Folio IV — The Fee, Computed

CREDIT LINE · POSTED AFTER THE SECOND NOVELTY AUDIT (JUNE 2026) This folio's measurement is a reproduction of settled literature, and its method carries a documented bias. The lattice extraction of the diamond temperature was published 2016–2019: Arias–Blanco–Casini–Huerta (arXiv:1611.08517) obtained 0.7826 vs π/4 = 0.7853 for the midpoint local term; Eisler–Peschel (1703.08126, 1805.00078) ran the identical sine-kernel computation and proved the nearest-neighbour amplitude is not the parabola but πx(1−x)·₃F₂(…) — an ~8% central enhancement, which explains why this folio's NN-parabola fit reads T·L ≈ 0.598, six percent shy of 2/π, and converges slowly. Eisler–Tonni–Peschel (1902.04474) proved the full lattice EH (long-range hoppings summed) converges exactly to the conformal parabola. The "broken bosonic thermometer" is also prior art with a sharper diagnosis: the zero mode is a regularizable divergence, not a persistent failure — Di Giulio–Tonni (1911.07188) recover the full CFT parabola from the harmonic chain with ωL = 10⁻⁵⁰⁰ at 6500-digit precision; our regulator (m = 10⁻³) was nowhere near small enough, so "failure" was a misdiagnosis of an under-regularized run. Even the arbitrary-precision necessity is standard published practice (40–6500 digits across that literature). The gate conclusion — the 1/L schedule — stands, as a reproduction, not a discovery. The Prior Art Ledger →

I · The Claim

Landauer's principle, billed at the horizon.

Postulate 4 asserts a fee: Γ ≥ (k_BT_L/ℏ)·Ṡ_ent, with T_L the temperature a causal region charges by virtue of its size alone. The fee has two factors, and they have different standing. The rate per nat is an argument. The temperature schedule is — as of this folio — a measurement.

The rate per nat. ARGUED

Every withdrawal (Folio III) writes an irreversible record: entanglement moved from the vacuum's books to matter's. Landauer's principle prices irreversible registration: one bit written at temperature T dissipates at least k_BT ln 2 — one nat, k_BT. The environment doing the registering here is the causal diamond itself, at its modular temperature T_L; the dissipation appears to the system as dephasing, at the rate the energy scale sets:

Γ ≥ (k_BT_L/ℏ) · Ṡ_ent ·· O(1) = 1 per nat, ln 2 per bit

This is dimensional analysis wearing Landauer's coat — the house said so on Folio I and repeats it here. What it is not anymore is unanchored: the temperature in it is now measured.

The temperature schedule. CHM THEOREM → MEASURED

Hislop–Longo and Casini–Huerta–Myers: the vacuum restricted to a causal diamond of size L is thermal with respect to its own modular flow, with a local inverse temperature profile β(x) = 2πf(x) peaking at the centre — so the centre temperature obeys

T(L) = (2/π) · ℏc/(k_BL) ·· the fee schedule: 1/L, constant 2/π ≈ 0.6366 at centre

A theorem of continuum QFT. The question this folio answers: does a vacuum actually assembled out of microscopic parts — a lattice, knowing nothing of Lorentz boosts — bill at this schedule? That is precisely the situation of the Ledger, where spacetime is emergent and the diamond is a derived object.

If the temperature extracted from the vacuum's own entanglement Hamiltonians does not fall as 1/L (power within 15% of −1), the fee schedule of Postulate 4 has the wrong distance law and the invoice of Folio I is mispriced.

II · The Lab Notebook

Two thermometers broke before one read true.

The house files its failures with the same ink as its results. Both of these are instructive.

Broken thermometer Nº 1 — the harmonic chain The first attempt read the entanglement Hamiltonian of the bosonic chain — and got nonsense: T ∝ L^−0.31, gate tripped at face value. Calibration on the one case where the answer is a theorem with no fitting (the Bisognano–Wichmann half-line, β(x) = 2πx exactly) showed the method, not the schedule, at fault: the lattice boson's EH refuses the local BW form, slopes off by factors of 5–20, worsening as m → 0. The culprit is the bosonic zero mode — one collective degree of freedom shared by every site — the same character that caused Folio V's critical anomaly and the 3D floor. One villain, three appearances, now positively identified. The zero mode's obstruction of local bookkeeping is hereby promoted from nuisance to named phenomenon in this programme.

Broken thermometer Nº 2 — double precision The second attempt used free fermions (no zero mode; sine-kernel vacuum, exact) in ordinary floating point. The EH is h = ln((1−C)/C): when C's eigenvalues reach machine epsilon, the logarithm silently saturates near 32 and fabricates a temperature plateau — a smooth, plausible, wrong law. It was caught because the one row still inside precision (L = 21: T·L = 0.606, profile correlation 0.9999) disagreed with the fabricated trend. Double-precision arithmetic lying plausibly is a known house hazard; this archive has caught it before. The cure: arbitrary precision, 30–60 digits.

III · The Measurement

The vacuum bills at 1/L. Read at sixty digits.

