Pregeometry and Gravitational Dynamics

7.1 Role and Scope

Chapter 7 defines the geometry and gravity projection channel of the TOE monograph. It receives record-side structures from Chapter 6 and formalizes the geometry-side readout objects: admissible record-events, dependency kernel, dependency partial order, record-distance structure, effective metric readout, curvature, effective gravitational dynamics, conservation compatibility, and the general relativity limit.

The geometry channel is the typed map

$$\boxed{ \Pi_G:\mathsf{Real}_0\to\mathcal{O}_G. } \tag{7.1}$$

The corresponding theorem problem is the quantum–gravity derivation R3 of Chapter 17.

Chapter 7 defines the geometry/gravity channel at a formally defined but partial claim grade. The theorem-grade derivation of metric geometry and Einstein-limit dynamics from record-dependency data is routed to Chapter 8 and Chapter 17.

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7.2 From Objective Records to Pregeometry and Metric Reconstruction

7.2.1 Intervention-derived influence

Geometry is reconstructed from objective record relations, not postulated at the source level. Fix a finite subarchive $I\Subset\mathcal A_Y^{\rm obj}$. The lineage of $a\in I$ determines a minimal source cylinder $\mathcal N_Y(a)\subseteq S_m$: the finite predecessor-closed subobject generated by the source support appearing in that lineage. This is localization in dependency and lineage, not localization in a spacetime that has yet to be reconstructed. Let $(\Theta_{Y,a},\mathcal B_{Y,a},P_{Y,a})$ be the standard-Borel family of realization variations supported on $\mathcal N_Y(a)$ and equal to the baseline realization on its source boundary. An admissible perturbation is a finite signed measure $\delta$ satisfying

$$\delta(\Theta_{Y,a})=0, \qquad P_{Y,a}+\delta\in\mathcal P(\Theta_{Y,a}), \tag{7.2}$$

with magnitude measured by total variation,

$$\|\delta\|_{\rm src} =d_{\rm TV}(P_{Y,a}+\delta,P_{Y,a}). \tag{7.3}$$

The perturbation changes only the realization coordinates carried by $\mathcal N_Y(a)$; the complement and the declared external setting variables remain fixed. No source-side notion of spatial separation is used. Operational spacelike independence is imposed only later, on branches for which a Lorentzian geometry has been reconstructed. Let

$$\operatorname{DepSig}_{Y,a\rightsquigarrow b}(P) \tag{7.4}$$

be the resulting probability distribution of redescription-invariant objective-deposition signatures at $b$, obtained by composing the physical response with the stages in (6.24). Thus (7.4) compares complete record formation under the baseline and perturbed realization; it does not apply a record-preparation map directly to a source object.

Define the influence capacity

$$\boxed{ \kappa_Y^{\rm inf}(a,b) = \sup_{\delta\ne0} \min\!\left\{ 1, \frac{ d_{\rm TV}(\operatorname{DepSig}_{Y,a\rightsquigarrow b}(P+\delta), \operatorname{DepSig}_{Y,a\rightsquigarrow b}(P)) }{\|\delta\|_{\rm src}} \right\}. } \tag{7.5}$$

Then $0\le\kappa^{\rm inf}\le1$. Correlation or mutual information may diagnose a relation, but only the intervention response (7.5) is used to define directed influence.

Set $a\prec_0b$ when $\kappa^{\rm inf}(a,b)>0$. A directed cycle may be collapsed only when its vertices are physically equivalent under the declared redescription relation. On the finite quotient $\mathsf P_{Y,I}=I/\!\sim_{\rm red}$, define

$$[a]\prec_D[b] \quad\Longleftrightarrow\quad [a]\ne[b]\ \text{and a directed path joins }[a]\text{ to }[b]. \tag{7.6}$$

If a cycle contains physically inequivalent records, the finite archive does not yet define an order and must not be forced into one. When all cycles are redescription cycles, $\prec_D$ is a strict partial order and its reflexive closure $\precsim_D$ is a partial order.

For nested finite subarchives, the inclusions are required to preserve redescription classes and the dependency order. The realization-wide carrier used by the arrow and observer chapters is the directed union

$$\mathsf P_Y=\varinjlim_{I\Subset\mathcal A_Y^{\rm obj}}\mathsf P_{Y,I}. \tag{7.7}$$

Thus the later temporal construction uses the same objective-event quotient and order obtained here, not a second independently defined carrier.

7.2.2 Relational distance, measure and dimension

For distinct quotient vertices define

$$w_{xy} = \sup_{a\in x,b\in y} \bigl(\kappa^{\rm inf}(a,b)+\kappa^{\rm inf}(b,a)\bigr), \qquad s_x=\sum_{y\ne x}w_{xy}, \tag{7.8}$$

and, when $w_{xy}>0$,

$$q_{xy}=\frac{w_{xy}}{\sqrt{s_xs_y}}, \qquad c_{xy}=1-\log q_{xy}. \tag{7.9}$$

Since $0\lt q_{xy}\le1$, every edge cost satisfies $c_{xy}\ge1$. The extended relational distance is

$$\boxed{ d_{\rm rel}(x,y) = \inf_{\substack{x=x_0,\ldots,x_m=y\\ w_{x_jx_{j+1}}>0\ (0\le j\lt m)}} \sum_{j=0}^{m-1}c_{x_jx_{j+1}}, } \tag{7.10}$$

with value $+\infty$ between disconnected components. This is a graph-derived comparison distance; it is not yet a Lorentzian interval.

