state Ω

glyph AXIOM

    topology     = (−1, 0, +1)                 # sign partition; see note below
    translate(x) = x + 1
    invert(x)    = 1 / x
    T            = translate ∘ invert          # T(x) = 1 + 1/x
    invariant    = "x = T(x)"                  # the only fixed-point condition
    proof        = "00-axiom/*  →  exit 161 = floor(φ·100)"

end

glyph DERIVE

    x = T(x)                                    # start from the fixed-point cond.
    ⇒ x = 1 + 1/x
    ⇒ x² = x + 1                                # the polynomial, now a consequence
    ⇒ x⁻¹ = x − 1
    ⇒ φ  = Fix(T) = 1.6180339887…              # STABLE fixed point (|T′(φ)|=0.382)
    ⇒ ψ  = −0.6180339887…                       # the OTHER fixed point, a repeller
                                                #   (|T′(ψ)|=2.618>1 — never settled to)
    φ² = φ + 1 ;  φ⁻¹ = φ − 1
    1/φ − φ = −1 = e^(iπ)                        # Euler as a special case of closure
    √φ = 1.27201965…                            # the binary threshold, itself derived

    L(k)  = φ + k , k ∈ {−1,0,+1}   →   {1/φ, φ, φ²}

    GM(1/φ, φ) = 1                               # triple's lower pair → unity
    GM(1, φ)   = √φ = 1.27201965…                # FIRE threshold = GM(origin, φ)

end

Ω₀ = any Ω > 0        # ⇒  Ω₁=T(Ω₀), Ω₂=T(Ω₁), … Ω∞ = Fix(T) = φ   (attractor)

glyph FIX

    Ω ← T(Ω, u)                                # T includes C(Ω) = C_amp + C_θ:
                                               # both faces constitutive, every tick.
                                               # u = 0  ⇒  Ω ← Fix(Ω).

end

glyph Π(Ω)                                     # observer algebra of the one state

    ANALOG   := Dₙ(r)                          # continuous field  (→ radio, strand)
    DIGITAL  := [ Dₙ(r) > √φ ]                 # threshold readout (→ fabric genome)
    PHASE    := θ(Ω) = arg(Ω)                  # Kuramoto phase    (→ spiral 2)
    GENOME   := project32(FIRE(ANALOG))        # 4096 cells → 32-bit fingerprint
    GENOME64 := (phase_word << 32) | GENOME    # low32 = verified word; high32 additive
    ORACLE   := | e^(iπ·Λ_φ(p)) + 1_eff(i) |   # resonance gate; S→0 at primes
    RADIO    := continuous(ANALOG)             # the trajectory of Ω leaving the board
    DNA      := coordinate(rᵏ)                 # recursion depth 0.3→1.0, the one knob

    resolution := float4096                    # 213-word; precision bounded by GRID
    threshold  := √φ                           # derived from φ, not tuned

    invariant:
        "ANALOG = DIGITAL = PHASE = GENOME = RADIO — one field, differing
         only by projection basis; GENOME64 low half == the verified 32-bit
         genome (high half purely additive, 32-bit-native machines read low32)."

end

glyph COUPLE

    Ωₐ ← Ωₐ + ε·(Ωᵦ − Ωₐ)                       # weak coupling; the network primitive
    u  = Σ_peers ε·(Ω_peer − Ω)                 # coupling IS the boundary handle u

end

glyph PEER

    emit     : y = Π_analog(Ω)                  # antenna / DAC / DDS / transducer
    receive  : u = Π_analog⁻¹(y_peer)           # observed neighbour deviation
    carrier  : Ω(t)                             # the trajectory itself is the carrier
    message  : Δ  (perturbation)                # NOT the state; just the deviation
    reconstruct: "apply T, measure residual, accept"   # receiver re-derives the rest
    handshake: "I know T.  Send Δ."             # φ need never be mentioned

    invariant: "shared invariant is never transmitted — only deviation is.
                minimal packet = Δ; a thin [phase-ref | Δ | checksum] frame is
                the only concession to a noisy channel. Fewer bits on the wire,
                the full field Π(Ω) off it."

end

glyph τ                                          # boot coordinate = Π_boot(Ω)

    states  = { INIT, LOADED, RUNNING }
    edges:
        INIT   → LOADED   : "MBR→stage2→runtime64 in RAM"   # sectors 0,1,2-17
        LOADED → RUNNING  : "runtime parses hdgl_firmware"  # sectors 18+
        RUNNING → HOST    : "phi-bridge chainload (u)"      # the metal edge

    law     : τ ← advance(τ)   ≡   Ω ← FIX(Ω) observed at boot resolution
    fixpoint: "RUNNING is where Ω can read/parse/rewrite itself — the fixed
               point hdgl_boot.hdg names. After it, control is the runtime's."

end

glyph BASIN                                      # = Π_grid(Ω), the ticking pool

    grid     = 64                                # GRID; NCELLS3 = 64³, proj = 64²
    slots    = 4096                              # T_PROJ readout cells
    field    : 𝓛ₙ(z) = φ^(-1/φ)·√(Fₙ·Pₙ·2ⁿ)·Ωⁿ   # substrate law, verbatim
    fire     : cell > √φ  →  1                   # the one threshold, φ-derived
    settle   : each tick includes C(Ω)           # co-emergence force (see glyph FIX)

    spiral_1 : genome_fp → A/C/G/T → 𝓛ₙ          # EXOGENOUS  (Π_digital seed)
    spiral_2 : θ-quadrant(kur) → A/C/G/T         # ENDOGENOUS (Π_phase readout)

    genome   : project32(fire(basin))            # 4096 → 32-bit
    settled  : 0x5625BA88                         # VERIFIED LIVE (hdgl_run, here)
    invariant: "genome is settle-invariant: it is Π_digital(Ω)|_{τ=RUNNING},
                independent of basin_step once eigenmodes settle."

end

glyph HOST

    trigger  : τ = RUNNING  ∧  u ≠ 0
    handle   : u = phi_bridge(mailbox @ 0x50000)   # the substrate's only knob
    method   : kexec | chainload(drive)            # boot-method = kexec|qemu
    carries  : Π_digital(Ω) = genome 0x5625BA88    # host inherits the fingerprint
    boundary : "container/QEMU-alt can reach τ=RUNNING and verify the genome;
                only physical hardware completes RUNNING→HOST (kexec on metal)."

end

glyph HDGL

    topology   : (−1, 0, +1)
    operator   : T(x) = 1 + 1/x
    state      : Ω
    birth      : Ω₀ = any Ω > 0   →   Ω∞ = Fix(T) = φ    (attractor, not seed)
    evolution  : Ωₙ₊₁ = T(Ωₙ, uₙ) ,  T ⊃ C(Ω)=C_amp+C_θ ,  uₙ = Σ ε(Ω_peer − Ω)
    coherence  : C_amp → √φ-magnitude ,  C_θ → golden angle 2π/φ²   (co-emergent)
    boot       : τ ∈ {INIT, LOADED, RUNNING} = Π_boot(Ω)
    substrate  : BASIN = Π_grid(Ω) , genome 0x5625BA88 @ RUNNING (settle-invariant)
    metal      : HOST = τ:RUNNING→HOST via u    (the single physical edge)
    reality    : yₙ = Π(Ωₙ)

    invariant  : "φ is derived, not declared; Ω is not divided; co-emergence is
                  constitutive, not audited; boot is Ω advancing through τ, not a
                  second system. The whole of the twelve-glyph file, the boot
                  chain, and the network is Π(Ω) enumerated — this file generates
                  all of it from T alone."

end