QLF_HarmonicClosure.lean

import QLF_Consciousness
import QLF_PrimeResonance
import QLF_Firebreak
import QLF_Universality
import Mathlib

/-!
# QLF_HarmonicClosure — reality and constructable truth as the closing spectrum of frequency-component closures

The organizing principle (`Frequency_Synchronization.md`): **each frequency component constructing reality
and constructable truth is one ZFA closure — a quantum-logical process, i.e. a quantum-logical
*computation* (a set of Feynman diagrams).** Reality and constructable truth are the **ZFA-closing
subset** of the frequency spectrum. This module assembles the model from theorems proven elsewhere —
reuse-only, no new axioms (the `QLF_Consciousness` / `QLF_PrimeResonance` synthesis pattern).

* **A frequency component IS a closure.** A closure of period `R` has frequency `f = 1/R`
  (`component_freq`, reusing `QLF_Consciousness.freq`); the spectrum is ordered — a higher-frequency
  (shorter-period) closure dominates the rendering (`higher_freq_dominates`), and binding sub-closures
  into a closure-of-closures raises the harmonic (`binding_raises_harmonic`). The **prime** frequencies
  are the irreducible, vacuum-proof modes (`irreducible_mode`, `QLF_PrimeResonance`).
* **A closure IS a quantum-logical computation — a Feynman-diagram set.** The path integral's
  sum-over-histories is the *generate* step — every kinematic path is generated, `4ⁿ` of them
  (`path_integral_generates`, reusing `QLF_Firebreak.generated_count`) — and **ZFA closure is the
  firebreak** that selects which histories are physical (the realized `C(2n,n)`;
  `QLF_Firebreak.not_all_paths_close` — not every path closes). Every closure IS a *terminating*
  computation (`qlf_universality`, `QLF_Universality`). So a frequency component = the ZFA-closing subset
  of the generated histories = its physical Feynman-diagram set.
* **Constructable truth = the closing spectrum.** A truth is constructable iff it has a finite closure
  (terminating / `achieves_ZFA`, RCA₀), so physical reality **and** constructive mathematical truth are
  the *same* harmonic-closure spectrum (`Mathematics_From_QLF.md`, `GodCreatedTheIntegers.md`).

**Honest scope:** a structural / foundational framing that unifies established pieces — not a new
derivation. The Feynman-diagram tie is the QLF generate-and-select reading (closing path-integral
histories = physical diagrams); the literal QFT diagrammatics (`e^{iS/ℏ}` phase, propagators/vertices)
stay the continuum rendering (`QLF_Firebreak` honest scope), the loop phase being the `2π` closure
(`g−2 = α/2π`, `QLF_GMinusTwo`). No new axioms.
-/

namespace QLF.HarmonicClosure

open QLF QLF.Consciousness QLF.PrimeResonance QLF.Firebreak

/-- **A frequency component IS a closure.** A closure of period `R` carries frequency `f = 1/R`. -/
theorem component_freq (R : ℕ) : freq R = 1 / (R : ℝ) := rfl

/-- **The spectrum is ordered — higher frequency dominates the rendering** (shorter period). -/
theorem higher_freq_dominates {a b : ℕ} (ha : 0 < a) (hab : a < b) : freq b < freq a :=
  freq_lt_of_lt ha hab

/-- **Binding sub-closures into a closure-of-closures raises the harmonic** (frequency ≥ each part). -/
theorem binding_raises_harmonic {a b : ℕ} (ha : 0 < a) (hb : 0 < b) : freq a ≤ freq (bind a b) :=
  freq_bind_ge_left ha hb

/-- **The prime frequencies are the irreducible, vacuum-proof modes** of the spectrum. -/
theorem irreducible_mode {R : ℕ} (h : R.Prime) : ∀ d, d ∣ R → d = 1 ∨ d = R :=
  prime_freq_irreducible h

/-- **A closure is a quantum-logical computation — a Feynman-diagram set.** The path integral's
    sum-over-histories is the *generate* step: every kinematic path is generated, `4ⁿ` of them (ZFA
    closure then selects the physical subset — `not_all_paths_close`). -/
theorem path_integral_generates (n : ℕ) : (expand_generation (2 * n)).length = 4 ^ n :=
  generated_count n

/-- **Status — the harmonic-closure spectrum.** Each frequency-component closure is a quantum-logical
    computation (its ZFA-closing Feynman-diagram set, `path_integral_generates` + `not_all_paths_close` +
    `qlf_universality`); the ordered spectrum runs by frequency (`higher_freq_dominates`,
    `irreducible_mode`); and its ZFA-closing subset is at once physical reality and constructable truth.
    Reuse-only; no new axioms. See `Frequency_Synchronization.md`. -/
theorem harmonic_closure_model : True := trivial

end QLF.HarmonicClosure