THE EINSTEIN RECOVERY PROGRAM — Exact Linear-Algebraic Reconstruction, First-Principles Lifts, and the Unfinished Unified-Field Ledger

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Objective. Reconstruct Einstein’s mathematical program as a machine-auditable chain of exact identities, variational derivations, linearizations, symmetry reductions, and explicitly labeled open branches. This is a recovery program, not a claim that Einstein’s unfinished unified-field theory has been solved.

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I. Recovery doctrine

The citizens will admit only statements that can be typed as one of four classes:

E0 — exact theorem/identity: derivable from declared axioms and standard differential/linear algebra.
E1 — machine reconstruction: symbolic or numerical witness reproducing the exact theorem with zero or controlled residual.
H — historical reconstruction: faithful restatement of an Einstein-era proposal, with provenance and no upgrade to modern truth.
O — open completion problem: mathematically explicit residual left unresolved by the historical program.

The master rule is

$$ \text{historical equation}\to\text{typed object}\to\text{first-principles derivation}\to\text{linearization}\to\text{machine witness}\to\text{boundary}\to\text{open residual}. $$

II. Einstein core to be fully reconstructed

Layer	Object	Proof vehicle	Machine target
Special relativity	Minkowski metric, Lorentz group, interval	bilinear-form invariance	$\Lambda^T\eta\Lambda=\eta$
Equivalence/covariance	metric, Levi-Civita connection, geodesics	uniqueness from torsion-free metric compatibility	$\nabla g=0$, torsion $=0$
Curvature	$R^\rho{}{\sigma\mu\nu},R{\mu\nu},R$	commutator of covariant derivatives	Bianchi residual $0$
Einstein tensor	$G_{\mu\nu}=R_{\mu\nu}-\frac12Rg_{\mu\nu}$	contracted Bianchi identity	$\nabla^\mu G_{\mu\nu}=0$
Field equations	$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G T_{\mu\nu}$	metric variation of Einstein-Hilbert action	Euler-Lagrange residual $0$
Weak field	$g=\eta+h$	Fréchet linearization	linearized gauge identity
Vacuum waves	TT modes	gauge quotient + wave operator	$\Box \bar h_{\mu\nu}=0$
Geodesic/Newtonian limit	$g_{00}\approx-(1+2\Phi)$	asymptotic expansion	recover $\nabla^2\Phi=4\pi G\rho$

III. First-principles linear-algebra skeleton

At every tangent space $T_pM$, the metric is a nondegenerate symmetric bilinear form. In matrix form,

$$ g_p\in\operatorname{Sym}^2(T_p^*M),\qquad \det g_p\neq0. $$

Coordinate change by Jacobian $J$ gives

$$ g' = J^{-T}gJ^{-1}, $$

while a Lorentz symmetry in flat spacetime satisfies

$$ \Lambda^T\eta\Lambda=\eta. $$

This is the first machine-level recovery: relativity begins as exact bilinear-form invariance.

The Levi-Civita connection is recovered from the simultaneous constraints