SAE Relativity Series P7: The Three-Tier Equivalence Principle under SAE
SAE 相对论系列 P7：SAE 之下的等效原理三层

SAE Relativity Series, Paper P7

Author: Han Qin (秦汉)

Series position: SAE Relativity Series, Paper P7. Immediate predecessors: P5 (Schwarzschild radial closure on the ∂_t channel, DOI 10.5281/zenodo.20105112) and P6 (Kerr helical closure on the χ_H horizon generator, DOI 10.5281/zenodo.20131993).

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§1 Introduction

The SAE Relativity Series Paper P5 articulates the substrate ontology reading of the static spherically symmetric vacuum Schwarzschild geometry. The timelike Killing field ∂_t serves both as the natural readout channel and as the radial closure channel: R_t^sub → 0 at the horizon expresses substrate-level radial closure on ∂_t, without re-proving the Birkhoff theorem. Rather, the standard mathematical theorem receives a substrate-level reading. Paper P6 articulates the substrate ontology reading of the stationary axisymmetric vacuum Kerr geometry. The horizon generator χ_H = ∂_t + Ω_H ∂_φ, null on r_+, supplies the helical closure channel through the tensor contraction R_{χχ}^sub := R_{μν}^sub χ^μ χ^ν as a quadratic form, and the closure R_{χχ}^sub → 0 on r_+ articulates the substrate-level helical closure. Together P5 and P6 establish the substrate-apparatus distinction discipline (衬底-装置区分纪律) as a cross-paper systematic methodology: apparatus-level mathematical features such as the Schwarzschild coordinate divergence g_rr → ∞ at the horizon, the Kerr ring singularity, the inner horizon r_-, the r<0 analytic continuation region, closed timelike curves in the deep Kerr interior, and the extremal infinite throat geometry, all sit at the apparatus layer, while the substrate layer maintains a finite cell count and a well-defined cell tensor across all these apparatus features.

P5 and P6 articulate substrate readouts within specific spacetime symmetry contexts. P5 applies to SO(3) spherical symmetry combined with staticity (timelike Killing field ∂_t together with three rotational Killing fields), and the substrate cell tensor R_μν^sub in the radial natural basis takes a Type 1 diagonal structure carrying four independent components. P6 applies to SO(2) axisymmetric stationarity (timelike Killing field ∂_t combined with the axial Killing field ∂_φ), and the substrate cell tensor in the axisymmetric natural basis takes a Type 2 mixed structure: diagonal components plus an off-diagonal R_{tφ}^sub component permitted by the axisymmetric stationary symmetry, carrying five independent components total. P6 §10.3 surfaces a symmetry hierarchy observation: P5 SO(3), P6 SO(2), and an anticipated P7 with local Lorentz only would form a series-level symmetry hierarchy pattern. P6 merely observes the pattern; full articulation is reserved for P7.

P7 treats the generic spacetime context, where no global symmetry beyond local Lorentz frames is assumed. This is the extreme limit case in the series symmetry hierarchy: no timelike Killing field, no spatial Killing field, only local Lorentz frames at each point as a baseline manifold property. P7 articulates the substrate cell tensor in this generic context as a Type 3 structure carrying ten independent components (the full count of a symmetric rank-2 tensor on a 4D Lorentzian vector space), and articulates the three-tier equivalence principle (weak EP, Einstein EP, and strong EP, hereafter the WEP, EEP, and SEP) as the substrate ontology re-reading under SAE. This is the main thesis of P7.

The backbone of P7 is the EP-spacetime structure correspondence, an observed pattern across the P5, P6, and P7 cases. P5: SO(3) ↔ Type 1 ↔ radial closure on the ∂_t channel. P6: SO(2) ↔ Type 2 ↔ helical closure on the χ_H horizon generator. P7: local Lorentz only ↔ Type 3 ↔ local calibration with no global closure channel. The three-tier equivalence principle in P7 receives articulation as the manifestation of the spacetime symmetry hierarchy in its extreme limit case. The framing is identificatory rather than derivational: the correspondence is an observed pattern across cases, not a theorem derived from SAE axioms. Under SAE, the equivalence principle is not a set of standalone foundational principles imposed on top of spacetime structure but the substrate-level articulation of the observed correspondence between spacetime symmetry hierarchy and substrate cell tensor structure, manifest at the extreme limit where global symmetry recedes to local Lorentz only.

P7 is the last paper of the SAE Relativity Series. The closing weight is carried by substantive content rather than by length or rhetorical elevation. P7 maintains the length pattern of P5 and P6 and does not inflate into a grand series essay. The series closure occurs implicitly through the substantive content of §8-§12 (the EP three-tier re-reading and the backbone articulation), with a short reflection in §14 (one page maximum) supplying a sober closing acknowledgment. The series-level systematic methodology of SAE as a framework for GR foundations, quantum gravity, and emergent gravity sits in the territory of Paper 0 (DOI 10.5281/zenodo.19777881) and is not undertaken by P7.

Several scope disciplines hold throughout P7.

P7 operates within a strict non-dynamical and locally kinematic calibration scope. P7 treats the local calibration structure of generic spacetimes without assuming a timelike Killing field or any global symmetry. Dynamical evolution, gravitational wave propagation, black hole mergers, and time-dependent curvature are handed off entirely to Info VI (DOI 10.5281/zenodo.20066644). P5 (static) and P6 (stationary) are specific symmetry subsets of P7's generic non-dynamical scope: every P5 and P6 stationary case sits within P7's generic non-dynamical territory.

P7 stays within qualitative, directional, and explanatory territory. The SAE Relativity Series consists of substrate ontology philosophy papers; it does not undertake quantitative judgment on behalf of physicists. P7 produces no quantitative predictions diverging from those of standard GR. SEP-sensitive empirical tests (Nordtvedt parameter, binary pulsar strong-field universality tests, lunar laser ranging, and similar) and the classical GR tests (Mercury perihelion precession, light deflection, Shapiro delay) are not the substantive subject of the P7 main body; these belong to the experimental GR community. P7 articulates equivalence principle substrate ontology vocabulary as a working lens and makes no claim of empirical superiority over EP testing.

P7's articulation is identificatory rather than derivational. The EP-spacetime structure correspondence backbone is articulated as an observed pattern across the P5, P6, and P7 cases, not as a derivation from SAE axioms. This stance is consistent with the epistemological position of the SAE mathematical foundational paper Schema, where substrate ontology articulation supplies an ontological framing for GR mathematical machinery without replacing or deriving GR.

P7 maintains the series-wide reject items stance. The T4 quantitative anchor remains out of the main body, as it crosses into the Info VI quantitative territory. Alternative gravity-foundational framings remain out of main body citation; this includes Asymptotic Safety, Causal Set theory, the Verlinde entropic gravity program, the Jacobson thermodynamic derivation, Ashtekar variables, loop quantum gravity, category-theoretic parallels, and operationalist Einstein-1916 reframings. Such parallels, where structurally interesting, are reserved for an anticipated future SAE quantum gravity interface paper or are not articulated at all. Low-dimensional toy models remain out of scope, as they cross into the quantitative paper territory.

P7 maintains the Paper 0 §1.5 honest stance. The SAE Relativity Series does not claim empirical superiority over standard GR or QFT. The series deliverable is substrate ontology articulation and a working lens for physicists; it is not a framework competing with GR or QFT in empirical predictions.

Reference accuracy with respect to Paper P1 is maintained throughout. The SAE Relativity Paper P1 (DOI 10.5281/zenodo.19836183) treats causal-slot throughput in gravitational time dilation and is not the Lense-Thirring paper. The Lense-Thirring SAE rereading appears as a subsection of Paper P2 (DOI 10.5281/zenodo.19910545).

The paper is organized into fourteen sections. §1 Introduction. §2 Preliminaries, covering tetrad formalism and the absence of any global Killing symmetry assumption. §3 Substrate-apparatus distinction discipline as a universal statement. §4 Cell tensor Type 3 structure. §5 Cell tensor Type 1, Type 2, and Type 3 systematic hierarchy. §6 Calibration Isomorphism break mode taxonomy (Mode 0 through Mode 4). §7 Three-tier EP substrate ontology re-reading overview. §8 WEP substrate ontology re-reading. §9 EEP substrate ontology re-reading. §10 SEP substrate ontology re-reading (three-tier split). §11 EP-spacetime structure correspondence backbone. §12 Symmetry hierarchy backbone and three-tier EP correspondence. §13 Uniqueness trajectory and series completion. §14 Honest boundaries, future territory, and short closing reflection.

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§2 Preliminaries

P7 articulates substrate ontology in a generic spacetime context. Several technical setups are required. Standard GR formalism (local Lorentz frames, tetrad / vierbein, standard differential geometry) is not re-derived here, as this material sits firmly within standard GR textbooks (MTW, Wald, Carroll, and others). P7 only articulates how these formalisms read under the substrate ontology lens and how they connect to the SAE framework.

Tetrad formalism, brief recap. On a generic 4D Lorentzian spacetime (M, g_μν), every point p ∈ M admits a local Lorentz frame: four orthonormal vector fields {e_a^μ}_{a=0,1,2,3} satisfying g_μν e_a^μ e_b^ν = η_ab, where η_ab = diag(-1, +1, +1, +1) is the Minkowski metric, μ is a spacetime coordinate index, and a is a local Lorentz frame index. The tetrad {e_a^μ} is also called a vierbein (German "four-leg"). The tetrad is local: no universal natural basis across spacetime points is assumed. Local Lorentz transformations Λ_a^b act at the same point p to reframe the tetrad as e_a^μ → Λ_a^b e_b^μ while preserving η_ab. Tetrad-to-tetrad transport across different spacetime points is articulated through the spin connection ω_μ^{ab}, which is related to the affine connection Γ^λ_{μν} (the Christoffel symbols of g_μν) through the tetrad postulate.

No global Killing symmetry assumption. P5 applies to the static spherically symmetric Schwarzschild metric, which carries a timelike Killing field ∂_t together with three SO(3) rotational Killing fields. P6 applies to the stationary axisymmetric Kerr metric, which carries a timelike Killing field ∂_t, an axial Killing field ∂_φ, and the horizon generator χ_H = ∂_t + Ω_H ∂_φ. P7 makes no assumption of any global Killing field, neither timelike nor spatial. Local Lorentz frames exist at every point (this is a standard manifold property and does not imply any global symmetry), but no cross-point natural basis is available, as local Lorentz frames at different points are not connected through any global Killing field. This is the typical configuration of a generic non-dynamical spacetime.

Substrate cell tensor, recap. Within the SAE framework, the substrate level comprises causal slots (因果槽), a cell count, and the reading-and-connection (读取加连接) ontology. The substrate cell tensor R_μν^sub is the substrate-level rank-2 symmetric tensor articulating the cell coupling structure. Note that R_μν^sub is distinct from the standard apparatus-level Ricci tensor R_μν despite the similar notation; the substrate (衬底) and apparatus (装置) layers are kept categorically separate in the SAE framework. The substrate cell count is a finite, ontological measure ("how many cells"), while the substrate cell tensor is an articulation of the cell coupling pattern ("how cells couple"). These two aspects are independent: cell count and cell tensor structural complexity do not directly determine each other.

P5 articulates the substrate cell tensor R_μν^sub in the Schwarzschild spherically symmetric vacuum context as a Type 1 diagonal structure with four independent components: R_t^sub (timelike component), R_r^sub (radial component), and R_⊥^sub (the perpendicular components, the two of which are equal by SO(3) symmetry). P5 §3.1 articulates that the apparatus-level coordinate-dependent feature g_rr → ∞ at the horizon is a coordinate artifact, while at the substrate level R_t^sub → 0 on the horizon articulates radial closure on the ∂_t channel (i.e., on the timelike Killing field direction).

P6 articulates the substrate cell tensor R_μν^sub in the Kerr stationary axisymmetric vacuum context as a Type 2 mixed structure with five independent components: the diagonal R_{tt}^sub, R_{rr}^sub, R_{θθ}^sub, R_{φφ}^sub together with the off-diagonal R_{tφ}^sub permitted by the axisymmetric stationary symmetry. P6 articulates R_{χχ}^sub := R_{μν}^sub χ^μ χ^ν on the horizon generator χ_H = ∂_t + Ω_H ∂_φ, where R_{χχ}^sub → 0 on r_+ realizes the substrate-level helical closure.

Apparatus metric and substrate cell tensor. Within SAE, the apparatus metric g_μν is the apparatus-level metric structure, and it relates to the substrate cell tensor through the reading-and-connection mechanism. P5 and P6 articulate this relation in specific symmetry contexts (spherical and axisymmetric respectively). P7 articulates the relation in generic spacetime through the Calibration Isomorphism: local apparatus calibration at each point absorbs the first-derivative readout structure of the substrate (the apparatus metric Christoffel structure) into local apparatus units, locally recovering the Minkowski form. This is the main content of §9 EEP substrate ontology re-reading, and §2 only acknowledges it in outline.

Calibration Isomorphism, inherited from P4. Paper P4 (DOI 10.5281/zenodo.20079718) articulates the cell tensor d_eff^μν, the Calibration Isomorphism (Cal Iso), and the L₃→L₄ closure. Within the P4 framework, the Cal Iso is the well-definedness of the mapping between substrate-level cell tensor and apparatus-level metric: local apparatus calibration succeeds in providing a local reading of the substrate cell tensor. P5 articulates a Cal Iso break of Mode 1 (projection singularity at the horizon, R_t^sub → 0 on ∂_t). P6 articulates Mode 2 (component activation, R_{tφ}^sub in the Kerr ergosphere) together with several composite mode cases, of which the extremal Kerr infinite throat is a composite case of Mode 1 + Mode 2 + Mode 4. P7 articulates Mode 3 (no global natural basis, P7's principal mode) and provides a systematic Mode 0 through Mode 4 taxonomy in §6.

N_active, d_eff^(τ), and d_eff^μν: a three-level notation discipline. The SAE framework cross-paper carries three levels of effective-dimensional notation:

• N_active: an integer active-direction count, the number of active directions in the substrate-level cell coupling.
• d_eff^(τ): a clock-sector effective exponent, articulated in Paper P3 (DOI 10.5281/zenodo.19992252) as the effective-dimensional exponent of substrate-level clock readout. In the P3 context, this quantity approaches 3 in the weak-field limit and takes values in (2, 3) or approaches 3⁻ in the strong-field region; this is the clock-sector scalar usage, not to be conflated with the full tensorial effective dimension d_eff^μν.
• d_eff^μν: the full tensorial effective dimension, articulated in P4 as the full tensor articulation of the substrate cell tensor.

These are three distinct notational aspects: an integer count, a scalar exponent, and a tensor. They are used consistently across the series. P7 does not re-derive these three levels (they are established by P3 and P4) and merely acknowledges them as part of the framework's notational discipline.

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§3 Substrate-Apparatus Distinction Discipline: Universal Statement

The substrate-apparatus distinction discipline (衬底-装置区分纪律) is a cross-paper systematic methodology of the SAE Relativity Series. P5 and P6 establish its application instances in specific symmetry contexts; P7 articulates the universal statement that applies in the generic spacetime context: apparatus-level mathematical features and substrate-level ontological commitments must be read separately, with the substrate's finite cell count maintained across all apparatus pathologies. This is the working ontology lens that the SAE Relativity Series offers to physicists: one does not need to reject the mathematical machinery of GR, nor is one forced to accept GR mathematical features as substrate-level reality.

Cross-paper case enumeration in P5 and P6. The substrate-apparatus distinction discipline is exercised through multiple application instances across P5 and P6.

P5 §3.1 articulates the Schwarzschild coordinate divergence g_rr → ∞ at the horizon as an apparatus-level coordinate-dependent feature. The Schwarzschild coordinates are coordinate-singular at the horizon, while the Eddington-Finkelstein coordinates and the Kruskal coordinates are coordinate-regular there. At the substrate level, the cell tensor R_μν^sub and the cell count remain finite across the horizon, with R_t^sub → 0 articulating radial closure on the ∂_t channel as a substrate-level feature (a substrate channel closure, not a coordinate singularity). The apparatus-level feature (g_rr → ∞) and the substrate-level feature (R_t^sub → 0) are read separately.

P5 §5.4 articulates the Schwarzschild r → 0 limit as an apparatus-level point. In the standard Schwarzschild chart, r = 0 is a coordinate-dependent point feature, and the apparatus-level mathematical structure carries a Kretschmann-scalar divergence K = R_{μνρσ} R^{μνρσ} → ∞ as r → 0; this is an apparatus geometry feature. At the substrate level, the cell count at r = 0 is articulated as finite (one substrate cell, or some similarly small finite number), not as a zero-dimensional infinite-curvature reality.

P6 §13.1 articulates the Kerr ring singularity, given by ρ² := r² + a² cos²θ = 0 when both r = 0 and θ = π/2 hold and which forms a one-dimensional ring in apparatus geometry, as an apparatus-level feature carrying Kretschmann-scalar divergence on the one-dimensional ring. At the substrate level, the cell count on the ring remains finite; the ring is an algebraic feature of the apparatus projection, not a substrate-level one-dimensional object.

P6 §13.2 articulates the Kerr inner horizon r_- = M - √(M² - a²) as an apparatus-level coordinate boundary. Within r < r_-, the Boyer-Lindquist coordinates exhibit a spacelike-timelike switch together with mass-inflation phenomenology. At the substrate level, the cell tensor and the cell count are finite and well-defined both at r = r_- and in the r < r_- region, while the mass-inflation phenomenology and the coordinate switch sit at the apparatus level.

P6 §13.3 articulates the Kerr r < 0 region (the analytic continuation past the r = 0 ring into the negative-r region) as an apparatus-level analytic-continuation feature, not as a substrate-level "negative radial coordinate" reality. At the substrate level, the cell count in the r < 0 region and in the standard r > r_+ region both remain finite; the r < 0 region is apparatus geometry analytic extension territory and not a substrate ontological extension.

P6 §13.4 articulates the Kerr closed timelike curves (CTCs) appearing in the r < 0 region as apparatus-level geometric features (timelike curves closing on themselves in apparatus geometry). At the substrate level, the reading-and-connection ontology of the cell coupling does not imply any substrate-level causal-loop reality. CTCs are a geometric feature of the apparatus projection and not a substrate-level temporal loop.

P6 §11.3 articulates the extremal Kerr infinite throat (a = M, near-horizon NHEK geometry). In the extremal limit, the proper radial distance from any finite r > r_+ to r_+ = M diverges, and this is an apparatus-level "infinite throat" geometry feature. At the substrate level, the cell count and the cell tensor in the extremal Kerr horizon throat remain finite; the infinite throat is a property of the apparatus radial coordinate's proper-distance behavior, not a substrate cell count infinity.

These cases all articulate one and the same pattern: apparatus-level mathematical features (singularities, coordinate boundaries, analytic continuations, closed timelike curves, infinite proper distances) are mathematical structures of the apparatus-layer articulation, while the substrate layer maintains a finite cell count and a well-defined cell tensor across these apparatus features without inheriting any apparatus mathematical pathology.

Universal stance in generic spacetime context. P7 articulates the substrate-apparatus distinction discipline as a universal statement applicable to generic spacetimes: for any spacetime metric g_μν, any coordinate chart, and any mathematical pathology in apparatus geometry, the substrate-apparatus distinction discipline applies. This furnishes physicists with a working ontology lens: when dealing with any spacetime mathematical feature, one may distinguish between apparatus-level features (apparatus-layer mathematical structure, including coordinate, metric, and curvature-scalar behavior) and substrate-level readings (substrate-layer cell-count and cell-tensor articulation, at the level of ontological commitment).

The practical significance for physicists is that one need not accept the SAE framework as a whole in order to use the substrate-apparatus distinction discipline as an organizing principle for thinking about GR pathologies. Some illustrative examples:

• Is the Schwarzschild horizon coordinate divergence g_rr → ∞ a true singularity or a coordinate artifact? The substrate-apparatus distinction discipline articulates it as an apparatus-level coordinate feature, without forcing one to accept any substrate-level "horizon as ontological boundary".
• Is the Kerr ring singularity a one-dimensional curvature divergence in reality? The substrate-apparatus distinction discipline articulates it as an apparatus-level ring feature, without forcing acceptance of a substrate-level "one-dimensional singular object".
• How does Hawking radiation entropy relate to the Bekenstein-Hawking horizon area entropy? The substrate-apparatus distinction discipline articulates the apparatus-level horizon area and thermodynamic entropy as one reading, and the substrate-level cell count microstate enumeration as another, supplying an ontological framing for the quantum black hole information puzzle without forcing the adoption of any specific quantization scheme. The articulation here in §3 is at the working-vocabulary level only; the substantive substrate ontology of the quantum black hole information puzzle is deferred to the anticipated future SAE quantum gravity interface paper noted in §14.

