Folio VIII — The Price at the Horizon

CREDIT LINE · POSTED AFTER THE NOVELTY AUDIT (JUNE 2026) The central mechanism of this folio — the DSW rate from the fluctuation-dissipation theorem with Ohmic friction at the Unruh/KMS temperature — was published before this house wrote it: Wilson-Gerow, Dugad & Chen, "Decoherence by warm horizons," arXiv:2405.00804 (PRD 110, 045002, 2024), with DSW's own local description in arXiv:2407.02567. What remains this programme's own here: the measured flat-space zero (Folio VII), the unified superohmic/Ohmic price list below, and the BMV-is-exactly-free corollary as a stated law. The Prior Art Ledger →

I · The Question

What exactly does a horizon charge?

Folio VII amended Postulate 4: no withdrawal is free whose record crosses a horizon — and promised the price list as the next gate. Independently, Danielson, Satishchandran and Wald proved (2022–2023, by continuum gravitational methods, no ledger anywhere) that a quantum superposition held static near any Killing horizon — black hole, Rindler, cosmological — decoheres at a constant rate, as soft quanta carry which-branch records through the horizon forever. If the amended fee is right, the DSW law should fall out of the Ledger's own measured machinery. This folio shows that it does — in mechanism and in scaling, with one honest gap left where the O(1) coefficient lives.

II · The Mechanism

The fee is the DC noise of the record.

Dephasing is zero-frequency spectroscopy. STANDARD

A superposition whose branches couple differently to the field — branch-distinguishing operator ΔV̂ — undergoes pure dephasing at the rate set by the environment's noise at zero frequency:

Γ = S_ΔV(ω → 0) / 2ℏ² ·· the fee is the DC noise of the record

Open-quantum-systems boilerplate. Everything that follows is about what S(0) is, where.

Flat space is superohmic: the DC noise is zero. EXPLAINS FOLIO VII

For the radiation field in Minkowski vacuum, the noise spectrum of any local source coupling vanishes cubically at low frequency — S(ω) ∝ ω³, the superohmic signature of a relativistic field in d = 3. Therefore S(0) = 0, therefore slow transactions are exactly free. This is no longer just a lattice measurement: Folio VII's six-decimal zero is the superohmic spectrum read at DC. The measurement and the spectrum confirm each other — and the 1D infrared catastrophe slots in too: in one dimension the same spectrum is subohmic (S(ω) ∝ 1/ω-class), the DC noise diverges, and the toy's D ∝ ε²t²/mN was that divergence arriving on schedule.

A horizon is an absorber: it makes the bath Ohmic. THE KEY STEP

A horizon is a one-way surface — the perfect absorber, the membrane with surface resistance that the membrane paradigm has priced since the 1980s. For a source held static in the wedge, the part of its field crossing the horizon never returns: the effective environment acquires a nonzero DC conductivity γ_hor — the absorptive, Ohmic response flat space lacks. And the state on the wedge is KMS at the modular temperature — Bisognano–Wichmann for the wedge, the same modular structure whose diamond version this house measured at sixty digits (T·L = 2/π, Folio IV). The fluctuation-dissipation theorem then fixes the DC noise that step 1 needs:

S_ΔV(0) = 2 k_BT_hor · γ_hor |Δp|² ·· KMS + absorption ⇒ Ohmic DC noise

Assemble, and read the label. THE AMENDED FEE

Insert step 3 into step 1:

Γ = (k_BT_hor/ℏ) · (γ_hor|Δp|²/ℏ)

This is the amended Postulate 4, with its two factors finally identified: the temperature of the horizon (the measured schedule) times the record current — the rate at which the branch-difference couples through the horizon's absorption. For a Rindler horizon at acceleration a, the temperature is Unruh's, k_BT = ℏa/2πc, and the absorptive coupling of a branch dipole Δp = qd at the horizon's natural scale c²/a carries γ|Δp|² ∝ (q²d²/4πε₀)·(a/c²)²·(1/c). The fee assembles to:

Γ_Rindler ∝ q² d² a³ / (4πε₀ ℏ c⁵)

— which is precisely the DSW decoherence law: linear in held time, cubic in acceleration, quadratic in the superposition's separation. Derived not from gravitational S-matrix methods but from a temperature this programme measured and an absorption the horizon's one-way causality forces.

III · The Price List

One mechanism, every setting.

Where the fee bills · spectrum at DC decides everything
Setting	Soft spectrum at DC	The bill
Minkowski, slow transaction	Superohmic, S(0) = 0	Free. Measured: 0.000000 (Folio VII)
Minkowski, quench	Finite-ω noise sampled by the switch	Bremsstrahlung fee. Measured: 0.00425 (Folio VII)
1D / soft-dominated vacua	Subohmic, S(0) divergent	Catastrophic. Measured: D ∝ ε²t²/mN (Folio IV notes)
Lattice vacuum, uniform motion	Lorentz-violating Ohmic channel (Cherenkov)	Motion itself bills. Measured: smooth in v, no threshold (Folio VII addendum)
Any Killing horizon	Ohmic: γ_hor > 0 by one-way causality, KMS at T_hor	Γ ∝ q²d²a³/ℏc⁵ — the DSW law, structure derived here
de Sitter (the universe itself)	Cosmological horizon, T = ℏH/2πk_B	Everything pays a Hubble-rate fee, forever. The arrow of time has a floor — and the records cross into the one account nobody can draw on (Folio VI)

The last row closes a loop the house did not plan: the cosmological horizon makes every superposition in the universe pay a small permanent fee, and its records cross into undrainable territory — the locked reserve. The fee (P4), the reserve (Folio VI), the arrow of time, and Λ meet at the same surface. One mechanism, audited at every scale the programme can reach.

IV · Audit

What is derived, what is owed.

Derived and measured: the mechanism (fee = DC noise of the record); the flat-space zero (superohmic — measured at six decimals); the subohmic catastrophe (measured); the kinematic split static/accelerated (measured); the KMS temperature that prices the horizon (measured as the diamond schedule, Folio IV).
Derived in structure: the DSW scaling Γ ∝ q²d²a³/ℏc⁵ from temperature × absorption. The Ledger and DSW agree on mechanism shape, time-linearity, and all three exponents.
Owed — the O(1): γ_hor was taken at its dimensional value at the horizon scale c²/a. The honest coefficient needs the actual low-frequency absorption cross-section of the Rindler horizon for a dipole's near field (membrane-paradigm calculation), compared digit-for-digit against DSW's published coefficient. This is a real calculation, weeks not hours, and it is the difference between "structure derived" and "derived."
Owed — gravity proper: the argument above is electromagnetic. The gravitational case replaces the dipole with the superposition's mass moment (monopole and dipole are gauged away; the leading record is higher-moment), changing exponents in known ways. DSW computed it; the Ledger's version must too.
Owed — the toy cannot reach this. An eternal horizon does not fit in a periodic box; the lattice gave the flat-space zero and the kinematic split, and that is as far as it goes. The horizon coefficient is continuum work.

The derivation chain, one line for the index:

fee = S(0)/2ℏ² → flat space superohmic ⇒ 0 (measured) → horizon = Ohmic absorber + KMS at measured T ⇒ Γ = (k_BT/ℏ)·γ|Δp|²/ℏ ∝ q²d²a³/ℏc⁵ = DSW the amended postulate 4, cashed at the horizon · O(1) coefficient owed