The normalized influence operator and Laplacian [76] are

$$Q_{xy}=q_{xy}\ (x\ne y),\qquad Q_{xx}=0, \qquad L_{\rm inf}=I-Q. \tag{7.11}$$

The heat return probability

$$P_I(t)=\frac{1}{|\mathsf P_{Y,I}|}\operatorname{Tr}e^{-tL_{\rm inf}} \tag{7.12}$$

defines a scale-dependent spectral dimension [34, 33] on a stable range,

$$d_s(t)=-2\frac{d\log P_I(t)}{d\log t}. \tag{7.13}$$

Independent volume-growth, nerve and walk-dimension estimators are compared rather than replaced by a declared dimension. A normalized support measure is obtained from nonnegative persistence/redundancy weights $r_I(x)$:

$$\nu_I(J) = \frac{\sum_{x\in J}r_I(x)}{\sum_{x\in\mathsf P_{Y,I}}r_I(x)}. \tag{7.14}$$

The finite pregeometric datum supplied by an archive is therefore

$$\mathfrak P_I = (\mathsf P_{Y,I},\prec_D,d_{\rm rel},L_{\rm inf},\nu_I, \{\operatorname{DepSig}^{\delta}\}_{\delta\in\Delta_I}),$$

where $\Delta_I$ is a fixed family of admissible localized perturbations. This datum contains order, comparison distance, spectral information, support measure and intervention response, but no presupposed continuum metric.

7.2.3 Set-valued metric reconstruction

Let $\{\mathfrak C_n^{\rm raw}\}_{n\ge1}$ be a directed, target-neutral family of finite candidate presentations. A raw candidate is only a finite relational, atlas, triangulation or cellular presentation with incidence data. It need not possess an order, volume, diffusion operator, response law, nonnegative comparison distance, dimension, signature or metric tensor. The union includes Riemannian, Lorentzian, degenerate, cone-only and nonmetric possibilities; four dimensions and one timelike direction are not encoded in its definition.

For an archive $I$, the metric-comparable subfamily $\mathfrak C_n^{\rm cmp}(I)\subseteq\mathfrak C_n^{\rm raw}$ consists of candidates $C$ equipped with all of the following:

$$\left( \phi_C,\ \prec_C,\ \mu_C,\ L_C,\ \{\operatorname{DepSig}_{C}^{\delta}\}_{\delta\in\Delta_I}, \ n_C,\ [g_C]_{\operatorname{Scale}},\ d_C^{\rm cmp} \right). \tag{7.15}$$

Here $\phi_C:\mathsf P_{Y,I}\to C$ is the comparison map, $\prec_C$ the candidate order, $\mu_C$ a finite positive comparison measure, $L_C$ a positive semidefinite diffusion generator whose heat semigroup is positivity-preserving, and $n_C$ the candidate dimension. The function $d_C^{\rm cmp}:C\times C\to[0,\infty]$ is a redescription-natural, symmetric, nonnegative comparison distance; it is not the signed Lorentzian interval. These structures are submitted with the candidate and cannot be inferred after a favourable fit, but submission alone is not enough for metric-comparable membership: all of them must be induced by, or certified compatible with, the same normalized candidate geometry.

Before any discrepancy is evaluated, fix the scale section on every metric orbit having a comparison region $A_C(\mathsf P_{Y,I})$ of finite, strictly positive $n_C$-volume:

$$s_I([g_C]_{\operatorname{Scale}}) =\widehat g_{C,I} =\lambda_{C,I}^{\,2}g_C, \qquad \lambda_{C,I} = \operatorname{Vol}_{|g_C|} \!\left(A_C(\mathsf P_{Y,I})\right)^{-1/n_C}. \tag{7.16}$$

Thus $\operatorname{Vol}_{|\widehat g_{C,I}|}(A_C(\mathsf P_{Y,I}))=1$. For Lorentzian candidates the absolute metric density is used. Put $A_C=A_C(\mathsf P_{Y,I})$. On a Lorentzian branch require $A_C$ to be compact, causally convex and causally distinguishing, with finite continuous time-separation function $\tau_C$ computed from $\widehat g_{C,I}$ within $A_C$. The latter property holds, in particular, when $A_C$ is a compact causally convex comparison region of a globally hyperbolic candidate. Define the strong, or Noldus, metric [74, 75]

$$d_{\rm str}^{C}(p,q) = \sup_{r\in A_C} \left| \tau_C(p,r)+\tau_C(r,p) -\tau_C(q,r)-\tau_C(r,q) \right|. \tag{7.17}$$

Causal distinction separates points in (7.17); compactness and continuity make the supremum finite. Hence $d_{\rm str}^{C}$ is a genuine symmetric metric on the admitted Lorentzian comparison region, rather than a Lorentzian time separation being used as though it were a metric. On a Riemannian branch let $d_{\rm geo}^{C}$ be the geodesic distance of $\widehat g_{C,I}$. Write

$$d_*^C = \begin{cases} d_{\rm str}^{C},&\widehat g_{C,I}\text{ Lorentzian},\\ d_{\rm geo}^{C},&\widehat g_{C,I}\text{ Riemannian}. \end{cases} \tag{7.18}$$

Let $B_x\subseteq A_C$ be the nonempty compact geometric support of the finite cell $x\in C$; for a point presentation take $B_x=\{p_x\}$. The comparison distance on cells is the Hausdorff metric induced by $d_*^C$:

$$\begin{aligned} d_C^{\rm cmp}(x,y) &=d_H^{\,d_*^C}(B_x,B_y)\\ &= \max\!\left\{ \sup_{p\in B_x}\inf_{q\in B_y}d_*^C(p,q), \sup_{q\in B_y}\inf_{p\in B_x}d_*^C(p,q) \right\}. \end{aligned} \tag{7.19}$$

The order, measure and dimension are induced by the same normalized geometry. Specifically,

$$\begin{aligned} x\prec_C y &\Longleftrightarrow \begin{cases} B_x\times B_y\subset I^+_{\widehat g_{C,I}}, &\widehat g_{C,I}\text{ Lorentzian},\\ \text{false},&\widehat g_{C,I}\text{ Riemannian}, \end{cases}\\ \mu_C &=(q_C)_* \overline{\operatorname{Vol}}_{|\widehat g_{C,I}|},\\ n_C&=\dim(A_C). \end{aligned} \tag{7.20}$$

Here $q_C:A_C\to C$ is the fixed measurable cell assignment associated with the presentation, defined up to volume-null cell boundaries. For a point presentation it is the metric Voronoi assignment for $d_*^C$. Its tie set must have zero normalized volume, so the push-forward measure does not depend on a labeling convention; a positive-volume tie set fails metric-comparable membership. Thus $\mu_C$ is not an independently fitted weight. The condition defining $\prec_C$ is irreflexive and transitive on nonempty cells and is robust against choosing a favourable representative point inside a cell.

