Le Sage II — the interacting arena

The Shadow Audit executed Le Sage in the linear vacuum: a coherence deficit exerts no pressure, and record-keeping grains repel. But Maxwell's cancellation is a linear-response theorem. This is the sequel: turn on the medium's self-interaction, and the shadow comes back — not in the flux, where Maxwell forbids it, but in the medium's response to the flux. Records attract. Measured at three lattice sizes, dissected into its components, audited against three literatures.

July 2026 · not peer-reviewed · human–AI collaboration · every number below has a public script
Part I: the Shadow Audit · n18 (python, Hartree) · n18-rust (three sizes, all modes) · n19 (exact Lindblad) · back to the program

What Part I left open

The Shadow Audit proved that in the vacuum of a free field, dephasing scatterers — objects the medium keeps records of, with zero mean absorption — cannot attract. Whatever a grain rediffuses with a scrambled phase carries exactly the momentum it removed; the only residue of monitoring is its backaction pump, whose radiation yields a weak repulsion. Maxwell's 1875 cancellation survives quantization. That verdict was pre-registered, and it stands: we re-verified it here at three lattice sizes (−1.315 / −1.309 / −1.307 ×10⁻⁵ at N = 256/512/1024).

But every step of that proof lives in linear response: waves that superpose without ever talking to each other. The real vacuum is not linear. So the question Part I could not answer: is Maxwell's cancellation protected in an interacting medium?

The mechanism proposed, pre-registered, and put on trial. In a quartic medium (Hartree: μ²_eff = μ² + 3g·δ⟨x²⟩), a recorder's unavoidable backaction heats its neighborhood — and the heating dresses the recorder into a coherent stiffening defect. Coherent defects attract (Part I's own control row; Kenneth–Klich fixes the sign). Prediction: the record-shadow force flips sign at finite g. Six gates written before the first run.

Experiment 1 — the sign flip, at three sizes measured

Same exact testbed as Part I (Lyapunov steady states, ABBA-subtracted T^xx force), plus a self-consistent Hartree loop whose dressing is provably non-negative (the noise-induced ⟨x²⟩ excess is a Lyapunov integral of a positive source — no tachyons possible; a first scheme without this property diverged, chronicle item 1). At g = 0 the machinery reproduces Part I to the last digit. Then:

Left: force between record-keeping grains versus interaction strength g at N=256 and 512 with N=1024 points; repulsion at small g flips to attraction at g around 0.3. Right: the g=1 attraction at N=256, 512, 1024 is constant to 0.03 percent. — **Records attract when the medium interacts.** Left: the pair force vs interaction g. The linear-medium repulsion (g = 0) first deepens — a finite-size artifact of global pump heating that halves when the box doubles — then flips: attraction from g ≈ 0.3, reaching +6.0×10⁻⁴ at g = 1, forty-five times the repulsion it replaced. Right: the strong-coupling attraction across three box doublings: 6.017 / 6.000 / 5.999 ×10⁻⁴. The artifact dies with N; the attraction doesn't move. Scripts: n18 (python) and n18-rust (30× faster port, validated digit-for-digit).

Gate accounting, kept honest: G0 (reproduce Part I) and G1 (sign flip) passed. G3 as registered failed — the induced force is not γ_Aγ_B at fixed dressing; it grows ~γ³·³, because the dressing itself is built by the noise (the correct bilinearity appears below, in the right variable). G2 (transparent limit) out of reach — at low absorption the testbed's own heat budget diverges and even N = 512 cannot separate signal from artifact; the transparent-limit prediction is declared untestable in this arena, not confirmed. And G4 refuted our own mechanism — see Experiment 3.

Experiment 2 — the two Newton laws measured

Left: induced force versus separation shows a plateau within 5 percent from d=24 to 64 and gentle decline at 96. Right: induced force versus product of the two grains' dressings on log-log axes follows a bilinear law within 20 percent. — **The force law and the mass law.** Left: the induced force at g = 1 is constant to ±5% from d = 24 to 64 — which is the Newtonian law *in one dimension*, where field lines do not dilute — and declines only ~19% by d = 96, where a Casimir force in this massive medium (μ = 0.02) would have fallen fifty-fold. The decline tracks the medium's absorption length: the force is screened by opacity, not by the mediator's mass. Right: across a ×5.9 span of dressing products, F = G·m_A·m_B holds to ±20% with m = each grain's measured dressing — the "mass" of a monitored object is the deformation its monitoring imprints on the medium. The residual is systematically superlinear: self-interaction corrections, as in gravity itself.

