10
◇ part II · applications

An honest ledger — what closes in the current hierarchy, and what still waits on the moving-throat branch

The point of this page is not to declare victory. It is to say, as cleanly as possible, which post-Newtonian rungs are already packaged inside the current 4D hierarchy, and which rungs still depend on one shared outgoing quadrupole-normalization problem.

◇ the ladder

How gravity gets stress-tested

Physicists test gravity rung by rung. Newton handles the first pass. Then come the post-Newtonian corrections: 1PN, 2PN, 2.5PN, 3PN, 4PN, and beyond. Each rung corresponds to a specific observable effect.
Newton
Inverse-square attraction
Static weak-field regime.
1PN
Perihelion precession
Mercury and the first relativistic orbital tests.
2PN
Higher conservative corrections
Shapiro delay, lensing, stronger two-body structure.
2.5PN
Quadrupole radiation loss
Binary pulsars and inspiral energy balance.
3PN
Full conservative binary ledger
Precision compact-binary modelling.
4PN
Local plus tail conservative sector
High-precision waveform modelling.
◇ the ledger

Where the current source record says the program stands

The right way to read the stack today is:
DERIVED
Newton
Exact projection identities plus a controlled Poisson regime give the Newtonian hook.
DERIVED
1PN
The full conservative 1PN sector is derived within the declared closure hierarchy and matched term-by-term to GR.
DERIVED
2PN
The full conservative 2PN sector is carried through the same hierarchy.
CONDITIONAL
2.5PN
The quadrupole radiation formula is structurally in place, but its final normalization still depends on the passive/outgoing branch.
DERIVED
3PN
The conservative 3PN ledger is fully assigned inside the same hierarchy. The open issue is not unfinished 3PN algebra.
CONDITIONAL
4PN
The local instantaneous 4PN sector is assembled, and the tail piece waits on the same normalization problem already exposed by 2.5PN.
OPEN
frontier
Realize the passive/outgoing quadrupole normalization on the moving-throat branch. That is the shared theorem gate.
◇ the one shared gap

2.5PN and 4PN are waiting on the same thing

The remaining gap is narrower than it may sound. The notes already assemble the local 4PN conservative part inside the hierarchy. What still has to be shown is that a moving throat sends away radiation with the right quadrupole strength. That same missing result controls both the 2.5PN radiation step and the open 4PN tail step.
Put plainly: 2.5PN and the open part of 4PN stand or fall together. They are not two unrelated gaps. They are waiting on the same moving-throat radiation calculation.
◇ why the ledger matters

The point is honesty about status

A research program earns trust by keeping exact identities, controlled reductions, and open branch data separate. The PN stack is where that distinction matters most, because the observational stakes are high and the language is easy to overstate.
◇ up next

Next: the frontier object

The next chapter is about the moving-throat PDE framework, which is where the shared passive/outgoing quadrupole-normalization question is supposed to be realized microscopically.
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