Fix a bandwidth $\epsilon_C>0$, measured in the normalized distance units of $\widehat g_{C,I}$, before reconstruction, and set

$$W_{xy} = \begin{cases} \exp\!\left[-\left( d_C^{\rm cmp}(x,y)/\epsilon_C \right)^2\right],&x\ne y,\quad d_C^{\rm cmp}(x,y)\lt \infty,\\ 0,&x=y\ \text{or}\ d_C^{\rm cmp}(x,y)=\infty, \end{cases} \qquad D_x=\sum_yW_{xy}. \tag{7.21}$$

On the subspace spanned by cells with $D_x>0$, define

$$L_C = \mathbf1-D^{-1/2}WD^{-1/2}; \qquad L_C|_{\{D_x=0\}}=0. \tag{7.22}$$

Each nonisolated connected component is treated separately, while an isolated cell is a one-dimensional zero block. Because $W$ is symmetric and nonnegative, $L_C$ is self-adjoint and positive semidefinite, and $\exp(-tL_C)$ is positivity-preserving. The spectral comparison therefore uses a specified operator induced by $d_C^{\rm cmp}$, not a Lorentzian d’Alembertian mislabeled as a positive diffusion generator.

Finally, the intervention response is calculated using the QFT and matched record dynamics on that same normalized geometry:

$$\operatorname{DepSig}_{C}^{\delta} = \operatorname{DepSig}^{\delta} \bigl[\widehat g_{C,I}; \Gamma_{\rm QFT,dyn}^{\rm ren}, \Gamma_{R,\rm CTP},\Gamma_{MR,\rm CTP},\mathfrak m_R, \mathcal I,B\bigr]. \tag{7.23}$$

Equations (7.17)–(7.23) permit a certified discretization or restriction that commutes with the comparison map $\phi_C$; they do not permit an independently fitted auxiliary order, measure, diffusion operator, distance or response law. All bandwidths, cell assignments and boundary data are fixed before discrepancies are evaluated. The resulting quantities are therefore those of one normalized representative and are unchanged when the original representative is rescaled. If the comparison region fails compactness, causal convexity, causal distinction, finite positive volume or continuity/finiteness of $\tau_C$ on the Lorentzian branch; if its cell supports or induced structures fail; if the dimension is absent; or if any datum in (7.15) is missing, the candidate remains in $\mathfrak C_n^{\rm raw}$ and every discrepancy requiring the missing or incompatible datum is assigned $+\infty$.

For $C\in\mathfrak C_n^{\rm cmp}(I)$, compare record and geometric data by dimensionless discrepancies:

$$D_{\rm ord}(C;I) = \frac1{N_2(I)} \sum_{x\ne y} \left| \mathbf1[x\prec_Dy] -\mathbf1[\phi_C(x)\prec_C\phi_C(y)] \right|, \tag{7.24}$$

for $|\mathsf P_{Y,I}|\ge2$, where

$$N_2(I)=|\{(x,y)\in\mathsf P_{Y,I}^2:x\ne y\}|.$$ $$D_{\rm vol}(C;I) = \sup_{\varnothing\ne J\subseteq\mathsf P_{Y,I}} \frac{\left|\nu_I(J)-\overline{\operatorname{Vol}}_C(A_C(J))\right|} {\nu_I(J)+\overline{\operatorname{Vol}}_C(A_C(J))}, \tag{7.25}$$

Here $A_C(J)$ is the union of candidate cells associated by $\phi_C$ with $J$, and $\overline{\operatorname{Vol}}_C$ is candidate volume divided by the total comparison volume. The fraction in (7.25) is assigned zero when both terms in its denominator vanish.

The remaining comparisons use data fixed before reconstruction. The edge costs (7.9) make $d_{\rm rel}$ dimensionless. For a finite set of registered scales $\mathcal E_I$, let $\mathcal N_I(\epsilon)$ be the nerve built from $d_{\rm rel}$, let $\mathcal N_C(\epsilon)$ be the corresponding nerve built from $d_C^{\rm cmp}$, and let $\boldsymbol\beta$ denote their finite vector of Betti numbers. Define

$$D_{\rm nerve}(C;I) = \max_{\epsilon\in\mathcal E_I} \frac{\|\boldsymbol\beta(\mathcal N_I(\epsilon)) -\boldsymbol\beta(\mathcal N_C(\epsilon))\|_1} {1+\|\boldsymbol\beta(\mathcal N_I(\epsilon))\|_1 +\|\boldsymbol\beta(\mathcal N_C(\epsilon))\|_1}, \tag{7.26}$$

and, on a registered diffusion interval $\mathcal T_I$,

$$D_{\rm dim}(C;I) = \sup_{t\in\mathcal T_I} \frac{|d_s^I(t)-d_s^C(t)|} {1+|d_s^I(t)|+|d_s^C(t)|}. \tag{7.27}$$

The intervention comparison is

$$D_{\rm context}(C;I) = \sup_{\delta\in\Delta_I} d_{\rm TV}\!\left( \operatorname{DepSig}^{\delta}_{I}, \operatorname{DepSig}^{\delta}_{C} \right), \tag{7.28}$$

where the second distribution is calculated from the causal response of the candidate geometry coupled to the same already-fixed QFT, record, intervention and boundary data. A candidate for which that response is undefined does not satisfy the comparison. Finally, if $I_n\subset I_{n+1}$, $\iota_n:\mathsf P_{I_n}\to\mathsf P_{I_{n+1}}$ is the induced inclusion and $r_{n+1,n}:C_{n+1}\to C_n$ is the declared common comparison map, set

$$D_{\rm refine}(C_{n+1},C_n) = \frac1{|\mathsf P_{I_n}|} \sum_{x\in\mathsf P_{I_n}} \min\!\left\{1, \frac{d_{C_n}^{\rm cmp}\!\left( \phi_n(x), r_{n+1,n}\phi_{n+1}\iota_n(x) \right)}{\ell_n} \right\}, \tag{7.29}$$

with a fixed positive dimensionless comparison scale $\ell_n$. These definitions make the topological, dimensional, contextual and refinement tests independent rather than allowing one fitted discrepancy to stand in for all of them.

