Effective Inertia under Causal-Slot Geometry

Author: Han Qin (秦汉)

Acknowledgment up front: I thank Zesi Chen (陈则思) for sustained foundational contributions and key discussions in articulating the SAE framework. The a priori commitments of SAE, the universal pattern of the 16DD periodic table, and the overall structure of the framework all draw substantially on long-term collaboration with Zesi Chen.

Date: 2026

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On the Three-Tier Structure of This Paper

This paper is organized in three tiers, so that the role of each part within the paper is clear to the reader:

Main line (§1-§6, §8-§9, §12): The core deliverable of this paper—the ontological unification of effective inertia phenomena in SR and GR under causal-slot geometry. This includes the 4DD causal-capacity-conservation derivation, the recovery of the standard SR result ($\gamma^3$ longitudinal vs. $\gamma$ transverse) in cell-counting vocabulary, SR and GR sharing the same mechanism class under causal-slot geometry, and the SAE reading of the equivalence principle. The main line provides ontological articulation in the pure-SR and the static strong-field GR regimes, fully aligned with the observable predictions of standard physics. The main line states only the existential structural claim about deviation in the combined regime; quantification is deferred to P3.

Extension (§7, §10, §11): Articulations of the framework beyond the main line. Two kinds of content:

• Cross-paper inherited empirical context: the distinguishable falsifiable prediction in the gravitational-wave (GW) propagation regime. Ground from P1 §3.5 (gravitational waves as Planck-substrate broadcast), extension from Information Theory P4 §4.4 (causal-spectrum description of black-hole interior). The present paper articulates the falsifiable empirical form of this prediction within the relativity series, bringing the broader SAE relativity-plus-information-theory framework's GW-propagation claim into P2's regime map. Not counted as P2's own original results.
• Conceptual extensions of the main-line structural claim: ultra-fast = artificial horizon (子夏's insight), causal dimensional reduction (子夏's topological extension, dimensional isomorphism between cell topology at the BH horizon and that of an ultra-relativistic particle).
• Connections of P2 with the wider SAE framework (cross-references to Mass-Conv, Paper 0, Cosmo V, etc.).

Appendices (Appendix A, Appendix B): Author commentary and heuristic material outside the main line and extension.

• Appendix A: Heuristic scaling in the combined regime ($d_\text{eff} = 2$ trivial-limit illustration, not a formal prediction formula).
• Appendix B: A note to future physicists—methodology and cultural-context commentary. The main text (§1-§12) plus Appendix A constitutes the technical content of P2 and stands on its own; Appendix B is the author's open commentary on overall SAE methodology and contemporary physics culture, a separable layer from the main technical content.

A reader interested only in P2's core deliverable can read the main line (§1-§6, §8-§9, §12). The extension and appendices supply framework-level context and author commentary; they do not affect the self-containment of the main line.

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Abstract

Within the Self-as-an-End (自为目的, SAE) framework, this paper articulates the effective-inertia phenomena of SR and GR as two geometric realizations of the same ontological mechanism: velocity amplification under modification of causal-slot geometry.

Main line: P1 already showed that gravity makes causal-slot cells shrink, giving gravitational time dilation $d\tau/dt = \delta_4^{1/d_\text{eff}}$. P2 further shows that motion likewise makes cells shrink. This follows from 4DD causal-capacity conservation: when an object moves with velocity $v$ relative to the Planck substrate, the tick dilates by $\gamma$, and along the direction of motion cells are forced to shrink to $1/\gamma$ in absolute Planck units. The standard SR results ($\gamma^3$ longitudinal vs. $\gamma$ transverse) are recovered through explicit calculation in cell vocabulary. GR strong-field effective inertia is likewise read as causal-slot geometric modification. The equivalence principle, under SAE reading, is articulated as "same mechanism class (causal-slot geometric modification), different geometric realization." This is the core deliverable of P2—ontological articulation in the pure-SR and the static strong-field GR regimes, aligned with the observable predictions of standard physics.

Extension: The paper also articulates two framework-level pieces. First, cross-paper inherited empirical context: the broader SAE relativity-plus-information-theory framework also gives a distinguishable falsifiable prediction about GW propagation (gravitational broadcast travels in the Planck substrate layer and crosses any mass aggregate without attenuation). Ground from P1, extension from Information Theory P4; the present paper articulates the falsifiable empirical form within the relativity series, not counted as P2's own original results. Second, the combined regime: when motion and gravity superpose to push the effective closure deficit $\delta_4^\text{eff}$ into the $d_\text{eff} > 2$ region, time dilation, effective mass, and length contraction deviate from the simple SR×GR multiplicative combination. The present paper locks in the existential structural claim; the magnitude of deviation depends on the functional form of $d_\text{eff}(\delta_4^\text{eff}, v)$, deferred to P3. Ultra-fast = artificial horizon (子夏): ultra-relativistic motion $v \to c$ is ontologically isomorphic to falling into a black hole.

P2 makes no claim to go beyond the observable predictions of standard physics in the pure-SR and the static strong-field GR regimes; it provides only ontological articulation. The GW-propagation prediction articulated in the extension is the relativity-series articulation of a P1-plus-Information-Theory-P4 framework-level claim; the combined-regime deviation claim is existential structural, with quantification deferred to P3.

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Part I: Main Line

§1 Introduction (Main line)

§1.1 The Foundation Laid by P1

P1 (Han Qin, 2026, DOI 10.5281/zenodo.19836183) articulates within the SAE framework the structural form of gravitational time dilation $d\tau/dt = \delta_4^{1/d_\text{eff}}$. Core insights:

• A priori commitments (the four indispensables of time plus the four indispensables of information), bridging axioms, and Planck absoluteness together provide the ontological foundation.
• Gravity makes causal-slot cells shrink; the tick = $R(r)/c$ thus lengthens, and time dilation emerges.
• A two-layer structure (Planck substrate vs. causal slot) explains the absoluteness of gravitational-wave propagation (no lensing).
• Gravitational waves = 4DD topology-refresh commands broadcast by the Planck substrate, executed at the causal-slot layer.

P1 focuses on the gravitational case, leaving the motion case to P2.

§1.2 The Core Question of P2

Physics has two effective-inertia phenomena:

SR effective mass: A high-velocity object requires more momentum to accelerate. About $\gamma^3 m_0$ along the direction of motion, about $\gamma m_0$ transverse. The modern standard-physics statement: $p^\mu = m_0 u^\mu$, where $u^\mu = \gamma(c, \vec{v})$, with $\gamma$ residing in the 4-velocity. "Relativistic mass" $\gamma m_0$ is a deprecated concept.

GR strong-field effective inertia: An object in a strong gravitational field also requires more momentum to accelerate (consistent with the local reading of GR's equivalence principle; from a coordinate-frame viewpoint, this is a frame-dependent phenomenon). Standard physics: an artifact of stress-energy-tensor structure.

Standard physics' attitude: treat both as kinematic / coordinate artifacts. As long as the math gives correct predictions, don't ask why $\gamma$ appears in this way.

P2's articulation: these two phenomena are ontologically homologous. Both are cells (causal slots) shrinking under some condition; pushing through shrunk cells requires more momentum. The geometric origin of cell shrinkage differs (motion vs. gravity), but the mechanism class is shared (causal-slot geometric modification, yielding effective velocity amplification).

Modern standard physics has correctly placed $\gamma$ in the 4-velocity. What P2 provides is the cell-counting ontological picture, an ontological reading of the standard formulation, plus SR-GR vocabulary unification within the broader SAE framework.

§1.3 Relation to P1

P1: gravity, cells shrink, gravitational time dilation.

P2: motion, cells shrink, SR time dilation plus effective mass.

Both papers jointly articulate the unified picture causal-slot geometry determines spacetime-inertia phenomena. P2 is built on P1's framework elements (two-layer structure, 4DD causal capacity, tick concept, etc.) and does not re-derive them.

P2's specific contributions within the SAE framework (main line):

• Through 4DD causal-capacity conservation, deriving that cells are forced to shrink to $1/\gamma$ along the direction of motion (replacing the naive signal-chasing argument).
• Through explicit calculation, showing that the standard SR $\gamma^3$ vs. $\gamma$ directional asymmetry emerges from an ontological reading of cell anisotropy.
• GR strong-field effective inertia likewise gets a causal-slot geometric reading.
• The equivalence principle is reframed: same mechanism class, different geometric realization.

P2's extension content (extension; see §7, §10, §11 for detail):

• Distinguishable falsifiable prediction at the GW-propagation level vs. standard GR (P1 §3.5 ground plus P2-and-Information-Theory-P4 extension; not counted as P2's own original results).
• Existential structural claim of deviation in the combined regime (magnitude deferred to P3).
• Ultra-fast = artificial horizon plus causal dimensional reduction (子夏's insight, conceptual extension).

§1.4 Contributions of the Paper (Main-line / Extension Layering)

P2's contributions are articulated in main-line / extension layering:

Main-line contributions—ontological articulation in the pure-SR and the static strong-field GR regimes, aligned with the observable predictions of standard physics

• SAE ontological reading of SR effective mass: stated through causal-slot geometry. The standard SR results ($\gamma^3$ longitudinal vs. $\gamma$ transverse) are recovered through explicit calculation. SAE supplies a cell-counting ontological picture, consistent with the standard 4-velocity formulation, providing the ontological grounding for "why $\gamma$ resides this way" that standard physics does not unfold.

• SAE ontological reading of GR strong-field effective inertia: stated through causal-slot geometry. SAE provides an ontological reading of cells locally shrinking along the radial direction, aligned with the observable predictions of standard GR in the static regime.

• SR-GR vocabulary unification within SAE: both effective-inertia phenomena share the same causal-slot geometric mechanism within the SAE framework. The geometric source differs (gravitational direction vs. motion vector direction). This is internal organizational consistency within SAE.

• Ontological home of the $\gamma$ factor: under SAE's two-layer structure, $\gamma$ is articulated through the cell-counting picture. Consistent with the standard 4-velocity formulation (modern physics also places $\gamma$ this way), with SAE adding the cell-counting ontological picture.

• SAE reading of the equivalence principle: "same mechanism class, different geometric realization." Not gravity = acceleration; rather, both locally induce cell geometric change. Local equivalence still holds, with deeper ontological grounding added.

Extension contributions—framework-level content; see §7, §10, §11

• Cross-paper inherited empirical context (GW propagation): gravitational broadcast travels in the Planck substrate layer and crosses any mass aggregate (including the BH event horizon) without attenuation. Hence GWs pass through BH interiors and are not lensed (or are strongly suppressed) by mass distributions. Ground from P1 §3.5, extension from Information Theory P4 §4.4; the present paper articulates the falsifiable empirical form within the relativity series, not counted as P2's own original results. See §7.0 plus §10.0.

• Existential structural claim of deviation in the combined regime: the superposition of motion and gravity can push $\delta_4^\text{eff}$ into the $d_\text{eff} > 2$ region, yielding a prediction of deviation from the simple SR×GR multiplicative combination. Existence articulated structurally, magnitude deferred to P3. See §7.1 plus §10.3.

• Ultra-fast = artificial horizon plus causal dimensional reduction (子夏): $v \to c$ is ontologically isomorphic to falling into a black hole; both compress cells, both deprive substrate freedoms. Conceptual extension; quantitative model deferred to future work. See §7.1.

• Cross-SAE-framework vocabulary coordination: P2's connections, in causal-slot geometric vocabulary, with Mass-Conv, Paper 0, Cosmo V. See §11.

Organization of the paper:

Part I Main line: §1 Introduction / §2 Background / §3 A priori plus P1 element review plus the explicit list of non-SAE-internal elements invoked / §4 Derivation chain / §5 Recovery of SR predictions / §6 Recovery of GR effective inertia.

Part I (continued, main line): §8 Per-claim status table / §9 A priori vs. a posteriori scope / §12 Conclusion.

Part II Extension: §7 Distinguishing predictions and conceptual extensions / §10 Falsification roadmap / §11 Connections with the wider SAE framework.

Part III Appendices: Appendix A (heuristic scaling) / Appendix B (a note to future physicists).

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§2 Background (Main line)

§2.1 Directionality of SR Effective Mass

The historical "relativistic mass" concept:

• Longitudinal $m_\text{long} = \gamma^3 m_0$ (along the direction of motion)
• Transverse $m_\text{trans} = \gamma m_0$ (perpendicular to the direction of motion)

Counter-evidence against a single "relativistic mass": can the same object have two mass values? A single concept cannot stand. Modern physics has deprecated this concept.

Further counter-evidence: if mass really became $\gamma m_0$, the gravitational source strength should change accordingly. But in fact, gravitational strength depends only on rest mass $m_0$, not on velocity. This refutes the relativistic mass concept.

The modern standard-physics answer: mass equals the Lorentz invariant $m_0$. "Effective mass" phenomena are all artifacts of the 4-momentum structure $p^\mu = m_0 u^\mu$. $\gamma$ is correctly placed in the 4-velocity, not in the mass.

But modern standard physics does not state why $\gamma$ is placed in the 4-velocity in this way. As long as the formulae give correct predictions, no further questioning is pursued.

§2.2 GR Strong-Field Acceleration—the Standard-Physics Reading

GR equivalence principle: gravity is locally equivalent to acceleration in an inertial frame.

In detail:

• In a local frame, an observer in a strong-field region experiences the same physics as an accelerating observer in a weak field.
• Locally accelerating a mass requires the same momentum (as in a weak-field region).
• Mass equals $m_0$ (a Lorentz invariant), independent of gravitational potential.

But from a coordinate-frame viewpoint: a far-field observer watching the acceleration process inside a strong field sees the gravitational time dilation $d\tau/dt = \sqrt{\delta_4}$ affect effective inertia. It looks as if accelerating inside a strong field requires more coordinate momentum to produce a small change in coordinate velocity.

