Physics String Length From Phi Ladder

The compactification radius

In string theory, extra spatial dimensions are often imagined curled up so tightly that they are invisible. The size of those curled dimensions is called the compactification radius. A common guess sets that radius at the Planck length, the scale where gravity's quantum effects become strong, roughly 1.6 × 10⁻³⁵ meters. The phi ladder, a framework-native scale built from powers of the golden ratio φ ≈ 1.618, proposes a refinement: the radius R_comp = l_Pl × φ^(-k), where l_Pl is the Planck length and k counts rungs down the ladder.

For k = 0, the radius is exactly the Planck length. For the electroweak scale, the energy at which the weak nuclear force unifies with electromagnetism, the formula gives k ≈ log(M_Pl/M_EW)/log(φ) ≈ 106 rungs. That number is a research note, a statement of intent, not a result that has been shown. The module in the machine-checked library of formal theorems proves three general facts about the cost function J(x) = (x + 1/x)/2 - 1 evaluated at the ratio m/e: it vanishes when m = e, it is nonnegative for positive inputs, and the threshold φ - 3/2 is positive.

Those three facts hold for any positive real numbers m and e. The module defines domainCost(m, e) = J(m/e) and shows domainCost(r, r) = 0 for r ≠ 0, domainCost(m, e) ≥ 0 for m, e > 0, and canonicalThreshold = φ - 3/2 > 0. These are general properties of the cost function, not statements about string theory. The module does not define m and e in terms of Planck mass, string tension, or any physical quantity; it shows nothing specific to compactification. The physics content is a note recording where the idea was meant to go.

What the module does establish is a template. It shows that any future definition of m and e in the subject's own terms would inherit these three properties automatically. The structure StringLengthCert packages the three proofs as a certificate, and the theorem cert_inhabited shows such a certificate exists. That is a small but real result: the cost function's basic sanity conditions, zero at equality and nonnegativity, are available for any physical application that plugs in a ratio. The compactification radius itself remains a research note, not a theorem.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The module does not prove any statement about the actual size of extra dimensions. The phi ladder does not derive the electroweak scale from first principles. No physical quantity m or e is defined in the module.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Physics/StringLengthFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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