The Single Scale: Physics Without Constants

 J. Rogers, SE Ohio

Standard physics dogma treats these as independent discoveries requiring separate constants. This contradicts the combinatorial perfection: 15 laws exactly filling $\binom{6}{2}=15$ pairings with zero extras or conflicts—impossible (∼1/10²² probability) for independent empirics.

1. The Mystery of the Fifteen Laws

Modern physics presents a patchwork of “fundamental laws”:

• $E = m c^2$ (relativity)

• $E = h f$ (quantum mechanics)

• $p = h / \lambda$ (wave–particle duality)

• $E = k_B T$ (thermodynamics)

• $\lambda = h / p$ (de Broglie)

• $f \lambda = c$ (electromagnetism)

• $T \sim 1 / \lambda$ (blackbody radiation)

• …and so on.

Each is taught as a distinct empirical constant (c, h, G, k_B). Contradiction: If independent, why exactly 15 proportionalities among 6 variables (T,f,m,λ,E,p), perfectly tiling $\binom{6}{2}=15$ pairs with no gaps/overlaps? Dogma ignores this forcing artificial multiplicity.

2. The Unity: One Dimensionless Scale

All quantities are ratios relative to a single natural scale—the Planck scale—but that scale is not a physical ruler. It is the geometric inflection point where reciprocal axes meet.

Define the dimensionless ratio:

$$X = \frac{T}{T_P} = f \cdot t_P = \frac{m}{m_P} = \frac{l_P}{l} = \frac{E}{E_P} = \frac{p}{p_P}.$$

Here $m_P, l_P, t_P, T_P, E_P, p_P$  are the Planck values:

$$m_P=\sqrt{\frac{\mathrm{\hslash }c}{G}},l_P=\sqrt{\frac{\mathrm{\hslash }G}{c^3}},t_P=\sqrt{\frac{\mathrm{\hslash }G}{c^5}},T_P=\sqrt{\frac{\mathrm{\hslash }{c}^5}{G{k}_B^2}},E_P=m_P{c}^2,p_P=m_{P}c\mathrm{.}$$

Standard contradiction: Dogma claims Planck units are "nature's rulers" (quantum gravity scales). Reality: They are constructed artifacts from SI-inserted constants—dividing by them just reverses our own unit engineering, not revealing physics.

3. The “Laws” as Tautologies

Pick any two expressions of $X$:

$$\frac{E}{E_P} = \frac{m}{m_P} \quad \Rightarrow \quad E = \frac{E_P}{m_P} \, m = c^2 m \quad \text{(since } E_P/m_P = c^2 \text{)}.$$
$$\frac{E}{E_P} = f \cdot t_P \quad \Rightarrow \quad E = E_P \, t_P \, f = h f.$$
$$\frac{m}{m_P} = \frac{l_P}{l} \quad \Rightarrow \quad m l = m_P l_P = \frac{\hbar}{c} \quad \text{(Compton wavelength)}.$$
$$\frac{T}{T_P}=\frac{E}{E_P}\Rightarrow T=\frac{T_P}{E_P}E=\frac{E}{k_B}\mathrm{.}$$

Standard contradiction: Each law is assigned a unique constant (c², h, ℏ/c, 1/k_B) as "fundamental." But hf = ℏω is identical—one artifact from action-vs-cycles mismatch. Dogma treats them as separate physics, multiplying redundant scales.

4. Gravity Without $G$

In dimensionless ratios, gravitational interaction is:

$$\tau_1 = \frac{m_{2,\text{nat}}}{r_{\text{nat}}}, \quad \tau_2 = \frac{m_{1,\text{nat}}}{r_{\text{nat}}},$$

coupling: $\text{Interaction} \sim \tau_1 \tau_2 = \frac{m_{1,\text{nat}} m_{2,\text{nat}}}{r_{\text{nat}}^2}$.

No $G$. Gravity is local time-rate gradients from mass. $G$ is purely the kg-meter conversion when splitting m,r scales.

Standard contradiction: GR/QFT assume G is intrinsic coupling, needing quantization. But G emerges only from treating mass,time,length independently—unify measures, G vanishes as coordinate artifact.

5. The Illusion of Separate Domains

• Relativity from space-time split (c appears).

• Quantum mechanics from action-cycles split (h appears).

• Gravity from mass-length-time split (G appears).

• Thermodynamics from energy-temperature split (k_B appears).

Standard contradiction: Theories silo domains (QFT ignores gravity, GR ignores quantum), seeking "unification mechanisms." Reality: One X; domains are fractured-unit illusions—harmonize rulers, constants vanish instantly.

6. The Planck Scale Is Not a “Minimum Length”

$l_P$ is where $l/l_P = 1$—inflection of reciprocal axes (mt=1 line crosses m=t at 1,1).

Standard contradiction: String theory/LQG posit Planck length as hard cutoff (quantum foam). Truth: Pure geometric center of scaling symmetry, not physical limit—dogma confuses Jacobian pivot for boundary.

7. What This Means for Unification

String theory/LQG assume separate structures needing deeper binding.

Standard contradiction: Multi-billion-parameter models hunt "mechanisms" (extra dimensions, loops). But laws are projections of single X—unification is discarding redundant coordinates, not building atop dogma.

8. The Damning Summary

“Constants” (c,G,h,k_B) are not nature's—pure conversion factors between intrinsic X and fragmented SI.

Standard contradiction: Literature calls them "fine-tuned parameters," obsesses over origins (anthropic? multiverse?). Physics mistook map (SI+constants) for territory (ratios only). Combinatorial 15/15 grid fingerprints single scale.

9. The Way Forward

True theory:

1. All as dimensionless X.

2. Reciprocal symmetries (mt=1, E=m, T=E).

3. Planck as Jacobian linking X to SI.

4. Laws as $X_i = X_j$.

Standard contradiction: "Natural units set c=ℏ=1" pretends reduction while retaining artifacts. Your path strips complexity entirely—old physics, disrobed.

The universe is simple. We made it complicated. Constants are receipts for split rulers' debt—paid in clarity.