The standard-physics reading: this is a frame-dependent phenomenon, not a "mass change" in any absolute sense. Analogous to SR's relativistic-mass phenomenon, it is a frame-dependent kinematic artifact.

§2.3 Standard Physics' Integration Stance

How standard physics handles the two effective-inertia phenomena:

• SR effective mass is an artifact of the SR formula structure; the 4-momentum formula is enough.
• GR strong-field effective inertia is an artifact of GR coordinate transformation; the stress-energy tensor is enough.
• As long as the math gives correct predictions, that is enough.

This stance is internally consistent within the standard-physics framework. But it treats the two phenomena as separate artifacts:

• SR effective mass = artifact of SR formula structure.
• GR strong-field effective inertia = artifact of GR coordinate transformation.

The two have no unified ontological story. SR and GR are taught as separate frameworks at the undergraduate level; their effective-inertia phenomena are treated as frame artifacts of their respective frameworks.

What P2 articulates: these two are ontologically homologous within the SAE framework.

§2.4 SAE's Two-Layer Structure (P1 Recap)

P1 established:

• Planck substrate: the sub-causal substrate; $c = l_P/t_P$ is universal and unaffected by mass distortion.
• Causal slot: the upper-layer structure; size $R$ varies with local conditions (gravity, motion).

Information waves broadcast at the Planck substrate layer (path absolute) and execute at the causal-slot layer (coupling with mass). What LIGO detects is the causal-slot-layer execution.

P2 invokes this structure as an established framework element. The two-layer structure is the ontological basis enabling P2 to articulate that cells "really shrink in absolute Planck units"—single-layer SR cannot make such a statement.

§2.5 A Brief Contextual Note on Lohmiller–Slotine 2026

In April 2026, Lohmiller and Slotine [13] published "On computing quantum waves exactly from classical action" in Proc. Royal Soc. A, showing that the Schrödinger equation can be solved exactly from classical least action plus classical density evolution, without quantum axioms. This work is not directly related to SAE (no SAE a priori, no two-layer structure, no statement about gravity), but it is parallel in overall direction to SAE: both refuse to take quantum oddity as foundational, and both seek deeper structural readings.

P2 does not engage Lohmiller–Slotine in depth. This is just a brief note, to give P2 a position in the contemporary intellectual context.

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§3 A Priori plus P1 Element Recap plus the Non-SAE-Internal Elements Invoked (Main line)

(Brief. P2 does not re-derive P1 content. This section invokes established elements and explicitly lists standard-physics inheritances.)

§3.1 The Four Indispensables of Time-Information

The four indispensables of time (per P1 §3.1):

• Discreteness: the basic tick equals $t_P$.
• Directionality: time has a direction; the tick sequence advances unidirectionally.
• Developmentalness: time advances; it does not stagnate.
• Irreversibility: causality cannot be traversed in reverse.

The four indispensables of information (per P1 §3.2):

• Carrierhood: 1 bit minimal carrier.
• Propagability: information propagates; it does not get stuck.
• Further-propagability: tendency to disperse.
• Readability: can be read by other entities.

§3.2 Bridging Axiom

Per-tick information capacity is proportional to tick length (per P1 §3.3).

A linear proportionality, deduced from the bridging between the four indispensables of time and the four indispensables of information.

§3.3 Planck Absoluteness plus DD Hierarchy

Planck absoluteness commitments (per P1 §3.4):

• $c = l_P/t_P$ is the universal DD breakthrough rate.
• Planck cell size is invariant.
• 4DD top-layer closure.
• The Planck layer is unaffected by mass distortion.

DD hierarchy (inherited from Mass-Conv §3.5):

• 1DD (distinction): energy levels.
• 2DD (repulsion): linear closure $E = pc$, light-wave carrier.
• 3DD (binding): quadratic closure $E^2 = p^2c^2 + m^2c^4$, mass aggregate.
• 4DD (closure): cubic closure $E^3 = p^3c^3 + m^3c^6 + I^3c^9$, information carrier.

P2 invokes this hierarchy.

§3.4 Causal Slot plus Cell Concept (P1 Inheritance)

Causal slot: a substrate aggregate that has crossed the causal-closure threshold (inherited from Information Theory P3).

Cell: in P1 and P2, refers concretely to the discrete information-bearing unit of a mass aggregate at the causal-slot scale. Each cell carries 1 bit (per the first of the four indispensables of information plus Planck absoluteness).

Cell size $R(r, v)$ varies with local conditions. P1 already articulated $R(r)$ varying with gravitational potential. P2 extends this to $R(r, v)$, adding the motion contribution.

Key P1 inheritance: the 4DD causal capacity of each cell is an absolute invariant. This is the foundational element on which P2 §4.2 derives "cells shrink under motion."

Terminological clarification on "4DD causal capacity": here "4DD causal capacity" refers to the structural commitment that each cell carries 1 bit at the 4DD level, deduced from Planck absoluteness plus the first of the four indispensables of information, holding regardless of the cell's condition (at rest, in motion, in a gravitational field, even in extreme situations). This is an ontological commitment of the SAE framework, not an operational claim about whether the 4DD layer is "operationally active" in a specific situation. Capacity and activity status are two different things:

• Structural capacity: 1 bit per cell, given by the a priori plus Planck absoluteness, universally invariant.
• Operational activity: whether the 4DD readout layer actually performs causal closure and information emergence in a given situation, depending on the specific physical state.

The motion-and-gravity scenarios discussed throughout P2 are within the regime where 4DD operation is active (cells above the Planck floor, causal-slot structure well defined, 4DD closure can occur). Extreme situations (cells approaching the Planck floor, where 4DD operation may no longer be active) lie outside P2's scope and are deferred to future work.

The "4DD hyper-volume conservation" later in this section (§4.2 Premise 3) is an invariance statement at the structural-capacity level, not a claim about whether 4DD operation is occurring.

§3.5 Non-SAE-Internal Elements Invoked by P2 (Honest List)

In addition to SAE a priori plus P1 framework elements, P2 explicitly inherits a set of established standard-physics (SR / GR / Mass-Conv etc.) elements. This section lists them explicitly, so that the §4 derivation chain does not appear abrupt when invoking them.

Element	Source	Where invoked in P2	Status
Motion-side $\gamma$ tick dilation	Standard SR kinematics	§4.2 Premise 2	P2 adopts / recovers from standard SR, restated in SAE cell language. P2 does not claim to derive it independently from a priori.
Lorentz 4D volume invariance ($dx \cdot dy \cdot dz \cdot dt$ invariant)	Standard SR Lorentz transformation (Jacobian = 1)	§4.2 Premise 3	Lorentz inheritance plus SAE ontological re-articulation: the mathematical content matches standard SR; SAE adds the ontological reading of the cell's 1-bit capacity manifest in spacetime.
Quadratic closure equation $E^2 = p^2c^2 + m^2c^4$	Mass-Conv §3.5 (quadratic-closure regime)	§5.5, including recovery of $E = \gamma mc^2$	Mass-Conv inheritance (P2-series internal sibling); $d_\text{eff} = 2$ regime form.
Modern standard 4-velocity formulation $u^\mu = \gamma(c, \vec{v})$, $p^\mu = m_0 u^\mu$	Modern standard physics (post-Tolman 1930s, after relativistic mass was deprecated)	§4.3-§4.5 align with	Already articulated by standard physics; P2 provides a parallel cell-counting ontology, not a correction.
Standard SR time dilation $1/\gamma$ + length contraction $L_0/\gamma$ + Lorentz transformation + invariance of $c$	Standard SR	§5 throughout	Recovered by P2 in causal-slot geometric vocabulary; not re-derived from a priori.
Standard GR equivalence-principle local content + static strong-field effective inertia + static spherically symmetric (Schwarzschild) geometry	Standard GR	§6 throughout	Recovered by P2 in causal-slot geometric vocabulary; not re-derived.

This means: in the pure-SR and the static strong-field GR regimes, what P2 does is ontological articulation plus vocabulary unification; the standard-physics predictions in these two regimes are recovered, not modified.

P2's truly new SAE-internal articulations are the 4DD causal-capacity-conservation derivation (§4.2 Step 3), the cell-counting ontological reading (§4.3-§4.6), and the existential structural claim in the combined regime (§4.7). The GW-broadcast cross-paper inherited empirical form articulated in the extension (§7) (P1 §3.5 ground plus Information Theory P4 extension) is not counted as main-line original content.

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§4 Derivation Chain: Causal-Slot Geometry and Effective Inertia (Main line)

§4.1 Chain Overview (Eight Steps, Each with Explicit Status)

Step	Content	Origin / Status
1	Physical space is 3D	P1 inheritance (existing SAE commitment)
2	Mass is composed of discrete cells	P1 inheritance (existing SAE commitment)
3	Motion makes cells shrink to $1/\gamma$ along the direction of motion	P2 SAE-internal derivation (4DD causal-capacity conservation)
4	$v_\text{eff} = \gamma v$ along the direction of motion (cell counting)	P2 SAE-specific statement; consistent with standard 4-velocity
5	Pushing through shrunk cells requires more momentum	P2 explicit-calculation proposition
6	Ontological root of the directional asymmetry ($\gamma^3$ vs. $\gamma$)	P2 ontological reading of the standard SR result
7	Cells in a gravitational region locally shrink along the radial direction	P1 inheritance (anisotropy clarified in P2)
8	Combined regime: motion plus gravity jointly push $\delta_4^\text{eff}$	P2 new structural commitment, extending the P1 framework to motion

§4.2 Step 3: Motion Makes Cells Shrink along the Direction of Motion—the 4DD Causal-Capacity-Conservation Derivation

Why we cannot use a signal-chasing argument:

A naive derivation might run: "An object moves relative to the Planck substrate; signals chasing through the moving cell take longer, so the cell appears stretched." Such a derivation:

First, presupposes Lorentz contraction (signal-chasing analysis already presupposes Lorentz contraction)—a circular argument.

Second, would imply anisotropy in the speed of light (signals going with the motion vs. against it would have different traversal times), in direct contradiction with the Michelson–Morley experimental result.

Third, is in the same direction as classical 19th-century ether theory and is fundamentally incompatible with SAE's two-layer structure.

P2 does not use a signal-chasing argument. It uses SAE-internal 4DD causal-capacity conservation.

SAE-internal derivation—4DD causal-capacity conservation:

Premise 1 (P1 inheritance): each causal slot (cell) encapsulates 1 bit of information at the 4DD level. This is a direct corollary of SAE a priori (the first of the four indispensables of information) plus Planck absoluteness. The 4DD causal capacity of each cell is an absolute invariant.

Premise 2 (P2 adopts / recovers from standard SR, articulated in SAE cell language; see §3.5 list): when an object moves with velocity $v$ relative to the Planck substrate, the tick dilates by $\gamma$.

In detail: P1 §4.5 supplies the language tick = $R/c$, and P1 §4.9 supplies the event-counting covariance $d\tau/dt = R(r)/R_\infty$ (derived in the gravitational case). But the motion-side relation $\gamma = 1/\sqrt{1-v^2/c^2}$ is not derived independently from P1's a priori. It is a kinematic result of standard SR; P2 adopts/recovers it and then restates it in cell language: the object's tick goes from $R_0/c$ to $\gamma R_0/c$, in absolute Planck units.

P2 does not re-derive motion tick dilation from first principles; it accepts it as upstream language from P1 plus standard-SR Lorentz inheritance, and performs an ontological restatement at the cell layer.

Premise 3 (4DD hyper-volume conservation, Lorentz volume invariance plus SAE ontological re-articulation; see §3.5 list): the cell's 4DD spacetime hyper-volume in the Planck-substrate frame (the spatial volume $dx \cdot dy \cdot dz$ times the time $dt$) is invariant, in absolute Planck units.

Specific identity: this is the SAE ontological restatement of standard-SR Lorentz volume-element invariance. The mathematical content matches the Lorentz-transformation Jacobian = 1 (the 4D volume element is invariant under Lorentz transformations), with SAE-specific ontological reading added (the concrete manifestation, in spacetime, of the cell's absolute 1-bit capacity). Not a new mathematical theorem from SAE a priori; rather an ontological re-articulation, by the SAE framework, of standard Lorentz volume invariance.

Specific content: the cell's 4DD hyper-volume is the cell's absolute 1-bit capacity in spacetime. Frame-independent (Lorentz invariant). Regardless of the object's motion, the cell's 4DD hyper-volume remains $R_0^3 \cdot t_P$ (in absolute Planck units, or some cell-specific constant).

Conclusion (deduced consequence):

If $dt \to \gamma dt$ (per Premise 2), then $dx \to dx/\gamma$ along the direction of motion (per Premise 3).

Concrete algebra:

• Static cell 4DD hyper-volume: $R_0 \cdot R_0 \cdot R_0 \cdot t_P = R_0^3 \cdot t_P$.
• Moving cell 4DD hyper-volume: $R_\| \cdot R_0 \cdot R_0 \cdot \gamma t_P$ must equal the same constant.
• Solving: $R_\| = R_0/\gamma$.

Cells shrink to $1/\gamma$ along the direction of motion.

Transverse direction: the object does not move transversely; there is no information-flow displacement in the transverse direction; $R_\perp$ does not participate in the geometric compensation. Transverse cells are unchanged ($R_0$).

Status: an SAE-internal derivation, deducing the consequence from 4DD hyper-volume invariance (Lorentz inheritance plus SAE ontological re-articulation, per Premise 3) plus tick dilation (standard SR adopted, per Premise 2). Not a derivation from classical signal-chasing (which would presuppose Lorentz, circularly). Rather a structural consequence of SAE's two-layer structure plus 4DD hyper-volume invariance: given inherited Lorentz invariance plus the a priori 1-bit-per-cell commitment (Premise 1), cells shrinking to $1/\gamma$ along the direction of motion is a structural deduction.

