Mathematical Bridges Between Levels
Abstract. This article shows that the grading by levels Tᵏ of ArXe follows from a single postulate: that the speed of light c and the gravitational constant G sit at the root level. From it, by algebra, come the levels of mass, length and action (ħ = T⁶), the levels of the Planck units, and the fact that the three partitions of T⁶ are three physical factorizations of ħ. The article then separates two kinds of bridge: those that pass between levels, which are algebra of c, G and ħ, and those that move within a level, which depend on dimensionless couplings such as α and α_G. With this, the empirical content ends up at level 0, which the grading does not distinguish. It contains no numerical predictions.
Author: Diego Luis Tentor · Independent researcher
Date: October 2026
Status: draft under review
License: CC BY-SA 4.0
Base text: ArXe — Foundations and Framework. References of the type “Foundations §12.1” point to that document. This article accompanies ArXe — Particles and Phenomena by Level and ArXe — Fundamental Particles.
0. How to read this article
| Label | Meaning |
|---|---|
| Proposition | Exact algebraic consequence of the definitions. Verified. |
| Verified | Numerical identity checked with CODATA values (scipy). |
| Reading | An interpretation within ArXe. It is evocation, not derivation. |
| Open | An unresolved question. |
Notation. “Level” is the exponent k of Tᵏ in the dimensional rule of Foundations §12.1: k = 3a + 2b + c for Mᵃ·Lᵇ·Tᶜ. It is not the arity n of the document (n = 2k for k > 0).
Scope. This article uses no empirical value to derive anything. CODATA values are only used to check identities. The results are algebra and counting. What they say about physics is reading, and is marked as such.
1. The grading and where it comes from
1.1 Three conditions fix it
(Proposition)
We ask for a linear function n on the dimensions (M, L, T) with three conditions: n(T) = 1, n(c) = 1, n(G) = 1. The solution is unique:
| Quantity | Level |
|---|---|
| T | 1 |
| L | 2 |
| M | 3 |
| ħ | 6 |
That is, T ≡ T¹, L ≡ T², M ≡ T³ and ħ = T⁶. The table of Foundations §12.1 is not an independent six-row assignment. It comes out of three conditions. ħ = T⁶ is a consequence, not an assumption.
1.2 The complete family
If n(T) = t, n(c) = a and n(G) = b are left free:
| Quantity | Level |
|---|---|
| n(L) | t + a |
| n(M) | t + 3a − b |
| n(ħ) | 2t + 5a − b |
The ArXe point is (t, a, b) = (1, 1, 1). Then n(L) = 2, n(M) = 3 and n(ħ) = 6. Any other point gives another grading, equally consistent with dimensional analysis. The choice “c and G are at the root level” is the only postulate. Everything else is algebra. (Proposition)
1.3 c = G is equivalent to a PIVOT
(Proposition)
n(c) − n(G) = n(M) + n(T) − 2·n(L).
Therefore: n(c) = n(G) if and only if n(L) = (n(M) + n(T))/2. That is, c and G are at the same level exactly when T² is the pivot of T¹ and T³ (Foundations §20). The PIVOT operator and the equality of the two conversion constants are the same condition. (Reading of why it matters: the “bridge” T¹ and the spatial pivot are one and the same thing.)
2. Planck units
2.1 Levels
(Proposition, verified)
| Planck unit | Formula | Level |
|---|---|---|
| t_P | √(ħG/c⁵) | 1 |
| l_P | √(ħG/c³) | 2 |
| m_P | √(ħc/G) | 3 |
| F_P | c⁴/G | 3 |
| p_P | m_P·c | 4 |
| P_P (power) | c⁵/G | 4 |
| E_P | √(ħc⁵/G) | 5 |
| ρ_P (density) | c⁵/(ħG²) | −3 |
Units with a square root (t, l, m, p, E) have as their level the average of the levels of their radicand. For example, m_P: (6 + 1 − 1)/2 = 3. It is an average, like PIVOT. One can write m_P² = ħ·(c/G): m_P is the pivot of T⁶ and T⁰. F_P and P_P carry no root.