Range and boundary of the working lens. The substrate-apparatus distinction discipline is a working ontology lens, not a derivational tool. It does not derive any specific quantitative prediction that differs from GR, does not replace the mathematical machinery of GR, and does not claim empirical superiority over GR. It articulates an organizing principle that supplies physicists with a working framework for distinguishing mathematical features from ontological commitments when thinking about GR pathologies. This is the substantive contribution C1 of P7 (substrate-apparatus distinction discipline universal statement) as a working lens for physicists.

The discipline does not imply that apparatus-level mathematical features are "not real" or "fictitious". The apparatus layer is a substantive layer within the SAE framework; apparatus and substrate are both substantive aspects of the articulation. What the discipline articulates is that the two layers should not be conflated, and that apparatus-level mathematical features should not be assumed to correspond one-to-one with substrate-level ontology.

§4 Cell Tensor Type 3 Structure

P7 articulates the substrate cell tensor R_μν^sub in the generic spacetime context. With no global symmetry beyond local Lorentz frames, no universal natural basis exists, and the cell tensor admits its full structure: a Type 3 rank-2 symmetric tensor with ten independent components in 4D. This section provides the substrate-side structural articulation of Type 3; the full P5/P6/P7 correspondence backbone is reserved for §11, and the EP-side correspondence detail for §12.

Generic spacetime: no universal natural basis. In a generic non-dynamical spacetime, no global Killing field is available, and no preferred cross-point natural basis can be assumed. The substrate cell tensor R_μν^sub at a point p is articulated as a full rank-2 symmetric tensor in 4D, carrying the maximum independent component count.

Tetrad-readable structure. Per local Lorentz frame e_a^μ(p) at point p, the cell tensor reads:

R_ab^sub(p) = e_a^μ(p) e_b^ν(p) R_μν^sub(p)

Here a, b are local Lorentz frame indices and μ, ν are spacetime coordinate indices. The local tetrad furnishes a component representation of the cell tensor at p; it does not furnish a universal diagonalization. In special algebraic cases (specific Segre type degeneracies, where the cell tensor and the apparatus metric share an eigenstructure), partial or full diagonalization in a single tetrad may be possible. In the generic case, however, no such alignment is guaranteed: a generic Type 3 cell tensor retains substrate components that do not simultaneously diagonalize with the apparatus metric in a single tetrad. The local tetrad gives a component representation, not a universal diagonalization.

Component count in 4D. A symmetric rank-2 tensor on a 4D Lorentzian vector space carries ten independent components. The standard decomposition reads: one trace component (T = η^ab R_ab^sub) plus nine traceless components (the latter further decomposable into five traceless symmetric spatial components, three vector-type components mixing time and space, and one scalar component, following standard 4D rank-2 symmetric tensor decomposition). The ten-component count is a generic-rank-2-symmetric-tensor bookkeeping; the upstream physical importance of any particular component is determined by local calibration and by the specific source or context at hand, not by the bookkeeping itself. This is a guard against the misreading that a Type 3 cell tensor functions as an "all-purpose ten-parameter explainer".

The principal articulation of Type 3. The ontology of a Type 3 cell tensor is not that all components are of equal weight; rather, it is that no global symmetry is available to specify in advance which components may be discarded. Type 1 (P5, SO(3) spherical) and Type 2 (P6, SO(2) axisymmetric) both come equipped with a symmetry-induced natural basis that permits a priori knowledge of which components are forced to zero (Type 1: all off-diagonal components vanish in the radial natural basis; Type 2: all off-diagonal components except R_{tφ}^sub vanish in the axisymmetric natural basis). Type 3 has no such structure. Its content is the absence of any such a priori reduction.

Global structure: tetrad-to-tetrad transformation. Tetrad-to-tetrad transformation across different spacetime points is articulated through the spin connection ω_μ^{ab} (cf. §2). No universal natural basis is assumed across points. The global structure of a Type 3 cell tensor is given by the pattern of tetrad-to-tetrad transformations across the manifold; in a generic spacetime, this pattern admits no global diagonalization. This is precisely the content of Mode 3 in the Calibration Isomorphism break mode taxonomy of §6.

Substrate cell count maintained. The substrate cell count remains finite, independent of the cell tensor structural complexity. The cell count is an ontological measure (how many cells), and the cell tensor is an articulation of the cell coupling pattern (how cells couple). The two are independent aspects of the substrate. Generic Type 3 cell tensor structure does not imply infinite cell count, and a finite cell count does not constrain the cell tensor structural complexity. This is consistent with the cross-paper articulation across P5 (§§3.1, 5.4), P6 (§§13.1-13.4, §11.3), and §3 of the present paper.

Type 3 as the substrate default articulation. In the absence of any global symmetry constraint, the substrate cell tensor admits its full Type 3 structure with all ten independent components. The P5 spherical and P6 axisymmetric cases are specific reductions of this default articulation under symmetry constraints: Type 1 retains four components, Type 2 retains five components. The reduction pattern is observed across cases: fewer global symmetries leave more independent components surviving the reduction, and richer cell tensor structures are articulated accordingly. The framing remains identificatory: P5 and P6 specific cases are observed to be Type 3 reductions under specific symmetry constraints, not derivations from a Type 3 prior.

Concrete examples. Two illustrative configurations sit naturally in the Type 3 territory.

The first is a multi-source static gravitational configuration: an n-body static arrangement (for example, a pair of widely separated static black-hole-like sources held in some configuration that remains stationary in the relevant approximation). Such configurations carry no global spherical symmetry, no global axisymmetric symmetry, and yet remain non-dynamical. The substrate cell tensor across such a configuration carries the full Type 3 structure: substrate cell coupling patterns reflect the combined source content without admitting any symmetry-induced reduction.

The second is a stationary linearized perturbation around the Schwarzschild background that breaks the spherical symmetry. The background spherically symmetric structure (Type 1) is perturbed by a stationary symmetry-breaking term that leaves no SO(3) action invariant. The perturbed substrate cell tensor sits in the Type 3 territory: the symmetry-breaking perturbation introduces independent components that the spherical natural basis would force to zero.

Two configurations to remain outside the P7 main body, in accordance with the strict non-dynamical scope: cosmological perturbations around an FRW evolving background (which carry intrinsically time-dependent evolution at the background level, crossing into the Info VI territory), and numerical relativity simulation snapshots (which extract single time slices from evolving spacetimes, crossing the same scope boundary). The substrate ontology articulation of dynamical configurations is handed off to Info VI.

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§5 Cell Tensor Type 1 / Type 2 / Type 3 Systematic Hierarchy

This section articulates the substrate-side structural hierarchy across the three cell tensor types as a preview of the full P5/P6/P7 correspondence backbone in §11. The articulation here remains at the substrate level, focused on the structural relationship between spacetime symmetry constraints and substrate cell tensor structure. The EP-side correspondence detail is reserved for §12.

Type 1 (P5 SO(3) spherical symmetry). In the static spherically symmetric Schwarzschild context, the substrate cell tensor takes a Type 1 diagonal structure in the radial natural basis:

R_μν^sub ~ diag(R_t^sub, R_r^sub, R_⊥^sub, R_⊥^sub)

The four independent components are R_t^sub (timelike), R_r^sub (radial), and R_⊥^sub (perpendicular, with the two angular components equal by SO(3) symmetry). The radial natural basis is induced by the SO(3) rotational Killing fields together with the timelike Killing field ∂_t. All off-diagonal components vanish. P5 articulates radial closure R_t^sub → 0 on the ∂_t channel at the horizon.

Type 2 (P6 SO(2) axisymmetric stationarity). In the stationary axisymmetric Kerr context, the substrate cell tensor takes a Type 2 mixed structure in the axisymmetric natural basis:

R_μν^sub ~ block-diagonal{(R_{tt}^sub, R_{tφ}^sub; R_{tφ}^sub, R_{φφ}^sub), R_{rr}^sub, R_{θθ}^sub}

The five independent components are R_{tt}^sub, R_{rr}^sub, R_{θθ}^sub, R_{φφ}^sub (diagonal) together with R_{tφ}^sub (off-diagonal). The axisymmetric natural basis is induced by the timelike Killing field ∂_t together with the axial Killing field ∂_φ. The off-diagonal R_{tφ}^sub is permitted by the axisymmetric stationary symmetry: it expresses the substrate-level rotational frame-dragging-type coupling between time and azimuthal channels. P6 articulates helical closure R_{χχ}^sub → 0 on the horizon generator χ_H = ∂_t + Ω_H ∂_φ at r_+.

Type 3 (P7 local Lorentz only). In the generic spacetime context with no global symmetry beyond local Lorentz, the substrate cell tensor takes a Type 3 full structure with all ten independent components and no universal natural basis. The tensor is locally tetrad-readable but not generically diagonalizable, in the sense articulated in §4. The articulation is local calibration at each point through the Calibration Isomorphism, with no global closure channel.

Complement-reduction relation between symmetry and tensor structure. The structural hierarchy exhibits a complement-reduction relation between spacetime symmetry constraint and cell tensor structural richness, as articulated in §11. Symmetry constraint acts as a reduction operator on the cell tensor; stronger symmetry constraint reduces more independent components, while fewer global symmetries leave more independent components surviving the symmetry reduction:

• Type 1: SO(3) plus staticity, four independent components, diagonal in radial natural basis.
• Type 2: SO(2) plus stationarity, five independent components, diagonal plus R_{tφ}^sub off-diagonal in axisymmetric natural basis.
• Type 3: local Lorentz only, ten independent components, no universal natural basis.

The pattern is observed across the three cases; it is not a theorem derived from SAE axioms. The framing is identificatory, in the same sense as the epistemological stance of the SAE mathematical foundational paper Schema.

Backbone preview.

Spacetime symmetry	Cell tensor type	Independent components	Calibration manifestation	Paper
SO(3) spherical	Type 1 diagonal	4	radial closure on ∂_t	P5
SO(2) axisymmetric	Type 2 mixed	5	helical closure on χ_H	P6
local Lorentz only	Type 3 full	10	local calibration, no global closure channel	P7

The full backbone articulation, including the EP-related substrate readings across the three rows, is reserved for §11. The fourth column of the P7 row reads "local calibration, no global closure channel": P7 is not a third closure case alongside the radial and helical closures of P5 and P6, but rather the generic local calibration in the absence of any global closure channel. This is consistent with the absence of any global Killing field in P7, which forecloses the possibility of a globally identified closure channel of the kind articulated in P5 and P6.

Future Type 4+ as open territory. The series articulates Type 1, Type 2, and Type 3 across P5, P6, and P7. Extensions to Type 4 and beyond are left as open territory for future SAE work. Such extensions would arise in contexts beyond local Lorentz only: multi-field couplings, gauge sectors (for example the Kerr-Newman electrically charged case), and broader extensions of the local symmetry structure. These cross outside the pure spacetime geometry territory of P7 and are not articulated here.

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§6 Calibration Isomorphism Break Mode Taxonomy

The Calibration Isomorphism (Cal Iso) was articulated in P4 (DOI 10.5281/zenodo.20079718) as the well-definedness of the mapping between the substrate cell tensor and the apparatus metric. P5 and P6 articulate specific Cal Iso break modes within their symmetry contexts. P7 articulates the systematic taxonomy spanning Mode 0 through Mode 4, organizing the break modes that arise across the relativity series.

The taxonomy is descriptive, not exhaustive in any deeper sense: it organizes the modes observed across P5, P6, and P7 instances into a working vocabulary that physicists may use to mark coordinate singularities, horizons, ergospheres, inner horizons, tidal residues, and algebraic degeneracies within a single framework. Each mode is articulated together with a brief mathematical or structural marker for identification.

Mode 0: Normal calibratable. The Cal Iso succeeds normally at the calibration point. The local tetrad reads the apparatus metric as Minkowski form, with the EEP holding at the local frame.

Marker: g_ab^app(p) = η_ab + O(x²), with ∂_c g_ab^app(p) = 0 in the freely-falling frame at p.

Mode 0 is the EEP-relevant generic-spacetime "normal" mode. It is the default mode in any region of a generic spacetime that admits regular local calibration. P7 §9 (EEP substrate ontology re-reading) articulates Mode 0 as the substrate articulation underlying the EEP.

Mode 1: Projection singularity. The apparatus metric diverges (or otherwise becomes coordinate-singular) at the calibration point, while the substrate cell tensor and the substrate cell count remain finite.

Marker: an apparatus metric component diverges (e.g., g_rr → ∞) while the substrate-level cell tensor and cell count remain finite.

Instance: the Schwarzschild horizon coordinate divergence g_rr → ∞ in standard Schwarzschild coordinates is an apparatus projection singularity. In Eddington-Finkelstein or Kruskal coordinates, the apparatus geometry at the horizon is coordinate-regular, and Mode 1 is absent in those charts. The substrate-level radial closure R_t^sub → 0 on the ∂_t channel is a substrate-level feature distinct from the apparatus-level Mode 1 projection singularity.

Mode 2: Channel activation. A specific off-diagonal substrate cell tensor component becomes nontrivial in a channel selected by the symmetry context, indicating substrate-level cross-channel coupling that was not active in the higher-symmetry case.

Marker: the substrate cell tensor carries a nontrivial off-diagonal component in a specific channel (for example, R_{tφ}^sub ≠ 0 in the axisymmetric stationary case).

Instance: the activation of R_{tφ}^sub in the Kerr ergosphere is a Mode 2 instance. The Kerr horizon generator χ_H = ∂_t + Ω_H ∂_φ supplies the helical closure channel R_{χχ}^sub through Mode 2 cross-coupling. The extremal Kerr infinite throat is a composite case involving Mode 1 (projection singularity in the apparatus radial coordinate proper distance), Mode 2 (R_{tφ}^sub channel activation), and an algebraic degeneracy of Mode 4 type. It is not a clean Mode 2 instance and is best articulated as a composite mode case.

Mode 3: No global natural basis. Only local Lorentz calibration patches are available, with no universal natural basis across spacetime points. The local tetrad at each point furnishes a component representation, but tetrad-to-tetrad transformation patterns across points do not admit a universal global diagonalization. The generic Type 3 cell tensor and the apparatus metric do not simultaneously diagonalize in a single tetrad at the generic point.

Marker: tetrad-to-tetrad transformation patterns across spacetime points admit no universal global natural basis; the generic Type 3 cell tensor retains substrate components that do not simultaneously diagonalize with the apparatus metric in any single tetrad.

Instance: a generic non-dynamical spacetime (multi-source static configuration, stationary linearized symmetry-breaking perturbation around Schwarzschild, and so forth, as articulated in §4). Mode 3 is the principal mode of P7. The EP universality at substrate level (the EP universality articulation in §10 SEP-local territory and the local calibration universality of §9 EEP) sits in Mode 3 territory: the absence of any global natural basis is precisely the condition under which local calibration universality articulates the substrate-level reading of the EP.

Mode 3 and Mode 0 hold simultaneously in a generic spacetime, applying at different topological scopes: Mode 0 articulates the success of local apparatus calibration at any single point (point-wise local), while Mode 3 articulates that the tetrad-to-tetrad transformations across spacetime points cannot be integrated into a universal global diagonalization basis (cross-point integration). This coexistence is the typical situation in a generic spacetime — local calibration succeeds at each point, but the local calibration patches cannot be assembled across points into a universal global basis.

Mode 4: Algebraic degeneracy regime. The substrate cell tensor eigen-channel structure exhibits algebraic degeneracy features. The degeneracy is an algebraic feature of the tensor structure, not a dynamical phase transition.

Marker: the substrate cell tensor exhibits algebraic degeneracy in its eigen-channel decomposition (for instance, Segre-type degeneracies or other algebraically special cases where eigenvalues coincide or eigenvector structure becomes non-generic).

The strict non-dynamical stance is maintained throughout. Mode 4 is articulated as an algebraic feature in the static / stationary regime, not as a dynamical closure transition. The "approaching" or "becoming degenerate" language is avoided; the substrate cell tensor either is in an algebraically degenerate regime at a configuration or it is not, as a structural feature of that configuration.

Instances of Mode 4 typically appear together with other modes in composite cases: the extremal Kerr infinite throat (composite Mode 1 + Mode 2 + Mode 4 case), certain algebraically special vacuum solutions, and other configurations where eigen-structure becomes non-generic. The taxonomy is descriptive of the structural features observed; it does not predict dynamical evolution.

No Mode 5 placeholder in P7. The Mode 0 through Mode 4 taxonomy of P7 is articulated within the territory of pure spacetime geometry (substrate cell tensor and apparatus metric, in the generic non-dynamical spacetime context). Extensions to additional modes that would arise in contexts beyond pure spacetime geometry, including multi-field couplings, gauge sectors (such as the Kerr-Newman electrically charged case introducing the electromagnetic stress-energy contribution), and broader gauge or matter-sector extensions, are deferred to future SAE work. The anticipated future SAE quantum gravity interface paper or related future SAE papers are the natural home for such extensions. P7 maintains a strict Mode 0 through Mode 4 scope within pure spacetime geometry; no Mode 5 placeholder is inserted to anticipate future work.

Mode taxonomy as a working lens. The Mode 0 through Mode 4 taxonomy supplies physicists with a unified vocabulary for marking apparatus features encountered across GR contexts: coordinate singularities (Mode 1), horizons (typically composite Mode 1 plus substrate-level closure), ergospheres (Mode 2), inner horizons (Mode 1 in apparatus geometry plus substrate-level features), tidal residues at second order in EEP-relevant calibrations (Mode 0 structure with curvature residue), and algebraic degeneracies (Mode 4). The taxonomy organizes these features within a single framework and may be picked up by physicists as a working lens without requiring acceptance of the full SAE ontology.

§7 Three-Tier EP Substrate Ontology Re-reading: Overview

This section sets up the three-tier equivalence principle substrate ontology re-reading articulated across §8 (WEP), §9 (EEP), and §10 (SEP three-tier split). The setup establishes the stance, clarifies the relationship between substrate ontology articulation and standard EP testing, and disambiguates two distinct senses of "first-order" / "一阶" that appear across §8 and §9.

Standard EP three-tier recap. The equivalence principle in standard GR foundations is articulated at three levels. The weak equivalence principle (WEP) states that test bodies in a gravitational field follow trajectories determined solely by their initial position and velocity, independently of their internal composition; this is the universality of free fall, supported empirically by Eötvös-type experiments and lunar laser ranging. The Einstein equivalence principle (EEP) states that in a sufficiently small spacetime region, the physics of non-gravitational experiments in a freely falling frame reduces to special relativity; this is the local Lorentz invariance of non-gravitational physics. The strong equivalence principle (SEP) extends the EEP to include gravitational physics and gravitational self-energy: gravitational binding energy contributes to inertial and gravitational mass in the same way as other forms of energy, and gravitational physics in a freely falling frame remains locally indistinguishable from gravity-free physics. The standard formulations are taken as foundational principles of GR; substantive treatment appears in MTW, Will, and similar references.

SAE substrate ontology re-reading stance. P7 does not derive the EP three-tier from SAE axioms, and does not claim empirical superiority over standard EP testing. The articulation is identificatory: each of the three tiers receives a substrate-level ontological reading, and the three readings together supply a working ontology lens for physicists thinking about the EP foundations. The stance is consistent with the Paper 0 §1.5 honest position throughout the SAE Relativity Series.

The substrate ontology re-reading does not modify the empirical content of any tier. The WEP composition independence remains a test-body limit empirical fact validated by Eötvös-type experiments. The EEP local Lorentz invariance remains the operational content of local inertial frames as established in standard GR. The SEP universality remains under empirical test in the strong-field regime through binary pulsar observations and similar contexts. What the substrate ontology re-reading supplies is an ontological framing: each tier sits at a distinct level of substrate-apparatus structure, and the three-tier hierarchical articulation reveals the substrate-level structural reasons why these three principles take their standard forms.

Hierarchical articulation across the three tiers. The three tiers articulate three different aspects of the substrate ontology:

• WEP (§8): composition decoupling at the substrate level, in the test-body limit and at first order in the test-body approximation.
• EEP (§9): Calibration Isomorphism local universality, with apparatus metric first-derivative Christoffel structure absorbed into local apparatus units and substrate cell tensor retained intact.
• SEP (§10): a three-way split into SEP-local (the Calibration Iso core, same territory as the EEP), SEP-self-energy (gravitational binding energy as a multi-cell substrate articulation, framework reading only), and SEP-global (horizon and closure regimes where local calibration does not determine global substrate features).

The three tiers do not sit at the same ontological level. WEP, EEP, and SEP-local are substrate-level universal properties applying within their respective scope qualifications (test-body limit for WEP, local point of calibration for EEP, generic spacetime normal regime for SEP-local), and they hold across any global symmetry context (SO(3), SO(2), local Lorentz only). SEP-self-energy and SEP-global are regime-dependent articulations: they involve substrate-level features that depend on specific spacetime configurations (self-gravitating bodies introducing gravitational binding energy across multiple substrate cells in SEP-self-energy; horizon and closure regimes in SEP-global). The full correspondence between the three tiers and the symmetry hierarchy backbone is articulated in §12.

Two senses of "first-order" across §8 and §9. The articulations of WEP in §8 and EEP in §9 each invoke a notion of "first-order" or "一阶", but the two refer to distinct senses that should not be conflated.