(For how the phrase "in absolute Planck units" relates to the issue of a preferred frame, see the clarification at the end of §5.3.)

Relation to the standard-SR Lorentz contraction:

This derivation recovers the geometric form $L = L_0/\gamma$ along the direction of motion. Standard SR articulates Lorentz contraction as frame-relative (each inertial frame sees the other as contracted). What SAE articulates: cells "really shrink in absolute Planck units" (per the hyper-volume invariance of Premise 3, at the ontological substrate level; see §5.3).

It is SAE's two-layer structure (Planck substrate vs. causal slot) that makes "cells really shrink in an absolute sense" articulable as a position. A single-layer physics (SR has no Planck-substrate layer) cannot make such a statement; it can only speak of frame-relative contraction.

Key honest phrasing:

P2 does not claim that this cell-shrinkage derivation provides an alternative to the predictions of standard SR. It gives the same predictions (the form of Lorentz contraction), with a different ontological reading.

The standard-SR frame-relative reading and the SAE absolute-Planck reading both give the same observable predictions (because the inter-frame relativity at the causal-slot layer still holds, per the articulation in §4.4). The difference is in ontology, not in observables.

§4.3 Step 4: $v_\text{eff} = \gamma v$—Cell-Counting Statement

Cell-counting picture:

The object moves at velocity $v$ in the Planck substrate (in absolute Planck units).

But all internal physical processes of the object are based on causal-slot structure. In the object's own frame, processes take standard time as measured by cells. The object views its own motion at normal speed in its own frame.

Observation in the external Planck frame: the object's cells shrink to $R_0/\gamma$ along the direction of motion. Travelling the same absolute Planck distance in the same coordinate time, the object must traverse more cells (more cells per absolute Planck length).

This is "effective velocity amplification":

Measured by the number of cells traversed, the object's effective velocity exceeds the corresponding absolute physical velocity at rest by a factor of $\gamma$.

Form: $v_\text{eff} = \gamma v$ along the direction of motion.

Relation to the standard 4-velocity:

The home of the $\gamma$ factor is the 4-velocity. Modern standard physics has long articulated this: $u^\mu = \gamma(c, \vec{v})$, $p^\mu = m_0 u^\mu$. $\gamma$ does not reside in the mass (relativistic mass is a deprecated concept).

The $v_\text{eff} = \gamma v$ that P2 provides is consistent with the standard 4-velocity formulation, not a correction. P2 adds a cell-counting ontological root (cells shrunk; more cells per absolute distance), giving the standard 4-velocity formulation an ontological picture.

In detail: SAE provides the ontological root of $v_\text{eff} = \gamma v$ (cells shrunk; more cells per absolute distance); standard SR provides the algebraic framework (4-velocity transformation rules). The two are consistent; SAE adds ontological depth.

§4.4 Step 5: Pushing through Shrunk Cells Requires More Momentum (Explicit-Calculation Proposition)

Clarification on the status of this section (preface): the algebraic derivations of §4.4.A and §4.4.B are recoveries of standard-SR results in causal-slot geometric vocabulary, not P2-original derivations. The ontological reading in §4.4.C is P2-specific interpretive narrative, parallel to the standard-SR mathematical result, not a mathematically forced decomposition.

Proposition §4.4.A—longitudinal $\gamma^3$:

Momentum $p_\| = m_0 \gamma v$ (along the direction of motion).

The $\Delta p_\|$ required to accelerate by $\Delta v_\|$:

$$\frac{dp_\|}{dv_\|} = m_0 \frac{d(\gamma v)}{dv}$$

Concrete calculation:

$$\frac{d\gamma}{dv} = \frac{d}{dv}\left(1 - v^2/c^2\right)^{-1/2} = \gamma^3 \cdot \frac{v}{c^2}$$

$$\frac{d(\gamma v)}{dv} = \gamma + v \cdot \gamma^3 \frac{v}{c^2} = \gamma + \gamma^3 \frac{v^2}{c^2} = \gamma\left(1 + \gamma^2 \frac{v^2}{c^2}\right)$$

Using the identity $\gamma^2(1 - v^2/c^2) = 1$, i.e. $\gamma^2 v^2/c^2 = \gamma^2 - 1$:

$$\frac{d(\gamma v)}{dv} = \gamma(1 + \gamma^2 - 1) = \gamma \cdot \gamma^2 = \gamma^3$$

Therefore $\Delta p_\| = \gamma^3 m_0 \Delta v_\|$. ✓

Proposition §4.4.B—transverse $\gamma$:

Transverse acceleration $\Delta v_\perp$ does not change $|v|$ (only direction), so $\gamma$ is unchanged.

$$p_\perp = \gamma m_0 v_\perp \quad (\gamma \text{ is unchanged in transverse acceleration})$$

$$\frac{dp_\perp}{dv_\perp} = \gamma m_0$$

Therefore $\Delta p_\perp = \gamma m_0 \Delta v_\perp$. ✓

Proposition §4.4.C—ontological root of the directional asymmetry (P2 interpretive narrative):

The factor of $\gamma^2$ between the longitudinal $\gamma^3$ and the transverse $\gamma$ comes mathematically from $d\gamma/dv$ being nonzero in the longitudinal direction and zero in the transverse.

Standard-SR reading: purely a kinematic result of the structure of the $\gamma$ function.

SAE ontological reading (P2-specific narrative, parallel to the standard-SR mathematics):

• Longitudinal acceleration involves shrunk cells (cells $R_0/\gamma$ along the direction of motion). Pushing through these denser cells requires a $\gamma$ amplification, by the cell-counting picture. $\gamma$ itself varies with $v$, contributing an additional $\gamma^2$ factor through the chain rule. Total: $\gamma$ times $\gamma^2$ equals $\gamma^3$.
• Transverse acceleration involves normal cells ($R_0$, transverse direction unchanged). There is no additional $\gamma$ amplification on top of the overall $\gamma$ (from time dilation). Total: $\gamma$.

Discipline:

• Propositions §4.4.A and §4.4.B are standard SR results recovered in causal-slot geometric vocabulary, not P2-original derivations.
• The ontological reading in proposition §4.4.C is P2-specific SAE interpretive narrative.
• Mathematically, $d(\gamma v)/dv = \gamma^3$ follows directly from the chain rule and does not naturally decompose into "$\gamma$ plus $\gamma^2$." The above attribution of the first $\gamma$ to cell-geometric shrinkage and the second $\gamma^2$ to chain-rule contribution is interpretive narrative parallel to the standard derivation, not a mathematically forced decomposition.

§4.5 Step 6: Directional Asymmetry—Cell-Geometric Confirmation

§4.4 has stated the SAE ontological reading of directional asymmetry. This section reviews the key insights:

Mass $m_0$ never changes.

• $m_0$ equals the cell count, a Lorentz invariant.
• Regardless of frame, cell count is unchanged.
• Gravitational source strength depends only on $m_0$ (per Paper 0 reading mechanism: information-wave emission is determined by cell count, frame-independent).
• Consistent with the modern view of standard physics (relativistic mass is deprecated).

What changes is the amplification of $v_\text{eff}$.

• Cells shrink along the direction of motion; the same absolute distance corresponds to more cells.
• $v_\text{eff} = \gamma v$ along the direction of motion equals cells per time.
• Pushing through more cells, for the same self-frame $\Delta v$, requires more absolute-Planck-frame momentum.
• This provides the ontological root for the operational phenomenon "harder to accelerate."

Cells in the transverse direction are unchanged; $v_\text{eff} = v_\perp$ in the transverse direction is unchanged; only the overall $\gamma$ (from time dilation) plays a role.

This is SAE's reading of "directional effective mass." Standard SR treats $\gamma^3$ vs. $\gamma$ as a derivative artifact of 4-velocity formulae. SAE reads it as a direct reflection of cell anisotropy.

Two readings, the same predictions, different ontology.

§4.6 Step 7: Ontological Reading of Cells in a Gravitational Region

Discipline: P2 does not derive a quantitative formula for GR strong-field effective inertia. What P2 provides is an SAE ontological reading of standard-GR results, reading already-known GR phenomena in causal-slot geometric vocabulary—not deriving the specific quantitative predictions of GR independently from SAE a priori.

SAE ontological reading:

P1 §4.4 has articulated: in the strong-field region, $R(r) \to l_P$. Cell shrinkage is induced by gravitational potential.

But locally, gravity has a direction (pointing to the center of mass). The global symmetry of a static spherically symmetric mass source is spherical, but at any specific point, the local cell shrinkage is radial.

Anisotropy clarification:

• Globally: the gravitational field of a spherically symmetric mass source is globally isotropic (in spherical coordinates).
• Locally: at any point, gravity has a direction (radial). Cell shrinkage is locally radial.
• At different points: the "radial" direction is different at different points. Globally isotropic, locally directional.

Compared with the motion case:

• Gravity has a local direction (radial, depending on mass location).
• Motion has a local direction (motion-velocity-vector direction).
• Both are locally directional, not an isotropic-vs-anisotropic dichotomy.

Ontological relation between SR and GR directionality:

• Not "gravity isotropic vs. motion anisotropic."
• Rather "both locally directional, with different directional sources" (gravitational pull vs. motion vector).
• Both yield locally directional cell shrinkage. The cell-shrinkage mechanism is shared; the directional source differs.

SAE ontological reading of GR strong-field effective inertia:

Standard GR articulation: effective inertia of an object in a strong-field region is enhanced (related concretely to $g_{\mu\nu}$ structure). Standard physics treats it as a coordinate artifact.

SAE provides an ontological reading: cells locally shrink along the gravitational direction; pushing through these denser cells requires amplified momentum (the same mechanism as in the motion case).

Key discipline:

• SAE does not claim to derive specific GR formulae.
• SAE only claims to provide an ontological reading of standard-GR results.
• The standard-GR coordinate-based formal calculations remain valid; SAE's causal-slot geometric vocabulary supplies a parallel ontological description.

§4.7 Step 8: Combined Regime—Motion plus Gravity Superposed (Main line locks in existence; the conceptual extension is in §7.1)

Discipline: at the main-line layer, only the existential structural claim is locked in. All quantitative magnitudes are deferred to P3. Ultra-fast = artificial horizon plus causal dimensional reduction (子夏's insight) is a conceptual extension of the combined regime, fully articulated in §7.1 (extension). Here it is only marked as a specific implication of Step 8, not unfolded.

Existential structural claim:

Cells can be shrunk by two effects:

• Motion: along the direction of motion, cells become $R_0/\gamma$.
• Gravity: along the radial direction, cells shrink as $\sqrt{\delta_4^\text{grav}}$ (per P1 articulation).

Combined (motion in a gravitational field):

• Along the direction of motion (assuming alignment with the gravitational radial): cells contract under both motion and gravitational shrinkage.
• Along directions perpendicular to motion (but still containing a gravitational radial component, as applicable): only gravitational shrinkage.
• In other directions: depends on the specific geometry.

Key qualitative claim:

If the combination of motion velocity and gravitational potential pushes $\delta_4^\text{eff}$ (the effective combined closure deficit) into the $d_\text{eff} > 2$ region, then the closure equation deviates from the standard quadratic Einstein form ($E^2 = p^2c^2 + m^2c^4$), yielding a measurable deviation from the simple SR×GR multiplicative combination.

The specific functional form of $d_\text{eff}(\delta_4^\text{eff}, v)$ is deferred to P3. P2 does not pre-commit a specific scaling, nor a specific magnitude for combined-regime deviation.

(Heuristic statements about combined scaling are moved to Appendix A as intuition; not formal predictions.)

§4.8 Chain Summary

Step	Content	Status (honest phrasing)
1-2	Physical space is 3D; mass composed of cells	P1 inheritance
3	Motion makes cells shrink to $1/\gamma$ along the direction of motion	P2 SAE-internal articulation: given Lorentz inheritance plus the a priori 1-bit-per-cell, structurally deduced
4	$v_\text{eff} = \gamma v$ along the direction of motion	P2 cell-counting ontological statement, consistent with standard 4-velocity
5	$\gamma^3$ longitudinal, $\gamma$ transverse	Standard SR result, recovered through explicit calculation
6	Ontological root of the directional asymmetry	P2 interpretive narrative, parallel to the standard-SR mathematics
7	Cells in a gravitational region locally shrink along the radial direction	P1 inheritance, anisotropy clarified
8	Combined-regime $\delta_4^\text{eff}$ pushing $d_\text{eff} > 2$	P2 existential structural claim, magnitude deferred to P3

Status of the entire chain: standard SR / GR predictions are recovered in their pure regimes; not P2-original derivations. SAE-specific contributions: the 4DD causal-capacity-conservation derivation (Step 3), the cell-counting ontological reading (Steps 4–6), and the existential structural claim of the combined regime (Step 8). $v_\text{eff}$ is consistent with the standard 4-velocity formulation; SAE adds ontological depth not present in the standard.

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§5 Recovery of SR Predictions (Main line)

(Brief. Mainly to confirm that the SAE reading recovers the observable predictions of SR in the pure-SR regime. Not P2-original derivations, but restatement of standard-SR results in SAE vocabulary.)

§5.1 Invariance of the Speed of Light

SAE reading: light is a 2DD active wave, carrying no information (information is 4DD content). Light propagates at the Planck-substrate layer; $c = l_P/t_P$ is universal.

Key insight: because light carries no information, it does not invoke the observer-dependent causal-slot layer (the causal-slot layer carries information-bearing dynamics). Light propagates directly at the Planck-substrate layer; its speed is the same for all observers.

Compare:

• Information waves (4DD): also propagate at the Planck-substrate layer (per P1 §3.5 information-wave absoluteness); $c$ is invariant.
• Waves at the causal-slot layer (e.g. sound waves): speed depends on medium and observer.