2.2 The three partitions of T⁶ are three ways of writing ħ
(Verified, with CODATA)
| Partition | Product of Planck units | Result |
|---|---|---|
| 1 + 5 | E_P · t_P | ħ (relative error 2·10⁻¹⁶) |
| 2 + 4 | p_P · l_P | ħ |
| 3 + 3 | m_P · (c · l_P) | ħ |
The first two are the canonically conjugate pairs: (t, E) and (x, p). The third is mass times (L²/T). Foundations §21.7 lists those three partitions of T⁶ as “pending”. All three are physical factorizations of the action.
3. The partitions of Foundations §21.7 are dimensional factorizations
(Proposition)
Each partition a + b = N of Foundations §21.7 corresponds to real products of quantities of level a and b:
| Level | Partition | Physical product | Result |
|---|---|---|---|
| T³ | 1 + 2 | t · L, or v · L (L²/T) | time·length |
| T⁴ | 2 + 2 | L · L | area |
| T⁴ | 1 + 3 | c · m, or F · t | momentum |
| T⁵ | 1 + 4 | p · v, or P · t | energy |
| T⁵ | 2 + 3 | F · L | energy (work) |
| T⁶ | 1 + 5 | E · t | action |
| T⁶ | 2 + 4 | L · p | action |
| T⁶ | 3 + 3 | m · (L²/T) | action |
Two coincidences with the base document:
- The 2 + 2 partition of T⁴ is area (length times length), the quantity that appears in the holographic principle (Foundations §21.3 and §21.6). (Reading)
- The 3 + 1 partition of T⁴ is mass times velocity (Foundations §21.3). Dimensionally it is momentum. (Reading)
4. Two bridges between mass and length
(Verified)
A mass m (T³) is converted into a length (T²) by two paths:
| Path | Formula | Factor | Operation |
|---|---|---|---|
| Gravitational | r_g = G·m/c² | G/c² (level −1) | direct PRODUCT |
| Quantum | λ̄_C = ħ/(m·c) | ħ/c (level 5) | COC-INV of m (level −3), then PRODUCT |
Levels: 3 − 1 = 2 in the first. 5 − 3 = 2 in the second.
The two paths coincide at a single point:
- r_g · λ̄_C = l_P², for every mass. Verified numerically: with r_s = 2Gm/c², the ratio r_s·λ̄_C/l_P² gives 2.0000000, which is the factor 2 of that definition.
- r_g / λ̄_C = G·m²/(ħc) = (m/m_P)², which is α_G (the dimensionless gravitational coupling constant).
- They are exactly 1 when m = m_P. There the two bridges coincide.
Verified values: α_G(electron) = 1.75·10⁻⁴⁵ and α_G(proton) = 5.91·10⁻³⁹.
What it says. The number α_G measures how much the level-1 bridge (G) and the level-6 bridge (ħ) differ for one and the same mass. The Planck mass is the point where both bridges cross. That is a property of the algebra of c, G and ħ. It is not empirical information. What is empirical is where m_p lies relative to m_P (α_G ≈ 6·10⁻³⁹). (Proposition + Verified)
5. Two kinds of bridge
(Reading, supported by §4 and by the following calculation)
| Kind | What it does | What expresses it | Is it empirical? |
|---|---|---|---|
| Between levels | Passes from one level to another | Monomials of c, G and ħ | No. It is algebra |
| Within a level | Passes from one scale to another of the same level | Powers of dimensionless couplings (level 0) | Yes |
Example, with a single mass (the electron) and three lengths, all of level 2: (Verified)
| Length | Value relative to λ̄_C |
|---|---|
| Classical radius r_e | α (checked: 0.0072973525…) |
| Compton λ̄_C | 1 |
| Bohr radius a₀ | 1/α (checked) |
And in energy (level 5): E_Hartree / (m_e c²) = α² (checked: 5.325135·10⁻⁵).
The three lengths are at the same level. A factor α separates them. Different levels are separated by c, G and ħ. The empirical lives at level 0.
6. What the grading does not see
6.1 The kernel has rank 2
(Proposition)
n: Z³ → Z is surjective. Its kernel (level-0 combinations) is a lattice of rank 2. Some elements:
| Quantity | Dimension | Level |
|---|---|---|
| Acceleration | L/T² | 0 |
| c/G | M·L⁻²·T | 0 |
| A pure number | dimensionless | 0 |
T⁰ does not mean “dimensionless”. It contains two independent dimensional directions. Acceleration and c/G have the same level as a pure number and are not one. Foundations §12.1 calls acceleration “dimensionless in ArXe”. That sentence is true only in the sense of level 0.
