05
◇ part II · applications

The Poisson hook: an exact projected identity plus a controlled Newtonian regime

The current 4D paper does not postulate Newton's law. It derives an exact projected continuity equation with leakage, then an exact longitudinal identity for the brane velocity potential. A Poisson equation and inverse-square scaling emerge only after clearly stated quasi-static and weak-correction assumptions.

Projected continuity + leakageLongitudinal identityPoisson regimesector: gravity4D · Action & Projections4D · 1PN bridge
◇ exact projected continuity

Projection introduces a leakage source; it does not close to a 3D fluid automatically

Given a normalized projection kernel W(w)W(w), the paper defines projected brane density and flux and derives the exact projected continuity identity
◇ projected continuity · exact
Exact projected continuity with the full leakage source.
∂tρbrane  +  ∇3 ⁣⋅ ⁣jbrane  =  Sleak,Sleak  =  −[W jw]−∞+∞+∫−∞+∞W′(w) jw dw\begin{aligned} \partial_t \rho_{\rm brane} \;+\; \nabla_3\!\cdot\!\mathbf j_{\rm brane} &\;=\; S_{\rm leak}, \\ S_{\rm leak} &\;=\; -\big[W\,j^w\big]_{-\infty}^{+\infty} + \int_{-\infty}^{+\infty} W'(w)\,j^w\,dw \end{aligned}
This is the paper's first gravity-side firewall: the brane does not inherit a closed 3D fluid theory for free. Leakage and correction terms are intrinsic to the projected observables.
◇ Helmholtz split

The Poisson hook starts as an exact identity for the brane velocity potential

Where ρbrane>0\rho_{\rm brane} > 0, define the brane velocity by vbrane=jbrane/ρbrane\mathbf v_{\rm brane} = \mathbf j_{\rm brane}/\rho_{\rm brane}and decompose it as
◇ Helmholtz decomposition · exact
The Poisson hook is built from the brane velocity potential φ\varphi, not from the parent matter field.
vbrane  =  ∇3φ  +  vT,∇3 ⁣⋅ ⁣vT  =  0\begin{aligned} \mathbf v_{\rm brane} &\;=\; \nabla_3 \varphi \;+\; \mathbf v_T, \\ \nabla_3\!\cdot\!\mathbf v_T &\;=\; 0 \end{aligned}
Substituting this split into projected continuity yields the exact longitudinal identity
◇ longitudinal identity · exact
This is exact. A Poisson equation follows only after a controlled regime reduction.
ρbrane ∇32φ  =  Sleak−∂tρbrane−∇3ρbrane ⁣⋅ ⁣(∇3φ+vT)\rho_{\rm brane}\,\nabla_3^2\varphi \;=\; S_{\rm leak} - \partial_t \rho_{\rm brane} - \nabla_3 \rho_{\rm brane}\!\cdot\!(\nabla_3\varphi + \mathbf v_T)
◇ Poisson regime · controlled

Newtonian behaviour appears when the explicit correction terms are parametrically small

In the quasi-static, longitudinal-dominant regime with slowly varyingρbrane\rho_{\rm brane}, the right-hand correction terms are suppressed and one gets the standard Poisson form
◇ Poisson hook · regime statement
Poisson and inverse-square are controlled regime statements, not exact laws.
∇32φ  ≈  1ρref Seff(x,t),vL  ≡  ∇3φ  ∼  1r2 r^\begin{aligned} \nabla_3^2 \varphi &\;\approx\; \frac{1}{\rho_{\rm ref}}\,S_{\rm eff}(\mathbf x,t), \\ \mathbf v_L \;\equiv\; \nabla_3\varphi &\;\sim\; \frac{1}{r^2}\,\hat{\mathbf r} \end{aligned}
◇ localized source regime

A throat enters the Newtonian limit through a localized effective source

In the localized-source regime one writes
◇ localized source schematic
Localized effective source in the Newtonian regime.
Seff(x)  ≈  S δ(3)(x−x0)S_{\rm eff}(\mathbf x) \;\approx\; \mathcal S\,\delta^{(3)}(\mathbf x - \mathbf x_0)
The carry-forward coefficient package used downstream isκρ=1\kappa_\rho = 1, κadd=1/2\kappa_{\rm add} = 1/2,κPV=3/2\kappa_{\rm PV} = 3/2, and β1PN=3\beta_{1{\rm PN}} = 3. These are derived within the declared hierarchy; they are not tuning knobs attached ad hoc to the Newtonian limit.
◇ scope · what this actually supports

The Poisson hook is the first rung of the PN story, not the whole ladder

The current source record supports the following reading:
  • Newtonian regime: exact projection identities plus a controlled Poisson reduction.
  • 1PN / 2PN / 3PN: conservative sectors handled downstream within one declared closure hierarchy.
  • 2.5PN / 4PN: conditional on the shared passive/outgoing quadrupole normalization.
  • Moving-throat frontier: the remaining theorem gap is branch realization, not another round of local Newtonian algebra.
Projection identitiesNewtonian regimeShared quadrupole-normalization gate
◇ forward reference

What uses this

Topic 10 takes the Newtonian hook and climbs the declared PN ladder. Topic 06 gives the analogous controlled brane reduction for electromagnetism. Topic 11 is where the remaining outgoing normalization question is meant to be settled.
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