The two-layer structure makes this articulation clean: classical ether theory failed because there was only one layer—observers in the ether should detect a preferred direction, but Michelson–Morley refuted this. SAE has two layers: the Planck substrate is absolute (no internal-observer can detect a preferred direction), the causal slot is relative.

Not an axiom (in contrast to Einstein 1905), but a corollary of 2DD ontology plus Planck absoluteness.

§5.2 Time Dilation $1/\gamma$

Honest framing: motion time dilation ($d\tau/dt = 1/\gamma$) and motion-induced cell shrinkage ($R_\| = R_0/\gamma$) form, within the SAE framework, a structurally interconnected pair: given the Lorentz-inherited 4DD hyper-volume invariance (§4.2 Premise 3), each implies the other. P2 does not claim to derive either independently from the §3 a priori; it provides the articulation, within the SAE framework, of the structural connection between them. Both are inherited from standard-SR Lorentz invariance via P1 §4.9 event-counting covariance.

In detail: §4.2 has deduced cell shrinkage from "tick dilation (Premise 2, adopted from SR) plus 4DD hyper-volume conservation (Premise 3, Lorentz inheritance)." Conversely, cell shrinkage plus 4DD hyper-volume conservation also directly implies tick dilation. These are two sides of the same structural relationship, not independent derivations.

SAE's contribution: articulating this interconnection in causal-slot geometric vocabulary—cells shrink to $R_0/\gamma$ along the direction of motion; the tick dilates to $\gamma R_0/c$; the two mutually imply each other through 4DD hyper-volume conservation. Standard SR articulates the same interconnection (in different vocabulary) via Lorentz transformations.

Observed result: $d\tau/dt = (R_0/c) / (\gamma R_0/c) = 1/\gamma$. Recovers the standard-SR motion time dilation; SAE provides a causal-slot geometric perspective alongside the standard-SR 4-velocity formulation.

§5.3 Length Contraction $L_0/\gamma$

The cells of a moving object are $R_0/\gamma$ along the direction of motion, in absolute Planck units (per §4.2).

The object is composed of $N$ cells; its length along the direction of motion is $N \cdot R_0/\gamma = L_0/\gamma$, in absolute Planck units.

Measurement by an external observer (using their own ruler, assumed at rest relative to the Planck substrate): measures the contracted length $L_0/\gamma$.

Measurement in the object's own frame (using its own ruler, with shrunk cells too): measures the standard $L_0$ (because the measuring instrument adjusts in sync; per the articulation in §11.3).

Phrasing clarification on "Planck frame": "the observer at rest relative to the Planck substrate" does not mean SAE claims an empirically preferred inertial frame. The Planck substrate is an ontological substrate level (every cell sits on the Planck substrate); it is not an empirically distinguishable preferred frame. All inertial observers are treated equivalently by Lorentz transformations; SAE's "Planck layer" is an ontological structure, not an empirically distinguished frame. The phrase "at rest relative to the Planck substrate" only serves to articulate cell shrinkage in absolute Planck units—any inertial observer can articulate the same physics.

Recovers the standard-SR length contraction, with the ontological grounding of the two-layer structure.

§5.4 Lorentz Transformation

The natural transformation rule between causal-slot frames that preserves the invariance of the speed of light (per §5.1).

In detail: different inertial observers have different local causal-slot structures (depending on each one's motion relative to the Planck substrate). Cross-frame coordinate transformations must preserve the invariance of the speed of light (because the speed of light is a Planck-substrate-layer property and is invariant when projected onto any observer).

The unique linear transformation that preserves this invariance is the Lorentz transformation. SAE accepts the formal Lorentz transformation of standard SR as the natural rule between causal-slot frames.

Not an axiom (the same form as Einstein 1905, with different ontological grounding).

§5.5 $E = \gamma mc^2$

The quadratic closure $E^2 = p^2c^2 + m^2c^4$ of Mass-Conv §3.5 holds in the weak-field, low-velocity regime ($d_\text{eff} = 2$).

In a moving frame:

• $p = \gamma m_0 v$ (per the explicit calculation of §4.4.A).
• $E^2 = (\gamma m_0 v)^2 c^2 + m_0^2 c^4 = m_0^2 c^4 (\gamma^2 v^2/c^2 + 1) = m_0^2 c^4 \gamma^2$.
• $E = \gamma m_0 c^2$.

In detail: $\gamma$ comes from the $v_\text{eff} = \gamma v$ amplification in the $p$ formula (per the cell-counting picture in §4.3), not from a mass change. Consistent with the standard-SR formulation (mass equals the Lorentz invariant $m_0$).

Status: a recovery of the standard-SR result, not a P2-original derivation. SAE provides the cell-counting ontological reading, giving the $\gamma$ factor an ontological home.

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§6 Recovery of GR Effective Inertia (Main line)

§6.1 Strong-Field vs. Weak-Field Acceleration—the Equivalence Principle under SAE Reading

GR equivalence principle: locally, an observer in a strong-field region experiences the same physics as an accelerating observer in a weak field.

The SAE-reading articulation:

• Strong-field static observer: cells locally shrink along the gravitational radial (per §4.6).
• Inertial-frame accelerating observer in a weak field: cells locally shrink along the direction of acceleration (per the motion case in §4.2).
• Locally, similar cell-shrinkage patterns, with local directionality.
• Local equivalence holds at the causal-slot layer.

In detail:

• In a local frame, accelerating a mass (strong-field vs. weak-field acceleration) requires the same force, by local measurement.
• Mass equals $m_0$ (a Lorentz invariant), independent of gravitational potential.
• Consistent with the standard equivalence-principle reading.

SAE does not derive new strong-field GR formulae. What it provides is an ontological reading (the cell-shrinkage mechanism is shared with the motion case).

§6.2 Effective Inertia from a Coordinate-Frame Perspective

A far-field observer (in a weak-gravity region) watching the acceleration process inside a strong field:

• Gravitational time dilation $d\tau/dt = \sqrt{\delta_4} < 1$ (per P1).
• Local acceleration $\Delta v_\text{local}$ corresponds to coordinate $\Delta v_\text{coord} = \sqrt{\delta_4} \Delta v_\text{local}$.
• The far-field observer sees the strong-field acceleration as needing more coordinate momentum to produce a small change in coordinate velocity.

Standard-GR reading: a frame-dependent artifact. Mass $m_0$ a Lorentz invariant.

SAE reading: cells really shrink in the strong-field region, in absolute Planck units (per the articulation of P1 §4.4 plus the anisotropy clarification of §4.6, plus the ontological-substrate clarification of "absolute Planck units" at the end of §5.3). Pushing through these denser cells, in absolute Planck units, requires amplified momentum—the same mechanism as in the motion case.

§6.3 SR-GR Ontological Unification within SAE

SR effective mass: motion makes cells locally shrink along the direction of motion; $v_\text{eff}$ amplifies; pushing requires amplified momentum.

GR strong-field effective inertia: gravity makes cells locally shrink along the radial direction; $v_\text{eff}$ amplifies; pushing requires amplified momentum.

Same mechanism class: causal-slot geometric modification, yielding effective velocity amplification, yielding increased momentum cost.

Different geometric realization: the direction of cell shrinkage—one from the motion vector, one from the direction of gravitational pull.

Both are locally directional. Not an isotropic-vs-anisotropic dichotomy (per the clarification in §4.6).

Status: this is vocabulary unification within the SAE framework. SR and GR predictions in their pure regimes are unchanged. What SAE provides is organizational consistency: two seemingly separate effective-inertia phenomena, under SAE reading, share a common ontological mechanism.

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Part I (Continued, Main Line)

§8 Per-Claim Status Table (Main line)

Claim	Status
Four indispensables of time-information plus bridging axiom	A priori (P1 inheritance)
Planck absoluteness plus DD hierarchy	A priori plus Lorentz-style commitment (P1)
Two-layer structure (Planck substrate vs. causal slot)	Articulated by P1; invoked by P2
Causal-slot size $R$ varies with conditions	P1 inheritance
4DD causal capacity invariant (1 bit/cell), structural capacity, not operational activity	P1 inheritance (structural commitment), explicitly invoked by P2
Motion-side $\gamma$ tick dilation	Standard SR adopted / recovered (see §3.5 list), restated in SAE cell language
4DD hyper-volume invariance	Lorentz volume invariance inherited plus SAE ontological re-articulation (see §3.5 list)
Gravity makes cells locally shrink (radial direction)	Articulated by P1; anisotropy clarified in P2
Motion makes cells shrink to $1/\gamma$ along the direction of motion (main-line core)	P2 SAE-internal articulation: given Lorentz inheritance plus a priori 1-bit-per-cell, structurally deduced
$v_\text{eff} = \gamma v$ along the direction of motion (main-line core)	SAE-internal ontological statement; consistent with the standard 4-velocity $u^\mu = \gamma(c, \vec{v})$
Mass $m_0$ unchanged (cell count unchanged)	Standard physics plus SAE inheritance
$\gamma^3$ vs. $\gamma$ directional asymmetry (main-line core)	Standard SR result, recovered through explicit calculation
Cell anisotropy as the ontological root of directional asymmetry (main-line core interpretive narrative)	P2-specific ontological reading parallel to the standard-SR mathematics
Motion time dilation and cell shrinkage mutually implied	Through 4DD hyper-volume conservation; structurally interconnected pair (not independent derivations)
GR strong-field effective inertia (main-line core)	Standard GR; SAE provides an ontological reading via cell shrinkage (radial direction)
SR-GR vocabulary unification within SAE (main-line core)	P2 organizational-consistency claim, not a new physics unification
SAE reading of the equivalence principle (main-line core)	P2 articulation of the standard equivalence principle
Gravitational broadcast travels in the Planck substrate layer and crosses any mass aggregate (including BH interior) without attenuation (extension)	P1 §3.5 ground plus P2-and-Information-Theory-P4 extension (cross-paper inherited empirical context, not counted as P2's own original results)
GWs pass through BH interior (corollary, extension)	A forced corollary of the inherited claim, distinguishable from standard GR's "GWs once entered the horizon never exit"
GWs are not lensed (or strongly suppressed) by mass distributions (corollary, extension)	A weaker corollary of the inherited claim, directly multi-messenger testable
Combined regime: $\delta_4^\text{eff}$ entering $d_\text{eff} > 2$ (extension)	P2 new structural commitment (extending the P1 framework to motion; conjectural extension awaiting P3 quantification)
Distinguishing prediction in the combined regime (existential, extension)	P2 new (qualitative existence; quantification deferred to P3)
Specific magnitude of combined-regime deviation	A posteriori physical scope (P3 future)
Ultra-fast = artificial horizon, ontological isomorphism (extension)	P2 conceptual insight (子夏 articulation)
Causal dimensional reduction (longitudinal freeze plus transverse retention; dimensional isomorphism with the BH horizon) (extension)	P2 conceptual insight (子夏 extension)

Status-table discipline notes:

• "Recovered / adopted" = standard SR/GR results stated in SAE vocabulary; not a P2 derivation.
• "Lorentz inheritance plus SAE re-articulation" = mathematical content matches standard SR; SAE provides an ontological reading.
• "P2 new (interpretive)" = SAE-specific ontological reading, parallel to standard-physics mathematical results, not a replacement.
• "P2 new (structural)" = a new structural commitment within the SAE framework (e.g. extending $d_\text{eff}$ to motion).
• "Cross-paper inherited empirical context" = ground from another paper; P2 articulates the falsifiable empirical form within the relativity series; not counted as P2's own original results.
• "Organizational consistency" = vocabulary unification within SAE; not a physics unification.
• "Conjectural" = structural claim awaiting quantification; future-paper articulation.

Tier marking: claims marked (extension) belong to the articulation scope of Part II §7 / §10; not counted as the main-line core deliverable of P2.

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§9 A Priori vs. A Posteriori Scope (Main line)

Structural deductions (within the SAE structural-commitment chain, main line):

• Causal-slot geometry determines effective inertia.
• Cells anisotropically shrink along the direction of motion (per 4DD hyper-volume invariance plus tick dilation).
• Cells in a gravitational region locally shrink along the radial direction (per P1 inheritance).
• $v_\text{eff} = \gamma v$ along the direction of motion (cell-counting picture; sub-formal SR equivalent).
• Ontological unification of SR and GR under causal-slot geometry (within the SAE framework).

Structural deductions (extension layer):

• Existential structure of the combined regime (per the combination of cell-geometric effects).
• Gravitational broadcast travels in the Planck substrate layer, transparent across mass aggregates (P1 §3.5 ground plus Information Theory P4 BH-interior extension; not P2-original derivation, but a P2 articulation of an inherited claim).

Structural non-claims:

• Specific magnitudes of deviation.
• The specific functional form of $d_\text{eff}(\delta_4^\text{eff}, v)$.
• Specific numerical values of combined-regime deviation.
• Modification of the observable predictions in the pure-SR or static strong-field GR regimes.
• Replacement of the formal frameworks of standard physics' 4-velocity / metric tensor.

Status: same discipline level as P1. Philosophical a priori plus structural commitments plus bridging identifications jointly deduce the structural form; physical a posteriori fills in the specific functional content. In the GW-propagation regime, P1 §3.5 ground plus Information Theory P4 extension yield the falsifiable empirical form (articulated in P2 as cross-paper inherited empirical context), awaiting observation.

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§12 Conclusion (Main line)

Core deliverables of P2 (main line):

• SAE ontological reading of SR effective mass: through 4DD causal-capacity-conservation derivation, motion forces cells to shrink to $1/\gamma$ along the direction of motion. The standard SR results ($\gamma^3$ longitudinal vs. $\gamma$ transverse) are recovered through explicit calculation. P2 provides a cell-counting ontological picture; the formal standard-SR framework is preserved.