6.2 The monomials of c, G and ħ form a lattice of index 2
(Proposition, verified by Smith normal form: diag(1, 1, 2))
An integer monomial in c, G and ħ can have any dimension (a, b, c) with a + b + c even. Those with odd sum need a square root. That is why the Planck units of energy, momentum, mass, length and time carry a root, and those of force and power (c⁴/G and c⁵/G) do not. Verified by exhaustive search in the range ±4.
6.3 Charge
With charge in Gaussian units, q² = ħc·α has level 7, and q has level 3.5: a half-integer. The integer grading does not see charge as a level. It matches the fact that charge was placed by its structural role (BCO) and not by dimension (ArXe — Fundamental Particles, §3.2). (Verified)
7. What they say together
- The integer grading is a consequence of a single postulate: c and G at the root level. Everything else is algebra. (§1)
- c = G is the same as “T² is the pivot of T¹ and T³”. (§1.3)
- The Planck units with a root are pivots of ħ and of a quotient of c and G. (§2)
- The partitions of Foundations §21.7 are real physical factorizations. For T⁶, all three give ħ with Planck units. (§2.2, §3)
- Between levels there is only the algebra of c, G and ħ. The empirical information is at level 0, in dimensionless couplings such as α and α_G. (§4, §5, §6)
- The number that measures how far two bridges of the same pair of levels separate is a dimensionless coupling. For mass and length, that number is α_G. (§4)
- With charge a third bridge appears, and the three couplings form a triangle with a single relation: α_G = α·β. The number between mass and charge is β = G·m²/q² = (m/m_S)². (§8)
- The empirical content is the number of independent dimensional constants minus 3. (§8.5)
8. The electromagnetic case
(Proposition and Verified, with CODATA. Gaussian units: q² = e²/(4πε₀).)
8.1 A third bridge between mass and length
A mass m (T³) has a third natural length, besides r_g and λ̄_C. It is the classical radius r_q = q²/(m·c²). For the electron it is r_e.
| Path | Formula | Factor | Operation on m | Level of the factor |
|---|---|---|---|---|
| Gravitational | r_g = G·m/c² | G/c² | direct PRODUCT | −1 |
| Quantum | λ̄_C = ħ/(m·c) | ħ/c | COC-INV | 5 |
| Electromagnetic | r_q = q²/(m·c²) | q²/c² | COC-INV | 5 |
Levels: 3 − 1 = 2 for the gravitational one. 5 − 3 = 2 for the other two.
The quantum bridge and the electromagnetic bridge have the same level (5) and the same operation (COC-INV). That is why their ratio is a level-0 number that does not depend on the mass. That number is α: r_q/λ̄_C = q²/(ħc) = α. Verified: 0.007297352564… for electron, muon and proton.
8.2 Triangle of couplings
There are three bridges and three ratios. For a mass m:
| Ratio | Formula | Dependence on m | Value |
|---|---|---|---|
| Gravitational / quantum | α_G = G·m²/(ħc) = (m/m_P)² | m² | electron 1.75·10⁻⁴⁵; muon 7.49·10⁻⁴¹; proton 5.91·10⁻³⁹ |
| Electromagnetic / quantum | α = q²/(ħc) | none | 7.2973525643·10⁻³ |
| Gravitational / electromagnetic | β = G·m²/q² = (m/m_S)² | m² | electron 2.40·10⁻⁴³; muon 1.03·10⁻³⁸; proton 8.09·10⁻³⁷ |
(Verified) Exact relation: α_G = α · β. Only two of the three are independent.
Dependence rule. When the two bridges use the same operation on m, the ratio does not depend on m. When one is PRODUCT and the other COC-INV, the ratio depends on m² (the exponent is 1 − (−1) = 2). (Proposition)
Crossing masses.
- Gravitational = quantum at m_P = √(ħc/G).
- Gravitational = electromagnetic at m_S = q/√G = 1.859·10⁻⁹ kg. This is the Stoney mass.
- Electromagnetic = quantum never cross: their ratio is α for every mass.