The working instrument: an interval of L sites in the exact sine-kernel vacuum of an infinite fermion chain (half filling, v_F = 1); entanglement Hamiltonian h = ln((1−C)/C) computed with mpmath at 30–60 significant digits; the centre temperature read from the nearest-neighbour EH amplitude profile fitted against the CHM parabola f(x):

The fee schedule, measured · sine-kernel vacuum · arbitrary precision
L	digits	β centre	T·L	profile corr
11	30	18.497	0.5947	0.9997
15	35	25.182	0.5957	0.9996
21	40	35.199	0.5966	0.9995
27	50	45.207	0.5973	0.9995
35	60	58.545	0.5978	0.9994

T(L) = 0.5883 · L^−0.995 the distance law: 1/L to half a percent · constant T·L = 0.5947 → 0.5978, rising monotonically toward 2/π = 0.6366 · gate passed

Three readings, in order of importance. The law is 1/L — exponent −0.995, well inside the gate, with no tuning. The constant is CHM's — T·L sits 6% below 2/π and climbs monotonically toward it as L grows, the signature of a lattice converging to its continuum. And the profile is the diamond's — the parabolic β(x) at correlation 0.999+, meaning the vacuum doesn't just have a temperature, it has the right temperature gradient: hottest books at the centre of the diamond, frozen at the edges.

For the laboratory (Folio VIII's territory): at the diamond centre the schedule reads T(L) = (2/π)·ℏc/(k_BL) — about 14.6 mK for a 10 cm causal region, four times the often-quoted wedge value ℏc/(2πk_BL) ≈ 3.6 mK. Where in the diamond the experiment sits is a factor-4 line item the error budget must carry.

IV · The Invoice, Priced

The fee with its numbers in.

Combining the argued rate (per nat) with the measured schedule (centre constant):

Γ ≥ (2/π) · (c/L) · Ṡ_ent dephasing rate ≥ 0.6366 · (c/L) · entanglement rate in nats — schedule measured, rate argued

This sharpens Folio I's invoice: the fee scales as 1/L with the causal size of the apparatus, the constant is 2/π at the diamond's centre, and the only unmeasured factor left is the Landauer rate itself — the claim that one nat of withdrawal costs exactly one quantum of dephasing action. That factor is this programme's next gate.

Cross-check owed: the DSW channel Danielson–Satishchandran–Wald established by independent (continuum, gravitational) methods that a horizon decoheres a maintained superposition at a constant rate — soft gravitons through the horizon, N_soft ∝ t. The fee predicts exactly that shape: a maintained transaction (constant Ṡ_ent) billed at a fixed T_L gives Γ constant in time. The shapes agree; the quantitative match — does DSW's rate equal (2/π)(c/L)·Ṡ_ent for their setup? — is a named debt, and the sharpest theory-to-theory test this folio leaves open.

V · Reproduce It

Sixty digits, sixty seconds.

# the thermometer that worked (mpmath required): curl -O https://emergent-gravity.com/n3_fermion.py python3 n3_fermion.py # the broken bosonic thermometer, kept for the record: curl -O https://emergent-gravity.com/n3_fee.py python3 n3_fee.py

Sources: n3_fermion.py · n3_fee.py (filed failure) · data: n3_fermion_hp.json

The rate per nat: three arenas now mapped (scripts n6_landauer.py, n6_bmv.py). Arena 1, in-band probes: noisy channel, no entanglement transacted — the Kafri–Taylor–Milburn regime exhibited in our own machinery. Arena 2, naive below-band probes: dressing catastrophe. Arena 3, the real one — the BMV experiment executed exactly in the lattice (branch-source qubits, coherent field response, 4×4 density matrix in closed form): mediated entanglement is real and never free — E_N up to 0.52 with the fee growing alongside; the fee-per-nat falls with separation (F = 3.99 → 1.75 for d = 10 → 20) but is transient-dominated under sudden switching (ε-dependent, exponent −0.2…−0.4, not universal); and the near-massless arm is an infrared catastrophe — the soft sector charges D ∝ ε²t²/mN and kills entanglement entirely, the zero mode's fourth and most violent appearance. Conclusion so far: the irreducible fee lives in the soft sector; isolating it needs adiabatic switching. The gate narrows; it has not yet been passed.
The thermometer is fermionic; the theory's arena was bosonic. The fee schedule is a statement about causal structure (any CFT), and c = 1 fermions are a legitimate vacuum — but the bosonic verification is blocked by the zero mode, and that obstruction is itself unexplained territory: does a compact boson (no zero mode) restore the schedule?
The constant is 6% shy of 2/π and rising monotonically; the continuum extrapolation is convincing but not performed. L ≤ 35 (arbitrary-precision cost grows fast).
DSW quantitative match: owed (see §IV).
What this folio establishes: the fee's distance law, 1/L to half a percent, read directly off the vacuum's own entanglement Hamiltonians, with the CHM constant emerging and the full diamond temperature profile at 0.999 correlation. The vacuum keeps a fee schedule, and it is the one Postulate 4 quoted.