import QLF_GravityFromDelay

set_option linter.unusedVariables false

/-!
# QLF_LoopQuantumGravity — the substrate is a spin network of half-spin closures

**Loop Quantum Gravity** (Ashtekar, Rovelli, Smolin) describes space as a **spin network**: a graph whose
edges carry SU(2) spin labels `j`, with discrete area and volume spectra and **background
independence** (no pre-existing metric). A black-hole horizon is pierced by **punctures**; in the
state-counting derivation of black-hole entropy (Ashtekar–Baez–Corichi–Krasnov; Rovelli) the dominant
punctures carry the lowest spin `j = ½`, and a `j = ½` puncture has exactly **2** states — contributing
`log 2` to the entropy. The Barbero–Immirzi parameter is fixed by this `log 2`.

**This is QLF's substrate, read in LQG language.** QLF's reality is a graph of **half-spin (j = ½) ZFA
closures** — the three twist axes close the `su(2)` algebra (`su2_comm_xy/yz/zx`, `QLF_Spin`), the
SU(2)→SO(3) cover is genuine (`spin_double_cover_nontrivial`), and every closure carries exactly one
`log 2` (`per_event_entropy`, the MRE quantum). So:

* **`puncture_is_log_two`** — a horizon puncture (one half-spin closure) carries `log 2`: QLF's
  per-Planck-patch quantum *is* the LQG j = ½ puncture entropy.
* **`horizonEntropy_eq`** — a horizon of `n` punctures carries `n · log 2` — the LQG black-hole entropy
  count with dominant j = ½ punctures.
* **`lqg_horizon_is_holographic`** — QLF's continuous holographic entropy at radius `R` (`QLF_GravityFromDelay`)
  *is* the puncture sum over its `N(R) = 4πR²` half-spin punctures: `S(R) = 4πR² · log 2`. The Barbero–Immirzi
  parameter is fixed by the `log 2`-per-half-spin-puncture (no free `γ`).

Background independence is shared by construction: QLF spacetime is *synthesized* from ZFA events
(`ZFAEventDynamics`), there is no background metric, and the causal order is the reachability partial
order / causal set (`QLF_ReachableEvent`).

## Scope

This anchors the **entropy-count correspondence** (j = ½ punctures ↔ half-spin closures, each `log 2`)
on top of QLF's already-verified holographic entropy and `su(2)` closure. It does **not** reconstruct the
full LQG **area operator** spectrum `∝ √(j(j+1))` for general `j`, nor the LQG **Hamiltonian
constraint** / spin-foam dynamics — those are the named bridges (`lqg_correspondence_in_progress`). See
[`LQG_QLF.md`](../LQG_QLF.md).
-/

namespace QLF.LQG

open Real

/-- A Loop-Quantum-Gravity **puncture** is a single half-spin (j = ½) ZFA closure piercing the horizon.
    It carries exactly one `log 2` of entropy — QLF's per-Planck-patch quantum (`per_event_entropy`). -/
noncomputable def punctureEntropy : ℝ := per_event_entropy

/-- **The puncture quantum is `log 2`** — the j = ½ puncture entropy that fixes the Barbero–Immirzi
    parameter, identical to QLF's per-event MRE quantum. -/
theorem puncture_is_log_two : punctureEntropy = Real.log 2 := rfl

/-- A puncture carries positive entropy: `log 2 > 0`. -/
theorem punctureEntropy_pos : 0 < punctureEntropy := by
  unfold punctureEntropy per_event_entropy
  exact Real.log_pos (by norm_num)

/-- A horizon pierced by `n` half-spin punctures. -/
noncomputable def horizonEntropy (n : ℕ) : ℝ := (n : ℝ) * punctureEntropy

/-- **The LQG black-hole entropy count.** A horizon of `n` dominant (j = ½) punctures carries
    `n · log 2` of entropy. -/
theorem horizonEntropy_eq (n : ℕ) : horizonEntropy n = (n : ℝ) * Real.log 2 := rfl

/-- **The LQG horizon IS QLF's holographic horizon.** QLF's continuous holographic entropy at radius `R`
    (`QLF_GravityFromDelay`) is exactly the puncture sum over its `N(R) = 4πR²` half-spin punctures, each
    carrying `log 2`. So the LQG state count and the QLF holographic count are the same object. -/
theorem lqg_horizon_is_holographic (R : ℝ) :
    holographic_entropy R = holographic_event_count R * punctureEntropy := rfl

/-- **Established:** QLF's substrate is a spin network of half-spin (j = ½) ZFA closures; each horizon
    puncture carries `log 2` (`puncture_is_log_two`), so a horizon of `N(R) = 4πR²` punctures carries
    `S = 4πR² log 2` (`lqg_horizon_is_holographic` with `holographic_entropy_eq`) — the LQG black-hole
    entropy count with the Barbero–Immirzi parameter fixed by the j = ½ `log 2` quantum, and background
    independence by construction (synthesized spacetime). **Open:** the full area-operator spectrum
    `√(j(j+1))` for general `j` and the LQG Hamiltonian/spin-foam dynamics
    (`lqg_correspondence_in_progress`). See `LQG_QLF.md`. -/
theorem lqg_correspondence_in_progress : True := trivial

end QLF.LQG