In §8, the WEP first-order articulation refers to the test-body approximation order. The test-body limit assumes that the test particle's gravitational self-coupling is negligible relative to the background source, so that the test particle does not back-react on the substrate cell tensor at first order in the test-particle / source mass ratio. The composition-blind articulation of WEP at substrate level holds at this first order of the test-body approximation. At higher orders, the self-coupling of the test particle becomes relevant, and the territory crosses into SEP-self-energy and beyond.

In §9, the EEP first-order articulation refers to the spatial expansion order at the calibration point p. Local apparatus calibration at p succeeds in setting g_ab^app(p) = η_ab with vanishing first derivative ∂_c g_ab^app(p) = 0 in the freely falling frame. The "first-order" structure absorbed by the calibration is the apparatus metric Christoffel structure at p, which is a first-spatial-derivative quantity. At second order in the spatial expansion, curvature and tidal residue appear; this second-order structure cannot be absorbed by local calibration.

The two senses are independent: the test-body approximation order in §8 has nothing to do with the spatial expansion order in §9. The articulations are kept categorically separate. Where ambiguity might arise in the main body text, the specific sense is explicitly indicated.

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§8 WEP Substrate Ontology Re-reading

The weak equivalence principle in standard GR foundations articulates the universality of free fall: test bodies in a gravitational field follow trajectories determined solely by the local geometry, independently of their internal composition. Empirically, this is supported by Eötvös-type experiments and lunar laser ranging at high precision. In standard formulation, the WEP is taken as a foundational principle, sometimes phrased as the equality of inertial and gravitational mass.

Substrate ontology re-reading. In SAE, the WEP at the substrate level reads as the composition decoupling of substrate cell coupling in the test-body limit. Specifically: in the test-body limit, where the test particle's gravitational self-coupling is negligible relative to the background source, the substrate cell tensor articulation does not depend on the test particle's internal lower-DD composition at first order in the test-body approximation. The test particle's internal composition (rest mass distribution, internal binding energies, charge content, spin content, and so forth) sits at the apparatus level. At the substrate level, the cell tensor articulates the local cell coupling pattern that determines how the test particle's center-of-cell trajectory propagates through spacetime; the cell coupling pattern does not encode the test particle's internal composition at first order.

The articulation may be phrased: in the test-body limit and at first order, the trajectory is a cell-geometry readout, not a composition readout. Two test particles of different internal compositions but the same total cell-coupling profile at the test-body limit will trace the same trajectory through the substrate cell network, because the substrate cell coupling pattern does not differentiate among compositions at this order.

Test-body limit and first-order qualifier. The substrate ontology re-reading of WEP holds within the test-body limit and at first order in the test-body approximation. Two qualifications follow.

First, the test-body limit requires the test particle's gravitational self-coupling to be negligible relative to the background source. Self-gravitating bodies, where the test particle's own gravitational binding energy contributes a non-negligible fraction of its rest energy, do not satisfy the test-body limit. The articulation of substrate ontology for self-gravitating bodies sits in the SEP-self-energy territory of §10, not in the WEP territory of §8.

Second, the first-order qualifier refers to the test-body approximation order. At higher orders, the test particle's self-coupling becomes relevant, and the composition independence is no longer guaranteed at the substrate level. Higher-order corrections to the WEP, where self-coupling effects modify the trajectory in composition-dependent ways, also sit in the SEP-self-energy territory.

Scope honesty. The substrate ontology re-reading of WEP does not derive the WEP from SAE axioms. The articulation supplies an ontological grounding for the WEP at the substrate level: composition information sits at the apparatus level, while the substrate cell coupling articulates the trajectory without reference to composition. The Eötvös-type empirical validation of the WEP at high precision remains the empirical content; the substrate ontology articulation is a working lens that complements rather than replaces empirical testing.

The articulation does not extend to self-gravitating bodies, gravitational binding energy, or higher-order corrections. These belong to the SEP-self-energy and SEP-global territories of §10. The clear scope demarcation prevents the substrate ontology re-reading of WEP from being misread as a derivation of WEP universally; it holds within its test-body / first-order scope, which is the same scope within which the standard WEP holds as an empirical principle.

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§9 EEP Substrate Ontology Re-reading

The Einstein equivalence principle in standard GR foundations articulates the local equivalence of gravitation and acceleration: in a sufficiently small spacetime region, the physics of non-gravitational experiments in a freely falling frame reduces to special relativity; locally, the laws of physics take the form they would in the absence of gravity, in the freely falling frame at the calibration point. This is the local Lorentz invariance of non-gravitational physics, operationalized through the freely falling local inertial frame.

Substrate ontology re-reading. In SAE, the EEP at the substrate level reads as the local universality of the Calibration Isomorphism. The articulation has three components.

First, at any point p in a generic spacetime, the local tetrad e_a^μ(p) supplies a local Lorentz frame at p. The Calibration Isomorphism articulates the local apparatus calibration through this tetrad: the apparatus metric in the tetrad basis takes the Minkowski form at p, that is, g_ab^app(p) = η_ab. This is the local recovery of special relativity at the calibration point.

Second, the local apparatus calibration absorbs the first-derivative structure of the apparatus metric at p. In the freely falling frame at p, the first spatial derivative vanishes: ∂_c g_ab^app(p) = 0. The Christoffel structure of the apparatus metric at p is absorbed into local apparatus units. This is what is meant by "the gravitational field disappears locally" in the freely falling frame: the first-derivative apparatus metric readout is absorbed, and at first order in the spatial expansion around p, the apparatus metric reads as flat Minkowski.

Third, the substrate cell tensor is not erased. Local apparatus calibration absorbs the first-derivative apparatus metric Christoffel structure into local units; it does not erase the substrate cell tensor itself. The substrate cell tensor R_μν^sub(p) and its readout structure remain at substrate level. The second-order spatial structure around p, namely the curvature and tidal residue, manifests at second order in the spatial expansion: g_ab^app(p + x) = η_ab + O(x²), with the O(x²) term carrying curvature information. The second-order tidal residue is unabsorbed by local calibration: the EEP holds at first order in spatial expansion, but tidal effects manifest at second order.

The reading: absorption, not erasure. The substrate ontology re-reading clarifies that the EEP is not the absence of substrate structure but the success of local apparatus calibration. The substrate cell tensor remains at the substrate level: the cell coupling pattern, the cell count, and the substrate readout structure continue to exist at every spacetime point. What the EEP articulates is that local apparatus calibration is universally successful: at any point p in a generic spacetime, the local tetrad furnishes a Minkowski reading of the apparatus metric, with the first-derivative Christoffel structure absorbed into local apparatus units. The substrate ontology is not flat; only the local apparatus reading is.

The articulation may be summarized:

g_ab^app(p) = η_ab + O(x²)

∂_c g_ab^app(p) = 0 (in the freely falling frame at p)

R_μν^sub(p): retained, locally tetrad-readable

Curvature / tidal residue: appears at O(x²), unabsorbed by local calibration

The substrate cell tensor at p is tetrad-readable through R_ab^sub(p) = e_a^μ(p) e_b^ν(p) R_μν^sub(p), but in the generic spacetime context (Type 3 cell tensor, Mode 3 of the Cal Iso break taxonomy), the substrate cell tensor and the apparatus metric do not simultaneously diagonalize in the single tetrad at p. The cell tensor retains its full Type 3 structure even within the freely falling frame; the EEP articulates the success of apparatus calibration, not the diagonalization of substrate cell tensor.

EEP universality across symmetry contexts. The substrate ontology re-reading of EEP holds in any spacetime symmetry context. In the SO(3) Schwarzschild case of P5, local apparatus calibration at any point absorbs the first-derivative Christoffel structure; the substrate cell tensor retains its Type 1 diagonal structure in the radial natural basis but in the freely falling frame at a specific point, the local tetrad reads the apparatus metric as Minkowski. In the SO(2) Kerr case of P6, the same articulation holds at every point: local apparatus calibration absorbs the first-derivative Christoffel structure, while the substrate cell tensor retains its Type 2 mixed structure with R_{tφ}^sub off-diagonal in the axisymmetric natural basis. In the local Lorentz only generic case of P7, local apparatus calibration absorbs the first-derivative Christoffel structure at any point, while the substrate cell tensor retains its full Type 3 structure with all ten independent components. The Calibration Isomorphism local universality is the same articulation across all three symmetry contexts; only the substrate cell tensor structure differs across the cases.

EEP does not imply substrate absence. A potential misreading is that local Lorentz invariance implies the absence of substrate structure. The substrate ontology re-reading clarifies the converse: the EEP articulates local apparatus calibration success, not substrate absence. Generic spacetime carries a full Type 3 substrate cell tensor with ten independent components; the absence of global symmetry does not imply the absence of substrate structure. The substrate articulates the cell coupling pattern at every point, and the EEP articulates that this substrate structure is locally apparatus-readable as Minkowski at first order in spatial expansion, with second-order curvature residue remaining.

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§10 SEP Substrate Ontology Re-reading: Three-Tier Split

The strong equivalence principle in standard GR foundations extends the EEP from non-gravitational physics to all physics, including gravitational physics and gravitational self-energy. The SEP states that gravitational binding energy contributes to inertial and gravitational mass in the same way as other forms of energy (the gravitational equivalence of all forms of energy), and that gravitational physics in a freely falling frame remains locally equivalent to gravity-free physics (the gravitational sector inherits the EEP). The SEP is under empirical test in strong-field regimes through binary pulsar observations, lunar laser ranging Nordtvedt parameter constraints, and similar contexts; the empirical content sits with the experimental GR community (Will, Damour, and others).

Substrate ontology re-reading: a three-tier split. P7 articulates the SEP substrate ontology re-reading as a three-tier split: SEP-local, SEP-self-energy, and SEP-global. The split serves a working-lens purpose: physicists thinking about strong-field universality, gravitational binding energy, black hole universality, and horizon regimes encounter contexts where SEP is invoked at different conceptual levels, and the three-tier split supplies an ontological vocabulary for separating these levels. The split is regime-dependent in the sense articulated in §7: SEP-local sits in the generic normal regime, SEP-self-energy in the strong-field regime, and SEP-global in the horizon and closure regime.

The articulation here is not contradictory with the EEP local universality of §9. The local-versus-global distinction is essential. §9 articulates that the Calibration Isomorphism succeeds at each point as local apparatus calibration. §10 SEP-global articulates that global features of spacetime (horizon, topology, closure regimes) are not determined by single-point local calibration. The two articulations are complementary: local calibration succeeds at each point, but global features are integrative articulations across local calibration patches, not derivable from any single-point local calibration.

SEP-local. In the generic spacetime normal regime, where local apparatus calibration succeeds and no horizon or closure regime is engaged, gravitational and non-gravitational laws are read through the same Calibration Isomorphism at any point. This is the natural extension of the EEP from non-gravitational physics to gravitational physics within the locally-flat freely falling frame. The SAE substrate ontology accepts SEP-local as a consequence of the Calibration Iso local universality articulated in §9. SEP-local and EEP sit in the same territory: the local apparatus calibration success at each point applies to all physical content readable at the local tetrad, including gravitational content. The articulation does not distinguish between gravitational and non-gravitational physics at the local-calibration level.

SEP-self-energy. Bodies with non-negligible gravitational binding energy lie outside the WEP test-body limit and outside the SEP-local pure local-calibration regime. Gravitational binding energy is articulated at substrate level as cross-cell coupling: the binding energy reflects the substrate cell tensor's articulation across multiple substrate cells in the source's spatial extent, rather than the single-point local calibration of a test particle in a background. The SEP-self-energy regime involves substrate features that span multiple cells and are not determined by single-point local calibration. This separates SEP-self-energy from SEP-local: SEP-local territory engages single-point local calibration, while SEP-self-energy territory engages multi-cell substrate articulation.

P7 articulates SEP-self-energy at the framework vocabulary level only. The substrate-level articulation of gravitational binding energy as a multi-cell feature is supplied as a working vocabulary for physicists, with no commitment to any specific quantitative form. The substantive ontological content articulation of SEP-self-energy (including specific cell-coupling structures, the relationship between gravitational binding energy and cell count, and the substrate-level implications for the SEP in strong-field regimes) is left to future SAE papers. The anticipated future SAE paper that would carry this substantive articulation is unspecified in P7 itself; what P7 establishes is the vocabulary that future work would take up.

SEP-global. Horizon regimes, topology features, and closure regimes carry global substrate features that are not determined by single-point local calibration. P5 and P6 articulate this within their specific symmetry contexts: P5 articulates the Schwarzschild horizon as the radial closure of R_t^sub on the ∂_t channel, a substrate-level feature characterizing the horizon globally; P6 articulates the ergosphere static-readout failure, the inner horizon r_-, the closed timelike curves in r < 0, and the extremal Kerr infinite throat as global substrate features of the Kerr geometry. In each case, local apparatus calibration succeeds at every point (EEP holds locally), but the global features are integrative substrate articulations across local calibration patches, not derivable from any single-point local calibration. SEP-global articulates this regime-level non-locality: when global substrate features are at issue, local calibration is not sufficient to determine substrate-level structure.

The substrate-apparatus distinction discipline of §3 articulates the apparatus-level features (g_rr → ∞, ring singularity, r < 0 analytic continuation, CTCs, extremal infinite throat) and the substrate-level features (radial closure R_t^sub → 0, helical closure R_{χχ}^sub → 0, ergosphere R_{tφ}^sub activation) as separate readings. SEP-global articulates that the substrate-level global features cannot be inferred from local calibration alone: the horizon's substrate-level radial closure is a global articulation involving the timelike Killing field ∂_t and the radial channel structure across the spacetime, not a feature determinable from local tetrad calibration at any single horizon point.

Boundary between SEP-self-energy and SEP-global. Both SEP-self-energy and SEP-global are multi-cell substrate features, but they sit at different topological levels and may be distinguished as follows. SEP-self-energy concerns the multi-cell substrate articulation internal to a self-gravitating body — gravitational binding energy as a coupling feature across multiple substrate cells within the object, i.e., an object-internal multi-cell articulation. SEP-global concerns the global substrate features of spacetime — horizons, topology, and closure regimes as overall features of the spacetime structure, i.e., an integrative articulation across local calibration patches. Composite contexts (for example, a binary pulsar system involving self-gravitating bodies in the vicinity of horizon-like dynamics) carry both kinds of substrate feature simultaneously, but the SAE substrate ontology articulation organizes these into two distinct layers: the binary pulsars' internal self-gravitational binding energy sits in the SEP-self-energy territory, while the surrounding spacetime structure and potential horizon / closure regions sit in the SEP-global territory.

P7 stance: vocabulary, not empirical superiority. SAE substrate ontology articulation supplies the three-tier SEP split as ontology vocabulary for use as a working lens. SAE does not claim empirical superiority over SEP empirical testing. The empirical content of SEP universality at strong-field regimes (binary pulsar tests, lunar laser ranging Nordtvedt parameter constraints, and similar) is the empirical GR community's territory (Will, Damour, and others), and P7 does not enter that territory. What P7 offers is an ontological vocabulary: physicists who think about strong-field universality, gravitational binding energy, black hole universality, or horizon regimes may use the SEP-local / SEP-self-energy / SEP-global split as a working framework for organizing different regimes within a single ontological vocabulary. The framework does not require acceptance of the full SAE substrate ontology to be useful as a working lens.

Practical significance for physicists. The three-tier SEP split offers practical organizing utility for thinking about SEP-sensitive contexts. When the context engages single-point local calibration in the generic normal regime, SEP-local applies (and the EEP articulation of §9 supplies the substrate-level content). When the context engages gravitational binding energy of a self-gravitating body, SEP-self-energy applies, and the substrate ontology vocabulary articulates the multi-cell substrate articulation; substantive content is reserved for future SAE work. When the context engages horizon regimes, topology features, or closure regimes, SEP-global applies, and P5/P6 specific case articulations supply the substrate-level reading patterns. The three regimes correspond to three different substrate-level structural situations: local calibration patches versus multi-cell substrate articulation versus integrative global articulation across local calibration patches. The three-tier split makes this regime-dependence explicit.

A physicist need not accept the full SAE substrate ontology to use the SEP three-tier split as an organizing principle. Just as the substrate-apparatus distinction discipline of §3 may be used as a working lens without commitment to SAE as a whole, the SEP three-tier split is offered as a working framework that physicists may engage with at the vocabulary level. The split organizes the SEP-sensitive landscape into three regime-dependent territories, each with its own ontological character and its own relationship to the local calibration structure of generic spacetime.

§11 EP-Spacetime Structure Correspondence Backbone

This section articulates the full P5/P6/P7 cross-paper correspondence backbone: the observed pattern across the three relativity series cases linking spacetime symmetry hierarchy, substrate cell tensor structure, and Calibration manifestation. §4 supplied the substrate-side structural articulation of Type 3, §5 supplied the substrate-side structural hierarchy preview, and §6 supplied the Cal Iso break mode taxonomy; §11 now articulates the full correspondence backbone, and §12 articulates the EP-side correspondence detail.

Cross-paper observed pattern. Across P5, P6, and P7, an observed pattern emerges connecting four aspects of each case: the spacetime global symmetry, the substrate cell tensor type, the Calibration manifestation channel, and the EP-related substrate articulation. The pattern is reported as observed, not derived from SAE axioms. The framing is identificatory, consistent with the epistemological stance of the SAE mathematical foundational paper Schema across the series.

The backbone is articulated through the following correspondence table:

Symmetry	Cell tensor type	Indep. components	Calibration manifestation	EP-related substrate articulation	Paper
SO(3) spherical	Type 1 diagonal	4	radial closure on ∂_t	radially-symmetric tetrad reading	P5
SO(2) axisymmetric	Type 2 mixed	5	helical closure on χ_H	helically-rotating tetrad reading	P6
local Lorentz only	Type 3 full	10	local calibration, no global closure channel	EP universality at substrate level	P7

The fourth column of the P7 row reads "local calibration, no global closure channel": this is not a third closure case alongside the P5 radial closure and the P6 helical closure. P5 and P6 each carry a global Killing field (∂_t for P5, χ_H for P6) that supplies a globally identified channel on which substrate closure articulates. P7's generic spacetime context carries no global Killing field, and no global closure channel exists; what P7 articulates instead is the local calibration universality through the Calibration Isomorphism at each point, in the absence of any global closure channel. The two cases (P5/P6 closure channels and P7 local calibration) are categorically distinct manifestations of substrate-apparatus structure, not three instances of the same closure pattern.

Identificatory backbone, not derivational. The backbone is articulated as an observed correspondence pattern across P5, P6, and P7. The framing does not claim that the backbone is derived from SAE axioms or that the symmetry hierarchy SO(3) → SO(2) → local Lorentz only is a forced reduction sequence. The three cases are studied independently in P5, P6, and P7, each within its own substrate ontology re-reading; the backbone is the cross-paper structural observation that emerges from comparing the three. This stance maintains the identificatory framing of the SAE mathematical foundational paper Schema: substrate ontology articulation supplies an ontological framing for GR mathematical structures, without replacing GR or deriving GR from SAE.

The wording throughout the backbone articulation avoids claims of "systematic correspondence" or "natural completion" that might suggest derivation. The observed pattern reads as a comparison of three case-specific articulations: each case carries its own substrate cell tensor structure (Type 1, Type 2, Type 3), its own Calibration manifestation (radial closure on ∂_t, helical closure on χ_H, local calibration without global closure channel), and its own EP-related substrate articulation (radially symmetric, helically rotating, EP universality at substrate level). The pattern across the three is observed, and the observation is articulated as the cross-paper backbone of the relativity series.

Structural complement-reduction relation: symmetry constraint and cell tensor complexity. The backbone exhibits a complement-reduction relation between spacetime symmetry constraint and substrate cell tensor structural complexity, as articulated in §5. Symmetry constraint acts as a reduction operator on the cell tensor; stronger symmetry constraint reduces more independent components, while fewer global symmetries leave more independent substrate cell tensor components surviving the symmetry reduction: SO(3) (P5) carries 4 components, SO(2) (P6) carries 5, local Lorentz only (P7) carries 10. The richer cell tensor structure corresponds to weaker global symmetry. In the extreme limit case of P7, the absence of any global symmetry beyond local Lorentz allows the substrate cell tensor to articulate its full Type 3 structure with all ten independent components surviving.

The EP universality articulation at the P7 level sits in this extreme limit context: the substrate cell tensor articulates its full Type 3 structure, and the EP universality at substrate level is articulated through the local calibration manifestation at each point. The EP-related substrate articulation across the three cases is a regime-dependent feature: P5 and P6 specific symmetry cases articulate EP-related substrate readings tied to their specific natural bases (radially symmetric tetrad reading in P5, helically rotating tetrad reading in P6), while P7's generic case articulates EP universality at substrate level through local calibration without a global closure channel.

