant (Reference)

fᴘ = 2.999 × 10⁴² Hz
h  = 6.62606868 × 10⁻³⁴ J·s

Brief Explanation

The presentation emphasizes three aligned elements used in the Extended Classical Mechanics (ECM) framework:

The correction term in k is approximately 3.5723×10⁻⁷⁷ J·s—about 43–44 orders of magnitude smaller than the leading term. While negligible in conventional calculations, it is essential in ECM for pre-Planck scaling and precise mass-energy bookkeeping.

Notation: superscripts denote powers of ten; compact and expanded forms are provided for ECM documentation clarity.

References

DOI-Sorted References

  1. ECM Executive Summary — Consolidated Framework for Time Distortion, Effective-Mass Dynamics, and Cosmological Interpretation
  2. A Journey into Existence, Oscillations, and the Vibrational Universe: Unveiling the Origin
  3. Speed Conditions and Mass Transition Clarified: Foundational Formulation of Extended Classical Mechanics
  4. Unified Quantum Cosmology: Exploring Beyond the Planck Limit with Universal Gravitational Constants
  5. Perturbations and Transformations in a Zero-Dimensional Domain
  6. Effective Mass of the Energetic Pre-Universe: Total Mass Dynamics from Effective and Rest Mass
  7. Appendix 12-II: Effective Acceleration vs. Effective Transformation Coefficient in ECM — Definitions and Applications
  8. Mass-Energy Transformations in ECM: −Mᵃᵖᵖ, Gravitational Interaction & Frequency-Driven Dynamics
  9. Gravity Beyond GR’s Spacetime Curvature: Mathematical Basis of the ECM Phase Kernel Formalism
  10. Gravitational Interactions and Energy-Force Relationships in 0ₜₕ-Dimensional Framework
  11. Theoretical Insights into Micro-Gravitational Forces: Potential Energy Dynamics in 0ₜₕ-Dimensional Abstractions
  12. ECM: A Unified Framework for Frequency-Mass Equivalence, Entropic Time Distortion, and Cosmological Self-Regulation
  13. Appendix 50: Effective Acceleration, NAM Dynamics, and Cosmic-Scale Motion in Extended Classical Mechanics (ECM)

Non-DOI Academic References

  1. Appendix 31: Frequency and Energy in Extended Classical Mechanics (ECM)
  2. Dark Energy as a Consequence of Gravitational and Kinetic Interactions
  3. Restoring Structural Energy Distinctions: Classical Mechanics, Planck Physics, and Their Integration in ECM
  4. Understanding Extended Classical Mechanics (ECM)
  5. ECM: A Post-Relativistic Interpretation of Early Universe Phenomena & Cosmic Expansion
  6. ECM Interpretation of Time Dynamics
  7. ECM Essence — telitnetwork archive

ECM Media Reference

  1. ECM Phase Kernel Formalism — Gravity Beyond Spacetime (Video Lecture)