- α = (m_S/m_P)² (Verified: 0.0854245431² = 0.0072973526). The two natural level-3 masses, m_P and m_S, are separated by √α.
8.3 The three couplings are at level 7
(Proposition)
| Coupling | Meaning | Level |
|---|---|---|
| ħ·c | Quantum scale (energy × length) | 7 |
| q² | Coulomb energy × length | 7 |
| G·m² | Newton energy × length | 7 |
All three are “energy times length”: 5 + 2 = 7. The pure numbers are their ratios (§8.2). Foundations §19 assigns T⁷ a compactification mnemonic. This article asserts no relation. (Open)
8.4 Planck and Stoney: two natural systems with the same levels
(Verified. It is generalized in an article on level-6 actions, in preparation: each system {c, G, X} defines an action ħ_X = c³·l_X²/G.)
Replacing ħ by q²/c (that is, by α·ħ) turns the Planck units into the Stoney ones.
| Unit | Level | Stoney / Planck ratio |
|---|---|---|
| Mass | 3 | √α = 0.0854245431 |
| Length | 2 | √α |
| Time | 1 | √α |
| Energy | 5 | √α |
| Force | 3 | 1 |
| Density | −3 | 1/α = 137.036 |
The exact rule: Q_Stoney/Q_Planck = α^γ, with γ the exponent of ħ in the Planck unit of Q. γ is not a function of the level. Mass and force are both at level 3, and their ratios are √α and 1.
Consequence: the jump between two natural systems of scales is blind to level. A weight per level xᵏ (Foundations §21.7) cannot represent it. The couplings live in the kernel of the grading (§6.1), and there no level-dependent factor sees them. (Proposition)
8.5 How many pure numbers there are
(Proposition)
- Independent dimensional constants: c, G, ħ, q². The dimension matrix has rank 3.
- Pure numbers: 4 − 3 = 1. It is α = q²/(ħc) (exponents (c, G, ħ, q²) = (−1, 0, −1, 1)).
- With the electron mass: 5 − 3 = 2. α and α_G(e).
- Each extra independent dimensional constant adds one pure number.
The empirical content is the number of independent dimensional constants minus 3. Fixing c, h and e as exact (SI 2019) does not change that count. The one that remains measured is α (through ε₀).
8.6 Charge and REG
(Reading, with exact arithmetic)
- Charge has half-integer level: q = 3.5; q² = 7 (§6.3).
- Coulomb’s law says r = q²/E. With levels: 7 + (−5) = 2.
- REG (T⁻⁵) is the open counterpart of energy (T⁵). The square of the charge (level 7) converts REG into length (level 2). It is the electromagnetic analogue of G/c², which converts mass into length.
8.7 What the case says
- Three level-7 constants define three bridges between mass and length. They are G·m², q² and ħc.
- The three pure numbers form a triangle with one relation: α_G = α·β.
- The number that separates charge and action is α. The one that separates mass and action is α_G. The one that separates mass and charge is β. With c fixed.
- α does not depend on the mass because both bridges are COC-INV. The other two do, through m².
- The separation between natural systems does not depend on the level. It is a power of α fixed by ħ.
- The empirical content of all this is a single number for each dimensional constant added.
9. Open questions
- Other interactions. The scheme of §8 works for mass and electric charge. It remains to be seen whether it holds for the strong and the weak interaction, with g_s² and g_w² as level-7 couplings.
- Why the postulate (t, a, b) = (1, 1, 1) and not another point of the family of §1.2? Today it is a postulate of the framework.
- Does the partition of §3 with a real physical product have any other consequence? Only levels 3 to 6 were examined.
- Does the rest of the kernel (acceleration and c/G) play any role? Today T⁰ confuses them with a pure number.
- Does the parity of §6.2 correspond to anything in the logic of modes (parity of the arity)?
- Coupling constants run with energy. α = 1/137 is the low-energy limit. At larger scales, α⁻¹ changes. Energy is level 5. How is that running expressed as a bridge between T⁵ and level 0?
- Is there a version of the dependence rule (§8.2) for bridges with a different COC-INV arity, for example m⁻² versus m⁻¹?
- Structural reading of T⁷, where ħc, q² and G·m² coincide (§8.3).
External sources
- Constant values: CODATA, via scipy.constants. The numerical identities of §2.2, §4, §5 and §8 were checked with those values.