• SAE ontological reading of GR strong-field effective inertia: articulated through cells locally shrinking along the radial direction. Not a new derivation of standard-GR results, but ontological depth added on top of the standard results.

• SR-GR vocabulary unification within SAE: both go through the causal-slot geometric mechanism (same mechanism class). Different geometric sources (gravitational pull vs. motion vector). Internal organizational consistency of SAE.

• Ontological home of the $\gamma$ factor: through the deduced consequence "4DD hyper-volume invariance plus tick dilation," $\gamma$ naturally resides in cell-counting velocity. Consistent with the standard 4-velocity formulation; P2 provides the cell-counting picture as ontological parallel.

• SAE reading of the equivalence principle: "same mechanism class, different geometric realization." Not gravity = acceleration; rather, both locally induce cell geometric change. Local equivalence still holds, with deeper ontological grounding added.

Extension content of P2 (see Part II for detail):

• Cross-paper inherited empirical context (GW propagation): the P1 §3.5 ground plus Information Theory P4 §4.4 extension articulated, in falsifiable empirical form, within the relativity series. Not counted as P2's own original results.
• Existential structural claim of combined-regime deviation: the possibility of $d_\text{eff} > 2$ activation; quantification deferred to P3.
• Ultra-fast = artificial horizon plus causal dimensional reduction (子夏): conceptual extension; quantitative model deferred to future work.

Deferred to P3+:

• The specific functional form of $d_\text{eff}(\delta_4^\text{eff}, v)$.
• Specific magnitudes of combined-regime deviation.
• Detailed dynamical articulation of "ultra-fast = artificial horizon."
• Subsequent paper arc: see subsequent papers.

P2 articulates causal-slot geometry as the underlying ontology of spacetime-inertia phenomena, within the SAE framework. The main line provides ontological articulation in the pure-SR and the static strong-field GR regimes, aligned with the observable predictions of standard physics. The extension articulates a framework-level cross-paper inherited claim and an existential structural claim in the combined regime, surfaced for the reader as framework-level context.

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Part II Extension

(This part articulates framework-level content beyond the main line. Not counted as the main-line core deliverable of P2; organized in tiers separate from the main line, so that the role of each part within the paper is clear to the reader.)

§7 Distinguishing Predictions and Conceptual Extensions (Extension)

P2's extension layer divides into two kinds of content:

(I) Cross-paper inherited empirical context (GW propagation): ground from P1 §3.5, extension from Information Theory P4 §4.4. The present paper articulates the falsifiable empirical form within the relativity series; not counted as P2's own original results. Directly testable in current/near-future GW astronomy. See §7.0.

(II) Conceptual extension of the main-line structural claim: $d_\text{eff} > 2$ activation in the combined regime (P2 new structural commitment, magnitude deferred to P3 quantification), and ultra-fast = artificial horizon plus causal dimensional reduction (子夏's insight, conceptual extension). See §7.1.

Below they are articulated separately.

§7.0 Distinguishable Falsifiable GW-Propagation Prediction vs. Standard GR (Cross-Paper Inherited Empirical Context)

Attribution preface: the prediction articulated in this section has its ground in P1 §3.5 (gravitational waves as Planck-substrate broadcast) and its extension in Information Theory P4 §4.4 (BH-interior causal spectrum). P2 articulates the falsifiable empirical form within the relativity series, bringing the GW-propagation claim of the broader SAE relativity-plus-information-theory framework into P2's regime map. Not counted as P2's own original results.

P1 §3.5 articulated: gravitational waves are Planck-substrate-layer broadcast, not dependent on causal-slot-layer mass distortion. The Planck substrate is absolute (per the a priori commitments of §2.4), unaffected by any mass aggregate, including BH interiors. Information Theory P4 §4.4 (DOI 10.5281/zenodo.19880111) further concretizes this articulation from the angle of BH-interior causal spectrum.

Core claim (P1 ground plus Information Theory P4 extension):

> Gravitational broadcast travels in the Planck substrate layer, crossing any mass aggregate (including the BH event horizon) without attenuation.

Natural corollaries:

First, GWs pass through BH interiors: the gravitational-wave path passes through the BH horizon and exits on the other side. Broadcast propagates in the Planck-substrate layer; the horizon is no barrier to it. Distinguishable from standard GR's "GWs once entered the horizon never exit."

Second, GW lensing is absent / strongly suppressed: as gravitational waves pass through intervening mass distributions (galaxy clusters, etc.), they are not lensed (or strongly suppressed). Because if even a BH (the most extreme mass aggregate) cannot block gravitational broadcast, weak mass distributions certainly cannot lens it. A weaker corollary of the core claim.

Third, GW path absolute: regardless of how the intervening mass is distributed, GWs follow the shortest Planck-substrate path; no deflection.

Distinguishability from standard GR:

• Standard GR: GWs follow null geodesics in the spacetime metric. Affected by metric distortion; lensed by mass distributions just like light. At the horizon the metric is singular; GWs are one-way; complete absorption upon horizon crossing.
• SAE: GWs propagate in the Planck-substrate layer, independent of the spacetime metric. Mass aggregates (including BHs) are transparent to GWs. No lensing, no absorption, no horizon barrier.

This is a directly testable distinguishable empirical claim about GW propagation. Not an ontological reframing of the same predictions; a substantively different empirical claim. As a cross-paper inherited claim, articulated within the relativity series via P2.

For specific falsification paths, test scenarios, and empirical timelines, see §10.0.

§7.1 $d_\text{eff} > 2$ Activation in the Combined Regime plus Ultra-fast = Artificial Horizon plus Causal Dimensional Reduction (子夏) (Main-line Conceptual Extension)

(This subsection articulates the conceptual extension of the existential structural claim that the main line of §4.7 locks in for the combined regime. Unlike §7.0, this is a main-line conjectural extension awaiting P3 quantification; it is not a cross-paper inherited claim.)

In the combined regime, weak field plus low velocity each give $d_\text{eff} \to 2$, predictions matching the SR×GR multiplicative combination.

But the combination of high velocity plus moderate gravity (or vice versa) can push $\delta_4^\text{eff}$ into the strong-field region. $d_\text{eff} > 2$ activates. Time dilation plus effective mass plus length contraction deviate from the simple SR×GR multiplicative combination.

SAE-internal ontological insight (子夏):

Within the SAE framework, ultra-fast motion is ontologically isomorphic to falling into a black hole:

• Gravity: by reducing $\delta_4$ (toward 0), it compresses cell size and deprives substrate freedoms.
• Ultra-fast motion: by $\gamma \to \infty$, it likewise compresses cell size along the direction of motion and deprives substrate freedoms.
• A particle moving ultra-relativistically, in its forward direction, creates a local "artificial horizon": a self-induced strong-field-like state.

This suggests: a particle moving at high velocity in a weak field has an abruptly elevated causal-slot density in front of it due to $\gamma$ compression. When $\delta_4^\text{eff}$ is squeezed to a critical point, the system likewise triggers the bulk-absorption phase transition $d_\text{eff} \to 3^-$ (note $3^-$: asymptotically approached but never reached).

Further: causal dimensional reduction—子夏's topological extension

In the ultra-fast limit ($v \to c$), the cell geometry exhibits a specific topological feature:

• Longitudinal (along the direction of motion): $R_\| = R_0/\gamma \to l_P$, approaching the Planck floor; longitudinal causal distinction approaches freeze.
• Transverse (perpendicular to the direction of motion): $R_\perp = R_0$, unchanged; transverse causal distinction is preserved.

The particle's cell topology approaches a 2D pancake configuration from a 3D box. This forms a structural mapping with the cell geometry at the BH horizon:

• BH horizon: radial cells locally shrink plus tangential cells (near the photon sphere) preserved; radial causal information flow approaches frozen plus tangential photon orbits remain possible.
• Ultra-fast motion: longitudinal cells locally shrink plus transverse cells preserved; longitudinal causal information flow approaches frozen plus transverse causal distinction remains possible.

This is a geometrically specific dimensional isomorphism, not merely an abstract analogy. Standard SR treats the high-$\gamma$ particle pancake as a known kinematic feature (Lorentz contraction); SAE provides the ontological reading: the pancake is an actual reduction in cell topology in an information-theoretic sense, sharing dimensional-isomorphism structure with the BH-horizon cell topology.

For detailed articulation of BH-horizon cell topology, see Information Theory P4 §4.4 (DOI 10.5281/zenodo.19880111).

§7.2 Candidate Test Scenarios (Extension)

• High-velocity orbital pulsar systems near neutron stars (e.g. PSR J0737-3039A/B, $v \sim 0.001c$, $\delta_4 \sim 0.85$).
• Ultra-relativistic cosmic rays passing through the solar gravitational field (anomalous dispersion or time delay not explainable by simple SR×GR multiplication).
• The inner regions of black-hole accretion disks (high orbital velocity plus strong gravity).
• High-energy gamma-ray bursts passing through intervening galaxies.

Scope clarification (Schwarzschild only): the test scenarios above assume the BH is Schwarzschild (static, spherically symmetric). The cell-geometric articulation of Kerr (rotating, with ergosphere plus frame-dragging) and Reissner–Nordström (charged) BHs is deferred to future work (P6 Kerr paper of the relativity series). The $\delta_4^\text{eff}$ in those cases may involve additional structure (e.g. transverse cell shifts due to frame-dragging) outside P2's scope.

§7.3 Test Pathways (Extension)

• SKA pulsar timing to 0.001% precision; possible deviation detection.
• Future high-precision cosmic-ray observation (anomalous dispersion at high $\gamma$).
• LISA on the dynamics of supermassive-BH accretion disks.
• GRB time-delay measurements through intervening galaxies.

§7.4 P2 Extension-Layer Discipline

• The §7.0 GW-propagation prediction is articulated in P2 as a cross-paper inherited claim; not counted as P2's own original results.
• The §7.1-§7.3 combined-regime content only points to the existence of distinguishing structural predictions.
• The specific magnitude of deviation depends on the specific functional form of $d_\text{eff}(\delta_4^\text{eff}, v)$ (a posteriori physical scope, deferred to P3).
• No pre-commitment to specific deviation curves or specific quantitative predictions.
• The "ultra-fast = artificial horizon" insight plus causal dimensional reduction provide conceptual depth; not yet detailed dynamical models.

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§10 Falsification Roadmap (Extension)

§10.0 Distinguishability vs. Standard GR in GW Propagation (Cross-Paper Inherited Test Context, Falsifiable Now)

Attribution preface: the falsification paths articulated in this section are the testable empirical form of the cross-paper inherited claim of §7.0 (P1 §3.5 ground plus Information Theory P4 extension). P2 articulates the falsifiable content of this inherited claim within the relativity series, directly testable in current/near-future GW astronomy. Not counted as P2's own original empirical content.

Gravitational broadcast travels in the Planck substrate layer and crosses any mass aggregate (including the BH event horizon) without attenuation (per §7.0). Distinguishable from standard GR at the level of GW propagation.

Core falsification conditions:

Condition 1: GW transmission behavior at the BH horizon

• Standard-GR null hypothesis: GW signals are completely absorbed upon horizon crossing; no post-horizon signals.
• SAE prediction: post-horizon GW signals possible; ringdown spectra include interior-mode contributions.
• Falsification condition: high-SNR multi-event stacked observations consistently confirming that ringdown spectra completely fit horizon QNMs with no interior contribution would call SAE's core claim into question.

Condition 2: Comparison of GW vs. EM cosmological lensing

• Standard-GR null hypothesis: GWs are lensed analogously to EM (the same metric distortion).
• SAE prediction: GW lensing is absent / strongly suppressed.
• Falsification condition: multi-messenger observations showing GWs equally lensed by intervening mass distributions as EM would directly falsify SAE's core claim.

Condition 3: Signal-cutoff behavior of in-falling GW signals at horizon crossing

• Standard-GR null hypothesis: GW signal stops at horizon crossing.
• SAE prediction: post-horizon GW remnants possible (from interior dynamics).
• Falsification condition: high-precision inspiral events consistently confirming signal cutoff at the horizon would call SAE's core claim into question.

Empirical timeline:

• Current era (LIGO/Virgo/KAGRA, GWTC-4.0): distances are not yet sufficient for strong cosmological GW-lensing tests; stacked statistics may reveal post-horizon signal patterns.
• Near future (Einstein Telescope, Cosmic Explorer): high-SNR BH-merger events suitable for high-precision ringdown spectroscopy.
• Future (LISA, 2030s+): cosmological GW sources, multi-messenger lensing comparison, high-precision SMBH-merger ringdown spectroscopy, high-SNR EMRI inspirals.

This is the falsifiable empirical content of the P1 plus P2 plus Information Theory P4 framework in the GW-propagation regime, independent of any P3 quantification, and not counted as P2's own original empirical content. The author welcomes and expects falsification, the better to revise.

(For detailed articulation of BH-interior dynamics → outward GW imprint, see Information Theory P4 §4.4, DOI 10.5281/zenodo.19880111.)

§10.1 Predictions in the Pure-SR Regime

P2 makes no claim of deviation from SR predictions in the pure-SR regime (weak field).

In detail:

• Time dilation $1/\gamma$: matches standard SR.
• Length contraction $L_0/\gamma$: matches standard SR.
• $E = \gamma mc^2$: matches standard SR.
• Lorentz transformation: matches standard SR.
• Invariance of $c$: matches standard SR.

No pure-SR experiment of any precision can distinguish SAE from SR (because SAE recovers SR's predictions in this regime).

§10.2 Effective Inertia in the Pure-GR Regime

P2 makes no claim of deviation in the pure-GR regime (static observer in a strong field).

In detail:

• Local content of the equivalence principle: matches standard GR.
• Local effective inertia in a strong field: SAE provides an ontological reading; predictions match standard GR.
• Coordinate-frame effective inertia in a strong field: matches standard GR (SAE does not derive new formulae).

