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The Planck Scale by Construction, Not Posited

Module: lean/QLF_PlanckScale.lean · Builds on: lean/QLF_QuantumBlackHole.lean, lean/QLF_GravityFromDelay.lean

The Planck length is the minimal coherent-closure length

QLF's "substrate event quantum" — one Planck length and one Planck tick, created together, per ZFA event — is not a posited input alongside ZFA. It follows by construction from QLF's own structure: a foundation does not assume its own granularity; the granularity is the smallest coherent closure.

QLF has the two ingredients this needs — emergent gravity (horizons form, QLF_GravityFromDelay) and Markov-blanket closure (reality is made of closures). Together they place the granularity, by construction, at the minimal coherent closure.

Work in Planck units (ℏ = c = 1). A coherent closure of spatial extent R confines an energy whose Compton radius is R — a mass μ = 1/R. That energy carries a gravitational horizon of radius r_s = 2μ (schwarzschild_radius, G = 1). The closure is a localized quantum object — not a self-engulfed horizon — exactly when its horizon does not exceed its extent. Three machine-checked steps (QLF_PlanckScale):

  1. coherent_iff_subplanck — schwarzschild_radius μ < compton_radius μ ↔ μ² < 1/2. A closure stays coherent iff it is sub-Planck in mass. (← reuses the already-proven sub_planck_compton_gt_schwarzschild; → is the one-line algebra 2μ < 1/μ ⟹ μ² < 1/2.)
  2. planck_length_floor — every coherent closure has Compton length > √2: 2 < (compton_radius μ)². There is no coherent closure below the Planck length. Try to confine a blanket tighter and its confinement energy's horizon grows larger than the blanket itself — it is inside its own horizon and cannot close.
  3. planck_self_dual — the floor is the unique self-dual point μ² = 1/2, where the quantum (Compton) and gravitational (Schwarzschild) closure radii coincide (reuses compton_eq_schwarzschild_iff). It is the fixed point of the length ↔ length duality.

So the substrate granularity is not arbitrary. The smallest coherent Markov blanket sits at the Compton–Schwarzschild self-dual point — and that point is the Planck scale. Given QLF's own commitments, the substrate is granular at the Planck length by construction; it is the closure floor, not a dial someone set.

One sentence: below the Planck length a closure would be inside its own gravitational horizon, so it cannot exist — the Planck length is the minimal coherent closure, hence the substrate quantum.

What is derived, and what is convention

Status
The Planck scale (a dimensionless fixed point μ²=1/2) ✅ Derived by construction as the closure floor (QLF_PlanckScale). It is not a free input — the quantum's scale is what the minimal coherent closure is.
The Planck length's SI value in metres Not a physics question (so not a flaw). "What a metre is" is a unit convention; you cannot derive it from logic. In Planck units the floor is the dimensionless √2; the O(1) factor is the Schwarzschild-2μ vs reduced-Compton convention.
Where matter sits above the floor (proton depth R_p) 🔵 Open, but separately tracked — the dimensional-transmutation hierarchy ln R_p = 14π (QLF_AlphaS, 0.07%) with a residual few-% M_Planck calibration. This is the hierarchy problem, not the granularity question settled here.

Why "by construction," not loose — and no circularity. The crossing uses both the Compton length (ℏ/mc) and the Schwarzschild radius (Gm/c²), which looks like it imports ℏ and G as external scales. It does not, because in QLF neither is an independent input:

  • G is emergent. QLF_GravityFromDelay derives G = L_P²c³/ℏ from holographic delay — G is not a fundamental constant, it co-varies with L_P. So using the Schwarzschild radius imports no scale independent of the substrate.
  • ℏ is just scaling. It is the unit of action (ℏ = 1 in natural units). Its only physical content is ℏ ≠ 0 — the substrate's discreteness, which is ZFA (Philosophy.md §1). The value of ℏ is a unit convention, not a physical scale.

So in natural units (ℏ = c = G = 1) the crossing μ² = 1/2 is a pure dimensionless fact that imports nothing external. "By construction" therefore means it falls out of QLF's own construction — emergent gravity + ℏ-as-discreteness + the closure ontology (the quantum = the minimal coherent closure) — not "derived from nothing." QLF does not get to choose its granularity; its own construction fixes it. The only thing left conventional is the SI value of a metre.

Bottom line

The Planck length is not assumed. It is the minimal coherent-closure length by construction — the self-dual point of the quantum/gravity length duality, machine-verified on top of QLF's own Compton–Schwarzschild crossing. The substrate is granular at the Planck scale because nothing smaller can close. The only residuals are a unit convention (not a flaw) and the already-logged few-percent hierarchy calibration.