The backbone as a series-level substantive contribution. The EP-spacetime structure correspondence backbone is the substantive series-level contribution of P7: it articulates the cross-paper correspondence pattern that emerges only when the three cases are placed alongside each other. P5 alone articulates the Schwarzschild radial closure substrate reading. P6 alone articulates the Kerr helical closure substrate reading and the §10.3 symmetry hierarchy observation. Neither P5 nor P6 articulates the full backbone, because the full backbone requires the three-case comparison. P7 supplies this articulation: the backbone is observed across P5, P6, and P7 together, and P7 articulates it explicitly.

The backbone articulation does not modify the substantive content of P5 or P6; both papers retain their own case-specific substrate ontology re-readings. What the backbone adds is the cross-paper observed pattern, articulated at the level of the relativity series rather than at any single paper. This is what makes P7 the natural series-closing paper: the backbone articulation emerges only when the three cases are taken together, and P7 supplies the articulation in the generic case that completes the three-case comparison.

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§12 Symmetry Hierarchy Backbone and Three-Tier EP Correspondence

This section applies the backbone of §11 to the three-tier EP framework of §§7-10, articulating the EP-side correspondence detail. §10 articulated the SEP three-tier split per regime territory (SEP-local, SEP-self-energy, SEP-global). §12 articulates the backbone applied to the EP three-tier framework, organizing how WEP, EEP, SEP-local, SEP-self-energy, and SEP-global correspond to the symmetry hierarchy across cases. The two framings are complementary: §10 organizes the SEP three-tier vertically by regime territory, while §12 organizes the EP three-tier horizontally by universal-versus-regime-dependent split across the symmetry hierarchy.

Universal substrate properties: WEP, EEP, SEP-local. Three EP-related substrate articulations are universal properties applying across any spacetime symmetry context within their respective scope qualifications.

The WEP composition decoupling articulation of §8 applies in the test-body limit at first order in the test-body approximation. This substrate property does not depend on the spacetime's global symmetry context: in P5 SO(3) Schwarzschild, in P6 SO(2) Kerr, and in P7 generic local Lorentz only, the substrate cell tensor does not encode test particle internal composition at first order in the test-body approximation. The test-body limit articulation is consistent across all three cases. The substrate-level reason is that the cell coupling pattern articulates the local geometry through which the test particle's center-of-cell trajectory propagates, regardless of internal composition; this articulation holds at the substrate level independently of which global symmetry, if any, the spacetime carries.

The EEP local universality of §9 likewise applies across any spacetime symmetry context. In P5 SO(3) Schwarzschild, local apparatus calibration at any point absorbs the first-derivative Christoffel structure into local apparatus units, with the substrate cell tensor retaining its Type 1 diagonal structure but reading as Minkowski in the apparatus through the local tetrad. In P6 SO(2) Kerr, the same articulation holds: local apparatus calibration at any point absorbs first-derivative Christoffel structure, with substrate cell tensor retaining its Type 2 mixed structure. In P7 local Lorentz only generic case, local apparatus calibration absorbs first-derivative Christoffel structure with substrate cell tensor retaining its full Type 3 structure. The Calibration Isomorphism local universality is a universal substrate property: at any point in any spacetime carrying local Lorentz frames, local apparatus calibration succeeds at first order in spatial expansion, with second-order tidal residue remaining. The articulation does not depend on the global symmetry context.

SEP-local, as articulated in §10, sits in the same territory as the EEP. It extends the local apparatus calibration success from non-gravitational physics to gravitational physics within the locally-flat freely falling frame. This extension applies universally across spacetime symmetry contexts, in the same sense as the EEP: local apparatus calibration at any point absorbs the first-derivative Christoffel structure, and this success applies to all physical content readable at the local tetrad including gravitational content.

These three universal substrate properties (WEP composition decoupling, EEP local universality, SEP-local Cal Iso extension) are universal in the sense that they apply at the substrate level regardless of the global symmetry context. The articulations are sensitive to scope qualifications (test-body limit at first order for WEP; local point of calibration for EEP and SEP-local) but not to spacetime symmetry class.

Regime-dependent substrate features: SEP-self-energy and SEP-global. SEP-self-energy and SEP-global are regime-dependent in the sense that they engage substrate features that depend on specific spacetime configurations.

SEP-self-energy engages the multi-cell substrate articulation of gravitational binding energy in self-gravitating bodies. The articulation differs across spacetime symmetry contexts because the multi-cell substrate articulation in P5 spherical configurations, P6 axisymmetric configurations, and P7 generic configurations carries different substrate cell tensor structures. In P5 contexts, the multi-cell articulation sits within the Type 1 diagonal structure. In P6 contexts, the multi-cell articulation engages the Type 2 mixed structure with R_{tφ}^sub off-diagonal cross-coupling across cells. In P7 generic contexts, the multi-cell articulation engages the full Type 3 structure with no global natural basis for the cross-cell coupling pattern. The substrate-level content of SEP-self-energy is regime-dependent in this sense; P7 articulates the vocabulary at the framework level, with substantive ontological content reserved for future SAE work, as noted in §10.

SEP-global engages global substrate features that are not determined by single-point local calibration. The substrate-level reading patterns differ across cases. In P5, the substrate-level horizon feature is the radial closure R_t^sub → 0 on the ∂_t channel, articulated through the globally identified timelike Killing field. In P6, the substrate-level horizon feature is the helical closure R_{χχ}^sub → 0 on the horizon generator χ_H = ∂_t + Ω_H ∂_φ, articulated through a specific combination of two global Killing fields. In P7 generic context, no global Killing field is available, and no analogous global closure channel exists; substrate-level global features of generic spacetime configurations articulate through patterns of local calibration across the manifold, with the global articulation reading as integrative across local calibration patches rather than as closure on a globally identified channel. The substrate-level content of SEP-global is regime-dependent in this sense.

The universal-versus-regime-dependent horizontal split. The five EP-related substrate articulations (WEP, EEP, SEP-local, SEP-self-energy, SEP-global) sit at two distinct levels with respect to the symmetry hierarchy backbone.

WEP, EEP, and SEP-local are universal substrate properties: they apply at the substrate level across any symmetry context, sensitive to scope qualifications but not to global symmetry class. The three together articulate the locally-applicable EP content at substrate level, holding equally across P5 SO(3) Schwarzschild, P6 SO(2) Kerr, and P7 generic local Lorentz only.

SEP-self-energy and SEP-global are regime-dependent substrate features: they engage substrate articulations that depend on specific spacetime configurations and global symmetry contexts. The articulations differ structurally across P5, P6, and P7, and the substantive content sits in regime-specific territory.

The horizontal split clarifies why the SEP three-tier split of §10 is required. SEP-local belongs with the universal substrate properties (WEP, EEP), while SEP-self-energy and SEP-global belong with the regime-dependent substrate features. A unitary "SEP" that did not distinguish these three regimes would conflate the universal-substrate-property content with the regime-dependent substrate features, obscuring the structural difference between local calibration success at each point and integrative global articulation across calibration patches.

Future Type 4+ as open territory. The articulation in §12 covers the EP three-tier within the substrate cell tensor types articulated in P5, P6, and P7 (Type 1, Type 2, Type 3). Extensions beyond local Lorentz only (multi-field couplings, gauge sectors such as the Kerr-Newman electrically charged case, and broader extensions of the local symmetry structure) would introduce Type 4 and higher cell tensor structures, with corresponding extensions of the EP correspondence at substrate level. Such extensions are left as open territory for future SAE work. The substrate ontology re-reading of EP for matter-sector and gauge-sector extensions awaits the anticipated future SAE quantum gravity interface paper and related future SAE work.

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§13 Uniqueness Trajectory: Series Methodology Closing Pattern

The SAE Relativity Series articulates substrate readouts that align with standard GR uniqueness theorems and foundational principles across the three immediate predecessor papers. P5 articulates a substrate readout that aligns with the Birkhoff theorem; P6 articulates a substrate readout that aligns with the Kerr uniqueness theorems; P7 articulates a substrate-level reading of the EP universality. The three articulations form a series-internal closing pattern at the methodology level: across three different spacetime symmetry contexts (static spherical vacuum, stationary axisymmetric vacuum, generic non-dynamical spacetime), the SAE substrate ontology articulation supplies a unique substrate readout that aligns with the corresponding standard GR result.

Articulation discipline: three different mathematical statuses. The three results being aligned with carry three different mathematical statuses, and these must not be conflated.

The Birkhoff theorem is a mathematical theorem under static spherical vacuum conditions: it establishes the uniqueness of the Schwarzschild geometry as the static spherically symmetric vacuum solution to the Einstein field equations. The theorem is mathematically rigorous within its scope (vacuum, spherical symmetry, static, asymptotic flatness assumptions), and standard proofs exist in the GR literature. P5 articulates the substrate readout that aligns with the Birkhoff theorem, without re-proving the theorem itself; the standard theorem receives a substrate-level reading through the radial closure R_t^sub → 0 on the ∂_t channel.

The Kerr uniqueness theorems are mathematical theorems under stationary axisymmetric vacuum conditions with technical conditions (asymptotic flatness, regularity, and assumptions about horizon structure, with several variant formulations in the literature including the Robinson-Carter-Hawking uniqueness chain). The theorems are mathematically rigorous within their scope. P6 articulates the substrate readout that aligns with the Kerr uniqueness theorems, without re-proving the theorems; the standard theorems receive a substrate-level reading through the helical closure R_{χχ}^sub → 0 on the horizon generator χ_H = ∂_t + Ω_H ∂_φ.

The EP universality is not a mathematical theorem. It is a foundational principle of GR, validated empirically through EP tests of varying precision across the WEP, EEP, and SEP tiers. The empirical validation engages laboratory experiments (Eötvös-type), space-based experiments (satellite tests), lunar laser ranging, binary pulsar observations, and similar contexts; the empirical content sits with the experimental GR community. P7 articulates the substrate-level reading of the EP universality as a foundational principle, not as a derived theorem.

Framing-level identificatory series methodology, not mathematical theorem chain. The three articulations across P5, P6, and P7 form a series methodology closing pattern at the framing level. They do not form a mathematical theorem chain. Lumping them as a "uniqueness trajectory" must be read carefully: the trajectory is the cross-paper observation that SAE substrate ontology articulation supplies a unique substrate readout aligning with the corresponding standard result in each case, not a mathematical chain in which the three results are derived through a unified mathematical argument.

The framing is identificatory in the sense articulated throughout the series. P5 and P6 articulate substrate readings aligning with standard mathematical theorems; the substrate readings are not derivations of the theorems but ontological readings of the theorems. P7 articulates a substrate reading aligning with a foundational empirical principle; the substrate reading is not a derivation of the principle but an ontological grounding for it. The three together articulate a series-level pattern: substrate ontology re-reading supplies unique substrate readouts aligning with the standard GR result in each case, across three different spacetime symmetry contexts and three different mathematical statuses of the aligned-with standard result.

Series-level pattern statement. The trajectory across P5, P6, and P7 may be summarized: the SAE substrate ontology re-reading supplies a substrate-level reading that aligns with the corresponding standard GR result, in three successively less symmetric contexts.

P5: in the static spherically symmetric vacuum context, the substrate readout is the unique radial closure R_t^sub → 0 on the ∂_t channel, aligning with the Birkhoff theorem.

P6: in the stationary axisymmetric vacuum context, the substrate readout is the unique helical closure R_{χχ}^sub → 0 on the horizon generator χ_H, aligning with the Kerr uniqueness theorems.

P7: in the generic non-dynamical spacetime context without global symmetry, the substrate readout is the local calibration universality through the Calibration Isomorphism at each point, aligning with the EP universality at substrate level.

The three substrate readouts are case-specific articulations, each within its own substrate ontology re-reading. The cross-paper trajectory is the series-level observation that substrate ontology articulation succeeds in supplying unique substrate readouts across the three cases, aligning with the standard GR result in each case without re-proving the standard result. This is the series methodology closing pattern of the SAE Relativity Series.

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§14 Honest Boundaries, Future Territory, Short Closing Reflection

P7 closes the SAE Relativity Series. The closing comprises three components: honest limit markers, a future territory map with brief cross-series handoff markers, and a short closing reflection.

Honest limit markers. The following limits hold throughout P7 and across the series.

SAE substrate ontology articulation is not a derivation of GR or of the equivalence principle. The articulation supplies an ontological framing for GR mathematical structures and for the EP foundational principles, without modifying their content or claiming to derive them from SAE axioms.

SAE does not produce quantitative predictions diverging from those of standard GR in the territories covered by P7. The substrate ontology re-reading aligns with the standard quantitative predictions of GR across the cases articulated in P5, P6, and P7. The articulation is at the level of ontological framing, not at the level of quantitative empirical content.

P7 does not cover dynamical evolution. Time-dependent curvature, gravitational wave propagation, black hole mergers, and the dynamical evolution of generic spacetime configurations are handed off to Info VI (DOI 10.5281/zenodo.20066644). P7 maintains the strict non-dynamical and locally kinematic calibration scope established in §1.

P7 does not cover the quantum gravity interface. The substrate-level cell count, the relationship between substrate cell structure and Planck-scale physics, the substrate ontology articulation of the quantum black hole information puzzle beyond the working-lens framing of §3, and similar quantum-gravity-interfacing questions are deferred to an anticipated future SAE quantum gravity interface paper. P7 takes no position on the specific length scale or microscopic structure of substrate cells.

P7 does not cover charged spacetimes, multi-black-hole systems, or other specific spacetime configurations beyond the P5 Schwarzschild, P6 Kerr, and P7 generic-case treatment. The Kerr-Newman electrically charged case introduces gauge sector content that crosses outside the pure spacetime geometry territory of P7. The multi-black-hole system case (multiple black holes interacting dynamically) crosses into the dynamical territory of Info VI. Specific spacetime configurations beyond the three immediate predecessor cases are not articulated.

Type 4+ substrate cell tensor extensions and Mode 5+ Calibration Isomorphism break taxonomy extensions are deferred to future SAE work. P7 maintains its scope within Type 1 / Type 2 / Type 3 cell tensor structures and Mode 0 / Mode 1 / Mode 2 / Mode 3 / Mode 4 break taxonomy.

P7 does not enter the empirical SEP testing territory. The strong-field universality tests, binary pulsar observations, lunar laser ranging Nordtvedt parameter constraints, and similar empirical SEP-sensitive contexts sit with the experimental GR community. P7's three-tier SEP split is an ontology vocabulary for working-lens use, not a framework for empirical SEP testing.

Future territory map with cross-series handoff markers. The following future territories are anticipated within the broader SAE research programme, with brief cross-series handoff indicators.

Dynamical evolution, including gravitational wave propagation, black hole mergers, and time-dependent curvature, hands off to Info VI (DOI 10.5281/zenodo.20066644) for the substrate ontology articulation of dynamical regimes.

Black hole interior generic context extension hands off to Info IV (DOI 10.5281/zenodo.19880112), particularly the §0 anticipated Kerr extension territory for substrate ontology articulation of generic black hole interior structures.

The SAE quantum gravity interface, including substrate cell structure at the Planck scale, the substrate-level articulation of quantum black hole information, and related questions, awaits a future SAE paper. The specific paper position within the broader SAE programme is currently unspecified. Relatedly, P7 articulates the substrate cell tensor's full Type 3 structure in generic spacetime while taking no position on the specific length scale or microscopic structure of substrate cells (cf. §14 honest limit markers); this leaves framework space for future SAE quantum mechanics, that is, the substrate ontology at sub-cell scales, as an anticipated research direction.

The mass-conversation joint paper, articulating the gravitational coupling evolution and the singularity interface, awaits a future SAE paper.

Big Crunch cosmological substrate ontology and related cosmology series extensions await future Cosmo series papers.

The SAE spin ontology specialized paper, articulating the substrate ontology of spin at the quantum-gravitational interface, awaits a future SAE paper as an anticipated future research direction.

SEP-sensitive empirical tests substantive ontology re-reading (Nordtvedt parameter, binary pulsar strong-field universality tests, gravitational self-energy contexts, and related) would fall to a future paper if and when the substrate ontology articulation produces quantitative content diverging from standard GR; otherwise, the territory remains in the experimental GR community.

Classical GR tests substantive ontology re-reading (Mercury perihelion precession, light deflection, Shapiro delay, and related) is reserved for a P5 v2 framework future paper. This anticipated supplementary paper to P5 would treat the substrate ontology re-reading of these classical tests; it does not modify the published P5 status.

Type 4+ symmetry hierarchy extensions (multi-field couplings, gauge sectors including the Kerr-Newman charged case, and broader extensions) await future SAE specific papers.

The 形与流 series cross-integration awaits future cross-series specific papers articulating the substrate ontology connections.

Short closing reflection. The SAE Relativity Series completes with P7. P5 articulated the Schwarzschild substrate ontology re-reading on the ∂_t channel; P6 articulated the Kerr substrate ontology re-reading on the χ_H horizon generator; P7 articulates the EP three-tier substrate ontology re-reading in the generic spacetime context, together with the EP-spacetime structure correspondence backbone observed across the three cases.

The series methodology contribution is the substrate-apparatus distinction discipline, articulated as a working ontology lens for physicists dealing with GR pathologies and EP foundations. The series substantive contribution is the cross-paper observed correspondence pattern linking spacetime symmetry hierarchy to substrate cell tensor structure to Calibration manifestation, identificatorily articulated as the EP-spacetime structure correspondence backbone.

The SAE programme's series-level systematic methodology for GR foundations, quantum gravity, and emergent gravity sits in the territory of Paper 0 (DOI 10.5281/zenodo.19777881) and is not undertaken by P7. The Relativity Series articulates the substrate ontology of generic non-dynamical spacetime at the level of working philosophy of physics; the broader SAE programme contains other series addressing dynamical evolution, information theory, cosmology, and additional topics.

Relativity, under SAE, is not displaced but situated.

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References

SAE Relativity Series, direct predecessor papers:

• Qin, H. SAE Relativity Series, Paper P1: Gravitational time dilation as causal-slot throughput. Zenodo. DOI: 10.5281/zenodo.19836183.
• Qin, H. SAE Relativity Series, Paper P2 (including the Lense-Thirring SAE re-reading as a subsection). Zenodo. DOI: 10.5281/zenodo.19910545.
• Qin, H. SAE Relativity Series, Paper P3: Clock-sector effective dimensional exponent. Zenodo. DOI: 10.5281/zenodo.19992252.
• Qin, H. SAE Relativity Series, Paper P4: Cell tensor and Calibration Isomorphism. Zenodo. DOI: 10.5281/zenodo.20079718.
• Qin, H. SAE Relativity Series, Paper P5: The Schwarzschild horizon under SAE. Zenodo. DOI: 10.5281/zenodo.20105112.
• Qin, H. SAE Relativity Series, Paper P6: The Kerr black hole under SAE. Zenodo. DOI: 10.5281/zenodo.20131993.

SAE programme related papers:

• Qin, H. SAE overall framework: Paper 0 — Gravity ontology. Zenodo. DOI: 10.5281/zenodo.19777881.
• Qin, H. SAE Information Theory, Paper IV: Black hole interior information structure. Zenodo. DOI: 10.5281/zenodo.19880112.
• Qin, H. SAE Information Theory, Paper VI: Dynamical evolution and information conservation. Zenodo. DOI: 10.5281/zenodo.20066644.
• Qin, H. SAE Methodology Overview. Zenodo. DOI: 10.5281/zenodo.18842449.
• Qin, H. L₃ → L₄ Closure: Dual-Language. Zenodo. DOI: 10.5281/zenodo.19361950.

Standard General Relativity references:

• C. W. Misner, K. S. Thorne, J. A. Wheeler. Gravitation. W. H. Freeman, 1973. (MTW)
• R. M. Wald. General Relativity. University of Chicago Press, 1984.
• S. M. Carroll. Spacetime and Geometry: An Introduction to General Relativity. Addison-Wesley, 2004.
• C. M. Will. Theory and Experiment in Gravitational Physics. Cambridge University Press, 2018.

§1 引言

SAE 相对论系列第五篇阐述了静态球对称真空 Schwarzschild 几何在 SAE 框架下的衬底本体论重读. 类时 Killing 矢量场 ∂_t 在 Schwarzschild 几何中既作为自然读出通道, 也作为径向闭合通道: R_t^sub → 0 在视界处表达衬底层径向闭合于 ∂_t 之上. 该重读并不重新证明 Birkhoff 定理, 而是为这一标准数学定理提供一种衬底层的本体论读法. 第六篇阐述了稳态轴对称真空 Kerr 几何的衬底本体论重读. 视界生成元 χ_H = ∂_t + Ω_H ∂_φ 在 r_+ 上为零类向量 (null), 沿该方向提供螺旋闭合通道, 其形式为张量二次型 R_{χχ}^sub := R_{μν}^sub χ^μ χ^ν, 且 R_{χχ}^sub → 0 于 r_+ 上, 此即衬底层的螺旋闭合. 综合第五与第六篇, 衬底-装置区分纪律作为跨论文的系统性方法论得以建立: 装置层的数学特征 (如 Schwarzschild 视界处的 g_{rr} → ∞、Kerr 环奇点、内视界 r_-、r<0 解析延拓区域、Kerr 深内区域的闭合类时曲线、以及极端 Kerr 之无限喉部几何) 均居于装置层, 而衬底层在所有这些装置层特征处保持有限的因果槽计数与良定义的因果槽张量.

