Source support, preparation and common witnesses

2. Admitted support and realized variables

2.1 A native source family

The source signature in monograph Chapter 2 has a dependency preorder, deposition-potential tokens, a marked subsort, finite supports, lineage, invariants and intrinsic boundary data. It contains no metric, coordinates, fields, action or preferred outcome. Admission is the least closure of a finite witness class under strong generated maps, generator-relative attachments, suitable retracts, stabilizing directed colimits and isomorphisms. Strong maps retain strict marked pairs and their lineages.

The walking marked dependency $X_\star$ is explicitly in the witness class. Its vertices satisfy $0\prec1$, with one marked token supported on $\{0,1\}$. Its proper predecessor-closed root face is $F_0=\{0\}$. An attachment domain need not have a marked token or itself be an admitted source candidate.

Proposition 2.1 (Admitted walking-cell support). For every nonempty small index set $I$, the rooted fan $X_I$ is admitted by the stated closure. It consists of a root $r$, distinct leaves $\ell_i$, dependencies $r\prec\ell_i$, and copies of the walking marked token on supports $\{r,\ell_i\}$. Invariant and lineage data are inherited from the walking cells.

Proof. Attach a fresh cell at its root face to the inherited root. The finite pushout consists exactly of the common root and distinct leaf and cell copies: maps agreeing on the root extend uniquely. Transitivity adds no dependency between leaves. Each marked support keeps its two distinct elements, its strict pair and inherited lineage. The root-face map is strong and mark-reflecting, and all old cell data are unchanged. These are the source conditions for these attachments, proving finite admission by induction.

Take the directed family of finite nonempty subsets of $I$. Once a finite cell appears, every later inclusion leaves its support, strict pair, lineage and invariant data unchanged. The allowed stabilizing directed colimit is precisely $X_I$. A permutation of $I$ is a source isomorphism. Only the existing walking cells are used, with no freely adjoined physical label.

The fan gives an admission presentation. Branch labels supply neither spatial nor temporal calibration, and the proof assumes no raw physical law for the support.

2.2 What the literal preparation contains

For a fixed marked token $m_\star$, take the literal preparation of monograph equations (3.1)–(3.2).

$$Y=J_{\mathrm{real}}(X_I,m_\star) =(X_I,m_\star,S_{m_\star},\preceq_{m_\star},\ell_{m_\star}, \mathrm{Inv}_{m_\star},\mathrm{Bnd}_{m_\star}). \tag{3}$$

$S_{m_\star}$ and its inherited data are the finite lineaged support, and $X_I$ is its admitted ambient source. Physical fields belong to a raw-sector relation instead.

$$K_{\mathrm{grav}}(Y)\subseteq O^{\mathrm{raw}}_{\mathrm{grav}}\times D_{\mathrm{grav}} . \tag{4}$$

The discrepancy column retains both zero and nonzero equation, composition and interface residuals. Einstein solutions do not define source admission.

A physical preparation supplies $U$, coordinate and tangent frames, the path family, reference density, coupling and boundary convention. A bare marked-occurrence assignment $\beta$ covers the prepared piecewise smooth paths, subarcs, reversals and compositions. Whole occurrences are assigned to paths. This assignment does not send the shared abstract root to every physical endpoint; endpoint coincidences and composition belong to the raw sector.

$$z=(\beta,T,B,\chi,\mathrm{preparation},\mathrm{residuals}),\qquad T_\gamma:T_xU\longrightarrow T_yU\quad(\gamma:x\to y). \tag{5}$$

The field $B$ is a smooth section of $\operatorname{Sym}^2TU\otimes\operatorname{Dens}(U)$ on a nondegenerate Lorentz branch. The chosen metric matter is denoted by $\chi$. Transport is invertible, multiplicative, reparameterization invariant and the identity on constant paths. The regularity chapter gives the precise smoothness contract.

This explicit realization rule retains off-shell histories with their residuals. The smooth manifold, transport class, density and action are inputs of (4). Their placement there preserves target neutrality at the level of the source signature and keeps physical selection visible.

2.3 Recover operations on one common witness

Let $\mathrm{Comp}_{\mathrm{GR}}$ have metric, connection, stress and matter configurations as objects with their compatibility equations. Its arrows are allowed gauge transformations. Let $E_{\mathrm{GR}}$ consist of the admitted fan, its preparation and histories (5), with arrows preserving these data and readouts. The projection $\pi$ reconstructs $A[T]$ and $g_B$ and evaluates the stress and matter maps. A witness for a physical configuration $O$ is a history $z$ together with an isomorphism $\phi:\pi(z)\simeq O$.

$$\operatorname{Lift}(O)=E_{\mathrm{GR}}^{\simeq} \times_{\mathrm{Comp}_{\mathrm{GR}}^{\simeq}}^{h}\{O\}. \tag{6}$$

This instantiates the monograph's common-witness definition in §1.3.3. Density, transport, matter, the action, its variations and the stress response all belong to the same history. The projection, inverse and variation calculations in §§3–6 establish complete solution coverage in the declared branch. They do not classify other possible source antecedents of $O$.