What the source architecture selects

7. Dependency reduction and physical selection

7.1 Trace every consequential premise

Native admission supplies the fan and compatible preparation. A raw relation supplies regular transport, nondegenerate density and matter. Smooth multiplicative transport yields the connection and its variations. The free spin-two law and horizontal response determine the stress; cubic total-current coupling closes the law. The affine equation eliminates distortion modulo projective shifts; the volume-restored observable returns metric transport. Density and matter variations then give (50) and (53).

In the table, source means the target-neutral admission and preparation architecture. Realization means additional physical or regularity structure in the raw sector. Admitting a representation and selecting a physical law are distinct operations.

Table 1. Premises, provenance and dependence

The reductions work together. Constant-path smoothness removes a separate jet postulate within the regular realization. The unrestricted affine law replaces torsion-free kinematics with a stationary equation and an observable quotient. Gauge invariance follows from the selected quadratic and covariant actions. Scalar minimality is fixed within the restricted natural class in Proposition 6.1. These reductions do not make the target-neutral architecture a unique selection of general relativity.

7.2 Competing realizations expose sharp obstructions

Proposition 7.1 (Target-neutral underdetermination). Native admission and literal preparation as used in §2, with the generic raw-relation schema, do not select dimension, signature, gravitational action or matter curvature coupling.

Proof. Keep the admitted fan, mark, order, lineage and invariant tuple. One raw sector has regular transport and Lorentzian densities in dimension four; another has Euclidean densities or a different dimension greater than two. The same schema retains either set of fields and nonzero residuals. Neither support nor literal preparation mentions the physical choices. Even within four-dimensional Lorentzian sectors, the affine Einstein action and its metric reduction augmented by $\alpha\int vR(g)^2$ have inequivalent Euler equations while fitting the same schema. The minimal scalar and its $\xi vR\phi^2$ extension also fit it. These explicit competitors satisfy the target-neutral premises.

This rules out deriving the selections from those premises. It does not refute all additional physical postulates in the full monograph. A stronger selection claim must identify which extra postulate excludes the competitor.

A chosen signature persists in a controlled neighborhood. If $\|\eta^{-1}\lambda h\|\lt1$ in a submultiplicative norm, the Neumann series gives invertibility of $B$. Along $\eta+s\lambda h$, $0\le s\le1$, the real symmetric matrix stays nondegenerate and inertia stays fixed. The argument preserves the selected signature; it does not select it.

Within the Einstein–Hilbert family, requiring the given flat vacuum to solve the completed equations forces the vacuum coefficient in $-2\Lambda v$ to vanish: $G_{\mu\nu}=0$ and the metric is nondegenerate. The same criterion does not exclude $R^2$ terms, whose first variation vanishes there. This constrains a parameter, not the whole action.

The competitors change concrete responses. In four-dimensional affine Palatini $f(R)=R+\alpha R^2$ with connection-independent matter, the trace equation gives $R=-\kappa_E T$. At a point with dust response $T=-\rho$ and orthonormal $T_{00}=\rho$,

$$R_{00}=\frac{\kappa_E\rho-f/2}{1+2\alpha\kappa_E\rho},\qquad R_{00}-\frac{\kappa_E\rho}{2} =-\frac{3\alpha\kappa_E^2\rho^2}{2(1+2\alpha\kappa_E\rho)} . \tag{58}$$

Here $R_{\mu\nu}$ is symmetric affine Ricci and $R=g^{\mu\nu}R_{\mu\nu}$. The connection equation uses $f'(R)B$, not $B$. For constant trace and $f'(R)\ne0$ the metric connection agrees with $\mathrm{LC}(g)$; it generally does not for varying trace. Equation (58) is a pointwise comparison, not a global dust solution. It distinguishes the equations wherever the denominator is nonzero.

For the scalar alternative take $g=e^{2t^2}\eta$ and constant massless $\phi=1$. The minimal scalar residual vanishes. The nonminimal residual in $(\Box_g-\xi R)\phi=0$ is $-\xi R$, with

$$R=12(1+2t^2)e^{-2t^2}. \tag{59}$$

Thus flat matching and common-density dependence do not fix minimal coupling without Proposition 6.1's extra restriction.

7.3 What the architecture explains and what comes next

The support, common preparation, explicit composition and map between transport and geometric dynamics give a complete source realization. Regularity provides reconstruction and error transfer. Projective reduction shows precisely which transport information is discarded and why torsion-free kinematics can be relaxed without changing the metric endpoint. The stress identity explains why the derivative current survives projection and why freezing the wrong variable fails. These are structural results, distinct from experimental evidence for a microscopic carrier.

The open problem is to select the raw law and observable class. A further source principle must distinguish the quadratic spin-two and total-current construction from the explicit alternative actions while preserving native admission. Requiring an Einstein readout at the outset would insert the desired conclusion. The realization instead declares its physical starting law and proves the exact recovery.