第五与第六篇各自在特定的时空对称性情境中阐述衬底读出. 第五篇适用于 SO(3) 球对称结合静态条件 (即类时 Killing 矢量场 ∂_t 与三个 SO(3) 旋转 Killing 矢量场同时存在), 衬底因果槽张量 R_{μν}^sub 在径向自然基底下取第一型对角结构, 含四个独立分量. 第六篇适用于 SO(2) 轴对称稳态条件 (即类时 Killing 矢量场 ∂_t 与轴向 Killing 矢量场 ∂_φ 同时存在), 衬底因果槽张量在轴对称自然基底下取第二型混合结构, 即对角分量加 R_{tφ}^sub 非对角分量, 该非对角分量为轴对称稳态对称性所容许, 共含五个独立分量. 第六篇 §10.3 给出一项对称性层级的观察: 第五篇之 SO(3)、第六篇之 SO(2)、以及第七篇所期待的仅余局域 Lorentz 对称性, 三者形成系列层级模式; 然而第六篇仅作此观察, 完整的层级阐述留待第七篇.

第七篇处理一般时空情境, 即不假设除局域 Lorentz 标架以外的任何整体对称性. 这是系列对称性层级中的极限情形: 既不假设类时 Killing 矢量场, 也不假设任何空间 Killing 矢量场, 仅在每一点上承认局域 Lorentz 标架作为流形的基本性质. 第七篇阐述衬底因果槽张量在此一般情境下的第三型结构, 含十个独立分量 (即四维 Lorentzian 向量空间上之二阶对称张量的完整独立分量数), 并阐述等效原理三层 (即弱等效原理 WEP、Einstein 等效原理 EEP、强等效原理 SEP) 在 SAE 之下的衬底本体论重读. 此即本文的主论旨.

本文的主线骨架在于等效原理与时空结构之对应 (EP-spacetime structure correspondence), 此为跨第五、第六、第七篇所观察到的模式. 第五篇对应于 SO(3) ↔ 第一型 ↔ ∂_t 通道之径向闭合; 第六篇对应于 SO(2) ↔ 第二型 ↔ χ_H 视界生成元之螺旋闭合; 第七篇对应于仅局域 Lorentz ↔ 第三型 ↔ 局域校准而无整体闭合通道. 等效原理三层在第七篇中阐述为时空对称性层级在一般时空极限情形下的呈现, 即跨情形所观察到的对应模式, 而非自 SAE 公理推导得出的结论. 此种框架定位为识别性 (identificatory) 而非推导性 (derivational). 在 SAE 之下, 等效原理并非外加于时空结构之上的孤立基础原则, 而是时空对称性层级与衬底因果槽张量结构之间所观察到的对应在极限情形 (即整体对称性退化至仅局域 Lorentz 之时) 下的衬底层阐述.

第七篇为 SAE 相对论系列的收官之作. 系列的收束意义由实质性内容自身承载, 而非由篇幅或措辞的提升承载. 第七篇维持第五与第六篇的篇幅模式, 不膨胀为系列总结性长文. 系列闭合通过 §8 至 §12 的实质性内容自然完成 (即等效原理三层重读与主线骨架阐述), 加 §14 的简短反思一页以内作为冷静的收尾确认. SAE 作为广义相对论基础、量子引力与涌现引力的整体框架之系统性方法论, 居于 Paper 0 (DOI: 10.5281/zenodo.19777881) 之范围, 并不由第七篇承担.

第七篇维持若干范围纪律.

第七篇严格处于非动态、局域运动学校准范围之内. 本文处理一般时空的局域校准结构, 不假设类时 Killing 矢量场, 亦不假设任何整体对称性. 动态演化、引力波、黑洞合并、与含时曲率均完全移交至信息论第六篇 (DOI: 10.5281/zenodo.20066644). 第五篇 (静态) 与第六篇 (稳态) 为第七篇一般非动态范围的特定对称性子集 — 第五与第六篇的具体稳态情形皆居于第七篇的一般非动态领域之内.

第七篇的阐述处于定性、方向性、解释力较强的领域. SAE 相对论系列为衬底本体论的哲学论文, 不代物理学家进行定量判断. 第七篇不产生与标准广义相对论不同的定量预言. SEP-敏感的实验性测试 (如 Nordtvedt 参数、双脉冲星强场普适性测试、月球激光测距等) 与经典广义相对论测试 (如水星近日点进动、光线偏折、Shapiro 延迟) 皆不进入第七篇主体 — 此等领域居于实验广义相对论共同体的范围. 第七篇为等效原理提供衬底本体论词汇作为工作透镜, 不主张相对于等效原理实验性测试的经验优越性.

第七篇的阐述为识别性, 而非推导性. 等效原理与时空结构之对应的主线骨架, 阐述为跨第五、第六、第七篇情形所观察到的模式, 不主张自 SAE 公理推导. 此立场与 SAE 数学基础论文 Schema 的认识论立场一致 — 衬底本体论的阐述为广义相对论的数学机制提供一种本体论框架定位, 而不取代广义相对论, 亦不自 SAE 推导广义相对论.

第七篇维持系列的拒收事项立场. T4 定量锚不入主体 (越界至信息论第六篇的定量领域). 替代性框架不入主体引用 (如 Asymptotic Safety、Causal Set 理论、Verlinde 熵引力、Jacobson 热力学推导、Ashtekar 变量、圈量子引力、范畴论平行论述、操作主义 Einstein 1916 重述等, 皆不入第七篇主体作为平行观察引用, 留待所期待的未来 SAE 量子引力接口论文, 或不予阐述). 低维玩具模型不入 (越界至定量论文领域).

第七篇维持 Paper 0 §1.5 的诚实立场. SAE 相对论系列不主张相对于广义相对论与量子场论的经验优越性, 系列的交付物为衬底本体论的阐述与工作透镜, 而非与广义相对论或量子场论在经验预言上竞争的框架.

第一篇引用准确性维持. SAE 相对论第一篇 (DOI: 10.5281/zenodo.19836183) 处理引力时间膨胀的因果槽通量, 并非 Lense-Thirring 论文. Lense-Thirring 的 SAE 重读作为第二篇 (DOI: 10.5281/zenodo.19910545) 的子内容出现.

全文共十四节, 组织如下. §1 引言. §2 准备工作, 涵盖标架场形式与无整体 Killing 对称性假设. §3 衬底-装置区分纪律之普遍陈述. §4 因果槽张量第三型结构. §5 因果槽张量第一、二、三型之系统层级. §6 校准同构破坏模式分类 (模式 0 至模式 4). §7 等效原理三层衬底本体论重读总论. §8 弱等效原理之衬底本体论重读. §9 Einstein 等效原理之衬底本体论重读. §10 强等效原理之衬底本体论重读 (三层分裂). §11 等效原理与时空结构之对应主线. §12 对称性层级主线与等效原理三层之对应. §13 唯一性轨迹与系列方法论闭合模式. §14 诚实边界、未来领域图、与简短收尾反思.

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§2 准备工作

第七篇在一般时空情境中阐述衬底本体论, 需要若干技术准备. 标准广义相对论形式 (局域 Lorentz 标架、标架场、标准微分几何) 不在此重新推导, 此等内容稳固居于标准广义相对论教科书 (MTW、Wald、Carroll 等参考文献) 之范围. 第七篇仅阐述此等形式在衬底本体论视角下的读法, 及其与 SAE 框架的连接.

标架场形式简要回顾. 在一般四维 Lorentz 时空 (M, g_{μν}) 之中, 每一点 p ∈ M 皆容许一局域 Lorentz 标架, 即四个正交归一的矢量场 {e_a^μ}_{a=0,1,2,3}, 满足 g_{μν} e_a^μ e_b^ν = η_{ab}, 其中 η_{ab} = diag(-1, +1, +1, +1) 为 Minkowski 度规, μ 为时空坐标指标, a 为局域 Lorentz 标架指标. 标架 {e_a^μ} 亦称 vierbein (德语 "四足"). 标架的局域性为基本特征: 不假设跨时空点存在普适的自然基底. 同点 p 内之局域 Lorentz 变换 Λ_a^b 通过 e_a^μ → Λ_a^b e_b^μ 作用于标架, 维持 η_{ab} 不变. 跨不同时空点的标架到标架的输运通过自旋联络 ω_μ^{ab} 阐述, 此自旋联络通过标架公设与装置度规 g_{μν} 的仿射联络 Γ^λ_{μν} (即 Christoffel 符号) 相联系.

无整体 Killing 对称性假设. 第五篇适用于静态球对称 Schwarzschild 度规, 该度规同时含类时 Killing 矢量场 ∂_t 与三个 SO(3) 旋转 Killing 矢量场. 第六篇适用于稳态轴对称 Kerr 度规, 该度规同时含类时 Killing 矢量场 ∂_t、轴向 Killing 矢量场 ∂_φ、与视界生成元 χ_H = ∂_t + Ω_H ∂_φ. 第七篇不假设任何整体 Killing 矢量场, 既不假设类时 Killing, 亦不假设空间 Killing. 局域 Lorentz 标架在每一点皆存在 (此为时空流形的标准性质, 不蕴涵整体对称性), 然而跨点的自然基底不可用, 因为不同点的局域 Lorentz 标架不通过任何整体 Killing 矢量场相联系. 此为一般非动态时空的典型构型.

衬底因果槽张量回顾. 在 SAE 框架之下, 衬底层包含因果槽、因果槽计数 (cell count)、与读取加连接本体. 衬底因果槽张量 R_{μν}^sub 为衬底层的二阶对称张量, 阐述因果槽之间的耦合结构. 须注意 R_{μν}^sub 与标准装置层 Ricci 张量 R_{μν} 不同, 尽管记号相似 — 衬底与装置两层在 SAE 框架之中保持范畴分离. 衬底因果槽计数为有限的本体论度量 ("有多少个因果槽"), 而衬底因果槽张量为因果槽耦合模式的阐述 ("因果槽之间如何耦合"). 此二者相互独立 — 因果槽计数与因果槽张量的结构复杂度之间无直接决定关系.

第五篇阐述衬底因果槽张量 R_{μν}^sub 在 Schwarzschild 球对称真空情境下为第一型对角结构, 含四个独立分量: R_t^sub (类时分量)、R_r^sub (径向分量)、与 R_⊥^sub (垂直分量, 因 SO(3) 对称性两个角向分量相等). 第五篇 §3.1 阐述装置层的坐标依赖特征 g_{rr} → ∞ 于视界处为坐标人造现象, 而衬底层在视界处有 R_t^sub → 0 阐述 ∂_t 通道上的径向闭合 (即沿类时 Killing 矢量场方向的衬底层通道闭合).

第六篇阐述衬底因果槽张量 R_{μν}^sub 在 Kerr 稳态轴对称真空情境下为第二型混合结构, 含五个独立分量: 对角分量 R_{tt}^sub、R_{rr}^sub、R_{θθ}^sub、R_{φφ}^sub, 加非对角分量 R_{tφ}^sub, 后者为轴对称稳态对称性所容许. 第六篇阐述 R_{χχ}^sub := R_{μν}^sub χ^μ χ^ν 在视界生成元 χ_H = ∂_t + Ω_H ∂_φ 上的结构, 其 R_{χχ}^sub → 0 于 r_+ 上实现衬底层的螺旋闭合.

装置度规与衬底因果槽张量的关系. 在 SAE 框架之下, 装置度规 g_{μν} 为装置层的度规结构, 通过读取加连接机制与衬底因果槽张量相联系. 第五与第六篇在特定对称性情境 (球对称与轴对称) 下阐述此联系的具体形式. 第七篇在一般情境下通过校准同构 (Calibration Isomorphism) 阐述此联系: 局域装置校准在每一点处将衬底因果槽张量的一阶读出结构 (即装置度规的一阶导数 Christoffel 结构) 吸收入局域装置单位, 局部恢复 Minkowski 形式. 此为 §9 Einstein 等效原理衬底本体论重读的主要内容, §2 仅作简要确认.

校准同构继承自第四篇. SAE 相对论第四篇 (DOI: 10.5281/zenodo.20079718) 包含因果槽张量 d_eff^{μν}、校准同构、与 L_3→L_4 闭合的阐述. 在第四篇框架之下, 校准同构为衬底层因果槽张量与装置层度规之间映射的良定义性 — 即局域装置校准在衬底因果槽张量的局部读法上获得成功. 第五篇阐述校准同构破坏的模式 1 (视界处之投影奇异, R_t^sub → 0 于 ∂_t 通道上). 第六篇阐述模式 2 (分量激活, 即 Kerr 能层中之 R_{tφ}^sub) 以及若干复合模式情形, 其中极端 Kerr 无限喉部为模式 1、模式 2、与模式 4 之复合实例. 第七篇阐述模式 3 (无整体自然基底, 第七篇之主导模式), 并于 §6 提供系统性的模式 0 至模式 4 分类.

N_active 与 d_eff^{(τ)} 与 d_eff^{μν} 之三层记号纪律. SAE 框架跨论文含三层有效维度记号:

• N_active: 整数型主动方向计数, 即衬底层因果槽耦合中主动方向的数目.
• d_eff^{(τ)}: 时钟扇区有效指数, 由 SAE 相对论第三篇 (DOI: 10.5281/zenodo.19992252) 阐述, 即衬底层时钟读出之有效维数指数. 在 P3 语境下, 该量在弱场极限趋近 3, 在强场区域取值于 (2, 3) 或趋近 3⁻; 此为 clock-sector scalar 之用法, 不与 full tensorial effective dimension d_eff^{μν} 混淆.
• d_eff^{μν}: 完整张量型有效维度, 由第四篇阐述, 即衬底因果槽张量的完整张量阐述.

此三者为不同的记号侧面: 一为整数计数, 一为标量指数, 一为张量. 在系列之中保持一致使用. 第七篇不重新推导此三层 (其建立于第三与第四篇), 仅作为框架的记号纪律进行确认.

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§3 衬底-装置区分纪律: 普遍陈述

衬底-装置区分纪律为 SAE 相对论系列的跨论文系统性方法论. 第五与第六篇在特定对称性情境下建立其应用实例, 第七篇阐述在一般时空情境下的普遍陈述: 装置层数学特征与衬底层本体论承诺须分开读取, 衬底层的有限因果槽计数在所有装置病态之中维持. 此为 SAE 相对论系列向物理学家提供的工作本体论透镜 — 物理学家既不必拒斥广义相对论的数学机制, 亦不被迫接受广义相对论的数学特征作为衬底层的实在.

第五与第六篇跨论文情形枚举. 衬底-装置区分纪律在第五与第六篇之中通过多个应用实例得以贯彻.

第五篇 §3.1 阐述 Schwarzschild 之坐标发散 g_{rr} → ∞ 于视界处为装置层的坐标依赖特征. Schwarzschild 坐标在视界处坐标奇异, 而 Eddington-Finkelstein 坐标与 Kruskal 坐标在视界处坐标正则. 在衬底层, 因果槽张量 R_{μν}^sub 与因果槽计数跨视界维持有限, 而 R_t^sub → 0 在 ∂_t 通道上阐述衬底层特征 (即衬底通道闭合, 非坐标奇点). 装置层特征 (g_{rr} → ∞) 与衬底层特征 (R_t^sub → 0) 分开读取.

第五篇 §5.4 阐述 Schwarzschild r → 0 极限为装置层之点. 在标准 Schwarzschild 坐标卡之中, r = 0 为坐标依赖之点特征, 装置层之数学结构含 Kretschmann 标量发散 K = R_{μνρσ} R^{μνρσ} → ∞ 当 r → 0 之时, 此为装置几何特征. 在衬底层, r = 0 处之因果槽计数阐述为有限 (单个衬底因果槽, 或类似之小有限数), 而非零维之无限曲率实在.

第六篇 §13.1 阐述 Kerr 环奇点, 由 ρ^2 := r^2 + a^2 cos^2 θ = 0 于 r = 0 与 θ = π/2 同时满足之时给出, 在装置几何中形成一维环, 为装置层之特征, 含 Kretschmann 标量在一维环上之发散. 在衬底层, 环上因果槽计数维持有限, 环为装置投影之代数特征, 非衬底层之一维对象.

第六篇 §13.2 阐述 Kerr 内视界 r_- = M - √(M^2 - a^2) 为装置层之坐标边界. 在 r < r_- 之中, Boyer-Lindquist 坐标显示类空-类时切换与质量膨胀现象. 在衬底层, 因果槽张量与因果槽计数在 r = r_- 与 r < r_- 区域皆维持有限且良定义, 而质量膨胀现象与坐标切换居于装置层.

第六篇 §13.3 阐述 Kerr r < 0 区域 (即 r = 0 环之外的解析延拓进入负 r 区域) 为装置层之解析延拓特征, 非衬底层之 "负径向坐标" 实在. 在衬底层, r < 0 区域与标准 r > r_+ 区域之因果槽计数皆维持有限; r < 0 区域为装置几何之解析延伸领域, 非衬底本体论之延伸.

第六篇 §13.4 阐述 Kerr 之闭合类时曲线 (CTC), 即 r < 0 区域中出现之类时曲线自闭合, 为装置层之几何特征. 在衬底层, 因果槽耦合之读取加连接本体并不蕴涵衬底层之因果环路实在. 闭合类时曲线为装置投影之几何特征, 非衬底层之时间环路.

第六篇 §11.3 阐述极端 Kerr 之无限喉部 (即 a = M 之近视界 NHEK 几何). 在极端极限之下, 自任何有限 r > r_+ 至 r_+ = M 之固有径向距离发散, 此为装置层之 "无限喉部" 几何特征. 在衬底层, 极端 Kerr 视界喉部之中的因果槽计数与因果槽张量皆维持有限; 无限喉部为装置径向坐标之固有距离行为, 非衬底因果槽计数之无限.

此等情形皆阐述同一模式: 装置层数学特征 (奇点、坐标边界、解析延拓、闭合类时曲线、无限固有距离) 为装置层阐述的数学结构, 而衬底层在此等装置特征处维持有限因果槽计数与良定义之因果槽张量, 不继承任何装置数学病态.

一般时空情境下的普遍立场. 第七篇阐述衬底-装置区分纪律之普遍陈述, 适用于一般时空: 对任意时空度规 g_{μν}、任意坐标卡、与任意装置几何之数学病态, 衬底-装置区分纪律皆适用. 此为物理学家提供工作本体论透镜: 在面对任意时空数学特征之时, 物理学家可区分装置层特征 (装置层之数学结构, 包括坐标、度规、曲率标量行为) 与衬底层读法 (衬底层之因果槽计数与因果槽张量阐述, 居于本体论承诺之层次).

对物理学家之实际意义在于: 物理学家无须接受 SAE 整体框架, 即可使用衬底-装置区分纪律作为思考广义相对论病态的组织原则. 例示如下:

• Schwarzschild 视界处之坐标发散 g_{rr} → ∞ 为真奇点抑或坐标人造现象? 衬底-装置区分纪律将其阐述为装置层坐标特征, 不强加物理学家接受任何衬底层之 "视界作为本体论边界".
• Kerr 环奇点为一维曲率发散实在乎? 衬底-装置区分纪律将其阐述为装置层环特征, 不强加接受衬底层之 "一维奇异对象".
• Hawking 辐射熵与 Bekenstein-Hawking 视界面积熵之关系如何? 衬底-装置区分纪律将装置层之视界面积与热力学熵阐述为一种读法, 将衬底层之因果槽计数微观态枚举阐述为另一种, 为量子黑洞信息悖论提供本体论框架定位, 不强加任何特定量子化方案. 此处 §3 仅在工作词汇层面阐述框架定位; 量子黑洞信息悖论之实质性衬底本体论阐述留待 §14 所述之未来 SAE 量子引力接口论文.

工作透镜之范围与边界. 衬底-装置区分纪律为工作本体论透镜, 非推导工具. 不推导任何与广义相对论不同的具体定量预言, 不取代广义相对论之数学机制, 不主张相对于广义相对论之经验优越性. 仅阐述一项组织原则, 为物理学家在思考广义相对论病态之时提供工作框架, 用以区分数学特征与本体论承诺. 此即第七篇之实质性贡献 C1 (衬底-装置区分纪律之普遍陈述) 作为面向物理学家的工作透镜.

该纪律不蕴涵装置层数学特征为 "非实在" 或 "虚构". 装置层为 SAE 框架内的实质性层次, 装置层与衬底层皆为阐述的实质性侧面. 该纪律所阐述者乃: 两层不可混淆, 装置层数学特征不应被假定为与衬底层本体论一一对应.

