# n204 — Is the force shape-dependent? Three sectors, three answers, and Birkhoff's silence broken

**Date:** 2026-07-11 · **Question (the human's).** Computation: equal-consumption ball vs rod potentials (128³ FFT solver, in-session); the rest assembles standing results.

## 1. The static force: shape-blind exactly where Newton is, by theorem

The long-range monopole is **fixed by the Gauss law itself** — total drainage = total consumption = mass, whatever the arrangement — so two bodies of equal mass and different shapes attract identically at long range, *exactly*, not approximately. Beyond monopole: the computed ball-vs-rod deviation scales like Newtonian multipoles (relative correction ~ Q/r²; verified within lattice artifacts, periodic images declared at large r). Composition-blindness is separately exact (η_N = 0, the Nordtvedt identity). Both are **exposures** — GR says the same; the theory can only lose here.

## 2. The short-range residual: yes — a shape- and arrangement-dependent force Newton lacks

The neutral-composite residual (r⁻³·⁵, parity-oscillating — results page) depends on the *internal arrangement* of the composite: which sublattice its constituents occupy, their relative parity. A genuinely shape-dependent static force, distinctive in form, with an amplitude fixed at the irrelevant scale by the sector's own arithmetic — [toy], and honest about it.

## 3. The dissipative sector: strongly shape-dependent — and here the theory breaks with GR structurally

The audit bills **local modulation per junction** (n181: Γ ∝ Σu_A², the source's own weights), not the far field. Two consequences:

- The billing is geometry-dependent where GR's radiation is too (quadrupole shapes) — the dumbbell/sphere band (~2–10, n181/reviewer sweep) lives here; not yet a universal number.
- **The sharp one: a radially breathing sphere pays.** In GR, Birkhoff's theorem makes spherically symmetric pulsation *exactly silent* — the exterior stays Schwarzschild, no radiation, no damping, zero. In the audited vacuum, breathing modulates T₀₀ locally, the trace channel is watched, and the bill is nonzero (n181 computed exactly this: g_sphere = 3.5×10⁻³ ≠ 0). **The audit's dissipation hierarchy starts at the monopole; GR's starts at the quadrupole.** The knee experiment (prediction 1) — energy-modulated superpositions — is precisely this monopole channel: the flagship prediction is, in relativists' language, a *Birkhoff-violation test in the dissipative sector*, while the static Birkhoff statement (exterior fixed by mass alone) is preserved by tier 1. Amplitude unarmed (n181's standing state); the zero-vs-nonzero contrast is shape-structural and needs no normalization.

## Summary sentence

Statically the force is shape-blind exactly as Newton's (and that is an exposure); at short range a parity-structured residual is shape-dependent in a way Newton never is; dissipatively the theory is *more* shape-sensitive than GR at low multipoles — it hears the breathing GR is deaf to — and less hierarchical about it.

**Class: tier 1 [theorem + A-model check]; tier 2 [toy, standing]; tier 3 [A-model for the zero/nonzero contrast; the Birkhoff framing is a reading of standing results, no new claim].**