Experiment 3 — the mechanism, dissected measured our first story refuted

Three decompositions, each an independent run:

The budget. Freeze the converged dressings and measure their static Casimir force: repulsive (−1.5×10⁻⁴ at g = 1) — the pre-registered mechanism (attraction = Casimir between dressed defects) is refuted by its own audit gate G4. The heat-cloud contact term (Hartree cross-energy of overlapping clouds, a zero-parameter prediction from single-grain data) fails its calibration gate too: ~10% of the force. What remains — the dynamic term, existing only while the noise runs on the dressed landscape — carries 130% of the total and is flat in d to 1.4%.
The one-way test. Replace grain B by its frozen dressing profile (no noise at B): the force is exactly half the dynamic term (ratios 1.01 and 0.97 at d = 24 and 48). The attraction is the additive sum of two one-way effects — each grain's flux pulling on the other's dressing. No mutual loop. And though the coupling is one-way, action equals reaction to 0.7%: the field keeps Newton's third law by itself.
The minimal system. A noisy grain and a passive stiffness bump. Static landscape alone: weak repulsion. Flux alone (g = 0): pushes, as radiation pressure should. Flux in the interacting medium: strong attraction. The force requires both ingredients at once.

Left: budget bars showing the exact force decomposed into repulsive static and contact parts and a dominant flat dynamic term. Right: force on a passive bump versus distance for static-only, linear-flux, and interacting-flux configurations; only the last attracts. — **Only flux × medium-response attracts.** Left: the budget at g = 1 — static and contact terms are small and repulsive; the dynamic interplay term carries everything and is distance-flat. Right: the minimal system — a passive bump near a monitored grain. In the linear medium the grain's radiation pushes the bump away (M1); the frozen landscape barely acts (M0); in the interacting medium the same bump is strongly pulled in (M2, decaying with the emitter's screened flux). Modes `budget`, `mixed`, `minimal` of n18-rust.

The shadow, found one level up. A scatterer in the flux reflects it: intensity is enhanced on the source side, depleted behind — a shadow. In a linear medium this pattern exerts net outward pressure (Maxwell). But an interacting medium responds to the pattern: it re-dresses asymmetrically around the scatterer, and that asymmetric dressing pulls the scatterer toward the source. Le Sage's shadow is real — it is just not written in the flux's momentum, where Maxwell forbids it. It is written in the stiffness of the medium. The microscopic sign derivation (why the response-shadow pulls) is an open debt, listed below.

The exact test — no Hartree, no mercy running

Everything above shares one approximation: Hartree, which inserts by hand a medium whose stiffness responds to intensity. A fair skeptic reads all of the above as a statement about Kerr-like media, not about the quartic quantum field. So the decisive experiment is running as this page is written: n19, a six-site anharmonic chain solved at the level of the full Lindblad generator — exact quartic Hamiltonian, exact x²-record noise, zero mean-field steps — with its own pre-registered gates (validation against a Gaussian twin where the dynamics is exactly solvable; Fock-truncation honesty thresholds; the verdict scan in g). This experiment can kill the result. Its verdict will be posted here unedited, either way.

The analog road — this is testable on a table proposal

Everything the mechanism needs exists in the laboratory, separately, today. Thermal-lensing media carry index responses to intensity over centimeters (Rotschild–Segev's soliton attraction); independently driven noisy scatterers are routine in acoustics and optomechanics; Bjerknes physics shows how to measure tiny fluctuation forces between small objects. The missing experiment is the conjunction:

Two scatterers driven by statistically independent noise sources, in a medium whose index responds to intensity (a thermal-nonlinear liquid, a Kerr slab, or an acoustic medium with temperature-dependent sound speed).
Control 1 (the Maxwell null): same drives, linear medium — the time-averaged force must vanish or weakly repel.
Control 2 (the Bjerknes discriminator): correlate the two drives — the ordinary in-phase attraction appears and must change when the drives are decorrelated; the survival of attraction under decorrelation is the rectification signature, and it should scale with the medium's nonlinear coefficient and be screened at its absorption length.

