Folio XV — The One Metric · Monogamy Gravity

I · The Question

A weighted world carries two distances. Which one is real?

Annex B's expedition built matter from graph topology and found the dictionary mass = Φ² × loop resistance. This house, all along, has read distance from correlation decay. Are these the same geometry? The question is sharper than it looks, because a chain with varying spring constants c carries two natural metrics that coincide only in the homogeneous case: the resistance distance d_R = Σ 1/c (the circuits' metric — the one that prices mass) and the time-of-flight distance d_T = Σ 1/√c (the waves' metric — the causal one, since the local signal speed is √c). Inside any lens of weakened springs they diverge at first order. So: when the vacuum's mutual information says "these two sites are far apart," which integral is it computing?

Pre-registered before any inhomogeneous run: WKB intuition says vacuum correlations propagate at the local wave speed, so MI collapses on d_T (time-of-flight), not d_R (resistance). Kill-gate: if neither metric collapses the lens data onto the homogeneous calibration curve, the dictionary thesis dies.

II · The Measurement

The house was wrong, eight lines out of eight.

Protocol: chain of 1200 sites; homogeneous calibration curve F(d) out to d = 560; a smooth Gaussian lens (adiabatic taper — reflections negligible, endpoints in the c = 1 region, so pure path-length physics survives); for each straddling pair, read the inferred distance d_inf = F⁻¹(MI) and compare. (A first run saturated its own calibration range and was discarded — the instrument failure is filed in the script header, per the Folio IV tradition.)

The lens arm · d_inf = what the entanglement says · pre-registered winner was d_T
lens depth	d_eucl	d_R (resistance)	d_T (flight)	d_inf (measured)	verdict
0.25	96	196	134	193	RESISTANCE
0.25	128	229	166	214	RESISTANCE
0.25	192	293	230	274	RESISTANCE
0.25	256	357	294	337	RESISTANCE
0.0625	96	389	174	375	RESISTANCE
0.0625	128	422	206	378	RESISTANCE
0.0625	192	486	270	428	RESISTANCE
0.0625	256	550	334	489	RESISTANCE

Read the highlighted row: through a deep lens, two sites 96 apart in label-space are 174 away by flight time and 389 by resistance — and the entanglement says 375. The disorder arm (random log-uniform weights, six independent pairs, the two calibration-saturated rows excluded by the rule learned in v1) agrees: rms relative error 0.092 against resistance, 0.163 against time-of-flight, 0.25 against the naive label distance. The deviations that remain sit systematically between the two metrics, a few percent below pure resistance — consistent with the conformal-weight correction the resistance-coordinate map leaves behind (in resistance coordinates the static operator is ∂²/c: flat geometry, position-dependent measure), and named as the folio's precision debt.

The vacuum's entanglement measures distance in ohms.
Mass = Φ² × loop resistance (Annex B). Distance = F(path resistance) (this folio).
Both sides of the field equation are statements about one object: the resistance of the vacuum.

III · What It Opens

The welding problem.

The refutation purchased a unification: in the toy, the static world — masses, entanglement, the Ledger's books — is governed entirely by the resistance metric. But the dynamical world — signal fronts, light cones — propagates by time-of-flight. The toy is, by measurement, bimetric: its entanglement geometry and its causal geometry are different integrals of the same weights, diverging at first order in any field.

Our universe is not. General relativity welds the two into one metric — matter's geodesics and light's cones read the same g_μν — and that welding is tested: gravitational waves and light from the same merger arrived within 10⁻¹⁵ of each other's speed. So the toy hands every emergent-gravity program a constraint nobody ordered:

The welding problem · new, named, posed Any model that builds geometry from weighted substrates (graphs, networks, tensor webs) generically produces two metrics — resistive (statics, entanglement, mass) and causal (propagation) — that differ at first order in the field. The observed universe has one. A viable emergent-gravity model must therefore contain a mechanism that welds them — a fixed point, a symmetry, or a self-consistency of back-reaction that locks Σ1/c to Σ1/√c in the IR. No such mechanism exists in the toy; finding one (or proving the weld impossible for a model class) is a sharp, exportable test that kills theories. The house notes one tantalizing scent before quarantining it: the two integrands differ by exactly the factor-2-in-the-potential pattern that separates Newtonian from GR light deflection — the welding mechanism, wherever it lives, is plausibly the same physics that fixes γ_PPN = 1. QUARANTINED

And one immediate dividend for the house's own books: the Folio V critical anomaly — force and account decaying with different exponents at criticality — was measured against Euclidean separation. Both laws should now be re-read against resistance distance; if the anomaly dissolves in ohms, the budget-gradient law of Folio V becomes exact where it seemed to fail. Named as the next re-audit.

IV · Audit

Stamps and debts.

The pre-registered prediction was refuted in public — the program's fourth open refutation (tail-curvature, the universal fee, the ¼ exchange rate, and now time-of-flight). The verdict is data-unanimous: 8/8 lens rows + disorder rms 0.092 vs 0.163.
Instrument failure filed: run v1 saturated its calibration (all inferences clamped to the curve's end) and produced meaningless verdicts; discarded, range tripled, two still-saturated disorder rows excluded. The clamp is the lesson: an inferred distance equal to the calibration maximum is a measurement of the instrument.
The collapse is resistance-dominated, not resistance-exact (residuals 2–11%, systematically toward d_T): the conformal-weight correction is unmodelled. Precision debt named.
One dimension, single-site probes, one disorder seed. The 2D/3D check (where resistance and flight metrics differ even more richly) is owed; so is the re-read of Folio V's anomaly in ohms.
Prior-art risk, declared: resistance distance is classical graph theory (Klein–Randić), and entanglement-vs-geometry is a crowded field; whether "vacuum MI collapses on resistance distance in inhomogeneous media" exists in print is unaudited. Round-3 sweep grows by one claim.
What this folio establishes: in the toy, one metric — the resistive one — underlies mass (Annex B) and entanglement distance (here), measured against a pre-registered alternative that lost; and the statics/causality split it exposes is promoted into a named, exportable constraint on all emergent-gravity models.

# the whole experiment (~1 min; v1's saturation failure documented in the header): curl -O https://emergent-gravity.com/n11_two_metrics.py python3 n11_two_metrics.py

Source: n11_two_metrics.py · data: n11_results.json