The refinement topology is fixed by common comparisons, not by an unspecified phrase “Cauchy under refinement.” Write $\mathbf C=(I,C,\phi)$ for a normalized metric-comparable candidate, and let $\mathfrak R(\mathbf C,\mathbf C')$ be the set of declared common refinements $R$ carrying structure-preserving maps $u:R\to C$ and $u':R\to C'$. The directed primitive comparison is

$$\delta_{\rm ref}(\mathbf C,\mathbf C') = \inf_{R\in\mathfrak R(\mathbf C,\mathbf C')} \max\!\left\{ D_{\rm ord}^{R},D_{\rm vol}^{R},D_{\rm nerve}^{R}, D_{\rm dim}^{R},D_{\rm context}^{R},D_{\rm refine}^{R} \right\}, \tag{7.30}$$

where every term is recomputed after transport to $R$, rather than merely copied from either endpoint. If there is no common refinement or a required transport/comparison is undefined, the value is $+\infty$. Symmetrize the primitive by

$$\delta_{\rm sym}(\mathbf C,\mathbf C') = \max\!\left\{ \delta_{\rm ref}(\mathbf C,\mathbf C'), \delta_{\rm ref}(\mathbf C',\mathbf C) \right\} \tag{7.31}$$

This removes any directional asymmetry of the primitive. In particular, $\delta_{\rm sym}=+\infty$ whenever either directed comparison is unavailable. Thresholding this quantity directly would not in general give a transitive relation. The reconstruction distance is therefore the extended path pseudometric

$$d_{\rm ref}(\mathbf C,\mathbf C') = \inf_{\substack{m\geq0,\ \mathbf C_0=\mathbf C,\ \mathbf C_m=\mathbf C'\\ \mathbf C_1,\ldots,\mathbf C_{m-1}}} \sum_{j=1}^{m} \min\!\left\{1, \delta_{\rm sym}(\mathbf C_{j-1},\mathbf C_j) \right\}. \tag{7.32}$$

The length-zero chain gives $d_{\rm ref}(\mathbf C,\mathbf C)=0$; reversal of chains gives symmetry; and concatenation gives the triangle inequality. Thus $d_{\rm ref}$ is a mathematically valid pseudometric without assuming that the primitive comparisons themselves satisfy a triangle inequality. Candidates at zero pseudodistance may equivalently be identified before taking the completion. The reconstruction uniformity is generated by

$$\mathcal U_\eta = \{(\mathbf C,\mathbf C'):d_{\rm ref}(\mathbf C,\mathbf C')\lt \eta\}, \qquad \eta>0, \tag{7.33}$$

together, on a regular metric branch, with the declared pointed $C^k$, Lorentzian Cheeger–Gromov or other specified geometric uniformity. This is a genuine uniformity by the pseudometric construction. A sequence is Cauchy precisely when for every $\eta>0$ there is $N$ such that all pairs with $m,n\ge N$ lie in $\mathcal U_\eta$ and in the chosen geometric entourage. R3 must still prove that the transport maps used in $\delta_{\rm ref}$ respect identities and composition and hence that this mathematical uniformity has the claimed presentation-independent physical interpretation.

A sequence $(I_n,C_n,\phi_n)$ is reconstructive when the archives refine one another, every $C_n$ is metric-comparable, the models are Cauchy in the uniformity (7.33), and

$$D_{\rm ord},D_{\rm vol},D_{\rm nerve},D_{\rm dim}, D_{\rm context},D_{\rm refine}\longrightarrow0. \tag{7.34}$$

Its limit must have a nondegenerate tensor, adequate regularity, a compatible spin structure when fermions occur, and the hyperbolicity required by the QFT and propagation constructions. The metric-reconstruction relation is

$$\boxed{ \begin{aligned} &\operatorname{Rec}_{\rm met}((\mathfrak P_n)_n)\\ &\quad= \left\{ \begin{aligned} &[M,g]_{\operatorname{Diff}\times\operatorname{Scale}}:\\ &\exists(I_n,C_n,\phi_n),\quad \mathfrak P_n=\mathfrak P_{I_n},\\ &\text{satisfying (7.34) and converging to }[M,g] \end{aligned} \right\}. \end{aligned} } \tag{7.35}$$

Equation (7.35) may be empty, a singleton or a non-singleton set. It is controlled nonunique relative to a declared observable family only when the reconstructed family $\mathcal M$ is compact in the declared metric topology and, for an observable vector fixed before reconstruction $\boldsymbol{\mathcal O}=(\mathcal O_1,\ldots,\mathcal O_N)$ containing every observable for which the control claim is made,

$$\operatorname{diam}\boldsymbol{\mathcal O}(\mathcal M) = \sup_{g,g'\in\mathcal M} d_{\rm obs}\!\left( \boldsymbol{\mathcal O}(g),\boldsymbol{\mathcal O}(g') \right) \le\varepsilon_{\rm deg}. \tag{7.36}$$

No bound for an unregistered observable is implied by (7.36). In particular, a small diameter for one component does not make the metric family globally controlled; all untested observables retain the full reconstruction degeneracy.