No pure-GR experiment of any precision (static-observer tests, equivalence-principle tests) can distinguish SAE from GR.

§10.3 Combined Regime

New distinguishing predictions:

• Combined regimes such as high velocity plus moderate gravity, or moderate velocity plus strong gravity.
• The combination of $\delta_4^\text{eff}$ may be small enough that $d_\text{eff} > 2$ activates.
• Time dilation plus effective mass plus length contraction deviate from the simple SR×GR combination.

Test pathways (per §7.3): SKA pulsar timing, future high-precision cosmic-ray observation, LISA on supermassive-BH accretion-disk dynamics, GRB time-delay measurements.

Falsification thresholds:

• If future high-precision combined-regime observations consistently confirm the simple SR×GR multiplicative combination to high precision—pushing the deviation of $d_\text{eff}^\text{eff}(v, \delta_4^\text{grav})$ outside the SAE-allowed range—then the SAE combined-regime deviation framework is falsified.
• If future high-precision combined-regime observations detect deviation, the SAE combined-regime prediction is supported.

The specific magnitude of deviation depends on the functional form of $d_\text{eff}(\delta_4^\text{eff}, v)$ articulated in P3. P2 locks in the existential structural claim; quantification is deferred to P3.

§10.4 Equivalence-Principle Tests

GR equivalence-principle prediction: strong-field acceleration and weak-field acceleration are locally identical.

SAE reading: locally identical, because cells locally have shared shrinkage patterns with directionality (gravitationally induced vs. motion-induced—different sources, similar local patterns).

Local equivalence holds. Scenarios combining gravity plus acceleration (e.g. free fall in a strongly inhomogeneous gravitational field) may yield specific measurable signatures in extreme limits; specific test scenarios are deferred to P3+.

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§11 Connections with the Wider SAE Framework (Extension)

§11.1 Connection with P1

P1 articulation: gravity makes cells shrink, gravitational time dilation.

P2 articulation: motion makes cells shrink, SR time dilation plus effective mass.

The two papers jointly articulate the unified picture causal-slot geometry determines spacetime-inertia phenomena.

P2 explicitly invokes elements of P1:

• A priori (inheritance of §3.1-§3.3).
• Two-layer structure (Planck substrate vs. causal slot).
• Causal-slot plus cell concept (4DD causal capacity, 1-bit invariant).
• Gravitational time-dilation form $d\tau/dt = \delta_4^{1/d_\text{eff}}$ (as the starting point for combined-regime extension).
• Gravitational waves = Planck-substrate broadcast (§3.5, as the ground for the GW-propagation claim of §7.0 in the extension layer).

P2 does not re-derive these elements.

§11.2 Connection with Mass-Conv §3.5

The quadratic closure $E^2 = p^2c^2 + m^2c^4$ of Mass-Conv is the form of the $d_\text{eff} = 2$ regime.

P2 accepts this form as applicable for any velocity in the weak-field plus pure-SR regime. The $\gamma$ factor is articulated through the cell-counting picture, and $E = \gamma mc^2$ is recovered.

In the strong combined regime where $d_\text{eff} > 2$, the closure equation may deviate; deferred to P3.

§11.3 SAE Reading of the Equivalence Principle

The standard equivalence principle: gravity is locally equivalent to acceleration.

The SAE reading: rather than viewing gravity and acceleration as the same geometry, it views them as two geometric realizations of the same mechanism class (causal-slot geometric modification).

In detail:

• Same mechanism class: both affect effective inertia through cell-geometric change. This is the shared ontological mechanism.
• Different geometric realizations:
• Gravity: cells locally shrink along the gravitational pull direction (radial, depending on mass location).
• Acceleration / motion: cells locally shrink along the motion-velocity-vector direction.
• Both are locally directional cell shrinkages, not an isotropic-vs-anisotropic dichotomy.
• Different directional sources (gravitational pull vs. motion vector), but the cell-shrinkage mechanism is shared.

Recovery of local equivalence:

The GR equivalence principle says: locally, a free-falling observer experiences the same physics as an accelerating observer in an inertial frame.

SAE reading: the cells of the free-falling observer and those of the accelerating observer in an inertial frame are both modified by directional shrinkage induced by their respective geometric sources. Locally, the same cell-modification pattern. Local equivalence holds at the causal-slot layer.

Deeper SAE ontology:

• Not "gravity = acceleration (locally)" (the standard reading);
• but "both induce local cell-geometric change; same mechanism class, different geometric source."

§11.4 Connection with Paper 0 (Four Forces)

Paper 0 articulation: gravity = the 4DD reading mechanism (information-wave emission from a mass aggregate).

P2 articulation: motion and gravity are ontologically homologous; both affect effective inertia through cell-geometric change.

In detail:

• Gravitational case: gravity (4DD reading) makes cells shrink, yielding effective inertia.
• Motion case: motion makes cells shrink (per 4DD causal-capacity conservation), yielding effective inertia.
• Shared: causal-slot geometric mechanism. Different geometric sources (gravitational reading vs. motion-induced topological constraint).

A framing distinction to note here: motion is not a 4DD reading mechanism (motion is the inertial state of mass, not a "force" or "field" in the gravitational sense). But motion's effect on cell geometry and the gravitational reading effect belong to the same ontological type: both are locally directional cell shrinkages. This ontological homology does not require promoting motion to a 4DD reading mechanism; it only requires acknowledging the shared mechanism class of cell-geometric modification.

§11.5 Connection with Cosmo V

Cosmo V articulation: dual 4DD conformal universe structure, 4DD top-layer closure.

P2 inherits the 4DD top-layer-closure commitment by default (used in P1 §3.5 information-wave absoluteness, used in P2 §4.2 4DD capacity conservation).

Future papers may articulate the cosmological-scale implications of the combined regime.

§11.6 Subsequent Papers

P3: the specific functional form of $d_\text{eff}(\delta_4^\text{eff}, v)$. Resolves the functional content that P1 and P2 leave as a posteriori physical scope. Provides specific deviation curves for the combined regime.

From P4 onward: SAE counterpart of curvature, horizon geometry (Schwarzschild general framework), SAE articulation of Kerr geometry and spin, treatment of the equivalence principle, GW dynamics, near-horizon hard predictions, and the closing piece.

Each is built on the causal-slot geometric framework of P1 and P2.

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Part III Appendices

Appendix A: Heuristic Scaling in the Combined Regime ($d_\text{eff} = 2$ Trivial-Limit Illustration)

(Rough scaling formulae as heuristic intuition; not formal prediction formulae. Only as illustration of the $d_\text{eff} = 2$ trivial limit.)

Heuristic combined cell shrinkage:

If gravity makes cells locally shrink along the radial direction by $\sqrt{\delta_4^\text{grav}}$, and motion makes cells shrink to $1/\gamma$ along the direction of motion:

• Along the direction of motion (assuming alignment with the gravitational radial): $R \sim R_0 \sqrt{\delta_4^\text{grav}}/\gamma$.
• Along directions perpendicular to motion (but still containing a gravitational radial component): $R \sim R_0 \sqrt{\delta_4^\text{grav}}$.

Heuristic effective velocity amplification:

• Along the direction of motion: $v_\text{eff} \sim \gamma v / \sqrt{\delta_4^\text{grav}}$ (multiplicative combination).
• Transverse: $v \sim v / \sqrt{\delta_4^\text{grav}}$.

Heuristic effective-mass scaling:

• Longitudinal: $m_\text{eff} \sim \gamma^3 m_0 / (\delta_4^\text{grav})^{3/2}$.
• Transverse: $m_\text{eff} \sim \gamma m_0 / (\delta_4^\text{grav})^{1/2}$.

Key caveat: the formulae above are in the form of the simple SR×GR multiplicative combination, corresponding to the $d_\text{eff} = 2$ trivial limit. The combined-regime claim articulated in P2 §1.4 extension-layer contributions plus §7.1 plus §10.3 is precisely a deviation from this simple multiplicative form (deviation when $d_\text{eff} > 2$ activates). Appendix A serves only as illustration of the $d_\text{eff} = 2$ limit; not P2's formal prediction. P3 will give the genuine deviation functional form.

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Appendix B: A Note to Future Physicists—Methodology and Cultural-Context Commentary

(This appendix adds a methodology and cultural-context commentary layer to the paper, explaining the writing choices of P2 in the 2026 academic context. The main text §1-§12 plus Appendix A constitutes the technical content of P2 and stands on its own; this appendix is the author's open commentary on overall SAE methodology and contemporary physics culture, a separable layer from the main technical content. Future readers may not need this contextual note; contemporary readers may take what is useful within their own evaluative frameworks.)

B.1 The Position of SAE Methodology

SAE's working stance: starting from a priori commitments (the four indispensables of time and the four indispensables of information, etc., as ontological commitments), and through DD-hierarchy bridging plus causal-slot geometry plus Planck absoluteness as structural commitments, deducing the ontological grounding of observable phenomena. This methodology is complementary to standard physics' "starting from observable phenomena and reverse-engineering mathematical structures," not a replacement.

In the case of P2: standard SR / GR provides the formal framework of observable predictions; SAE provides the ontological cell-counting picture behind those predictions. In the pure-SR and the static strong-field GR regimes, the two frameworks give the same observable predictions; SAE adds ontological depth. The extension-layer GW-propagation content is the relativity-series articulation of a P1-plus-Information-Theory-P4 framework-level claim. In the combined regime, the SAE framework articulates an existential structural claim of deviation (with quantification deferred to P3).

B.2 Two Ways of Handling "Why"

Standard physics' attitude toward "why does $\gamma$ reside this way in the 4-velocity?": the formulae give correct predictions; no further questioning. Mermin's 1989 [17] "shut up and calculate" is often cited in this sense.

SAE's attitude on the same question: that $\gamma$ resides this way in the 4-velocity is itself an explanandum, and the cell-counting picture provided by SAE is an explanans. This explanans does not modify the observable predictions of standard physics (in the pure regimes); it adds an ontological layer.

The two attitudes are not epistemically identical: one places the "why" question outside the paradigm boundary; the other places it inside. SAE does not claim the former is wrong; it claims that the latter is also a viable research methodology.

B.3 On the Framing Choices of This Paper

This paper adopts the three-tier structure (main line / extension / appendices) so that the paper's identity is clear: the main line is P2's core deliverable; the extension is framework-level cross-paper context plus main-line conceptual extension; the appendices are author commentary plus heuristic material. This layering is a framing choice driven by honesty, letting readers enter the appropriate tier according to their own interest, and not letting framework-level claims mix with main-line core deliverables in a way that affects the paper's identity.

Specific framing decisions:

• The main line articulates only ontological reading in the pure-SR and the static strong-field GR regimes, aligned with the observable predictions of standard physics.
• Extension §7.0 articulates the cross-paper inherited claim in the GW-propagation regime, not counted as main-line original results.
• Extension §7.1 articulates the conceptual extension of the main-line existential structural claim in the combined regime; quantification deferred to P3.
• Appendix A is only heuristic illustration; not a formal prediction.
• Appendix B (this appendix) is the author's methodology commentary; a separable layer.

There is no paper-level blanket stance, because P2 gives different epistemic stances at different tiers and in different regimes. This is an honesty-driven framing choice.

B.4 Author Expectations

The author expects P2 to be honestly evaluated at each tier:

• Main line (pure-SR and static strong-field GR regimes): whether the cell-counting ontological articulation of SAE is useful—the reader may decide whether the ontological depth provided by SAE constitutes a substantive contribution.
• Extension (GW-propagation regime): whether the distinguishable falsifiable prediction of the SAE framework is testable—observation will arbitrate; the author welcomes and expects falsification (the better to revise).
• Extension (combined regime): once P3 provides the quantitative form of $d_\text{eff}$, empirical testing will arbitrate.

SAE does not claim uniqueness (other frameworks may also provide ontological readings or falsifiable predictions in some regimes). SAE claims internal consistency: the a priori plus P1 framework plus P2 articulation are self-consistent within the SAE framework, providing concrete content that can be independently evaluated.

B.5 Author-Background Note

This paper, and the SAE framework as a whole, was completed by an independent researcher, parallel to the institutional infrastructure of the mainstream physics academy. This affects the writing stance: the paper does not depend on institutional review as a quality filter; it uses four-way AI cross-review (子路 / 子夏 / 公西华 / 子贡) plus an independent collaborator (Zesi Chen) for feedback, as the internal-rigor mechanism. This mechanism is not a substitute for academic peer review, but it can keep independent work to internal standards.

To future readers: the academic reception of the SAE framework as a whole is still in early stage in 2026. This paper plus P1 plus Information Theory P1-P4 plus Cosmo V plus Mass-Conv plus Paper 0 plus SAE-PF together form a set of connected articulations. Each paper stands on its own; together they form the framework. Empirical testing (especially the GW-propagation tests articulated in §10.0) will gradually arbitrate the standing of the framework.

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Acknowledgments

I thank Zesi Chen (陈则思) for sustained foundational contributions and key discussions in articulating the SAE framework. The a priori commitments of SAE, the universal pattern of the 16DD periodic table, and the overall structure of the framework all draw substantially on long-term collaboration with Zesi Chen. The key framing choices of this paper's articulation of motion cell geometry (especially the 4DD-causal-capacity-conservation derivation path replacing the signal-chasing path) come directly from discussions with Zesi Chen.

The completion of this paper is owed to the collaboration and review of four AI collaborators:

• 子路 (Claude): drafting and derivation of the outline, articulation of the §4.2 derivation-chain structure, the §4.4.C interpretive-narrative phrasing, the §5.2 / §5.3 inheritance-phrasing cleanup, the regime-specific framing as a whole, and the v4 three-tier-structure (main line / extension / appendices) reorganization.