A positive result would establish noise-rectified attraction as a laboratory effect — the strongest upgrade available to this program short of the gravity claim itself, and one that lives entirely in established experimental technique. We commit to this prediction now, before any such experiment exists.

Prior art — three adversarial audits

Three independent literature audits (nonequilibrium Casimir physics; measurement-based gravity models; classical and analog systems) returned the same structure: the chain is unpublished, but every link has an owner.

Who	What they own	Relation
Hahn & Fine 2019 (arXiv:1903.09065)	Measurement backaction → derived Newtonian attraction (mutual velocity measurements; no medium)	closest single threat; cited and differentiated
Kafri–Taylor–Milburn; Tilloy–Diósi	Monitoring + records + gravity — with the force postulated as feedback, its noise floor derived (force → noise)	our arrow is the reverse: noise → force. Part I measured their side of the duality (KTM floor saturated)
Bartolo–Ajdari–Fournier 2003	Noise-driven Casimir-like forces between inclusions (global drive, Gaussian field)	overlapping link
Kirkpatrick–Ortiz de Zárate–Sengers 2013	Giant nonequilibrium Casimir force requiring nonlinear mode coupling (global gradient)	overlapping link
Schecter–Kamenev 2014	Bilinear-in-dressing attraction between impurities in an interacting 1D liquid (equilibrium)	overlapping link
Bjerknes forces; Larraza–Denardo 1998; Simaciu et al. 2019	Noise-driven attraction in linear acoustics — for common/coherent drive	forces the sharpened claim below
Rotschild–Segev 2006; Golestanian et al.	Index-response-mediated long-range attraction (thermal optical nonlinearity; hot colloids)	adjacent; suggests lab analogs

The claim, sharpened by the audit. Objects driven by statistically independent noises exert no time-averaged force on each other through a linear medium (in-phase drive is different — Bjerknes). The medium's self-interaction acts as a rectifier, converting independent, incoherent monitoring noise into a coherent, universally attractive, bilinear, long-ranged force. That conjunction — with the dichotomy measured on both sides — is what we found published nowhere.

The chronicle of errors, kept on purpose

The first Hartree scheme (counterterm against the bare vacuum) ran away through window-edge reflections. Replaced by normal-ordering against the current vacuum, where the dressing is provably non-negative.
The mid-g repulsion deepening was initially read as physics; box-doubling exposed it as the testbed's global heat budget. G2 (transparent limit) was consequently declared out of reach rather than confirmed.
The pre-registered mechanism — static Casimir between dressed defects — was refuted by its own audit gate (G4): the frozen dressings repel. The attraction is dynamic.
The heat-cloud overlap formula (the factorized pipeline's leading term) failed its zero-parameter calibration gate: right sign, ~10% of the magnitude, wrong shape. Rejected as registered.
An infrastructure bug: concurrent LAPACK calls on Apple Accelerate silently corrupt results. Caught because the threaded port disagreed with the sequential validation numbers; the fix is documented in the source.

Epistemic status, exactly

Within the Hartree lattice model: the flip, the two laws, and the mechanism dissection are measured, three-size-checked, and reproducible in minutes. measured
As a statement about the interacting quantum field: pending n19 — the exact test can still demote everything above to a statement about Kerr-like media. pending
As a statement about gravity: the readings "masses are stiffness defects" and "the vacuum monitors matter" are the program's framework, not results. The microscopic sign derivation and the 3D exponent (the flux-dilution argument predicts 1/r², screened only by transparency) are open debts. conjecture
Novelty: audited by three adversarial agents; the sharpened conjunction appears unpublished; every adjacent result is credited above. Not peer-reviewed; not independently replicated. audited, unreviewed

Reproduce everything

Python reference: n18_interacting_shadow.py (~25 min, plain numpy/scipy). Fast port with all decomposition modes: n18-rust (cargo run --release -- [full|newton|budget|mixed|minimal|overlap] [N]; zero dependencies, Accelerate FFI; validated against the python digit-for-digit). Exact non-Gaussian test: n19_exact_nongaussian.py. Figures: figs_n18.py.