The quotient by global scale in (7.35) records that the dimensionless archive data need not fix an absolute unit. At every finite stage the representative is fixed before comparison by $\widehat g_{C,I}=s_I([g_C]_{\operatorname{Scale}})$ in (7.16). A reconstructive sequence is admitted only when these sections are compatible under the transports in (7.30), so that $s_\infty=\lim_n s_{I_n}$ is defined and $[M,\widehat g]_{\operatorname{Diff}} =s_\infty([M,g]_{\operatorname{Diff}\times\operatorname{Scale}})$. Only after reconstruction does an independent dimensional calibration supply a positive scale $\ell$ from a specified set $\mathcal L_{\rm dim}$. The dimensional metric family is

$$\mathcal M_{\rm dim} = \{[M,\ell^2\widehat g]_{\operatorname{Diff}}: [M,\widehat g]_{\operatorname{Diff}} =s_\infty([M,g]_{\operatorname{Diff}\times\operatorname{Scale}}),\ [M,g]_{\operatorname{Diff}\times\operatorname{Scale}} \in\operatorname{Rec}_{\rm met},\ \ell\in\mathcal L_{\rm dim}\}. \tag{7.37}$$

If the calibration set is non-singleton, that scale ambiguity is propagated into the gravitational and observable calculations rather than fixed by convention.

Finite search without a convergence or obstruction theorem establishes neither existence nor nonexistence. No metric is chosen from a non-singleton $\mathcal M$ unless a further physical principle distinguishes it.

The Lorentzian four-dimensional physical branch is the subset of (7.35) for which dimension estimators stabilize at four, the limiting tensor has signature $(-,+,+,+)$, the order agrees with its causal cones [29, 30, 31, 32], and the spin, QFT and hyperbolicity conditions hold. The monograph selects this branch as the target relevant to observed Tier-1 physics; R3 must determine whether it is nonempty and how unique it is.

7.3 Gravitational Dynamics and QFT–Geometry Consistency

7.3.1 Effective action, record dynamics and boundary data

For a dimensional metric in $\mathcal M_{\rm dim}$, causal backreaction is derived from the doubled closed-time-path functional

$$\begin{aligned} \Gamma_{\rm tot,CTP} ={}& S_{\rm grav}^{\rm ren}[g^+]-S_{\rm grav}^{\rm ren}[g^-] +S_M^{\rm dyn}[g^+,\Phi_M^+]-S_M^{\rm dyn}[g^-,\Phi_M^-]\\ &+\Gamma_{\rm QFT,dyn}^{\rm ren} [g^+,\Phi_q^+;g^-,\Phi_q^-\mid\omega] +\Gamma_{R,\rm CTP}[g^\pm,R^\pm;\mathcal A^{\rm obj}]\\ &+\chi_{MR}\Gamma_{MR,\rm CTP} +\chi_{\rm DM}\bigl(\Gamma_{DM}[g^+,\Phi_{DM}^+] -\Gamma_{DM}[g^-,\Phi_{DM}^-]\bigr)\\ &+\chi_{\rm DE}\bigl(\Gamma_{DE}[g^+,\Phi_{DE}^+] -\Gamma_{DE}[g^-,\Phi_{DE}^-]\bigr). \end{aligned} \tag{7.38}$$

The matter functional $S_M^{\rm dyn}$ contains the classical matter action and those matter effective operators not already contained in $\Gamma_{\rm QFT,dyn}^{\rm ren}$. The record term in (7.38) is not an automatic additional matter sector: $\Gamma_{R,\rm CTP}$ is the matched, record-exclusive remainder defined in (7.47)–(7.49), and $\Gamma_{MR,\rm CTP}$ contains only separately owned mixed interactions. The matching basis and renormalized operator decomposition are fixed before the gravitational solution relation is evaluated. The renormalized gravitational action is

$$S_{\rm grav}^{\rm ren} =S_G^{\mathfrak b_G}[g,\Psi_G] +\Gamma_{\rm high}^{\rm grav}[g], \tag{7.39}$$

where the constant and Einstein–Hilbert parts of $\Gamma_{\rm grav,loc}^{\rm ren}$ in (5.22) have been absorbed into $\Lambda_{\rm br}$ and $G_R$, and its remaining pure-metric higher-curvature terms occur once in $\Gamma_{\rm high}^{\rm grav}$. They do not also occur in the QFT stress.

The two allowed gravitational branches are

$$S_G^{\mathfrak b_G}[g,\Psi_G] = \begin{cases} \displaystyle \frac{1}{16\pi G_R}\int_M(R-2\Lambda_{\rm br})\,\mathrm{vol}_g +S_{\partial M}^{\rm Ord}[g], &\mathfrak b_G=\mathrm{Ord},\quad\Psi_G=\varnothing,\\[1.1ex] \displaystyle \begin{aligned}[t] \frac{1}{16\pi G_R}\int_MR\,\mathrm{vol}_g -\frac{1}{8\pi G_R}\int_M\lambda(\mathrm{vol}_g-dA_3)\\ +S_{\partial M}^{\rm HT}[g,\lambda,A_3], \end{aligned} &\mathfrak b_G=\mathrm{HT},\ \Psi_G=(\lambda,A_3). \end{cases} \tag{7.40}$$

On a manifold without boundary the boundary terms vanish. Otherwise $S_{\partial M}^{\rm Ord}$ contains the Gibbons–Hawking–York term [72, 73] on every non-null boundary component, the appropriate Hayward joint terms, and the standard null-boundary and null-joint terms (including the reparametrization counterterm when the null generators are not affinely fixed):

$$S_{\partial M}^{\rm Ord} =\frac{1}{8\pi G_R} \left[ \sum_{\Sigma}\epsilon_\Sigma\!\int_\Sigma K\,\mathrm{vol}_h +\sum_J\int_J\eta_J\,\mathrm{vol}_\sigma +\sum_N\int_N\kappa_N\,d\lambda\,\mathrm{vol}_\gamma \right]+S_{\rm null,ct}. \tag{7.41}$$