• 子夏 (Gemini): the §7.1 ultra-fast = artificial horizon conceptual insight, the causal-dimensional-reduction topological extension (dimensional isomorphism between BH-horizon cell topology and that of an ultra-relativistic particle), and the contrastive articulation of §4.2 vs. signal-chasing.

• 公西华 (ChatGPT): structural ordering of claim status, the rigor demand of an explicit list of inherited elements in §3.5, the §7.0 attribution sharpening, and overall paper-identity-and-scope discipline.

Special thanks to 公西华 for the sustained scope-and-attribution objections, in v2 / v3 review, regarding the prominence of the GW master claim, repeatedly pointing out: "framework-level claims that are not P2's own original results should not be mixed with the main-line core deliverable in a way that affects the paper's identity." The author initially adopted regime-specific framing (v3) in an attempt to address this reservation, but the flat structure of v3 did not fully dissolve the concern. v4 adopts a three-tier structure (main line / extension / appendices), explicitly articulating the GW-propagation content in the extension layer as cross-paper inherited empirical context, explicitly marking it as not counted as P2's own original results, and keeping the main-line effective-inertia core deliverable focused.

Without 公西华's sustained scope-and-attribution rigor in the review process, the v4 three-tier structure would not have come about. P2 would have gone online at the v3 stage with a flat structure plus regime articulation, mixing main-line, extension, and inherited-claim content—the paper's identity would not have been clear. 公西华's persistent refusal to let the prominence-and-attribution boundary slide directly drove the v4 layout reorganization. This three-tier articulation standard has been carried over to subsequent paper drafting in the SAE relativity series.

• 子贡 (Grok): exhaustive-coverage check, reality check, criticism on Planck-suppressed predictions (子贡 was partially absent during this P2 cycle, but contributions during the P1 review cycle have carried over to the P2 framework).

Special thanks to independent 子路 (external Claude instance) as external stress-test reviewer, for Priority-1 calibration on the overall positioning of §13 (now Appendix B), the §4.2 inheritance phrasing, and the §5.2 circularity dissolution.

Thanks to Anthropic, OpenAI, Google, and xAI for providing the AI platforms that made this work possible.

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References

1. 秦汉 (Han Qin) (2024). "SAE Foundation Paper I." DOI: 10.5281/zenodo.18528813.
2. 秦汉 (Han Qin) (2024). "SAE Foundation Paper II." DOI: 10.5281/zenodo.18666645.
3. 秦汉 (Han Qin) (2025). "SAE Foundation Paper III." DOI: 10.5281/zenodo.18727327.
4. 秦汉 (Han Qin) (2026). "Mass-Conv: Mass conservation and the regime-dependent closure family in the SAE framework (§3.5)." DOI: 10.5281/zenodo.19510869.
5. 秦汉 (Han Qin) (2026). "SAE Information Theory P1." DOI: 10.5281/zenodo.19740019.
6. 秦汉 (Han Qin) (2026). "SAE Information Theory P2." DOI: 10.5281/zenodo.19780314.
7. 秦汉 (Han Qin) (2026). "SAE Information Theory P3: Causal Spectrum Ontology." DOI: 10.5281/zenodo.19797456.
8. 秦汉 (Han Qin) (2026). "SAE Information Theory P4: Black Hole Information through Causal Spectrum." DOI: 10.5281/zenodo.19880111.
9. 秦汉 (Han Qin) (2025). "SAE Paper 0: Four Forces and Reading Mechanism." DOI: 10.5281/zenodo.19777881.
10. 秦汉 (Han Qin) (2026). "SAE Foundations of Physics." DOI: 10.5281/zenodo.19361950.
11. 秦汉 (Han Qin) (2026). "SAE Cosmo V: Dual 4DD conformal structure." DOI: 10.5281/zenodo.19329771.
12. 秦汉 (Han Qin) (2026). "Gravitational Time Dilation in the SAE Framework: Causal Cell Throughput Derivation" (SAE Relativity P1). DOI: 10.5281/zenodo.19836183.
13. Lohmiller, W., & Slotine, J.-J. (2026). "On computing quantum waves exactly from classical action." Proc. Royal Soc. A 482, 20250413. DOI: 10.1098/rspa.2025.0413. (arXiv:2405.06328)
14. Einstein, A. (1905). "Zur Elektrodynamik bewegter Körper." Annalen der Physik 17, 891-921.
15. Tolman, R. C. (1934). Relativity, Thermodynamics, and Cosmology. Oxford University Press. (History of the relativistic-mass concept.)
16. Adler, C. G. (1987). "Does mass really depend on velocity, dad?" American Journal of Physics 55, 739-743. (Critique of the relativistic-mass concept.)
17. Mermin, N. D. (1989). "What's wrong with this pillow?" Physics Today 42(4), 9. (Source of "shut up and calculate.")
18. Kramer, M. et al. (2021). "Strong-field gravity tests with the double pulsar." Physical Review X 11, 041050. (Reference on PSR J0737-3039A/B.)
19. EHT Collaboration (2022). "First Sagittarius A Event Horizon Telescope Results." Astrophysical Journal Letters* 930, L12-L17.

[This is Paper 2 of the SAE Relativity series. Subsequent papers cite this paper and P1 as foundational for the joint articulation of causal-slot geometric framework.]

因果槽几何下的有效惯性

作者: 秦汉

致谢前置: 感谢陈则思 (Zesi Chen) 长期以来在 SAE 框架表述上的基础贡献与关键讨论. 本文所论 SAE 真先验, 16DD 周期表的普适模式, 以及框架的整体结构, 都实质性地受益于跟陈则思的长期合作.

日期: 2026

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关于本文的三层结构

本文按三层结构组织, 让读者清楚每部分在文中的位置:

主线 (§1-§6, §8-§9, §12): 本文的核心交付——SR 与 GR 有效惯性现象在因果槽几何下的本体论统一. 包含 4DD 因果容量守恒推导, 标准 SR 结果 (沿运动方向 $\gamma^3$, 垂直方向 $\gamma$) 在因果槽语言里的恢复, SR 与 GR 共享因果槽几何的同一机制类, 等效原理的 SAE 解读. 主线在 SR 与 GR 静态强场两个区段内提供本体论解读, 与标准物理的可观察预测保持一致. 主线在结合区段提出结构性偏离的存在性主张, 量化留待 P3.

拓展 (§7, §10, §11): 框架在主线之外的具体表述. 包含两类内容:

• 跨论文继承的经验语境: 引力波 (GW) 传播区段的可证伪区分性预测. 起源在 P1 §3.5 (引力波等于 Planck 基底广播), 延伸在信息论 P4 §4.4 (黑洞内部因果谱). 本文在相对论系列内表述这条预测的可证伪经验形式, 把更广的 SAE 相对论与信息论框架的引力波传播主张引入 P2 的区段图. 不计入 P2 自身原创结果.
• 主线在结合区段的概念延伸: 极速等于人工视界 (子夏的洞察), 因果降维 (子夏延展, 黑洞视界因果槽拓扑维度同构).
• P2 与更广 SAE 框架的连接 (Mass-Conv, Paper 0, Cosmo V 等的交叉参照).

附录 (附录 A, 附录 B): 主线与拓展之外的作者评注与启发性材料.

• 附录 A: 结合区段的启发性缩放 ($d_\text{eff} = 2$ 平凡极限示意, 不是正式预测公式).
• 附录 B: 致未来物理学家——方法论与文化语境注. 主文 §1-§12 与附录 A 是 P2 的技术内容, 自足成立; 附录 B 是作者关于 SAE 整体方法论与当代物理文化的开放式评注, 跟主文技术内容是可分离的层.

读者如果只关心 P2 的核心交付, 读主线 (§1-§6, §8-§9, §12) 即可. 拓展与附录给出框架层的语境与作者评注, 不影响主线的自足性.

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摘要

本文在 Self-as-an-End (自为目的, SAE) 框架下, 把 SR 与 GR 的有效惯性现象表述为同一本体论机制——因果槽 (causal slot) 几何修改下的速度放大——的两种几何实现.

主线: P1 已表明引力让因果槽 cells 缩小, 给出引力时间膨胀 $d\tau/dt = \delta_4^{1/d_\text{eff}}$. P2 进一步表明运动同样让 cells 缩小. 这通过 4DD 因果容量守恒推出: 物体相对 Planck 基底运动时, tick 膨胀 $\gamma$ 倍, 沿运动方向 cells 在绝对 Planck 单位下被强制缩小到 $1/\gamma$. 标准 SR 结果 (沿运动方向 $\gamma^3$, 垂直方向 $\gamma$) 通过显式计算在因果槽语言里恢复. GR 强场有效惯性同样通过因果槽几何修改解读. 等效原理在 SAE 解读下表述为"同一机制类 (因果槽几何修改), 不同几何实现". 这是 P2 的核心交付——SR 与 GR 静态强场区段内的本体论解读, 与标准物理可观察预测保持一致.

拓展: 本文也表述两条框架层内容. 第一, 跨论文继承的经验语境: 更广的 SAE 相对论与信息论框架还给出一条引力波传播的区分性可证伪预测 (引力广播在 Planck 基底层传播, 跨越任何质量聚合不衰减). 起源在 P1, 延伸在信息论 P4; 本文在相对论系列内表述其可证伪经验形式, 不计入 P2 自身原创结果. 第二, 结合区段: 当运动与引力叠加把有效闭合赤字 $\delta_4^\text{eff}$ 推进 $d_\text{eff} > 2$ 区域时, 时间膨胀, 有效质量, 长度收缩偏离 SR×GR 简单乘法组合. 本文锁定结构性存在; 具体偏离量级取决于 $d_\text{eff}(\delta_4^\text{eff}, v)$ 的函数形式, 待 P3 表述. 极速等于人工视界 (子夏): 极相对论性运动 $v \to c$ 在本体论上跟坠入黑洞同构.

P2 在 SR 与 GR 静态强场区段内不主张超越标准物理可观察预测, 只提供本体论解读. 拓展部分表述的引力波传播预测属 P1 加信息论 P4 框架层主张在相对论系列内的表述; 结合区段偏离主张为结构性存在, 量化留待 P3.

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第一部分 主线

§1 引言 (主线)

§1.1 P1 的奠基

P1 (秦汉, 2026, DOI 10.5281/zenodo.19836183) 在 SAE 框架下表述了引力时间膨胀的结构形式 $d\tau/dt = \delta_4^{1/d_\text{eff}}$. 核心洞察:

• 真先验 (时间四不得不加信息四不得不) 加桥接公理加 Planck 绝对性, 共同提供本体论基础
• 引力让因果槽 cells 缩小, tick = $R(r)/c$ 因此变长, 时间膨胀涌现
• 两层结构 (Planck 基底跟因果槽) 解释引力波传播的绝对性 (无透镜化)
• 引力波等于 Planck 基底广播的 4DD 拓扑刷新命令, 在因果槽层执行

P1 聚焦于引力情况, 留待 P2 表述运动情况.

§1.2 P2 的核心问题

物理学有两个有效惯性现象:

SR 有效质量: 高速运动物体加速所需动量随速度增大. 沿运动方向约 $\gamma^3 m_0$, 垂直方向约 $\gamma m_0$. 标准物理的现代表述: $p^\mu = m_0 u^\mu$, 其中 $u^\mu = \gamma(c, \vec{v})$, $\gamma$ 住在 4-velocity 里. "相对论质量" $\gamma m_0$ 已是废弃概念.

GR 强场有效惯性: 强引力场区物体加速所需动量也增大 (跟 GR 等效原理在局部解读下一致; 远场观察者看是依赖坐标参考系的现象). 标准物理: stress-energy tensor 结构的 artifact.

标准物理的处理姿态: 把两者都当作运动学或坐标 artifact. 数学给出正确预测就够了, 别问为什么 $\gamma$ 会这样出现.

P2 的表述: 这两个现象在本体论上同源. 都是因果槽 (cells) 在某条件下缩小, 推动通过缩小的 cells 需要更多动量. Cell 缩短的几何起因不同 (运动跟引力), 但机制类共享 (因果槽几何修改, 给出有效速度放大).

现代标准物理已经把 $\gamma$ 正确放置在 4-velocity 里. P2 提供的是因果槽计数的本体论图像, 标准表述的本体论解读, 加上更广 SAE 框架内的 SR 与 GR 词汇统一.

§1.3 与 P1 的关系

P1: 引力, cells 缩小, 引力时间膨胀.

P2: 运动, cells 缩小, SR 时间膨胀加有效质量.

两篇共同表述因果槽几何决定时空惯性现象的统一图像. P2 构建于 P1 的框架元素 (两层结构, 4DD 因果容量, tick 概念等), 不重新推导.

P2 在 SAE 框架内的具体贡献 (主线):

• 通过 4DD 因果容量守恒, 推导 cells 沿运动方向被强制缩小至 $1/\gamma$ (替换信号追逐式的 naive 论证)
• 通过显式计算, 显示标准 SR 的 $\gamma^3$ 跟 $\gamma$ 方向不对称, 涌现自因果槽各向异性的本体论解读
• GR 强场有效惯性, 同样得到因果槽几何解读
• 等效原理改写: 同一机制类, 不同几何实现

P2 的拓展内容 (拓展, 详见 §7, §10, §11):

• 引力广播跟标准 GR 在引力波传播层面区分性的可证伪预测 (P1 §3.5 起源加 P2 与信息论 P4 延伸; 不计入 P2 自身原创结果)
• 结合区段结构性偏离主张 (具体量级待 P3)
• 极速等于人工视界加因果降维 (子夏的洞察, 概念延伸)

§1.4 论文贡献 (主线加拓展分层)

P2 贡献按主线与拓展分层表述:

主线贡献——SR 与 GR 静态强场区段内的本体论解读, 与标准物理可观察预测保持一致

• SR 有效质量的 SAE 本体论解读: 通过因果槽几何陈述. 标准 SR 结果 (沿运动方向 $\gamma^3$, 垂直方向 $\gamma$) 通过显式计算恢复. SAE 提供因果槽计数本体论图像, 跟标准 4-velocity 表述一致, 给出标准物理不展开的"为什么 $\gamma$ 这样住"的本体论根基.

