Geometric Realization of Completed Source Relations Descent, Orbit Spaces, Invariant Relations, and Variational Response in Shadow Theory

Geometric realization of the minimal completion

Jeremy RodgersSection 6 of 12Original PDF ↗Publication DOI ↗

Geometric realization of the minimal completion

The geometric evaluation map

Definition 5.1 (Declared readout and relation family).

Let $p_{\mathrm{geo}}\colon S_{\mathrm{geo}}\longrightarrow T\tag{31}$ be a declared Tier-1 readout: a map, defined on the reduced source, recording exactly the data retained by the Tier-1 description of the model. Let $A$ be an index set and let $J_\alpha\colon S_{\mathrm{geo}}\longrightarrow V_\alpha, \qquad \alpha\in A,\tag{32}$ be invariant geometric relations in the sense of Definition 4.1. Write $J_A=(J_\alpha)_{\alpha\in A}$ and $j_A^{\mathrm{geo}}\;=\;\bigl(p_{\mathrm{geo}},\;J_A\bigr)\colon S_{\mathrm{geo}}\longrightarrow T\times\prod_{\alpha\in A}V_\alpha .\tag{33}$

Definition 5.2 (Geometrically realized completion).

The geometrically realized completion of $p_{\mathrm{geo}}$ relative to $\{J_\alpha\}_{\alpha\in A}$ is $E_A^{\mathrm{geo}}\;:=\; \operatorname{Im}j_A^{\mathrm{geo}}\;\subseteq\; T\times\prod_{\alpha\in A}V_\alpha ,\tag{34}$ with corestriction $e_A^{\mathrm{geo}}\colon S_{\mathrm{geo}}\twoheadrightarrow E_A^{\mathrm{geo}}$ and forgetful map $u_A\colon E_A^{\mathrm{geo}}\longrightarrow T, \qquad u_A\bigl(t,(v_\alpha)\bigr)=t ,\tag{35}$ so that $u_A\circ e_A^{\mathrm{geo}}=p_{\mathrm{geo}}$ by construction.

Note that $E_A^{\mathrm{geo}}$ is the image, not the ambient product. This restriction performs the essential mathematical work: it enforces all the compatibility relations among metric, bundle, connection, matter, boundary, and invariant data that Section 3 exhibited. The Cartesian product of independently written coordinate values is in general not realizable (Corollary 3.7, Remark 3.8).

The restricted realization theorem

Theorem 5.3 (Restricted geometric realization).

Fix an admissibility class $\mathcal{C}_{\mathrm{geo}}$ , a readout $p_{\mathrm{geo}}$ , and a family $\{J_\alpha\}_{\alpha\in A}$ of invariant geometric relations on $S_{\mathrm{geo}}$ . Then:

(Realizability.) For every $\xi\in E_A^{\mathrm{geo}}$ there exists $[\Psi]\in S_{\mathrm{geo}}$ with $e_A^{\mathrm{geo}}([\Psi])=\xi$ .
(Factorization.) $p_{\mathrm{geo}}=u_A\circ e_A^{\mathrm{geo}}$ and $J_\alpha=\mathrm{pr}_\alpha\circ e_A^{\mathrm{geo}}$ for every $\alpha$ .
(Terminality.) If $\pi\colon S_{\mathrm{geo}}\twoheadrightarrow E$ is any surjection through which $p_{\mathrm{geo}}$ and every $J_\alpha$ factor, say $p_{\mathrm{geo}}=u\circ\pi$ and $J_\alpha=\kappa_\alpha\circ\pi$ , then there is a unique map $\theta\colon E\to E_A^{\mathrm{geo}}$ with $\theta\circ\pi=e_A^{\mathrm{geo}}$ ; $\theta$ is surjective and satisfies $u_A\circ\theta=u$ and $\mathrm{pr}_\alpha\circ\theta=\kappa_\alpha$ .

Thus $E_A^{\mathrm{geo}}$ is the canonical minimal completion of $p_{\mathrm{geo}}$ relative to $\{J_\alpha\}$ , in the sense of Paper 3 [18], computed on the geometric source $S_{\mathrm{geo}}$ .

Proof. (1) is immediate: $E_A^{\mathrm{geo}}$ is by definition the image of $j_A^{\mathrm{geo}}$ . (2) holds by definition of $j_A^{\mathrm{geo}}$ and of the projections; well-definedness of $p_{\mathrm{geo}}$ and each $J_\alpha$ as maps on $S_{\mathrm{geo}}$ is exactly the invariance hypothesis of Definition 4.1, which is where gauge and diffeomorphism invariance are consumed.