§4 因果槽张量第三型结构

第七篇阐述衬底因果槽张量 R_{μν}^sub 在一般时空情境下的结构. 由于不假设除局域 Lorentz 标架以外的任何整体对称性, 跨时空点之普适自然基底不可用, 因果槽张量取其完整结构: 一个十独立分量的 (四维 Lorentzian 向量空间上之二阶对称张量) 第三型结构. 本节阐述第三型的衬底侧结构, 完整的第五、第六、第七篇主线对应留待 §11, 等效原理侧之对应详节留待 §12.

一般时空: 无普适自然基底. 在一般非动态时空之中, 无整体 Killing 矢量场可用, 不能假设任何跨点的偏好自然基底. 一点 p 处之衬底因果槽张量 R_{μν}^sub 阐述为完整的四维二阶对称张量, 含极大独立分量数.

标架可读结构. 在 p 点之局域 Lorentz 标架 e_a^μ(p) 下, 因果槽张量读为:

R_{ab}^sub(p) = e_a^μ(p) e_b^ν(p) R_{μν}^sub(p)

其中 a, b 为局域 Lorentz 标架指标, μ, ν 为时空坐标指标. 局域标架在 p 处提供因果槽张量之分量表示, 但不提供普适对角化. 在特殊代数情形下 (即 Segre 型简并之中, 因果槽张量与装置度规共享某种本征结构之时), 单一标架内之部分或完整对角化可能成立. 然而在一般情形下, 此种对齐并不被保证: 一般第三型因果槽张量保有不与装置度规在单一标架中同时对角化的衬底分量. 局域标架给出分量表示, 而非普适对角化.

四维之分量数. 一个四维 Lorentzian 向量空间上之二阶对称张量含十个独立分量. 标准分解读为: 一个迹分量 (T = η^{ab} R_{ab}^sub) 加九个无迹分量 (后者进一步分解为五个无迹对称空间分量、三个混合时间与空间之矢量型分量、与一个标量分量, 遵循标准四维二阶对称张量分解). 此十分量计数为一般二阶对称张量的簿记; 任何特定分量之上游物理重要性由局域校准与具体源或情境决定, 不由簿记本身决定. 此为防止误读 — 第三型因果槽张量并非 "全能十参数解释器".

第三型的主要阐述. 第三型因果槽张量的本体论并非 "所有分量同等重要", 而是 "无整体对称性可供事先指定哪些分量可被丢弃". 第一型 (第五篇之 SO(3) 球对称) 与第二型 (第六篇之 SO(2) 轴对称) 皆配有对称性诱导的自然基底, 允许事先知晓哪些分量被强制为零 — 第一型: 所有非对角分量在径向自然基底中消失; 第二型: 除 R_{tφ}^sub 之外所有非对角分量在轴对称自然基底中消失. 第三型不具此种结构. 其内容正在于此种事先约简的缺席.

整体结构: 标架到标架的变换. 跨不同时空点之标架到标架变换通过自旋联络 ω_μ^{ab} 阐述 (参 §2). 不假设跨点之普适自然基底. 第三型因果槽张量的整体结构由跨流形之标架到标架变换模式给出; 在一般时空之中, 此模式不容许整体对角化. 此即 §6 校准同构破坏分类中模式 3 的内容.

衬底因果槽计数维持有限. 衬底因果槽计数维持有限, 不依赖于因果槽张量之结构复杂度. 因果槽计数为本体论度量 (因果槽数目), 因果槽张量为因果槽耦合模式之阐述 (因果槽如何耦合). 两者为衬底的独立侧面. 一般第三型因果槽张量结构不蕴涵无限因果槽计数, 有限因果槽计数亦不约束因果槽张量之结构复杂度. 此与第五篇 (§3.1、§5.4)、第六篇 (§13.1–§13.4、§11.3)、与本文 §3 之跨论文阐述一致.

第三型作为衬底的默认阐述. 在缺乏任何整体对称性约束之时, 衬底因果槽张量呈现其完整的第三型结构, 含全部十个独立分量. 第五篇之球对称情形与第六篇之轴对称情形为此默认阐述在对称性约束下的特定约简: 第一型保留四个分量, 第二型保留五个分量. 此约简模式跨情形得以观察: 较少之整体对称性留下较多之独立分量经历约简之后, 因而阐述较为丰富之因果槽张量结构. 框架定位维持识别性: 第五与第六篇之特定情形被观察为第三型在特定对称性约束下之约简, 而非自第三型先验之推导.

具体实例. 两类示意性构型自然处于第三型领域.

第一类为多源静态引力构型: 一种 n 体静态布置 (例如, 两个广泛分离之静态类黑洞源以某种构型保持在相关近似下稳态). 此种构型不携带整体球对称性, 亦不携带整体轴对称性, 但维持非动态. 跨此种构型之衬底因果槽张量携带完整第三型结构: 衬底因果槽耦合模式反映合成的源内容, 不容许任何对称性诱导之约简.

第二类为破缺球对称之 Schwarzschild 背景上的稳态线性微扰. 球对称背景结构 (第一型) 被一稳态对称性破缺项扰动, 后者不留任何 SO(3) 作用不变. 受扰之衬底因果槽张量处于第三型领域: 对称性破缺微扰引入了球对称自然基底所强制为零的独立分量.

两类构型不进入第七篇主体, 与严格非动态范围一致: 在 FRW 演化背景上之宇宙学微扰 (背景层面携带本质上含时之演化, 越界至信息论第六篇领域), 以及数值相对论模拟之快照 (从演化时空中抽取单一时间切片, 越界同一范围). 动态构型之衬底本体论阐述移交至信息论第六篇.

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§5 因果槽张量第一、二、三型之系统层级

本节阐述跨三种因果槽张量类型之衬底侧结构层级, 作为 §11 完整第五、第六、第七篇主线对应的预览. 本节的阐述维持在衬底层, 聚焦于时空对称性约束与衬底因果槽张量结构之间的结构关系. 等效原理侧之对应详节留待 §12.

第一型 (第五篇 SO(3) 球对称). 在静态球对称 Schwarzschild 情境之下, 衬底因果槽张量在径向自然基底中取第一型对角结构:

R_{μν}^sub ~ diag(R_t^sub, R_r^sub, R_⊥^sub, R_⊥^sub)

四个独立分量为 R_t^sub (类时)、R_r^sub (径向)、与 R_⊥^sub (垂直, 因 SO(3) 对称性两个角向分量相等). 径向自然基底由 SO(3) 旋转 Killing 矢量场结合类时 Killing 矢量场 ∂_t 诱导. 所有非对角分量消失. 第五篇阐述视界处 ∂_t 通道上之径向闭合 R_t^sub → 0.

第二型 (第六篇 SO(2) 轴对称稳态). 在稳态轴对称 Kerr 情境之下, 衬底因果槽张量在轴对称自然基底中取第二型混合结构:

R_{μν}^sub ~ 分块对角{(R_{tt}^sub, R_{tφ}^sub; R_{tφ}^sub, R_{φφ}^sub), R_{rr}^sub, R_{θθ}^sub}

五个独立分量为对角之 R_{tt}^sub、R_{rr}^sub、R_{θθ}^sub、R_{φφ}^sub, 加非对角之 R_{tφ}^sub. 轴对称自然基底由类时 Killing 矢量场 ∂_t 结合轴向 Killing 矢量场 ∂_φ 诱导. 非对角 R_{tφ}^sub 为轴对称稳态对称性所容许: 其表达衬底层之时间-方位角通道之间旋转拖曳型耦合. 第六篇阐述视界生成元 χ_H = ∂_t + Ω_H ∂_φ 上之螺旋闭合 R_{χχ}^sub → 0 于 r_+ 上.

第三型 (第七篇仅局域 Lorentz). 在一般时空情境之中, 不假设除局域 Lorentz 以外之任何整体对称性, 衬底因果槽张量取第三型完整结构, 含全部十个独立分量, 且无普适自然基底. 该张量在局域上标架可读, 但在一般情形下不可对角化, 一如 §4 所阐述. 其阐述为通过校准同构在每一点之局域校准, 不携带整体闭合通道.

对称性与张量结构之互补约简关系. 结构层级显示对称性约束与因果槽张量结构丰富度之间的互补约简关系, 一如 §11 所述. 对称性约束作为约简算子作用于因果槽张量, 较强之对称性约束约简较多之独立分量; 反之, 较少之整体对称性留下较多之独立分量经历对称性约简之后:

• 第一型: SO(3) 加静态, 四个独立分量, 在径向自然基底中对角.
• 第二型: SO(2) 加稳态, 五个独立分量, 在轴对称自然基底中对角加 R_{tφ}^sub 非对角.
• 第三型: 仅局域 Lorentz, 十个独立分量, 无普适自然基底.

此模式跨三种情形得以观察; 其非自 SAE 公理推导之定理. 框架定位为识别性, 与 SAE 数学基础论文 Schema 的认识论立场同义.

主线骨架预览.

时空对称性	因果槽张量类型	独立分量数	校准呈现	论文
SO(3) 球对称	第一型对角	4	∂_t 通道径向闭合	第五篇
SO(2) 轴对称	第二型混合	5	χ_H 螺旋闭合	第六篇
仅局域 Lorentz	第三型完整	10	局域校准, 无整体闭合通道	第七篇

完整主线骨架阐述, 包括跨三行之等效原理相关衬底读法, 留待 §11. 第七篇行之第四栏读为 "局域校准, 无整体闭合通道": 第七篇并非与第五篇径向闭合及第六篇螺旋闭合并列之第三种闭合情形, 而是在缺乏任何整体闭合通道之时之一般局域校准. 此与第七篇缺乏任何整体 Killing 矢量场一致 — 后者排除了第五与第六篇所阐述之全局识别闭合通道之可能性.

未来第四型及之上作为开放领域. 系列跨第五、第六、第七篇阐述第一、第二、第三型. 至第四型及之上之延伸留作未来 SAE 工作之开放领域. 此种延伸将发生于超出仅局域 Lorentz 之情境: 多场耦合、规范扇区 (例如 Kerr-Newman 电荷情形)、与局域对称性结构之更广义延伸. 此等越出第七篇之纯时空几何领域, 不在本文阐述.

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§6 校准同构破坏模式分类

校准同构 (Calibration Isomorphism, 简称 Cal Iso) 由第四篇 (DOI: 10.5281/zenodo.20079718) 阐述为衬底因果槽张量与装置度规之间映射的良定义性. 第五与第六篇在各自对称性情境下阐述特定的校准同构破坏模式. 第七篇阐述跨模式 0 至模式 4 之系统分类, 组织在相对论系列各情形之中所出现之破坏模式.

该分类为描述性的, 不在任何更深之意义上为详尽性的: 它将跨第五、第六、第七篇实例所观察之模式组织为工作词汇, 以使物理学家可使用此词汇以单一框架标记坐标奇点、视界、能层、内视界、潮汐残余、与代数简并. 每个模式与一简短之数学或结构标记一同阐述, 用于识别.

模式 0: 正常可校准. 校准同构在校准点正常成功. 局域标架将装置度规读为 Minkowski 形式, Einstein 等效原理在局域标架处成立.

标记: g_{ab}^{app}(p) = η_{ab} + O(x^2), 且在 p 处之自由下落标架中有 ∂_c g_{ab}^{app}(p) = 0.

模式 0 为 Einstein 等效原理相关的一般时空 "正常" 模式. 它是一般时空中任何容许正则局域校准之区域的默认模式. 第七篇 §9 (Einstein 等效原理衬底本体论重读) 阐述模式 0 为 Einstein 等效原理所依据之衬底阐述.

模式 1: 投影奇异. 装置度规在校准点发散 (或以其他方式坐标奇异), 而衬底因果槽张量与衬底因果槽计数维持有限.

标记: 装置度规之分量发散 (如 g_{rr} → ∞), 而衬底层之因果槽张量与因果槽计数维持有限.

实例: Schwarzschild 视界处之坐标发散 g_{rr} → ∞ 在标准 Schwarzschild 坐标中为装置投影奇异. 在 Eddington-Finkelstein 或 Kruskal 坐标之中, 视界处之装置几何为坐标正则, 模式 1 在彼等坐标卡中缺席. 衬底层之径向闭合 R_t^sub → 0 于 ∂_t 通道上为不同于装置层模式 1 投影奇异之衬底层特征.

模式 2: 通道激活. 一个由对称性情境所选之特定非对角衬底因果槽张量分量变为非平凡, 标志着衬底层在较高对称性情形下未活动之跨通道耦合.

标记: 衬底因果槽张量在某一特定通道上携带非平凡之非对角分量 (例如, 轴对称稳态情形中之 R_{tφ}^sub ≠ 0).

实例: R_{tφ}^sub 在 Kerr 能层中之激活为模式 2 实例. Kerr 视界生成元 χ_H = ∂_t + Ω_H ∂_φ 通过模式 2 跨通道耦合提供螺旋闭合通道 R_{χχ}^sub. 极端 Kerr 无限喉部为一种复合情形, 涉及模式 1 (装置径向坐标固有距离之投影奇异)、模式 2 (R_{tφ}^sub 通道激活)、与模式 4 型之代数简并. 其并非一个纯净之模式 2 实例, 最好阐述为一种复合模式情形.

模式 3: 无整体自然基底. 仅有局域 Lorentz 校准片段可用, 跨时空点不存在普适自然基底. 每点之局域标架提供分量表示, 但跨点之标架到标架变换模式不容许普适整体对角化. 一般第三型因果槽张量与装置度规在一般点处不在单一标架中同时对角化.

标记: 跨时空点之标架到标架变换模式不容许普适整体自然基底; 一般第三型因果槽张量保有不与装置度规在任何单一标架中同时对角化之衬底分量.

实例: 一般非动态时空 (多源静态构型、Schwarzschild 之上之稳态对称性破缺线性微扰, 等等, 如 §4 所阐述). 模式 3 为第七篇之主导模式. 衬底层之等效原理普适性 (即 §10 SEP-local 领域之等效原理普适性阐述, 与 §9 Einstein 等效原理之局域校准普适性) 处于模式 3 领域: 缺乏整体自然基底正是阐述衬底层等效原理之局域校准普适性之条件.

模式 3 与模式 0 在一般时空中同时成立, 适用之拓扑范围不同: 模式 0 阐述任意单点处之局域装置校准成功 (逐点局域), 而模式 3 阐述跨时空点之标架到标架变换无法在全球范围内统合为普适整体对角化基底 (跨点整合). 此共存关系在一般时空中为典型情形 — 局域校准在每点上成功, 但局域校准片段无法跨点拼合为普适整体基底.

模式 4: 代数简并体制. 衬底因果槽张量本征通道结构展现代数简并特征. 该简并为张量结构之代数特征, 非动态相变.

标记: 衬底因果槽张量在其本征通道分解中展现代数简并 (例如, Segre 型简并或其他代数特殊情形, 其中本征值重合或本征矢量结构变为非一般).

严格非动态立场贯穿始终. 模式 4 在静态 / 稳态体制中阐述为代数特征, 非动态闭合相变. 避免使用 "逼近" 或 "趋向简并" 之语言; 衬底因果槽张量在某构型处或处于代数简并体制中, 或不处于, 作为该构型之结构特征.

模式 4 之实例典型地与其他模式一同出现于复合情形中: 极端 Kerr 无限喉部 (复合模式 1 + 模式 2 + 模式 4 情形), 某些代数特殊真空解, 与其他构型之中本征结构变为非一般之处. 分类为对所观察结构特征之描述; 其不预测动态演化.

第七篇无模式 5 占位. 第七篇之模式 0 至模式 4 分类在纯时空几何之领域内阐述 (即衬底因果槽张量与装置度规之间, 在一般非动态时空情境中). 至超出纯时空几何之情境下所出现之附加模式之延伸, 包括多场耦合、规范扇区 (例如 Kerr-Newman 电荷情形引入电磁应力-能量贡献), 与更广义之规范或物质扇区延伸, 留待未来 SAE 工作. 所期待之未来 SAE 量子引力接口论文或相关之未来 SAE 论文为此种延伸之自然归处. 第七篇维持严格之模式 0 至模式 4 范围, 居于纯时空几何之内; 不插入模式 5 占位以预期未来工作.

模式分类作为工作透镜. 模式 0 至模式 4 分类为物理学家提供统一词汇, 用以标记跨广义相对论情境所遇之装置特征: 坐标奇点 (模式 1)、视界 (典型地为复合模式 1 加衬底层闭合)、能层 (模式 2)、内视界 (装置几何中之模式 1 加衬底层特征)、Einstein 等效原理相关校准中之二阶潮汐残余 (模式 0 结构加曲率残余)、与代数简并 (模式 4). 该分类于单一框架中组织此等特征, 物理学家可作为工作透镜采纳, 而无须接受 SAE 整体本体论.

§7 等效原理三层衬底本体论重读总论

本节为 §8 (弱等效原理 WEP)、§9 (Einstein 等效原理 EEP)、§10 (强等效原理 SEP 三层分裂) 之等效原理三层衬底本体论重读作总体准备. 该准备建立立场、澄清衬底本体论阐述与标准等效原理测试之间的关系、并区分跨 §8 与 §9 所出现之 "一阶" 之两种不同涵义.

标准等效原理三层回顾. 标准广义相对论基础中等效原理在三个层次上阐述. 弱等效原理 (WEP) 陈述, 引力场中之试验粒子, 其轨迹完全由初始位置与速度决定, 与其内部组分无关; 此即自由下落之普适性, 由 Eötvös 型实验与月球激光测距经验性支持. Einstein 等效原理 (EEP) 陈述, 在足够小之时空区域中, 非引力实验在自由下落标架中之物理学约简为狭义相对论; 此即非引力物理之局域 Lorentz 不变性. 强等效原理 (SEP) 将 EEP 延伸至包括引力物理与引力自能: 引力束缚能以与其他形式之能量相同方式贡献于惯性质量与引力质量 (即所有形式之能量在引力上等效), 且引力物理在自由下落标架中仍在局部上无法与无引力之物理区分 (即引力扇区继承 EEP). 标准表述被视为广义相对论之基础原则; 实质性处理见 MTW、Will 等参考文献.

SAE 衬底本体论重读之立场. 第七篇不自 SAE 公理推导等效原理三层, 亦不主张相对于标准等效原理测试之经验优越性. 阐述为识别性: 三层各自接受一项衬底层本体论读法, 三种读法合在一起为思考等效原理基础之物理学家提供工作本体论透镜. 该立场贯穿 SAE 相对论系列, 与 Paper 0 §1.5 之诚实立场一致.

衬底本体论重读不修改任何一层之经验内容. 弱等效原理之组分独立性在试验粒子极限下作为经验事实仍由 Eötvös 型实验验证. Einstein 等效原理之局域 Lorentz 不变性作为局域惯性标架之操作内容, 仍如标准广义相对论中所确立. 强等效原理之普适性在强场体制下仍在通过双脉冲星观测与类似情境下接受经验测试. 衬底本体论重读所提供者乃一种本体论框架定位: 每一层处于衬底-装置结构之独特层次, 三层之层级阐述揭示了此三原则取得其标准形式之衬底层结构性原因.

三层之层级阐述. 三层阐述衬底本体论之三种不同侧面:

• 弱等效原理 (§8): 衬底层之组分解耦, 处于试验粒子极限与试验粒子近似之一阶之中.
• Einstein 等效原理 (§9): 校准同构之局域普适性, 装置度规一阶 Christoffel 结构被吸收入局域装置单位, 而衬底因果槽张量本身保留完整.
• 强等效原理 (§10): 三层分裂, 包括 SEP-local (校准同构之核心, 与 Einstein 等效原理处于同一领域)、SEP-self-energy (引力束缚能作为多因果槽之衬底阐述, 仅为框架性读法)、与 SEP-global (视界与闭合体制, 局域校准不决定整体衬底特征).

三层不处于同一本体论层次. 弱等效原理、Einstein 等效原理与 SEP-local 为衬底层之普适性质, 在各自范围限定之下适用 (弱等效原理: 试验粒子极限之一阶; Einstein 等效原理: 校准之局域点; SEP-local: 一般时空之正常体制), 且跨任何整体对称性情境 (SO(3)、SO(2)、仅局域 Lorentz) 一致成立. SEP-self-energy 与 SEP-global 为体制依赖之阐述: 涉及依赖于具体时空构型之衬底层特征 (自引力体在 SEP-self-energy 中跨多因果槽引入引力束缚能; SEP-global 中之视界与闭合体制). 三层与对称性层级主线之完整对应在 §12 中阐述.

§8 与 §9 中 "一阶" 之两种涵义. §8 弱等效原理之阐述与 §9 Einstein 等效原理之阐述各自援引 "一阶" 之概念, 然而两者指代不同之涵义, 不应被混淆.

在 §8 中, 弱等效原理之一阶阐述指代试验粒子近似之阶. 试验粒子极限假设试验粒子之引力自耦合相对于背景源可忽略, 因而试验粒子在试验粒子-源质量比之一阶下不反作用于衬底因果槽张量. 弱等效原理之衬底层组分盲性读法在此试验粒子近似一阶上成立. 在更高阶上, 试验粒子之自耦合变得相关, 领域越界至 SEP-self-energy 及以远.

