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◇ part II · applications

Newton through 3PN sit inside one declared hierarchy; 2.5PN and 4PN share one outgoing quadrupole-normalization gate

The current 4D program does not phrase the PN story as a fresh set of projected Einstein equations. It builds a carried two-body ledger inside a declared closure hierarchy: Newtonian matching, the 1PN bridge package, the full conservative 1PN/2PN/3PN sectors, then the conditional 2.5PN and 4PN bridges. The remaining open datum is not a new 4PN constant; it is the same passive/outgoing quadrupole normalization already isolated by 2.5PN.

1PN · 2PN · 3PN within hierarchy2.5PN · 4PN share one gate4D · Action4D · 1PN bridge4D · 1PN4D · 2PN4D · 2.5PN4D · 3PN4D · 4PN
◇ setup

The ledger lives inside a declared closure hierarchy

The bridge and full 1PN papers freeze the lower-order data carried by the rest of the PN stack:
◇ carry-forward closure package
The bridge / 1PN package freezes the lower-order data used throughout the PN stack.
n  =  5,κρ  =  1,κadd  =  12,κPV  =  32,β1PN  =  3\begin{aligned} n &\;=\; 5, \\ \kappa_\rho &\;=\; 1, \\ \kappa_{\rm add} &\;=\; \tfrac12, \\ \kappa_{\rm PV} &\;=\; \tfrac32, \\ \beta_{1{\rm PN}} &\;=\; 3 \end{aligned}
That is the sense in which the site should use words like derived and closed: exact within the named hierarchy, not assumption-free theorems of a fully solved moving-throat PDE.
◇ the ledger · coefficients

One row per PN rung

EFFECT
Newtonian regime
Poisson / inverse-square hook
4D · Action + 4D · 1PN bridge
Exact projection identities plus a controlled Poisson regime; downstream matching fixes kappa_rho = 1 in the bridge package.
REDUCTION
1PN (full)
Full first post-Newtonian sector
4D · 1PN full · zenodo.19450102
Full conservative 1PN two-body ledger derived within the declared hierarchy; term-by-term match to GR.
REDUCTION
2PN (conservative)
Full conservative 2PN sector
4D · 2PN conservative · zenodo.19450284
All conservative terms at O(v⁴/c⁴) fixed within the same hierarchy and matched to the standard target.
CLOSURE
2.5PN (conditional)
Point-particle quadrupole radiation
4D · 2.5PN conditional · zenodo.19492270
Structure and coefficient match GR subject to the passive/outgoing quadrupole normalization.
REDUCTION
3PN (conservative)
Full conservative 3PN sector
4D · 3PN conservative · zenodo.19501724
The conservative 3PN ledger is fully assigned in the fixed ADM chart inside the declared hierarchy.
CLOSURE
4PN (conditional)
Local 4PN plus inherited tail bridge
4D · 4PN conditional · zenodo.19561056
The local instantaneous 4PN sector is assembled inside the hierarchy; the remaining gate is the same quadrupole-normalization / tail bridge already isolated by 2.5PN.
OPEN
frontier
Passive/outgoing quadrupole normalization
moving-throat branch realization
This is the live theorem target carried jointly by 2.5PN and 4PN. 4PN does not open a second independent normalization datum.
◇ the shared gate

2.5PN and 4PN wait on the same outgoing quadrupole normalization

The 2.5PN paper writes the odd sector as the standard quadrupole branch with one open normalization factor, while the 4PN paper shows that the hereditary tail coefficient is controlled by that same scalar:
◇ shared normalization gate · conditional
2.5PN odd radiation and the 4PN tail coefficient wait on the same outgoing quadrupole normalization.
γquadeff  =  m^02Γ5,γquadeff∣GR  =  2G5c5,Ctail  =  GM2c3 γquadeff\begin{aligned} \gamma_{\rm quad}^{\rm eff} &\;=\; \hat m_0^2\Gamma_5, \\ \gamma_{\rm quad}^{\rm eff}\big|_{\rm GR} &\;=\; \frac{2G}{5c^5}, \\ C_{\rm tail} &\;=\; \frac{GM}{2c^3}\,\gamma_{\rm quad}^{\rm eff} \end{aligned}
◇ experimental cross-checks

What this ledger must eventually answer to

Each rung is anchored to familiar tests:
  • Mercury perihelion advance — 1PN.
  • Shapiro delay and light-bending — bridge / 1PN / 2PN carry-forward sector.
  • Hulse–Taylor and double-pulsar timing — 2.5PN normalization test.
  • LIGO/Virgo inspirals — 3PN and conditional 4PN waveform coefficients.
◇ forward reference

What actually remains to be done

Topic 11 is not waiting to invent a new 4PN constant. It is waiting to realize the passive/outgoing quadrupole normalization on the moving-throat branch. That is the one gate this page should keep in view.
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