Every higher-curvature term carries the boundary completion required by the selected boundary data. In the HT branch, $S_{\partial M}^{\rm HT}$ contains the same metric terms together with the HT boundary completion, and the variational problem fixes the pullback/flux class of $A_3$:

$$\delta\!\left(\int_{\partial M}A_3\right)=0, \qquad \delta g|_{\partial M}=0 \quad\text{or the corresponding canonical boundary data.} \tag{7.42}$$

Variation with respect to $A_3$ and $\lambda$ then gives $d\lambda=0$ and the fixed-volume relation; call the integration constant $\lambda_{\rm HT}$. After constant vacuum pieces have been removed from the dynamical CTP functional and assigned once to the gravitational constant term, the HT branch uses

$$\Lambda_{\rm br} =\lambda_{\rm HT}+8\pi G_R\rho_{\rm vac}^{\rm const}. \tag{7.43}$$

In the ordinary branch $\Lambda_{\rm br}$ is the corresponding renormalized parameter. No copy of $\rho_{\rm vac}^{\rm const}$ remains in $T_{\mu\nu}^{\rm QFT,dyn}(\omega)$.

The objective archive enters dynamics through a specified covariant record realization. For each admitted metric $g$, define

$$\operatorname{Emb}_g(\mathcal A^{\rm obj}) = \left\{ (R^A,J_A^\mu): \begin{array}{l} \operatorname{supp}(R,J)\subseteq \operatorname{supp}_g(\mathcal A^{\rm obj}),\\ \nabla_\mu J_A^\mu =\sigma_A^{\rm dep}-\sigma_A^{\rm term},\\ \displaystyle \int_\Sigma J_A^\mu n_\mu\,\mathrm{vol}_\Sigma =Q_A(\mathcal A^{\rm obj};\Sigma) \end{array} \right\}. \tag{7.44}$$

The endpoint densities $\sigma_A^{\rm dep},\sigma_A^{\rm term}$, support and charges $Q_A$ are fixed by the actual-deposition and persistence data; they are not fitted to the gravitational solution. The embedding is natural under archive redescription. For an element of (7.44), first construct the unrestricted archive-induced closed-time-path influence functional

$$\begin{aligned} \Gamma_{\rm arch,CTP}^{\rm full} ={}&S_R[g^+,R^+,J^+]-S_R[g^-,R^-,J^-]\\ &+\frac12\int J_A^\Delta(x) K_{\rm ret}^{AB}(x,x';g)\, J_B^\Sigma(x')\,\mathrm{vol}_g(x)\mathrm{vol}_g(x')\\ &+\frac{i}{2}\int J_A^\Delta(x) N^{AB}(x,x';g)\, J_B^\Delta(x')\,\mathrm{vol}_g(x)\mathrm{vol}_g(x'), \end{aligned} \tag{7.45}$$

where $J^\Delta=J^+-J^-$, $J^\Sigma=(J^++J^-)/2$, $K_{\rm ret}$ has retarded support and $N$ is a positive noise kernel. The local part is drawn from the covariant class

$$S_R[g,R,J] = \int_M\!\sqrt{-g}\, \left[ -\frac12K_{AB}(R)\nabla_\mu R^A\nabla^\mu R^B -V_R(R)+J_A R^A \right]d^4x, \tag{7.46}$$

with $K_{AB}$ positive on propagating record modes and with boundary conditions inherited from the archive endpoints. The kernels and coefficients are fixed before solving the gravitational equation, are natural under archive redescription, and admit a well-defined physical-limit metric variation. Equations (7.45)–(7.46) define the full archive-induced functional before operator matching; they do not yet authorize adding all of it beside the matter and QFT actions.

Fix, at renormalization scale $\mu$, a renormalized CTP operator basis $\mathcal B_{\rm op}$ modulo integration by parts, field redefinitions and the admitted equations-of-motion redundancies. The matching datum

$$\mathfrak m_R = \bigl( \mu,\mathcal B_{\rm op}, \Pi_M,\Pi_{\rm QFT},\Pi_R,\Pi_{MR} \bigr), \qquad \Pi_a\Pi_b=\delta_{ab}\Pi_a,\qquad \sum_{a\in\{M,\rm QFT,R,MR\}}\Pi_a=\mathbf1, \tag{7.47}$$

is a fixed renormalized operator decomposition of the 1PI operator kernels. Its $M$ and QFT ranges consist precisely of terms already owned, after coefficient matching, by $S_M^{\rm dyn}$ and $\Gamma_{\rm QFT,dyn}^{\rm ren}$; its $R$ range contains independent record operators; and its $MR$ range contains genuinely mixed matter–record or QFT–record operators that belong to neither separate sector. Define

$$\begin{aligned} \operatorname{Match}_M(\Gamma_{\rm arch,CTP}^{\rm full}) &:=\Pi_M\Gamma_{\rm arch,CTP}^{\rm full},& \operatorname{Match}_{\rm QFT}(\Gamma_{\rm arch,CTP}^{\rm full}) &:=\Pi_{\rm QFT}\Gamma_{\rm arch,CTP}^{\rm full},\\ \Gamma_{R,\rm CTP} &:=\Pi_R\Gamma_{\rm arch,CTP}^{\rm full},& \Gamma_{MR,\rm CTP} &:=\Pi_{MR}\Gamma_{\rm arch,CTP}^{\rm full}, \end{aligned} \tag{7.48}$$

so that, equivalently,

$$\Gamma_{R,\rm CTP}+\Gamma_{MR,\rm CTP} = \Gamma_{\rm arch,CTP}^{\rm full} -\operatorname{Match}_M(\Gamma_{\rm arch,CTP}^{\rm full}) -\operatorname{Match}_{\rm QFT}(\Gamma_{\rm arch,CTP}^{\rm full}). \tag{7.49}$$

The two matched projections in (7.48) alter the already-owned coefficients in $S_M^{\rm dyn}$ and $\Gamma_{\rm QFT,dyn}^{\rm ren}$ and are not added again in (7.38). The partition is transported with the same renormalization-group and threshold-matching prescription as those coefficients and cannot be changed after inspecting a gravitational solution. If the record fields merely reparameterize degrees of freedom already owned by the matter or QFT sectors and no record-exclusive remainder survives, then $\Gamma_{R,\rm CTP}=0$. If there is also no separately owned mixed interaction, $\Gamma_{MR,\rm CTP}=0$. In that branch the record stress and current defined below vanish, while $\operatorname{Emb}_g(\mathcal A^{\rm obj})$ remains the kinematical map that relates objective archive support to the reconstructed spacetime.