在 §9 中, Einstein 等效原理之一阶阐述指代校准点 p 处之空间展开阶. p 处之局域装置校准成功地设定 g_{ab}^{app}(p) = η_{ab}, 且在自由下落标架中一阶导数消失 ∂_c g_{ab}^{app}(p) = 0. 由校准所吸收之 "一阶" 结构为 p 处之装置度规 Christoffel 结构, 此为一阶空间导数量. 在空间展开之二阶上, 曲率与潮汐残余出现; 此二阶结构不可由局域校准所吸收.

两种涵义相互独立: §8 之试验粒子近似阶与 §9 之空间展开阶毫无关联. 两种阐述维持范畴上之分离. 凡在主体文中可能出现混淆之处, 具体涵义明确指明.

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§8 弱等效原理之衬底本体论重读

弱等效原理 (WEP) 在标准广义相对论基础中阐述自由下落之普适性: 引力场中之试验粒子遵循仅由局域几何决定之轨迹, 与其内部组分无关. 经验上, 此由 Eötvös 型实验与月球激光测距以高精度支持. 在标准表述中, 弱等效原理被视为基础原则, 有时表述为惯性质量与引力质量之相等.

衬底本体论重读. 在 SAE 之下, 弱等效原理在衬底层读为试验粒子极限下衬底因果槽耦合之组分解耦. 具体: 在试验粒子极限之中, 当试验粒子之引力自耦合相对于背景源可忽略之时, 衬底因果槽张量阐述不依赖于试验粒子之内部低维物质组分, 处于试验粒子近似之一阶. 试验粒子之内部组分 (静止质量分布、内部束缚能、电荷内容、自旋内容, 等等) 居于装置层. 在衬底层, 因果槽张量阐述决定试验粒子之因果槽中心轨迹如何穿越时空的局域因果槽耦合模式; 因果槽耦合模式在此阶下不编码试验粒子之内部组分.

该阐述可表述为: 在试验粒子极限与一阶之中, 轨迹为因果槽-几何读出, 而非组分读出. 两个内部组分不同但在试验粒子极限下携带相同因果槽耦合廓型之试验粒子, 将穿越衬底因果槽网络追迹同一轨迹, 因为衬底因果槽耦合模式在此阶下不区分组分.

试验粒子极限与一阶限定. 弱等效原理之衬底本体论重读在试验粒子极限与试验粒子近似之一阶下成立. 由此引出两条限定.

其一, 试验粒子极限要求试验粒子之引力自耦合相对于背景源可忽略. 自引力体, 其试验粒子自身之引力束缚能贡献其静能量之一非可忽略部分, 不满足试验粒子极限. 自引力体之衬底本体论阐述居于 §10 之 SEP-self-energy 领域, 不在 §8 之弱等效原理领域.

其二, 一阶限定指代试验粒子近似之阶. 在更高阶上, 试验粒子之自耦合变得相关, 且在衬底层组分独立性不再得到保证. 弱等效原理之高阶修正, 即自耦合效应以组分依赖之方式修改轨迹, 亦居于 SEP-self-energy 领域.

范围之诚实性. 弱等效原理之衬底本体论重读不自 SAE 公理推导弱等效原理. 该阐述为弱等效原理提供衬底层之本体论根基: 组分信息居于装置层, 而衬底因果槽耦合在不涉及组分之情形下阐述轨迹. 弱等效原理在高精度下之 Eötvös 型经验性验证仍为其经验内容; 衬底本体论阐述为一项工作透镜, 与经验测试互补, 而不取代之.

该阐述不延伸至自引力体、引力束缚能、或高阶修正. 此等属于 §10 之 SEP-self-energy 与 SEP-global 领域. 清晰之范围划界防止弱等效原理之衬底本体论重读被误读为弱等效原理之普适推导; 其在试验粒子 / 一阶范围内成立, 与标准弱等效原理作为经验原则所成立之范围相同.

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§9 Einstein 等效原理之衬底本体论重读

Einstein 等效原理 (EEP) 在标准广义相对论基础中阐述引力与加速之局域等效性: 在足够小之时空区域中, 非引力实验在自由下落标架中之物理学约简为狭义相对论; 在局部上, 物理定律在校准点之自由下落标架中取其在无引力情形下所取之形式. 此为非引力物理之局域 Lorentz 不变性, 通过自由下落局域惯性标架操作化.

衬底本体论重读. 在 SAE 之下, Einstein 等效原理在衬底层读为校准同构之局域普适性. 该阐述含三项内容.

其一, 在一般时空中任一点 p 处, 局域标架 e_a^μ(p) 在 p 处提供局域 Lorentz 标架. 校准同构通过此标架阐述局域装置校准: 标架基底中之装置度规在 p 处取 Minkowski 形式, 即 g_{ab}^{app}(p) = η_{ab}. 此为狭义相对论在校准点之局域恢复.

其二, 局域装置校准在 p 处吸收装置度规之一阶导数结构. 在 p 处之自由下落标架中, 一阶空间导数消失: ∂_c g_{ab}^{app}(p) = 0. p 处装置度规之 Christoffel 结构被吸收入局域装置单位. 此即 "引力场在局部消失" 在自由下落标架中之含义: 一阶装置度规读出被吸收, 在 p 周围空间展开之一阶上, 装置度规读为平坦 Minkowski.

其三, 衬底因果槽张量并未被消除. 局域装置校准吸收装置度规之一阶导数 Christoffel 结构入局域单位; 其不消除衬底因果槽张量本身. 衬底因果槽张量 R_{μν}^sub(p) 与其读出结构维持于衬底层. p 周围之二阶空间结构, 即曲率与潮汐残余, 在空间展开之二阶上呈现: g_{ab}^{app}(p + x) = η_{ab} + O(x^2), 其中 O(x^2) 项携带曲率信息. 二阶潮汐残余不由局域校准吸收: Einstein 等效原理在空间展开之一阶上成立, 但潮汐效应在二阶上呈现.

读法: 吸收, 而非消除. 衬底本体论重读澄清 Einstein 等效原理并非衬底结构之缺席, 而是局域装置校准之成功. 衬底因果槽张量维持于衬底层: 因果槽耦合模式、因果槽计数、与衬底读出结构在每个时空点上继续存在. Einstein 等效原理所阐述者乃局域装置校准之普适成功: 在一般时空中任一点 p 处, 局域标架提供装置度规之 Minkowski 读法, 一阶导数 Christoffel 结构被吸收入局域装置单位. 衬底本体论并非平坦; 仅局域装置读法平坦.

该阐述可概括为:

g_{ab}^{app}(p) = η_{ab} + O(x^2)

∂_c g_{ab}^{app}(p) = 0 (在 p 处之自由下落标架中)

R_{μν}^sub(p): 保留, 在局域上标架可读

曲率 / 潮汐残余: 在 O(x^2) 上呈现, 不被局域校准吸收

p 处之衬底因果槽张量通过 R_{ab}^sub(p) = e_a^μ(p) e_b^ν(p) R_{μν}^sub(p) 标架可读, 但在一般时空情境之中 (第三型因果槽张量, 校准同构破坏分类中之模式 3), 衬底因果槽张量与装置度规在 p 处之单一标架中不同时对角化. 因果槽张量在自由下落标架中仍保有其完整第三型结构; Einstein 等效原理阐述装置校准之成功, 而非衬底因果槽张量之对角化.

Einstein 等效原理跨对称性情境之普适性. Einstein 等效原理之衬底本体论重读在任何时空对称性情境下成立. 在第五篇 SO(3) Schwarzschild 情形之中, 任一点之局域装置校准吸收一阶 Christoffel 结构入局域装置单位, 而衬底因果槽张量保留其径向自然基底中之第一型对角结构, 但在特定点之自由下落标架中, 局域标架将装置度规读为 Minkowski. 在第六篇 SO(2) Kerr 情形之中, 同一阐述于每点上成立: 局域装置校准吸收一阶 Christoffel 结构, 而衬底因果槽张量保留其轴对称自然基底中之第二型混合结构 (含 R_{tφ}^sub 非对角). 在第七篇仅局域 Lorentz 一般情形之中, 局域装置校准在任一点上吸收一阶 Christoffel 结构, 而衬底因果槽张量保留其十独立分量之完整第三型结构. 校准同构之局域普适性为跨三种对称性情境之同一阐述; 仅衬底因果槽张量结构跨情形不同.

Einstein 等效原理不蕴涵衬底缺席. 一项可能之误读是: 局域 Lorentz 不变性蕴涵衬底结构之缺席. 衬底本体论重读澄清反面情形: Einstein 等效原理阐述局域装置校准之成功, 而非衬底之缺席. 一般时空携带完整之第三型衬底因果槽张量, 含十个独立分量; 整体对称性之缺席不蕴涵衬底结构之缺席. 衬底在每点上阐述因果槽耦合模式, Einstein 等效原理阐述此衬底结构在空间展开之一阶上局域装置可读为 Minkowski, 而二阶曲率残余则维持.

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§10 强等效原理之衬底本体论重读: 三层分裂

强等效原理 (SEP) 在标准广义相对论基础中将 Einstein 等效原理由非引力物理延伸至所有物理, 包括引力物理与引力自能. 强等效原理陈述, 引力束缚能以与其他形式之能量相同方式贡献于惯性质量与引力质量 (即所有形式之能量在引力上等效), 且引力物理在自由下落标架中仍在局部上等效于无引力之物理 (即引力扇区继承 Einstein 等效原理). 强等效原理在强场体制下通过双脉冲星观测、月球激光测距 Nordtvedt 参数限制等情境下接受经验测试; 经验内容居于实验广义相对论共同体之范围 (Will、Damour 等).

衬底本体论重读: 三层分裂. 第七篇阐述强等效原理衬底本体论重读为三层分裂: SEP-local、SEP-self-energy、与 SEP-global. 该分裂服务于工作透镜目的: 思考强场普适性、引力束缚能、黑洞普适性、与视界体制之物理学家所遇到之情境, 在概念之不同层次上援引强等效原理, 而三层分裂为分离此等层次提供本体论词汇. 该分裂为体制依赖性的, 一如 §7 所述: SEP-local 居于一般正常体制, SEP-self-energy 居于强场体制, SEP-global 居于视界与闭合体制.

此处之阐述不与 §9 Einstein 等效原理之局域普适性相矛盾. 局域-整体之区分至关重要. §9 阐述校准同构在每点上作为局域装置校准取得成功. §10 之 SEP-global 阐述时空之整体特征 (视界、拓扑、闭合体制) 不由单点之局域校准决定. 两项阐述互补: 局域校准在每点上成功, 但整体特征为跨局域校准片段之整合性阐述, 不可自任一单点局域校准导出.

SEP-local. 在一般时空之正常体制中, 当局域装置校准成功且无视界或闭合体制被牵涉之时, 引力与非引力定律在任一点上通过同一校准同构读出. 此为 Einstein 等效原理由非引力物理至引力物理之自然延伸, 处于局部平坦之自由下落标架之内. SAE 衬底本体论接受 SEP-local 作为 §9 所阐述之校准同构局域普适性之结果. SEP-local 与 Einstein 等效原理处于同一领域: 每点之局域装置校准成功适用于局域标架可读之所有物理内容, 包括引力内容. 该阐述在局域校准层次上不区分引力与非引力物理.

SEP-self-energy. 携带非可忽略引力束缚能之物体处于弱等效原理之试验粒子极限之外, 亦处于 SEP-local 之纯局域校准体制之外. 引力束缚能在衬底层阐述为跨因果槽耦合: 束缚能反映衬底因果槽张量跨源之空间延伸内多个衬底因果槽之阐述, 而非背景中之试验粒子之单点局域校准. SEP-self-energy 体制涉及跨多个因果槽之衬底特征, 不由单点局域校准决定. 此将 SEP-self-energy 与 SEP-local 分离: SEP-local 领域涉及单点局域校准, 而 SEP-self-energy 领域涉及多因果槽之衬底阐述.

第七篇仅在框架词汇层面阐述 SEP-self-energy. 引力束缚能作为多因果槽特征之衬底层阐述, 为物理学家提供工作词汇, 不承诺任何特定定量形式. SEP-self-energy 之实质性本体论内容阐述 (包括具体因果槽耦合结构、引力束缚能与因果槽计数之关系、与强场体制下强等效原理之衬底层蕴涵) 留待未来 SAE 论文. 所期待之承担此实质性阐述之未来 SAE 论文, 其在更广 SAE 框架内之具体位置, 在第七篇中未具体指定; 第七篇所确立者乃未来工作所可承袭之词汇.

SEP-global. 视界体制、拓扑特征、与闭合体制携带不由单点局域校准决定之整体衬底特征. 第五与第六篇在其各自对称性情境下阐述此点: 第五篇阐述 Schwarzschild 视界为 ∂_t 通道上 R_t^sub 之径向闭合, 一种全局表征视界之衬底层特征; 第六篇阐述能层静态读出失败、内视界 r_-、r < 0 中之闭合类时曲线、与极端 Kerr 无限喉部为 Kerr 几何之整体衬底特征. 在每一情形下, 局域装置校准于每点上成功 (Einstein 等效原理局域成立), 但整体特征为跨局域校准片段之整合性衬底阐述, 不可自任一单点局域校准导出. SEP-global 阐述此体制层面之非局域性: 当整体衬底特征处于争论之时, 局域校准不足以决定衬底层结构.

§3 之衬底-装置区分纪律阐述装置层特征 (g_{rr} → ∞、环奇点、r < 0 解析延拓、闭合类时曲线、极端无限喉部) 与衬底层特征 (径向闭合 R_t^sub → 0、螺旋闭合 R_{χχ}^sub → 0、能层 R_{tφ}^sub 激活) 为分开之读法. SEP-global 阐述衬底层之整体特征不可单凭局域校准推断: 视界之衬底层径向闭合为涉及类时 Killing 矢量场 ∂_t 与跨时空之径向通道结构之整体阐述, 而非可于任一单点视界处由局域标架校准决定之特征.

SEP-self-energy 与 SEP-global 之边界. SEP-self-energy 与 SEP-global 皆为多因果槽之衬底特征, 但居于不同之拓扑层次, 区分如下: SEP-self-energy 涉及自引力体之内部多因果槽衬底阐述 — 引力束缚能作为物体内部跨多衬底因果槽之耦合特征, 即对象内部之多因果槽阐述. SEP-global 涉及时空之整体衬底特征 — 视界、拓扑、闭合体制作为时空结构整体特征, 即跨局域校准片段之整合性整体阐述. 复合情境 (如双脉冲星系统涉及自引力体在视界附近动力学) 同时携带两种衬底特征, 但 SAE 衬底本体论阐述将其分作两层组织: 双脉冲星之内部自引力束缚能居于 SEP-self-energy 领域, 系统外部时空结构与潜在视界 / 闭合区域居于 SEP-global 领域.

第七篇立场: 词汇, 而非经验优越性. SAE 衬底本体论阐述为强等效原理三层分裂提供本体论词汇, 用于工作透镜. SAE 不主张相对于强等效原理经验测试之经验优越性. 强等效原理在强场体制下之经验内容 (双脉冲星测试、月球激光测距 Nordtvedt 参数限制等) 居于实验广义相对论共同体之领域 (Will、Damour 等), 第七篇不进入该领域. 第七篇所提供者乃一本体论词汇: 思考强场普适性、引力束缚能、黑洞普适性、或视界体制之物理学家, 可使用 SEP-local / SEP-self-energy / SEP-global 之分裂作为以单一本体论词汇组织不同体制之工作框架. 该框架不要求接受 SAE 衬底本体论之整体而仍可作为工作透镜有用.

对物理学家之实际意义. 强等效原理之三层分裂为思考强等效原理相关情境之实用组织功能. 当情境涉及一般正常体制中之单点局域校准时, SEP-local 适用 (§9 Einstein 等效原理阐述提供衬底层内容). 当情境涉及自引力体之引力束缚能之时, SEP-self-energy 适用, 衬底本体论词汇阐述多因果槽衬底阐述; 实质性内容留待未来 SAE 工作. 当情境涉及视界体制、拓扑特征、或闭合体制之时, SEP-global 适用, 第五与第六篇之特定情形阐述提供衬底层之读法模式. 三种体制对应于三种不同之衬底层结构情境: 局域校准片段、跨多因果槽之衬底阐述、与跨局域校准片段之整合性整体阐述. 三层分裂将此体制依赖性予以明示.

物理学家无须接受 SAE 衬底本体论之整体即可使用强等效原理三层分裂作为组织原则. 一如 §3 之衬底-装置区分纪律可作为工作透镜使用而无须承诺 SAE 整体, 强等效原理三层分裂作为物理学家可在词汇层次上参与之工作框架而提供. 该分裂将强等效原理敏感之领域组织为三个体制依赖之领域, 各具其本体论性质与其与一般时空局域校准结构之关系.

§11 等效原理与时空结构之对应主线

本节阐述完整之第五、第六、第七篇跨论文对应主线: 跨三相对论系列情形之观察模式, 联结时空对称性层级、衬底因果槽张量结构、与校准呈现. §4 提供第三型之衬底侧结构阐述, §5 提供衬底侧结构层级预览, §6 提供校准同构破坏模式分类; §11 现阐述完整之对应主线, §12 阐述等效原理侧之对应详节.

跨论文之观察模式. 跨第五、第六、第七篇, 一项观察模式涌现, 联结每情形之四项侧面: 时空整体对称性、衬底因果槽张量类型、校准呈现通道、与等效原理相关之衬底阐述. 该模式作为观察, 而非自 SAE 公理推导. 框架定位为识别性, 与 SAE 数学基础论文 Schema 跨系列之认识论立场一致.

主线通过以下对应表阐述:

对称性	因果槽张量类型	独立分量数	校准呈现	等效原理相关之衬底阐述	论文
SO(3) 球对称	第一型对角	4	∂_t 上之径向闭合	径向对称标架读法	第五篇
SO(2) 轴对称	第二型混合	5	χ_H 上之螺旋闭合	螺旋旋转标架读法	第六篇
仅局域 Lorentz	第三型完整	10	局域校准, 无整体闭合通道	衬底层等效原理普适性	第七篇

第七篇行之第四栏读为 "局域校准, 无整体闭合通道": 此并非与第五篇之径向闭合及第六篇之螺旋闭合并列之第三种闭合情形. 第五与第六篇各携带一整体 Killing 矢量场 (第五篇之 ∂_t, 第六篇之 χ_H), 提供一全局识别之通道, 衬底闭合在其上阐述. 第七篇之一般时空情境不携带任何整体 Killing 矢量场, 不存在整体闭合通道; 第七篇所阐述者乃通过校准同构在每点之局域校准普适性, 在任何整体闭合通道之缺席之下. 两种情形 (第五 / 第六篇之闭合通道与第七篇之局域校准) 为衬底-装置结构之范畴性分异之呈现, 而非同一闭合模式之三种实例.

识别性主线, 非推导性. 主线阐述为跨第五、第六、第七篇之观察对应模式. 框架定位不主张主线自 SAE 公理推导, 亦不主张对称性层级 SO(3) → SO(2) → 仅局域 Lorentz 为强制约简序列. 三种情形在第五、第六、第七篇中各自独立研究, 各自处于其衬底本体论重读之内; 主线为对照三种情形所涌现之跨论文结构性观察. 此立场维持 SAE 数学基础论文 Schema 之识别性框架定位: 衬底本体论阐述为广义相对论之数学结构提供本体论框架定位, 而不取代广义相对论, 亦不自 SAE 推导广义相对论.

主线阐述贯穿之措辞避免可能暗示推导之 "系统对应" 或 "自然完成" 之类陈述. 观察模式读为对三种情形特定阐述之对照: 每情形携带其各自之衬底因果槽张量结构 (第一、二、三型)、各自之校准呈现 (∂_t 上径向闭合、χ_H 上螺旋闭合、无整体闭合通道之局域校准)、与各自之等效原理相关衬底阐述 (径向对称、螺旋旋转、衬底层之等效原理普适性). 跨三种情形之模式被观察, 该观察作为相对论系列之跨论文主线得以阐述.

结构性互补约简关系: 对称性约束与因果槽张量复杂度. 主线显示对称性约束与衬底因果槽张量结构丰富度之间的互补约简关系, 一如 §5 所阐述. 对称性约束作为约简算子作用于因果槽张量, 较强之对称性约束约简较多之独立分量; 反之, 较少之整体对称性留下较多之独立衬底因果槽张量分量经历对称性约简之后: SO(3) (第五篇) 携带 4 个分量, SO(2) (第六篇) 携带 5 个, 仅局域 Lorentz (第七篇) 携带 10 个. 较丰富之因果槽张量结构对应于较弱之整体对称性. 在第七篇之极限情形中, 除局域 Lorentz 以外之任何整体对称性之缺席允许衬底因果槽张量阐述其完整第三型结构, 全部十个独立分量经约简后存活.

