The Recognition Ledger of a Black Hole: Horizon Records, Entropy Counts and Lattice Hawking Radiation

A black hole of horizon area A carries the entropy A/(4 l_P²), radiates at the temperature ħκ_s/(2π) set by its surface gravity κ_s, and, if quantum mechanics holds, gives back the information that fell in. All three concern what a horizon records, and the recognition ledger's events are records. On the ledger's lattice G = 1/(8πs) is fixed by the action scale s and ħ = 1/a by the phase scale a, so the Bekenstein–Hawking quarter holds exactly when a horizon holds 2πas per unit area, the imaginary-time action of a full turn. The ledger's phase premise sets this to one bit per record area λ_rec, the quarter at a single face, so a count that assumes it returns the quarter by construction. The lattice's order of postings holds no black hole. In the order of a geometry's interval a trapped region needs a strong field, and postings that have turned spacelike leave it. The configurations the path sum integrates over keep the time diagonal timelike and hold none, so a black hole needs a larger configuration space and an extended action, or the ledger has none. A horizon hides the postings that cross it, each with its side, debit or credit; a ledger with one state per record hides a single bit behind an observer horizon, whatever its size. One bit per crossing gives a coefficient that depends on the horizon's orientation, 0.108, 0.325 or 0.1875 at the identified spacing, none of them the quarter. At the identified spacing a quarter that is the same on every horizon can therefore only come from a count that correlates the records, as the ledger's balance conditions do. The lattice scalar field disperses like a superluminal medium at short wavelengths. On a test horizon, and on the exact geometry of a lattice horizon with a test interior (not a lattice solution), it radiates a Planck spectrum that tends to the Hawking temperature as the spacing shrinks, its late quanta born inside the horizon (numerical). Where the information goes depends on a unitary evolution in record time, which the identified Regge path sum lacks, since a growing mode leaves no positive norm preserved; the ledger's measure weight gives one only in the continuum limit. A lattice black hole, the quarter as a count of the ledger's own records, the radiation after a collapse and the fate of the information are open, and each is stated as a precise target. Where a proof is complete but its independent verification is not, the text says so.