The switches $\chi_{MR},\chi_{\rm DM},\chi_{\rm DE}\in\{0,1\}$ state whether an independent matter–record, dark-matter or dynamical-dark-energy action sector is present. Consistency requires $\chi_{MR}=1$ exactly when the separately owned mixed projection is retained; otherwise $\Pi_{MR}\Gamma_{\rm arch,CTP}^{\rm full}=0$. A field already included in $S_M^{\rm dyn}$ or $\Gamma_{\rm QFT,dyn}^{\rm ren}$ has the corresponding independent-sector switch set to zero. A pure cosmological-constant branch has $\chi_{\rm DE}=0$ and represents its constant solely by $\Lambda_{\rm br}$.

At the physical limit $g^+=g^-=g$, define

$$T_{\mu\nu}^{M,\rm dyn} =-\frac{2}{\sqrt{-g}}\frac{\delta S_M^{\rm dyn}}{\delta g^{\mu\nu}}, \qquad T_{\mu\nu}^{\rm QFT,dyn}(\omega) =\langle T_{\mu\nu}^{\rm QFT,dyn}\rangle_\omega \tag{7.50}$$

with the second expression given by (5.24), and

$$\begin{aligned} T_{\mu\nu}^{R} &=-\frac{2}{\sqrt{-g}} \left. \frac{\delta\Gamma_{R,\rm CTP}} {\delta g^{+\mu\nu}}\right|_{+=-},\\ T_{\mu\nu}^{MR} &=-\frac{2}{\sqrt{-g}} \left. \frac{\delta\Gamma_{MR,\rm CTP}} {\delta g^{+\mu\nu}}\right|_{+=-},\\ \mathcal Q_{\mu\nu}^{\rm grav} &=\frac{16\pi G_R}{\sqrt{-g}} \frac{\delta\Gamma_{\rm high}^{\rm grav}}{\delta g^{\mu\nu}} . \end{aligned} \tag{7.51}$$

For later sector equations, use the unambiguous shorthand

$$T_{\mu\nu}^{R,\rm tot} := T_{\mu\nu}^{R}+\chi_{MR}T_{\mu\nu}^{MR}. \tag{7.52}$$

If $\Pi_R\Gamma_{\rm arch,CTP}^{\rm full}=0$, then $\Gamma_{R,\rm CTP}=0$ and (7.51) gives $T_{\mu\nu}^{R}=0$ identically. The mixed stress $T_{\mu\nu}^{MR}$ survives only when the independently owned interaction projection $\Pi_{MR}\Gamma_{\rm arch,CTP}^{\rm full}$ is nonzero and $\chi_{MR}=1$. Neither conclusion removes the archive embedding (7.44).

For independent dark action sectors,

$$T_{\mu\nu}^{\rm DM} =-\frac{2}{\sqrt{-g}} \frac{\delta\Gamma_{DM}}{\delta g^{\mu\nu}}, \qquad T_{\mu\nu}^{\rm DE} =-\frac{2}{\sqrt{-g}} \frac{\delta\Gamma_{DE}}{\delta g^{\mu\nu}}. \tag{7.53}$$

The gravitational equation is

$$\boxed{ G_{\mu\nu}[g]+\Lambda_{\rm br}g_{\mu\nu} +\mathcal Q_{\mu\nu}^{\rm grav} = 8\pi G_R \left( T_{\mu\nu}^{M,\rm dyn} +T_{\mu\nu}^{\rm QFT,dyn}(\omega) +T_{\mu\nu}^{R,\rm tot} +\chi_{\rm DM}T_{\mu\nu}^{\rm DM} +\chi_{\rm DE}T_{\mu\nu}^{\rm DE} \right). } \tag{7.54}$$

This allocation counts every action and stress contribution once. Cosmological specialization decomposes the terms in (7.54) but does not add a second copy of any field, vacuum term or local gravitational counterterm.

7.3.2 Solution relation and conservation

For a dimensional metric family $\mathcal M_{\rm dim}$ obtained from $\operatorname{Rec}_{\rm met}$, QFT state data $\omega$, objective archive $\mathcal A^{\rm obj}$, admissible initial/boundary data $B$, and matching datum $\mathfrak m_R$ fixed as in (7.47) before solving, define

$$\begin{aligned} &\operatorname{Sol}_G( \mathcal M_{\rm dim},\omega,\mathcal A^{\rm obj},B;\mathfrak m_R)\\ &\quad= \left\{(g,\Phi): \begin{aligned} &g\in\mathcal M_{\rm dim},\\ &(g,\Phi)\text{ satisfies (7.54) and }B,\\ &\left. \frac{\delta\Gamma_{\rm tot,CTP}}{\delta\Phi^+} \right|_{+=-}=0,\\ &\text{the Ward/Bianchi identities hold} \end{aligned} \right\}. \end{aligned} \tag{7.55}$$

This relation may be empty or nonunique. The matching datum is part of the input, not an output of $\operatorname{Sol}_G$; its operator ranges, subtractions, scale and threshold prescription are therefore identical for every metric sibling tested by the relation. If the reconstructed or dimensionally calibrated family is controlled nonunique, every member is propagated into (7.55); a preferred solution is not chosen by notation.

