From completion to Tier-1 representation

From completion to Tier-1 representation

The completed geometric architecture established by Theorems 3.4, 5.3 and 7.8 is the factorization $$S_{\mathrm{geo}} \;\xrightarrow{\;\;e_A^{\mathrm{geo}}\;\;}\; E_A^{\mathrm{geo}} \;\xrightarrow{\;\;u_A\;\;}\; T , \qquad u_A\circ e_A^{\mathrm{geo}}=p_{\mathrm{geo}}.\tag{93}$$ The first map retains the readout together with the selected invariant geometric relations; the second forgets those completion coordinates and returns the original Tier-1 state. Neither map chooses a coordinate chart, a gauge potential, or a preferred representative of a physical orbit: $S_{\mathrm{geo}}$ is already the orbit space, and $e_A^{\mathrm{geo}}$ is built from $\mathfrak{G}_{\mathrm{geo}}$-invariant functions. This is what it means for the completion to be geometric rather than notational.

When a relation-dependent action is specified in the sense of Definition 7.2, the completed state enters Tier-1 dynamics through the derived response fields $$T^{\mathrm{rel}}_{\mu\nu}, \qquad J^{\mu}_{\mathrm{rel}}, \qquad \mathcal{E}^{\mathrm{rel}}_{\Phi} .\tag{94}$$ A distinction must be drawn here, since the two halves of the construction carry invariance in different senses, and conflating them would misdescribe the responses.

Invariance. What is invariant are the scalars and the orbit-defined information: the action $S_{\mathrm{rel}}$ is a $\mathfrak{G}_{\mathrm{geo}}$-invariant functional by hypothesis (R2); the relation coordinates $r_\alpha$ and the readout $p_{\mathrm{geo}}$ are functions on the orbit space $S_{\mathrm{geo}}$ by Definition 4.1; and the completion (93) is assembled from these.

Covariance. The responses (94), by contrast, are not invariant pointwise, and are not scalars. They are the functional derivatives of an invariant action with respect to the fields, hence tensorial objects on $M$ carrying spacetime and representation indices: $T^{\mathrm{rel}}_{\mu\nu}$ is a symmetric $2$-tensor, $J^{\mu}_{\mathrm{rel}}$ an $\mathrm{ad}P$-valued spacetime vector field, and $\mathcal{E}^{\mathrm{rel}}_{\Phi}$ a section of the dual matter bundle. Under an arrow $(\hat F,f)$ of $\mathfrak{G}_{\mathrm{geo}}$ they transform covariantly, by pushforward of tensor indices along $f$ and by the adjoint or dual representation along $\hat F$, rather than being fixed. Thus, for instance, $J^{\mu}_{\mathrm{rel}}\mapsto \operatorname{Ad}_{h^{-1}}J^{\mu}_{\mathrm{rel}}$ under a gauge transformation $h$, and $T^{\mathrm{rel}}_{\mu\nu}\mapsto (f^{-1})^{*} T^{\mathrm{rel}}_{\mu\nu}$ under an allowed diffeomorphism $f$.

The correct statement is therefore the following. What descends to the orbit space is the invariant action and the relation data; what the variation produces is a family of covariant response tensors, one per gauge and coordinate representative, transforming into one another under the arrows of $\mathfrak{G}_{\mathrm{geo}}$. It is precisely this covariance, not pointwise invariance, that makes the field equations (76)–(78) well posed as equations between tensors, and that makes the Ward identities of Section 8 meaningful: $\nabla^{\mu}T^{\mathrm{tot}}_{\mu\nu}=0$ is a covariant equation, and is preserved under the arrows of $\mathfrak{G}_{\mathrm{geo}}$ rather than being invariant under them in the sense of Definition 4.1. By Theorem 7.8 and Section 8, the responses (94) are correctly normalized, Ward-consistent, covariant objects obtained by functional variation of a $\mathfrak{G}_{\mathrm{geo}}$-invariant action.

This is the whole of what the present paper establishes about dynamics, and the boundary of the argument should be stated precisely. Paper 4 constructs covariant response fields from $\mathfrak{G}_{\mathrm{geo}}$-invariant actions. It does not determine whether the resulting Tier-1 evolution closes autonomously on $T$, which completion coordinates (if any) must be retained for a closed description, what happens when unresolved variables are eliminated, or whether such elimination produces memory or other history-dependent effective behaviour. In particular, no inference of Tier-1 autonomy may be drawn merely from the observation that some response functional happens to descend to a function of $t$ at the kinematic level considered here: descent of a response at one level does not establish closure of an evolution, and the two questions are not the same. Autonomous closure, retained variables, elimination, memory, and projected evolution require the separate analysis of Paper 5, and are not treated here.