Force is the gradient of the shared account.
Folio II derived the field equations from bookkeeping. This folio descends to the oldest fact in physics — two bodies attract — and asks the ledger to produce it from scratch. The claim: when two masses sit in the vacuum, the interaction energy between them and the depletion of the vacuum's correlation account between them are the same function of distance. The force is not transmitted; it is accounted. Bodies fall along the gradient of what their presence costs the vacuum.
If the interaction energy Eint(r) and the connected account depletion δI(r) carry manifestly different r-dependences — one flat where the other falls — the budget-gradient reading of force fails, and this folio cannot ship as physics.
Two masses on the books, both columns read.
A static body, in the simplest arena that stays exactly solvable: a pinned site — a local mass term M² added to the vacuum's coupling matrix at one lattice point. Two pins, separation r. Then read both columns of the ledger:
The energy column.
Ground-state energy with both pins, minus twice the single-pin energy, plus the vacuum: Eint(r) = E₂(r) − 2E₁ + E₀. The connected part — the energy that exists only because both bodies do. Its gradient is the force.
The account column.
The mutual-information account between the two bodies' sites, with the same connected subtraction: δI(r) = I₂(r) − Ia(r) − Ib(r) + Ivac(r). The correlation that exists only because both bodies do. This is the ledger's entry for the pair.
The first lesson of the run was that the comparison is meaningless without this symmetry: energy is always quoted connected, and early attempts comparing it to the raw account gave drifting ratios. Connected against connected, the books can be compared line by line.
Same decay constant, measured twice.
In a gapped vacuum (mass m), textbook field theory makes a sharp prediction for the energy column: Yukawa, Eint ~ e−2mr. The Ledger adds its own: the account column must decay with the same constant. Chain of 400 sites, m = 0.1 (so 2m = 0.200), weak pins:
| r | E_int(r) | δI(r) | ratio |
|---|---|---|---|
| 8 | −1.122e−03 | +6.036e−03 | −0.186 |
| 12 | −4.137e−04 | +2.077e−03 | −0.199 |
| 16 | −1.615e−04 | +7.585e−04 | −0.213 |
| 20 | −6.506e−05 | +2.875e−04 | −0.226 |
| 24 | −2.674e−05 | +1.118e−04 | −0.239 |
| 28 | −1.115e−05 | +4.428e−05 | −0.252 |
While each column falls through two orders of magnitude, their ratio moves by a third of itself — slowly varying prefactors, one exponential law. To leading order, the Yukawa interaction and the account depletion are the same object. Note also the signs: the energy is negative (attraction) precisely where the connected account is in surplus. Bodies are pulled toward the configuration where their joint books exceed the sum of their separate ones — gravity as the preference for shared accounting.
The power-law test.
In three spatial dimensions with a near-massless field, weak quadratic pins interact through the van-der-Waals sector of the vacuum: perturbation theory predicts both columns fall as a power of r, and the Ledger again demands the same power for both. Lattice 18³ = 5,832 sites, m = 0.02, weak pins, separations r = 3…8.
| r | E_int(r) | δI(r) | E above floor | δI above floor | ratio |
|---|---|---|---|---|---|
| 3 | −8.368e−05 | +1.321e−04 | −1.486e−05 | +6.151e−05 | −0.242 |
| 4 | −7.709e−05 | +1.018e−04 | −8.267e−06 | +3.125e−05 | −0.265 |
| 5 | −7.402e−05 | +8.878e−05 | −5.200e−06 | +1.819e−05 | −0.286 |
| 6 | −7.234e−05 | +8.218e−05 | −3.515e−06 | +1.159e−05 | −0.303 |
| 7 | −7.137e−05 | +7.863e−05 | −2.547e−06 | +8.043e−06 | −0.317 |
| 8 | −7.086e−05 | +7.685e−05 | −2.038e−06 | +6.256e−06 | −0.326 |
The raw columns barely fall with distance — and the fit explains why: the interaction has two components. Fitting each column as floor + power, y(r) = y∞ + c·r−p:
Eint = −6.88e−5 − 1.43e−4 · r−2.06 ·· δI = +7.06e−5 + 8.41e−4 · r−2.38 floor ratio E/δI = −0.975 ≈ −1 · power exponents 2.06 vs 2.38 · above-floor ratio ≈ −0.25…−0.33, matching the 1D armAbove the floor, the genuinely Newtonian sector: both columns fall as a power law, exponents 2.06 and 2.38 — the same law within the reach of an 18-site box (separations r ≤ 8 with periodic images pressing from 10 sites away), with the exchange ratio drifting slowly through the same band (−0.24 → −0.33) as the massive 1D arm's (−0.19 → −0.25). Two dimensions, two regimes, one recurring exchange rate of order −¼. Whether that number is the ¼ of Area/4Gℏ wearing lattice clothes is exactly the kind of question the house does not answer on excitement alone — it is filed in the audit.
At criticality, the simple books fail — and the house says so.
The near-critical chain (m = 10⁻³, correlation length far beyond the system) refuses the proportionality: over the same range, Eint falls like r−0.75 while the connected account falls like r−0.38. The ratio is not constant; the gate condition is not met there in its leading-order form.
The proximate cause is identifiable: at near-zero mass the chain's zero mode — one collective degree of freedom shared by every site — floods the single-site accounts with slowly-decaying correlation that has no counterpart in the energy column. The honest options, both named: either the budget-gradient law holds only against the right account (the modular structure proper, not raw mutual information — connecting this folio to the modular machinery of the archive era), or the law genuinely fails at criticality and the Ledger's force is a gapped-phase phenomenon. Distinguishing these is a named debt in the General Ledger. This is what an anomaly looks like when it is filed instead of hidden.
Run both columns yourself.
Sources: n4_newton.py · n4_3d.py · data: n4_connected.json · n4_3d.json
- Scope: quadratic (mass-type) pins in a free scalar vacuum. This probes the van-der-Waals / Casimir sector, not the linear-source Coulomb sector; classical (coherent) sources displace the field without touching any entanglement account, and how the Ledger reads them is an open conceptual debt, filed.
- The critical anomaly (§V) is unresolved. It constrains the theory: either the account must be modular, or the law is gapped-phase only.
- Ratio drift: the E/δI ratio varies ~35% across the Yukawa range — consistent with subleading prefactors, but the asymptotic constancy is extrapolated, not proven.
- 3D caveats: the box is small (18³, r ≤ 8, images at 18−r); the above-floor exponents agree to ~15% (2.06 vs 2.38), not exactly; and whether the recurring ≈ −¼ exchange rate relates to the ¼ of Area/4Gℏ is an open question, recorded — not a claim.
- Prior art (posted after the novelty audit): the two halves of this comparison exist separately in print — the connected two-excitation Yukawa potential measured in a MERA network by Sahay–Lukin–Cotler (arXiv:2401.13595, PRX 2025), and the matching analytic mutual-information decay by Casini–Huerta (arXiv:0905.2562). The side-by-side measurement with equal decay constants was not found anywhere; that comparison is this folio's contribution. Prior Art Ledger →
- What this folio establishes: in the gapped vacuum, the force law and the account law decay with the same constant, measured independently, two orders of magnitude deep. The force between bodies tracks what their coexistence does to the vacuum's books — Newton's law read as a bank statement.