The first law of entanglement δS = δ⟨K_A⟩ holds in an interacting transverse-field Ising ground state: the ratio reaches 0.9945 at ε=10⁻³ and relative entropy is positive and quadratic (log-log slope 2.05). This is the identity from which linearized Einstein equations emerge in holographic derivations (Faulkner et al. 2013), verified exactly in a spin chain.
The hinge that turns entanglement bookkeeping into gravity's linearized field equations is an exact quantum-information identity, and it holds in a real interacting many-body ground state, not just free fields.
Falsify-box — how to kill this claim
Verification record — every quoted number, re-run 2026-07-26
| quantity | measured | verdict |
|---|---|---|
| δS/δ⟨K⟩ at ε=1e-3 | 0.9945 (approaches 1 monotonically across 5 ε values) | MATCH |
| relative-entropy log-log slope | 2.050 (quadratic, as the first law demands) | MATCH |
| exact identity δK − δS = S(ρ′‖ρ) | residual ≤ 1.1e-15 at every ε | MATCH |
| discriminator audit | NO DISCRIMINATOR — reproduces GR/QFT by construction | MATCH |
Provenance
script status: PRESENT
datasets: information-seam-scripts (downloadable from /data — run it yourself)
re-run: 2026-07-26
Status, honestly
verified by adversarial re-run (workflow, 2026-07-26); admitted from MCP intake INTAKE-7d61c3a532 | DISCRIMINATOR AUDIT 2026-07-26 (emergent_discriminator.py): this claim belongs to the thermodynamic-derivation wing, which reproduces general relativity or quantum field theory BY CONSTRUCTION and therefore carries NO observational discriminator — a computed and an uncomputed universe answer identically. Same category as USR-2026-0037/0038: verified machinery, not evidence about nature. Recorded on the claim rather than left implicit.. Status here is computed from evidence — the author cannot set it, and neither can we. Independent reproduction would move it; nothing else will.
The founder's own run of this script, captured verbatim. A match proves the result is reproducible; it is still R0 on this registry's independence rings — same code, so it cannot move a status. Only an outside run does that.
========================================================================
TEST 3: FIRST LAW OF ENTANGLEMENT dS = d<K_A> [REPRODUCED result]
========================================================================
numpy 1.26.4 | scipy 1.17.1 | seed 42 (unused: exact linear algebra)
Model: TFIM, N=10, open boundary, h=1.05, A = first 4 spins
Eigenvalue clip threshold: 1e-14
Unperturbed ground energy E0 = -12.754966705808
Unperturbed spectral gap = 0.372618717476 (nondegenerate ground state)
S(rho_A) unperturbed = 0.333616955856
Smallest KEPT eigenvalue of rho_A = 1.222637e-14 (clipped below 1e-14: 5 of 16)
eps | dS | dK | dS/dK | S(rho_e||rho_0) | dK-dS-relent
--------------------------------------------------------------------------------------------------
1e-01 | +0.0878107952 | +0.1412285175 | 0.6217639101 | 5.3417722224e-02 | -5.00e-16
3e-02 | +0.0237195729 | +0.0277336031 | 0.8552647446 | 4.0140301213e-03 | -7.36e-16
1e-02 | +0.0076565968 | +0.0080804695 | 0.9475435546 | 4.2387270840e-04 | -5.13e-16
3e-03 | +0.0022710731 | +0.0023085536 | 0.9837645184 | 3.7480479110e-05 | -4.28e-16
1e-03 | +0.0007545700 | +0.0007587136 | 0.9945386605 | 4.1435926579e-06 | -7.83e-16
HEADLINES
[H1] dS/dK at smallest eps (eps=1e-03) = 0.9945386605 (first law predicts -> 1)
[H2] fitted log-log slope of S(rho_e||rho_0) vs eps = 2.050318 (expect ~2, quadratic)
[H3] positivity: all relative entropies >= 0 = True
[H4] |1 - dS/dK| at smallest eps = 5.461e-03
INTERPRETATION (honesty rules apply)
Exact reproduction of the first law of entanglement in a real
many-body ground state: to first order in the perturbation,
dS = d<K_A>, with the quadratic remainder equal to the (always
non-negative) relative entropy. This is the identity the
entanglement->gravity derivations (Faulkner et al. 2013) rest on.
Machinery check only. It does NOT show that spacetime or gravity
are emergent in nature.
This script needs data files, a heavy dependency, or more time than a browser tab should take. Download it from /data and run it locally.