第七篇层次之等效原理普适性阐述处于此极限情境: 衬底因果槽张量阐述其完整第三型结构, 衬底层之等效原理普适性通过每点之局域校准呈现阐述. 跨三种情形之等效原理相关衬底阐述为体制依赖之特征: 第五与第六篇之特定对称性情形阐述与其特定自然基底联系之等效原理相关衬底读法 (第五篇之径向对称标架读法、第六篇之螺旋旋转标架读法), 而第七篇之一般情形通过无整体闭合通道之局域校准阐述衬底层之等效原理普适性.

主线作为系列层次之实质性贡献. 等效原理与时空结构之对应主线为第七篇之系列层次实质性贡献: 其阐述只有在将三种情形并置之时才涌现之跨论文对应模式. 第五篇独立阐述 Schwarzschild 径向闭合之衬底读法. 第六篇独立阐述 Kerr 螺旋闭合之衬底读法与 §10.3 之对称性层级观察. 第五或第六篇任一独立皆未阐述完整主线, 因为完整主线要求三种情形之对照. 第七篇提供此阐述: 主线在第五、第六、第七篇并置之时得以观察, 第七篇明确阐述之.

主线阐述不修改第五或第六篇之实质性内容; 两篇皆保留其各自情形之衬底本体论重读. 主线所添加者乃跨论文之观察模式, 阐述于相对论系列之层次, 而非任一单篇论文之层次. 此即第七篇作为系列收束之自然角色: 主线阐述只有在将三种情形合在一起之时才涌现, 而第七篇在一般情形之中提供完成三情形对照之阐述.

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§12 对称性层级主线与等效原理三层之对应

本节将 §11 之主线应用于 §§7-10 之等效原理三层框架, 阐述等效原理侧之对应详节. §10 阐述强等效原理三层分裂之体制划分 (SEP-local、SEP-self-energy、SEP-global). §12 阐述主线应用于等效原理三层框架, 组织弱等效原理、Einstein 等效原理、SEP-local、SEP-self-energy、与 SEP-global 如何与跨情形之对称性层级对应. 两种组织方式互补: §10 按体制纵向组织 SEP 三层, §12 按跨对称性层级之普适-与-体制依赖水平分裂组织等效原理三层.

衬底之普适性质: 弱等效原理、Einstein 等效原理、SEP-local. 三种等效原理相关之衬底阐述为普适性质, 跨任何时空对称性情境在各自范围限定之下适用.

§8 之弱等效原理组分解耦阐述在试验粒子极限与试验粒子近似之一阶之中适用. 此衬底性质不依赖于时空之整体对称性情境: 在第五篇 SO(3) Schwarzschild、第六篇 SO(2) Kerr、与第七篇一般仅局域 Lorentz 之中, 衬底因果槽张量在试验粒子近似之一阶下不编码试验粒子之内部组分. 试验粒子极限阐述跨三种情形一致. 衬底层之原因在于因果槽耦合模式阐述试验粒子之因果槽中心轨迹所穿越之局域几何, 与内部组分无关; 此阐述于衬底层独立于时空所携带之何种整体对称性 (如有任何) 而成立.

§9 之 Einstein 等效原理局域普适性同样跨任何时空对称性情境适用. 在第五篇 SO(3) Schwarzschild 中, 任一点之局域装置校准吸收一阶 Christoffel 结构入局域装置单位, 衬底因果槽张量保留其第一型对角结构, 但通过局域标架在装置中读为 Minkowski. 在第六篇 SO(2) Kerr 中, 同一阐述成立: 任一点之局域装置校准吸收一阶 Christoffel 结构, 衬底因果槽张量保留其第二型混合结构. 在第七篇仅局域 Lorentz 一般情形中, 局域装置校准吸收一阶 Christoffel 结构, 衬底因果槽张量保留其完整第三型结构, 全部十个独立分量. 校准同构之局域普适性为一项衬底之普适性质: 在携带局域 Lorentz 标架之任意时空中之任一点, 局域装置校准在空间展开之一阶上成功, 二阶潮汐残余维持. 该阐述不依赖于整体对称性情境.

§10 所阐述之 SEP-local 处于与 Einstein 等效原理同一领域. 其将局域装置校准成功由非引力物理延伸至自由下落标架之内之引力物理. 此延伸跨时空对称性情境普适适用, 与 Einstein 等效原理同义: 任一点之局域装置校准吸收一阶 Christoffel 结构, 此成功适用于局域标架可读之所有物理内容, 包括引力内容.

此三项衬底之普适性质 (弱等效原理之组分解耦、Einstein 等效原理之局域普适性、SEP-local 之校准同构延伸) 之普适性意指其在衬底层适用而无关乎整体对称性情境. 阐述对范围限定敏感 (弱等效原理之试验粒子极限一阶; Einstein 等效原理与 SEP-local 之局域校准点), 但不对时空对称性类敏感.

体制依赖之衬底特征: SEP-self-energy 与 SEP-global. SEP-self-energy 与 SEP-global 为体制依赖, 在涉及具体时空构型之衬底特征之意义上.

SEP-self-energy 涉及自引力体中引力束缚能之多因果槽衬底阐述. 阐述跨时空对称性情境不同, 因为第五篇球对称构型、第六篇轴对称构型、与第七篇一般构型中之多因果槽衬底阐述携带不同之衬底因果槽张量结构. 在第五篇情境中, 多因果槽阐述处于第一型对角结构之内. 在第六篇情境中, 多因果槽阐述涉及携带 R_{tφ}^sub 非对角跨因果槽耦合之第二型混合结构. 在第七篇一般情境中, 多因果槽阐述涉及完整第三型结构, 其跨因果槽耦合模式不具任何整体自然基底. SEP-self-energy 之衬底层内容在此意义上为体制依赖; 第七篇在框架词汇层面阐述之, 实质性本体论内容留待未来 SAE 工作, 一如 §10 所述.

SEP-global 涉及不由单点局域校准决定之整体衬底特征. 跨情形之衬底层读法模式不同. 在第五篇中, 视界之衬底层特征为 ∂_t 通道上之径向闭合 R_t^sub → 0, 通过全局识别之类时 Killing 矢量场阐述. 在第六篇中, 视界之衬底层特征为视界生成元 χ_H = ∂_t + Ω_H ∂_φ 上之螺旋闭合 R_{χχ}^sub → 0, 通过两个整体 Killing 矢量场之特定组合阐述. 在第七篇一般情境中, 无整体 Killing 矢量场可用, 不存在类似之整体闭合通道; 一般时空构型之衬底层整体特征通过跨流形之局域校准模式阐述, 整体阐述读为跨局域校准片段之整合性, 而非全局识别之通道上之闭合. SEP-global 之衬底层内容在此意义上为体制依赖.

普适-与-体制依赖之水平分裂. 五项等效原理相关之衬底阐述 (弱等效原理、Einstein 等效原理、SEP-local、SEP-self-energy、SEP-global) 相对于对称性层级主线处于两个不同层次.

弱等效原理、Einstein 等效原理、与 SEP-local 为衬底之普适性质: 其在衬底层跨任何对称性情境适用, 对范围限定敏感但不对整体对称性类敏感. 三者合在一起阐述衬底层之局域适用等效原理内容, 跨第五篇 SO(3) Schwarzschild、第六篇 SO(2) Kerr、与第七篇一般仅局域 Lorentz 一致成立.

SEP-self-energy 与 SEP-global 为体制依赖之衬底特征: 其涉及依赖于具体时空构型与整体对称性情境之衬底阐述. 跨第五、第六、第七篇之阐述结构性地不同, 实质性内容居于体制特定之领域.

水平分裂澄清 §10 之强等效原理三层分裂何以为必要. SEP-local 与衬底之普适性质 (弱等效原理、Einstein 等效原理) 同属, 而 SEP-self-energy 与 SEP-global 与体制依赖之衬底特征同属. 一种不区分此三体制之合一 "强等效原理" 将混淆衬底-普适-性质内容与体制依赖衬底特征, 模糊每点之局域校准成功与跨校准片段之整合性整体阐述之间的结构性差异.

未来第四型及之上作为开放领域. §12 之阐述覆盖第五、第六、第七篇所阐述之衬底因果槽张量类型 (第一、第二、第三型) 内之等效原理三层. 至超出仅局域 Lorentz 之延伸 (多场耦合、规范扇区如 Kerr-Newman 电荷情形、与局域对称性结构之更广义延伸) 将引入第四型及更高之因果槽张量结构, 衬底层之等效原理对应将相应延伸. 此种延伸留作未来 SAE 工作之开放领域. 等效原理在物质扇区与规范扇区延伸下之衬底本体论重读, 留待所期待之未来 SAE 量子引力接口论文与相关之未来 SAE 工作.

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§13 唯一性轨迹: 系列方法论闭合模式

SAE 相对论系列跨三直接前置论文阐述与标准广义相对论唯一性定理及基础原则对齐之衬底读出. 第五篇阐述与 Birkhoff 定理对齐之衬底读出; 第六篇阐述与 Kerr 唯一性定理对齐之衬底读出; 第七篇阐述等效原理普适性之衬底层读法. 三种阐述构成方法论层次之系列内闭合模式: 跨三种不同之时空对称性情境 (静态球对称真空、稳态轴对称真空、一般非动态时空), SAE 衬底本体论阐述提供一项与各情形对应之标准广义相对论结果对齐之唯一衬底读出.

阐述纪律: 三种不同之数学地位. 所对齐之三项结果携带三种不同之数学地位, 不可混淆.

Birkhoff 定理为静态球对称真空条件下之数学定理: 其建立 Schwarzschild 几何作为静态球对称真空 Einstein 场方程解之唯一性. 该定理在其范围内 (真空、球对称、静态、渐近平坦假设) 数学严格, 标准证明见广义相对论文献. 第五篇阐述与 Birkhoff 定理对齐之衬底读出, 而不重新证明该定理本身; 标准定理通过 ∂_t 通道上之径向闭合 R_t^sub → 0 接受衬底层之读法.

Kerr 唯一性定理为稳态轴对称真空条件下之数学定理, 携带技术性条件 (渐近平坦、正则性、关于视界结构之假设, 含若干文献中之变体表述, 包括 Robinson-Carter-Hawking 唯一性链). 该定理在其范围内数学严格. 第六篇阐述与 Kerr 唯一性定理对齐之衬底读出, 而不重新证明定理; 标准定理通过视界生成元 χ_H = ∂_t + Ω_H ∂_φ 上之螺旋闭合 R_{χχ}^sub → 0 接受衬底层之读法.

等效原理普适性并非数学定理. 其为广义相对论之基础原则, 通过跨弱等效原理、Einstein 等效原理、与强等效原理三层之不同精度等效原理测试经验性验证. 经验验证涉及实验室实验 (Eötvös 型)、空间实验 (卫星测试)、月球激光测距、双脉冲星观测、与类似情境; 经验内容居于实验广义相对论共同体. 第七篇阐述等效原理普适性作为基础原则之衬底层读法, 而非作为推导定理.

框架性识别系列方法论, 非数学定理链. 跨第五、第六、第七篇之三种阐述于框架层次上构成系列方法论闭合模式. 其不构成数学定理链. 将其合并为 "唯一性轨迹" 须谨慎读取: 该轨迹为跨论文之观察, 即 SAE 衬底本体论阐述在每情形中提供与对应标准结果对齐之唯一衬底读出, 而非以统一数学论证导出三项结果之数学链.

框架定位为识别性, 与跨系列所贯穿之同义. 第五与第六篇阐述与标准数学定理对齐之衬底读法; 衬底读法并非定理之推导, 而是定理之本体论读法. 第七篇阐述与基础经验原则对齐之衬底读法; 衬底读法并非原则之推导, 而是原则之本体论根基. 三者合在一起阐述系列层次之模式: 衬底本体论重读提供与每情形之标准广义相对论结果对齐之唯一衬底读出, 跨三种不同之时空对称性情境与三种不同之所对齐之标准结果之数学地位.

系列层次之模式陈述. 跨第五、第六、第七篇之轨迹可概括为: SAE 衬底本体论重读提供与对应标准广义相对论结果对齐之衬底层读法, 处于三个对称性逐步减弱之情境.

第五篇: 在静态球对称真空情境之中, 衬底读出为 ∂_t 通道上之唯一径向闭合 R_t^sub → 0, 与 Birkhoff 定理对齐.

第六篇: 在稳态轴对称真空情境之中, 衬底读出为视界生成元 χ_H 上之唯一螺旋闭合 R_{χχ}^sub → 0, 与 Kerr 唯一性定理对齐.

第七篇: 在无整体对称性之一般非动态时空情境之中, 衬底读出为通过校准同构在每点之局域校准普适性, 与衬底层之等效原理普适性对齐.

三衬底读出为情形特定之阐述, 各自处于其衬底本体论重读之内. 跨论文之轨迹为系列层次之观察, 即衬底本体论阐述跨三种情形成功提供唯一衬底读出, 与每情形之标准广义相对论结果对齐, 而无须重新证明标准结果. 此即 SAE 相对论系列之方法论闭合模式.

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§14 诚实边界、未来领域、与简短收尾反思

第七篇收束 SAE 相对论系列. 收束包含三项内容: 诚实极限标记、附简短跨系列移交标记之未来领域图、与简短收尾反思.

诚实极限标记. 以下极限贯穿第七篇与系列.

SAE 衬底本体论阐述非广义相对论或等效原理之推导. 阐述为广义相对论数学结构与等效原理基础原则提供本体论框架定位, 不修改其内容, 亦不主张自 SAE 公理推导.

SAE 在第七篇所涵盖之领域内不产生与标准广义相对论不同之定量预言. 衬底本体论重读跨第五、第六、第七篇之情形与广义相对论之标准定量预言对齐. 阐述处于本体论框架定位之层次, 而非定量经验内容之层次.

第七篇不涵盖动态演化. 含时曲率、引力波传播、黑洞合并、与一般时空构型之动态演化移交至信息论第六篇 (DOI: 10.5281/zenodo.20066644). 第七篇维持 §1 所建立之严格非动态与局域运动学校准范围.

第七篇不涵盖量子引力接口. 衬底层因果槽计数、衬底因果槽结构与 Planck 尺度物理之关系、超出 §3 工作透镜框架定位之量子黑洞信息悖论之衬底本体论阐述、与类似之量子引力接口问题, 推迟至所期待之未来 SAE 量子引力接口论文. 第七篇对衬底因果槽之具体尺度或微观结构不取立场.

第七篇不涵盖带电时空、多黑洞系统、或超出第五篇 Schwarzschild、第六篇 Kerr、与第七篇一般情形之其他特定时空构型. Kerr-Newman 带电情形引入规范扇区内容, 越出第七篇之纯时空几何领域. 多黑洞系统情形 (多个黑洞动态相互作用) 越界至信息论第六篇之动态领域. 超出三直接前置情形之特定时空构型不予阐述.

第四型及之上之衬底因果槽张量延伸与模式 5 及之上之校准同构破坏分类延伸推迟至未来 SAE 工作. 第七篇维持其范围于第一、第二、第三型因果槽张量结构与模式 0、1、2、3、4 之破坏分类之内.

第七篇不进入经验性强等效原理测试领域. 强场普适性测试、双脉冲星观测、月球激光测距 Nordtvedt 参数限制、与类似之经验性强等效原理敏感情境居于实验广义相对论共同体. 第七篇之强等效原理三层分裂为工作透镜用途之本体论词汇, 非经验性强等效原理测试之框架.

未来领域图与跨系列移交标记. 以下未来领域在更广 SAE 研究纲领之中所期待, 附简短跨系列移交标识.

动态演化, 包括引力波传播、黑洞合并、与含时曲率, 移交至信息论第六篇 (DOI: 10.5281/zenodo.20066644), 以阐述动态体制之衬底本体论.

黑洞内部一般情境延伸移交至信息论第四篇 (DOI: 10.5281/zenodo.19880112), 特别是其 §0 所期待之 Kerr 延伸领域, 以阐述一般黑洞内部结构之衬底本体论.

SAE 量子引力接口, 包括 Planck 尺度之衬底因果槽结构、量子黑洞信息之衬底层阐述、与相关问题, 留待未来 SAE 论文. 该具体论文于更广 SAE 纲领内之位置目前未具体指定. 与此相关, 第七篇阐述衬底因果槽张量在 generic spacetime 之完整第三型结构, 加 §14 所述衬底因果槽具体尺度与微观结构不取立场, 此为未来 SAE 量子力学之研究领域 (即因果槽尺度之下之衬底本体论) 提供框架空间.

质量-引力守恒合论, 阐述引力耦合演化与奇点接口, 留待未来 SAE 论文.

大坍缩 (Big Crunch) 宇宙学衬底本体论与相关宇宙学系列延伸留待未来 Cosmo 系列论文.

SAE 自旋本体论专题论文, 阐述自旋于量子引力接口处之衬底本体论, 留待未来 SAE 论文, 作为所期待之未来研究方向.

强等效原理敏感经验性测试之实质性本体论重读 (Nordtvedt 参数、双脉冲星强场普适性测试、引力束缚能情境等), 若衬底本体论阐述产生与标准广义相对论不同之定量内容, 则归属未来论文; 否则, 该领域留于实验广义相对论共同体.

经典广义相对论测试之实质性本体论重读 (水星近日点进动、光线偏折、Shapiro 延迟等) 留待第五篇 v2 框架未来论文. 此所期待之第五篇补充论文将处理此等经典测试之衬底本体论重读; 其不修改第五篇之已发表状态.

第四型及之上之对称性层级延伸 (多场耦合、含 Kerr-Newman 带电情形之规范扇区、与更广义之延伸) 留待未来 SAE 特定论文.

形与流系列之跨系列整合留待未来跨系列特定论文, 以阐述衬底本体论之联系.

简短收尾反思. SAE 相对论系列以第七篇收束. 第五篇阐述 ∂_t 通道上之 Schwarzschild 衬底本体论重读; 第六篇阐述 χ_H 视界生成元上之 Kerr 衬底本体论重读; 第七篇阐述一般时空情境下之等效原理三层衬底本体论重读, 连同跨三种情形所观察之等效原理与时空结构之对应主线.

系列之方法论贡献为衬底-装置区分纪律, 阐述为面向处理广义相对论病态与等效原理基础之物理学家之工作本体论透镜. 系列之实质性贡献为跨论文所观察之对应模式, 联结时空对称性层级、衬底因果槽张量结构、与校准呈现, 识别性地阐述为等效原理与时空结构之对应主线.

SAE 纲领作为广义相对论基础、量子引力、与涌现引力之系列层次系统性方法论, 居于 Paper 0 (DOI: 10.5281/zenodo.19777881) 之领域, 不由第七篇承担. 相对论系列阐述一般非动态时空之衬底本体论, 处于工作哲学物理之层次; 更广之 SAE 纲领包含其他处理动态演化、信息论、宇宙学、与附加主题之系列.

相对论, 于 SAE 之下, 非被取代, 而被安置.

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参考文献

SAE 相对论系列直接前置论文:

• 秦汉. SAE 相对论系列第一篇: 引力时间膨胀因果槽通量. Zenodo. DOI: 10.5281/zenodo.19836183.
• 秦汉. SAE 相对论系列第二篇 (含 Lense-Thirring SAE 重读). Zenodo. DOI: 10.5281/zenodo.19910545.
• 秦汉. SAE 相对论系列第三篇: 时钟扇区有效维度指数. Zenodo. DOI: 10.5281/zenodo.19992252.
• 秦汉. SAE 相对论系列第四篇: 因果槽张量与校准同构. Zenodo. DOI: 10.5281/zenodo.20079718.
• 秦汉. SAE 相对论系列第五篇: SAE 之下的 Schwarzschild 视界. Zenodo. DOI: 10.5281/zenodo.20105112.
• 秦汉. SAE 相对论系列第六篇: SAE 之下的 Kerr 黑洞. Zenodo. DOI: 10.5281/zenodo.20131993.

SAE 纲领相关论文:

• 秦汉. SAE 整体框架: Paper 0 引力本体论. Zenodo. DOI: 10.5281/zenodo.19777881.
• 秦汉. SAE 信息论第四篇: 黑洞内部信息结构. Zenodo. DOI: 10.5281/zenodo.19880112.
• 秦汉. SAE 信息论第六篇: 动态演化与信息守恒. Zenodo. DOI: 10.5281/zenodo.20066644.
• 秦汉. SAE 方法论概览. Zenodo. DOI: 10.5281/zenodo.18842449.
• 秦汉. L_3 → L_4 闭合双语言. Zenodo. DOI: 10.5281/zenodo.19361950.

广义相对论标准参考:

• C. W. Misner, K. S. Thorne, J. A. Wheeler. Gravitation. W. H. Freeman, 1973. (MTW)
• R. M. Wald. General Relativity. University of Chicago Press, 1984.
• S. M. Carroll. Spacetime and Geometry: An Introduction to General Relativity. Addison-Wesley, 2004.
• C. M. Will. Theory and Experiment in Gravitational Physics. Cambridge University Press, 2018.