(3) Define $\theta(e):=e_A^{\mathrm{geo}}(s)$ for any $s\in\pi^{-1}(e)$ ; the fibre is nonempty since $\pi$ is surjective. This is well defined: if $\pi(s)=\pi(s')=e$ then $p_{\mathrm{geo}}(s)=u(\pi(s))=u(\pi(s'))=p_{\mathrm{geo}}(s')$ and likewise $J_\alpha(s)=\kappa_\alpha(\pi(s))=\kappa_\alpha(\pi(s'))=J_\alpha(s')$ for every $\alpha$ ; hence $j_A^{\mathrm{geo}}(s)=j_A^{\mathrm{geo}}(s')$ , i.e. $e_A^{\mathrm{geo}}(s)=e_A^{\mathrm{geo}}(s')$ . By construction $\theta\circ\pi=e_A^{\mathrm{geo}}$ , and since $e_A^{\mathrm{geo}}$ is surjective onto $E_A^{\mathrm{geo}}$ , so is $\theta$ . Uniqueness: any $\theta'$ with $\theta'\circ\pi=e_A^{\mathrm{geo}}$ agrees with $\theta$ on $\pi(S_{\mathrm{geo}})=E$ , hence everywhere. The compatibilities follow by evaluating on $\pi(s)$ and using surjectivity of $\pi$ .

This is the terminal universal property established in Paper 3 [18], applied to the source set $S_{\mathrm{geo}}$ , the readout $p_{\mathrm{geo}}$ , and the relation family $\{J_\alpha\}$ ; the additional content here is that $S_{\mathrm{geo}}$ is the reduced geometric source and that the $J_\alpha$ are invariant geometric relations, so that all maps involved are functions of physical orbits rather than of gauge or coordinate representatives. ◻

Remark 29 (Relative, not absolute).

Theorem 5.3 is restricted in two independent senses, both essential. First, it realizes the completion only relative to the declared family $\{J_\alpha\}$ : a different family gives a different $E_A^{\mathrm{geo}}$ , generally neither finer nor coarser. Second, it realizes only what geometry attains: $E_A^{\mathrm{geo}}$ is an image, and Corollary 3.7 shows that this image can be a proper subset of the abstract product. Neither the theorem nor anything else in this paper asserts that an arbitrarily nominated abstract completion $E_A$ of Paper 3 is geometrically realizable. It is realizable exactly when it is isomorphic, over $T$ , to $E_A^{\mathrm{geo}}$ for some admissible $\mathcal{C}_{\mathrm{geo}}$ , readout, and invariant relation family, and establishing that in a given model is work.

When does the completion reconstruct the source?

Proposition 5.5 (Orbit reconstruction criterion).

The completion map $e_A^{\mathrm{geo}}\colon S_{\mathrm{geo}}\to E_A^{\mathrm{geo}}$ is injective, equivalently, bijective onto $E_A^{\mathrm{geo}}$ , if and only if the family $\{p_{\mathrm{geo}}\}\cup\{J_\alpha : \alpha\in A\}\tag{36}$ separates points of $S_{\mathrm{geo}}$ : for all $[\Psi],[\Psi']\in S_{\mathrm{geo}}$ , equality of the readouts and of every $J_\alpha$ forces $[\Psi]=[\Psi']$ . In that case $S_{\mathrm{geo}}\cong E_A^{\mathrm{geo}}$ as sets over $T$ .

Proof. By construction $e_A^{\mathrm{geo}}([\Psi])=e_A^{\mathrm{geo}}([\Psi'])$ if and only if the readouts agree and every $J_\alpha$ agrees. Injectivity of $e_A^{\mathrm{geo}}$ is therefore literally the separation property. Surjectivity onto $E_A^{\mathrm{geo}}$ always holds, by Theorem 5.3(1). The identification is over $T$ because $u_A\circ e_A^{\mathrm{geo}}=p_{\mathrm{geo}}$ . ◻

Proposition 5.5 converts a question about completions into a question of invariant theory: does the declared family separate the orbits? No general answer exists, and none is asserted (Remark 4.6). Separation must be proved on the restricted domain relevant to the model. Section 6 carries out exactly such a proof in a case where the answer is affirmative and complete.