# The Forces from α — the minimal input set

**The question (Jim):** *Can the four forces be rooted in just α — or in one empirical input like the
mass of the electron or proton?*

> **α itself** — the one derived structure everything below roots in — is derived from first principles
> (the IR / fully-rendered-3D value `1/137`, proven time-invariant) in [**Alpha.md**](Alpha.md).

**The answer:** Both, and they are different halves of the same statement. The **dimensionless force
strengths** root in **α** (which QLF *derives*, so they need no empirical coupling input at all); the
**absolute scale** needs **exactly one empirical mass** (`m_e` or `m_p`). So the four forces — couplings
*and* the scale they act on — reduce to **one derived structure (α) + one empirical mass.**

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## Two things to root: couplings vs. scale

A "force" has a *dimensionless strength* (how hard it pulls, e.g. α ≈ 1/137) and acts at an *absolute
scale* (set by the masses, in kg·m·s). These root differently.

## Part A — the dimensionless couplings root in α, and α roots in the substrate

α is **not an input** in QLF. It is derived from the substrate's own dimension:

- **α = 1/137 from `N = 3²`** — the 9 independent components of the 3×3 directional-coupling tensor of
  3-D space (`alpha_QLF_eq`, `only_3d_substrate_gives_137`: 2D→1/132, 4D→1/144). Zero free parameters.
  **α is the fine structure of three-dimensionality.**

Every other gauge coupling is α plus a *group-theory ratio* fixed by the same `6+2` / three-axis split:

- **Weak/EM mixing:** `sin²θ_W = 3/8` at unification (`sin2_weinberg_substrate_eq`) — relates the weak
  `g₂` and hypercharge `g₁` to each other and to α (the SU(5) normalization).
- **Strong:** `α_s = 1/b₀²` with `b₀ = 11N_c/3 − 2n_f/3 = 7`, from `N_c = 3` (axes) and `n_f = 6`
  (3 generations × 2 flavours) (`beta_coefficient_eq_seven`, `substrate_alpha_s`) — the same `3`.
- **Gravity's relative strength:** the Planck/proton hierarchy `ln(M_P/m_p) = 14π = 2π·b₀`
  (`hierarchy_log_eq_fourteen_pi`, 0.07%) — again the integer `7`.

At the unification scale the three gauge couplings **meet** (one coupling), split below it by exactly
these ratios. So all four force strengths trace to **one root: `N = 3`** (the dimension of space) and
the `6+2` alphabet split — which is *also* the root of α. **Rooting the forces "in α" and "in the
substrate" are the same statement**, because α *is* `N = 3²`.

## Part B — the absolute scale needs exactly one empirical mass

Part A fixes every dimensionless number. To convert to SI you need **one** scale, and one is enough,
because the whole mass spectrum is one scale times verified ratios:

- **`spectrum_one_scale`** — every mass = `m_p` × a verified ratio. `m_e = m_p/6π⁵`
  (`electron_mass_from_proton_eq`, `mass_ratio_QLF_eq`, 0.002%); `m_p = M_Planck · e^{−14π}`.
- So fixing **any one** mass (`m_e` *or* `m_p`) fixes **all** masses, and — with the Planck scale
  (`G = L_P²c³/ℏ`, by construction) — the absolute `G`.

That one mass is the single irreducible **empirical** input. Everything else is structure.

## The synthesis

> **The four forces root in α (derived, so the couplings are pure structure) + one empirical mass (the
> scale).** Because α is `N = 3²`, there is **no empirical coupling input at all**; the lone empirical
> number is one mass, which sets the size of everything.

This is the QLF answer to "how few inputs": **one derived constant family (α and the integers `2, 3, 7`
it shares a root with) + one mass.** Newton, Maxwell, Einstein, and the Standard Model each *assume*
their couplings; QLF derives them from the dimension of space, and asks only for the scale.

## Honest scope

- **Solid:** α (`N=3²`), `sin²θ_W=3/8`, `b₀=7`, the `14π` hierarchy, and `spectrum_one_scale` are
  machine-verified; the couplings' *roots* are the substrate integers.
- **Open (the value-level frontier):** the **RG running** of the couplings from the unification scale to
  observed energies (`QLF_RunningCouplings` anchors the structure, not the full β-functions); the W/Z
  masses / Higgs VEV; `α_s`'s value-level precision; and the absolute-`G` / Planck-mass calibration
  (the same single-scale residual). So the **structure** — forces from α + one mass — is established;
  the **last few-percent of each value** is the open calculational sector, not a missing input.

See [`Forces_From_Three_Axes.md`](Forces_From_Three_Axes.md) (the 3-axis origin), [`Beyond_Standard_Model.md`](Beyond_Standard_Model.md) (the free-parameter ledger), [`Per_Qubit_Mass_Quantum.md`](Per_Qubit_Mass_Quantum.md) §3.3 (one-scale spectrum), [`Planck_Scale.md`](Planck_Scale.md) (the scale by construction).