The Bianchi identity requires the total right-hand side of (7.54) to be covariantly consistent. With typed exchange currents,

$$\begin{aligned} \nabla^\mu T_{\mu\nu}^{M,\rm dyn} &=J_\nu^M,\\ \nabla^\mu T_{\mu\nu}^{\rm QFT,dyn} &=J_\nu^{\rm QFT},\\ \nabla^\mu T_{\mu\nu}^{R} &=J_\nu^R,\\ \nabla^\mu T_{\mu\nu}^{MR} &=J_\nu^{MR},\\ \nabla^\mu T_{\mu\nu}^{\rm DM} &=J_\nu^{\rm DM},\\ \nabla^\mu T_{\mu\nu}^{\rm DE} &=J_\nu^{\rm DE}. \end{aligned} \tag{7.56}$$

and therefore

$$J_\nu^M+J_\nu^{\rm QFT}+J_\nu^R+\chi_{MR}J_\nu^{MR} +\chi_{\rm DM}J_\nu^{DM}+\chi_{\rm DE}J_\nu^{DE} =0. \tag{7.57}$$

On the record-reparameterization branch $\Gamma_{R,\rm CTP}=0$, one has $T_{\mu\nu}^{R}=0$ and $J_\nu^R=0$; a nonzero $J_\nu^{MR}$ is possible only for the separately owned mixed interaction. This identity holds on the equations of motion for fields that exchange energy and momentum. If nondynamical backgrounds are present, their explicitly displayed force densities are included. Independently, diffeomorphism invariance of the pure-metric higher-curvature action gives the interior Noether identity

$$\nabla^\mu\mathcal Q_{\mu\nu}^{\rm grav}=0, \tag{7.58}$$

with boundary fluxes cancelled by the boundary terms and boundary conditions in (7.41)–(7.42). Conservation is therefore not obtained by moving an arbitrary effective-action divergence between the two sides of the gravitational equation.

The pure-gravity and matter effective operators are kept distinct [35]:

$$\Gamma_{\rm high}^{\rm grav} = \int d^4x\sqrt{-g} \left( c_1R^2+c_2R_{\mu\nu}R^{\mu\nu} +c_3R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} +\sum_{d_i>4}\frac{c_i^G}{M_*^{d_i-4}}\mathcal O_i^G \right), \tag{7.59}$$

with the boundary completions required above. Matter operators occur only in the matched $S_M^{\rm dyn}$; QFT operators occur only in the matched $\Gamma_{\rm QFT,dyn}^{\rm ren}$; record-exclusive operators occur only in $\Gamma_{R,\rm CTP}$; and independently owned mixed operators occur only in $\Gamma_{MR,\rm CTP}$, according to (7.47)–(7.49). A curvature-squared bulk combination is topological in four dimensions only together with its boundary contribution. At characteristic energy $E$, an operator of dimension $d_i>4$ carries the stated power of $E/M_*$, while a curvature-squared term is smaller than the Einstein–Hilbert term by order $c_iE^2/M_P^2$, where $M_P^2=(8\pi G_R)^{-1}$, unless matching produces an anomalously large coefficient. Classical GR is recovered when the higher-curvature and record-exclusive corrections are negligible and the matter stress has a classical limit. Semiclassical gravity retains the state-dependent $\langle T_{\mu\nu}^{\rm QFT,dyn}\rangle_\omega$ on a classical metric. Flat-spacetime QFT is the weak-curvature, negligible-backreaction limit of the coupled system.

7.3.3 R3 starting point

Quantum-record-geometry theorem target (R3). Starting from an objective archive and the source-cylinder perturbations (7.2)–(7.5), prove that the influence capacity is well defined and natural under refinement and that the cycle quotient yields the partial order (7.6). Establish the well-definedness, redescription naturality and refinement stability of the strong metric, cell Hausdorff distance, induced order and volume measure, and normalized graph Laplacian in (7.17)–(7.22), including the stated Lorentzian regularity hypotheses. Prove the identity and composition compatibility of the common-refinement transports in (7.30), and prove that the scale section (7.16), the path-pseudometric uniformity (7.30)–(7.33) and the reconstruction relation (7.35) are independent of presentation and stable under archive refinement. Characterize precisely when raw candidates fail to become metric-comparable. Establish conditions under which the Lorentzian four-dimensional branch is nonempty and unique or controlled, relative only to the complete observable vector fixed in advance in (7.36), and propagate every untested or surviving scale and metric multiplicity through (7.37). Finally, prove existence and stability conditions for the same-branch system (8.11), including the operational QFT completion, regenerated objective archive, the matched record-exclusive influence functional and its zero-remainder branch (7.45)–(7.49), the pre-solution matching decomposition $\mathfrak m_R$, gravitational boundary problem and fixed-flux branch (7.40)–(7.43), solution relation (7.55), Noether identities (7.57)–(7.58), and the classical, semiclassical and effective-field-theory limits.

The constructions to be analysed are fixed here. Proof of completion existence for interacting QFT, reconstruction convergence, physical nonemptiness, numerical coefficients and global gravitational solutions remains work for the R3 and R9 companion papers.

7.4 Recovery limits

The required conditional limits are:

• classical GR when $\Gamma_{\rm EFT}\to0$ and the stress data admit a classical regime;

• semiclassical gravity with classical $g$ and renormalized quantum expectation stress;

• flat-spacetime QFT on a branch approaching Minkowski geometry with negligible backreaction;

• EFT/QG suppression $c_i\mathcal O_i/M_*^{d_i-4}$ for $d_i>4$ at $E\ll M_*$.

These limits are theorem conclusions under stated hypotheses, not source assumptions.

7.5 Reconstructed Geometry Downstream

The reconstructed metric families and gravitational dynamics of this chapter are consumed by the quantum–record–geometry bridge of Chapter 8, the matter and QFT couplings of Chapter 9, and the cosmological branch of Chapter 12. The reconstruction, convergence and same-branch existence questions are collected in the R3 programme of Chapter 17.