The Planck Inflection: Reciprocal Symmetry Between Mass and Time in Natural Ratios

J. Rogers, SE Ohio

Abstract

This paper redefines the Planck scale as the inflection point of reciprocal scaling between mass and time. Using $h = 1$ as the invariant of physical action, all dimensional constants ($G, c, h$) appear as coordinate transformations rather than fundamental quantities. The Planck domain represents the equilibrium of mass–time reciprocity — the unique dimensional identity where the ratio of mass to time equals unity.

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1. Introduction

Conventional physics treats the Planck scale as a physical limit, defined through the constants $G, c, \hbar$. Yet these constants are not intrinsic to the universe; they encode the observer’s choice of arbitrary unit definition. When the we harmonize units to align with the physical scale of the universe, we reveal a natural equilibrium between mass and time. This symmetry transforms the Planck scale from a boundary condition into a geometric identity.

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2. The Reciprocal Law of Mass and Time

Mass and time behave as reciprocal measures of a single invariant substrate. Their fundamental relation arises from the dimensional consistency:

$$h=m{c}^{2}t=1.\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}$$

Harmonizing units gives:

$$mt=1.\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}$$

Rearranging gives:

$$t=\frac{1}{m{\mathrm{<br>}}^{}}\mathrm{.}$$

Thus, as mass increases, the corresponding characteristic time decreases, preserving constant action. In this frame, mass and time form conjugate dimensions of one invariant product.

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3. The Inflection Point as Dimensional Unity

Plotting the reciprocal curve $t=1$ in logarithmic space produces a line of slope –1. Only at the value $m = t = 1$ does the curve cross itself — the inflection point, where proportionality between mass and time is perfectly balanced. At this point:

$$m_P = \frac{1}{t_P} = 1, \quad m_P \, t_P = 1.$$

This state defines the Planck equilibrium: dimensional symmetry, where physical scaling collapses to pure identity. The Planck scale is not a threshold but the origin of metric coherence.

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4. Constants as Coordinate Offsets

When physical systems are measured away from this equilibrium, scale offsets appear as constants:

• $c$.

• $h$.

• $G$.

Viewed from the inflection frame where the axis of measurement align to the on physical scale of the unified universe, these constants vanish because there were always just artifacts of our unit chart. They are simply illusions arising from how units are projected, not properties of nature itself.

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5. The Action Geometry

Mass and time can be interpreted as orthogonal coordinates whose product defines invariant action. In full form:

  T/T_P = f·t_P = m/m_P = l_P/l = E/E_P = p/p_P = X  

If one axis (mass) is extended by a scale factor, the other (time) contracts proportionally to maintain constant area in a unit chart. This geometric conservation mirrors fundamental invariances across both classical and quantum regimes.

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6. Implications

1. Natural ratios are not theoretical simplifications but  the elimination of arbitrary coordinate realignments against a single physical scale in a unified universe.

2. The Planck “limit” marks the return to pure symmetry, not the collapse of physics.

3. Quantization  does not exist, Energy is frequency is mass is 1/length in the natural ratios of the universe.

Thus, all our descriptions are confused because we do not see that the universe is already trivially unified.

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7. Conclusion

The Planck inflection defines reality’s invariant equilibrium, where mass and time reciprocate around constant action. Harmonizing units to a single physical scale identifies this unity explicitly: $m \, c^2 \, t = 1$. All dimensional constants emerge from relative displacement to this equilibrium. The Planck jacobians between si unit scaling and unit free natural ratios is not where physics ends, but where its geometry begins — the axis of dimensional unity connecting all measurement.

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Appendix: Contrast with Standard Presentation of Planck Units

A. Standard View of Planck Units

In mainstream physics:

• Definition by constants
  Planck units are defined by algebraically combining four constants $𝐺,𝑐,ℎ$ (or $\mathrm{\hslash }$) and ${𝑘}_{𝐵}$ into base quantities (Planck length, time, mass, temperature, etc.). One then chooses units so that these constants are numerically equal to 1.

• Interpretation of the Planck scale
  The “Planck scale” is commonly described as:

  • A regime where quantum effects of gravity are expected to be strong.

  • A limit of applicability of current theories (GR + QFT), beyond which “new physics” is required.

  • Often loosely treated as a “smallest meaningful length/time” in popular explanations.

• Status of constants
  The constants $𝐺,𝑐,ℎ$ are treated as:

  • Empirically given, universal properties of nature.

  • Deep, but unexplained, numbers whose values must be measured.

  • Setting them to 1 in “natural units” is framed as a convenient convention for theorists, not as a clue to underlying geometry.

In short: the standard view defines Planck units as a clever rescaling based on four “fundamental constants,” and the Planck scale as a suspected breakdown/transition regime.

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B. Inflection-Point View (This Framework)

In your framework:

• Harmonizing units, not setting constants to 1.
  We take harmonize the unit scaling to a single physical scale of a unified universe.   We do not set $\mathrm{any\; constant}"=\mathrm{1"}$. Mass, time, and other quantities are conjugate projections of a single unified substrate.

• Reciprocity of mass and time
  Mass $𝑚$ and time $𝑡$ are related by an inverse law under this constraint (schematically $𝑚\mathrm{<br>}\sim \text{<br>}$), so that increasing one necessarily decreases the other to preserve the invariant product. They are not independent axes, but reciprocal coordinates on the same underlying unified object.

• Planck as inflection, not limit
  The “Planck point” is:

  • The unique inflection point where reciprocal axes harmonize: $𝑚𝑡=1$ in the dimensionless ratios.

  • The equilibrium point of dimensional identity, where the substrate has no bias toward “mass-ness” or “time-ness.”

  • A center of symmetry, not a smallest scale or breakdown boundary.

• Constants as offsets, not essence
  In this view:

  • $𝐺,𝑐,ℎ$ are coordinate offsets describing how far our arbitrary unit choices are displaced from the inflection point in this single physical scale.

  • When we stand at the inflection point (natural units), these constants collapse to 1 because the geometry is in its identity frame.

  • Their “mystery values” are artifacts of measuring from the corner, not from the center.

In short: our view defines "Planck units" as the geometrically necessary harmony point of reciprocal dimensions under constant action, and reinterprets the “constants of nature” as byproducts of our misaligned coordinates.

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C. Side-by-Side Summary

Aspect	Standard View	Inflection-Point View
Core defining idea	Combine $𝐺,𝑐,ℎ,{𝑘}_{𝐵}$ into base units	Mass–time reciprocity under $\mathrm{natural\; ratios}$
Role of Planck scale	Threshold, possible breakdown/limit	Equilibrium, symmetry/identity point
Status of constants	Fundamental properties of nature	Coordinate offsets from the symmetry center
Meaning of “natural units”	Mathematical convenience	Standing at the geometric inflection
Geometry intuition	Smallest scales, extreme regime	Inflection of hyperbolic reciprocity (mass–time)
Why constants = 1	By choice of units	Because the identity frame erases offsets

This appendix makes explicit that out contribution is not just “using natural units,” but reinterpreting why Planck units exist at all: as a geometric necessity arising from the reciprocal structure of the substrate, rather than a clever convention built around unexplained constants.