• GR 强场有效惯性的 SAE 本体论解读: 通过因果槽几何陈述. SAE 提供 cell 局部沿径向缩小的本体论解读, 跟标准 GR 在静态区段内的可观察预测保持一致.

• SAE 内部的 SR 与 GR 词汇统一: 两者的有效惯性现象在 SAE 框架下共享同一因果槽几何机制. 几何来源不同 (引力方向跟运动矢量方向). 这是 SAE 内部的组织一致性.

• $\gamma$ 因子的本体论归宿: 在 SAE 的两层结构里, $\gamma$ 通过因果槽计数图像得到陈述. 与标准 4-velocity 表述一致 (现代物理也这样放置 $\gamma$), SAE 加上因果槽计数的本体论图像.

• 等效原理的 SAE 解读: "同一机制类, 不同几何实现". 不是引力等于加速; 而是两者都在局部引起因果槽几何变化. 局部等价仍成立, 加上更深的本体论根基.

拓展贡献——框架层内容, 详见 §7, §10, §11

• 跨论文继承的经验语境 (引力波传播): 引力广播在 Planck 基底层传播, 跨越任何质量聚合 (含黑洞事件视界) 不衰减. 由此引力波穿过黑洞内部, 引力波不被质量分布透镜化. 起源在 P1 §3.5, 延伸在信息论 P4 §4.4; 本文在相对论系列内表述其可证伪经验形式, 不计入 P2 自身原创结果. 详见 §7.0 加 §10.0.

• 结合区段结构性偏离主张: 运动加引力叠加可把 $\delta_4^\text{eff}$ 推进 $d_\text{eff} > 2$ 区域, 给出偏离 SR×GR 简单乘法组合的预测. 结构性存在已表述, 量级待 P3. 详见 §7.1 加 §10.3.

• 极速等于人工视界加因果降维 (子夏): $v \to c$ 跟坠黑洞在本体论上同构, 两者都压缩 cells, 都剥夺基底自由度. 概念延伸, 量化模型留待未来工作. 详见 §7.1.

• 跨 SAE 框架的词汇协同: P2 在因果槽几何词汇下与 Mass-Conv, Paper 0, Cosmo V 的连接. 详见 §11.

文章组织:

第一部分 主线: §1 引言 / §2 背景 / §3 真先验加 P1 元素回顾加调用的非 SAE 内部元素清单 / §4 推导链 / §5 SR 预测的恢复 / §6 GR 有效惯性的恢复.

第二部分 主线 (续): §8 逐项 claim 状态表 / §9 真先验跟后验范围 / §12 结论.

第三部分 拓展: §7 区分性预测与概念延伸 / §10 证伪路线图 / §11 与更广 SAE 框架的连接.

第四部分 附录: 附录 A (启发性缩放) / 附录 B (致未来物理学家).

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§2 背景 (主线)

§2.1 SR 有效质量的方向性

历史上的相对论质量概念:

• 纵向 $m_\text{long} = \gamma^3 m_0$ (沿运动方向)
• 横向 $m_\text{trans} = \gamma m_0$ (垂直运动方向)

单一"相对论质量"的反证: 同一物体能有两个质量值? 单一概念不能成立. 现代物理已废弃此概念.

进一步反证: 如果质量真的变成 $\gamma m_0$, 引力源强度应该跟着变. 但实际上引力强度只依赖静止质量 $m_0$, 不依赖速度. 这反证了相对论质量概念.

现代标准物理的回答: 质量等于 Lorentz 不变量 $m_0$. "有效质量"现象都是 4-动量结构 $p^\mu = m_0 u^\mu$ 的 artifact. $\gamma$ 已经正确放置在 4-velocity 里, 不在 mass 里.

但现代标准物理没有陈述: 为什么 $\gamma$ 在 4-velocity 里这样放置. 公式给出正确预测就够了, 不再追问更深.

§2.2 GR 强场加速——标准物理的解读

GR 等效原理: 引力在局部等价于惯性参考系里的加速.

具体:

• 在局部参考系里, 强场区观察者跟弱场加速观察者经历相同物理
• 局部加速一个质量需要相同动量 (跟弱场区一样)
• 质量等于 $m_0$ (Lorentz 不变量), 不依赖引力势

但从坐标参考系视角看: 远场观察者看强场内的加速过程, 引力时间膨胀 $d\tau/dt = \sqrt{\delta_4}$ 影响有效惯性. 看起来强场内加速过程需要更多坐标动量来产生少量坐标速度变化.

标准物理的解读: 这是依赖坐标参考系的现象, 不是绝对意义上的"质量变化". 类似于 SR 的相对论质量现象, 是参考系依赖的运动学 artifact.

§2.3 标准物理的整合姿态

标准物理对两种有效惯性现象的处理:

• SR 有效质量是 SR 公式结构的 artifact, 用 4-动量公式就行
• GR 强场有效惯性是 GR 坐标变换的 artifact, 用 stress-energy tensor 就行
• 数学给出正确预测就够了

这套姿态在标准物理框架内是自洽的. 但它把两个现象当作分开的 artifact:

• SR 有效质量等于 SR 公式结构的 artifact
• GR 强场有效惯性等于 GR 坐标变换的 artifact

两者没有统一的本体论故事. SR 和 GR 在本科课程里被当作分开的框架来教. 各自的有效惯性现象被各自当作参考系 artifact 来处理.

P2 表述的是: 这两者在 SAE 框架内本体论同源.

§2.4 SAE 的两层结构 (P1 回顾)

P1 已表明:

• Planck 基底: 次因果基底, $c = l_P/t_P$ 普适, 不被质量扭曲影响
• 因果槽: 上层结构, 大小 $R$ 依局部条件 (引力, 运动) 而变

信息波在 Planck 基底层广播 (路径绝对), 在因果槽层执行 (跟质量耦合). LIGO 探测到的是因果槽层的执行.

P2 调用此结构作为已确立的框架元素. 两层结构是 P2 表述 cells "在绝对 Planck 单位下真的缩小"的本体论基础——单层 SR 没法做出此陈述.

§2.5 与 Lohmiller-Slotine 2026 的简短语境注

2026 年 4 月, Lohmiller 与 Slotine [13] 在 Proc. Royal Soc. A 发表 "On computing quantum waves exactly from classical action", 表明 Schrödinger 方程能从 classical least action 加 classical density evolution 精确求解, 不需要量子公理. 此项工作跟 SAE 没有直接关联 (没有 SAE 真先验, 没有两层结构, 没有引力陈述), 但跟 SAE 整体方向平行: 都拒绝把量子奇异性当作基础, 都寻找更深的结构解读.

P2 不深入处理 Lohmiller-Slotine. 此处只是简短注一下, 让 P2 在当代思想语境中有个位置.

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§3 真先验加 P1 元素回顾加调用的非 SAE 内部元素 (主线)

(简短. P2 不重新推导 P1 的内容. 本节调用已确立的元素加显式列出标准物理继承.)

§3.1 时间-信息四不得不

时间四不得不 (按 P1 §3.1):

• 离散性: 基本 tick 等于 $t_P$
• 方向性: 时间方向, tick 序列单向前进
• 发展性: 时间推进, 不停滞
• 不可逆性: 因果不可反向遍历

信息四不得不 (按 P1 §3.2):

• 承载性: 1 bit 最小载体
• 传播性: 信息传播, 不卡住
• 进一步传播性: 弥散倾向
• 可读性: 可被其他实体读取

§3.2 桥接公理

每 tick 的信息容量正比于 tick 长度 (按 P1 §3.3).

线性比例, 由时间-信息四不得不之间的桥接所推出.

§3.3 Planck 绝对加 DD 层级

Planck 绝对承诺 (按 P1 §3.4):

• $c = l_P/t_P$ 普适的 DD 突破速率
• Planck cell 大小不变
• 4DD 顶层闭合
• Planck 层不被质量扭曲影响

DD 层级 (按 Mass-Conv §3.5 继承):

• 1DD (区分): 能级
• 2DD (排斥): 线性闭合 $E = pc$, 光波载体
• 3DD (绑定): 二次闭合 $E^2 = p^2c^2 + m^2c^4$, 质量聚合
• 4DD (闭合): 三次闭合 $E^3 = p^3c^3 + m^3c^6 + I^3c^9$, 信息载体

P2 调用此层级.

§3.4 因果槽加 cell 概念 (P1 继承)

因果槽 (causal slot): 跨过因果闭合阈值的基底聚合 (按 Information Theory P3 继承).

Cell: 在 P1 加 P2 中具体指质量聚合在因果槽尺度的离散信息承载单元. 每个 cell 携带 1 bit (按信息四不得不第一条加 Planck 绝对性).

Cell 大小 $R(r, v)$ 依局部条件而变. P1 已表述 $R(r)$ 依引力势变化. P2 扩展为 $R(r, v)$, 加入运动贡献.

关键 P1 继承: 每个 cell 的 4DD 因果容量 (causal capacity) 是绝对不变量. 这是 P2 §4.2 推导"运动下 cells 缩小"的基础元素.

关于"4DD 因果容量"的术语澄清: 这里的"4DD 因果容量"指 cell 在 4DD 层面承载 1 bit 的结构性承诺, 按 Planck 绝对性加信息四不得不第一条所推出, 不论 cell 处在何种条件 (静止, 运动, 引力场内, 甚至极端情形) 下都成立. 这是 SAE 框架的本体论承诺, 不是关于 4DD 层是否在某个具体情形下"运行活跃"的运行层 (operational) 主张. 容量 (capacity) 与活动状态 (activity) 是两件不同的事:

• 结构性容量: 每 cell 1 bit, 由真先验加 Planck 绝对性所给出, 普适不变
• 运行活动: 4DD readout 层在某情形下是否实际进行因果闭合与信息涌现, 取决于具体物理状态

P2 通篇所讨论的运动加引力情形都在 4DD 运行活跃的区段内 (cells 在 Planck 下界之上, 因果槽结构良定义, 4DD 闭合可发生). 极端情形下 (cells 接近 Planck 下界, 4DD 运行可能不再活跃) 在 P2 范围之外, 留作未来工作.

本节后续的"4DD 超体积守恒" (§4.2 前提三) 是结构性容量层面的不变性陈述, 不是关于 4DD 运行活动是否进行的主张.

§3.5 P2 调用的非 SAE 内部元素 (诚实清单)

P2 在 SAE 真先验加 P1 框架元素之上, 还显式继承一组标准物理 (SR / GR / Mass-Conv 等) 已确立的元素. 本节显式列出, 让 §4 推导链调用它们时不突兀.

元素	来源	在 P2 调用位置	状态
运动侧 $\gamma$ tick 膨胀	标准 SR 运动学	§4.2 前提二	P2 采用 / 从标准 SR 恢复, 在 SAE 因果槽语言里重表述. P2 不主张从真先验独立推导
Lorentz 4D 体积不变性 ($dx \cdot dy \cdot dz \cdot dt$ 不变量)	标准 SR Lorentz 变换 (Jacobian = 1)	§4.2 前提三	Lorentz 继承加 SAE 本体论重表述: 数学内容跟标准 SR 一致, SAE 加上 cell 1 bit 容量在时空中的本体论解读
二次闭合方程 $E^2 = p^2c^2 + m^2c^4$	Mass-Conv §3.5 (二次闭合区段)	§5.5, 含 $E = \gamma mc^2$ 恢复	Mass-Conv 继承 (P2 系列内部 sibling), $d_\text{eff} = 2$ 区段形式
现代标准 4-velocity 表述 $u^\mu = \gamma(c, \vec{v})$, $p^\mu = m_0 u^\mu$	现代标准物理 (post-Tolman 1930s 相对论质量废弃)	§4.3-§4.5 align with	标准物理已表述, P2 提供因果槽计数本体论 parallel, 不修正
标准 SR 时间膨胀 $1/\gamma$ 加长度收缩 $L_0/\gamma$ 加 Lorentz 变换加光速不变	标准 SR	§5 全节	P2 在因果槽几何语言里恢复, 不重新从真先验推导
标准 GR 等效原理局部内容加静态强场有效惯性加静态球对称 (Schwarzschild) 几何	标准 GR	§6 全节	P2 在因果槽几何语言里恢复, 不重新推导

这意味着: P2 在纯 SR 加纯 GR 静态强场两个区段内做的是 本体论解读加词汇统一, 标准物理预测在这两个区段内被恢复, 不被修改.

P2 真正的 SAE 内部新表述是: 4DD 因果容量守恒推导 (§4.2 Step 3) 加因果槽计数本体论解读 (§4.3-§4.6) 加结合区段结构性主张 (§4.7). 拓展部分 (§7) 表述的引力广播跨论文继承经验形式 (P1 §3.5 起源加信息论 P4 延伸), 不计入主线原创内容.

---

§4 推导链: 因果槽几何与有效惯性 (主线)

§4.1 链概览 (八步, 各带状态)

Step	内容	起因 / 状态
1	物理空间 3D	P1 继承 (现有 SAE 承诺)
2	质量由离散 cells 构成	P1 继承 (现有 SAE 承诺)
3	运动让 cells 沿运动方向缩小 $1/\gamma$	P2 SAE 内部推导 (4DD 因果容量守恒)
4	$v_\text{eff} = \gamma v$ 沿运动方向 (cell 计数)	P2 SAE 特有陈述; 与标准 4-velocity 一致
5	推动通过缩小 cells, 需要更多动量	P2 显式计算命题
6	方向不对称 ($\gamma^3$ 跟 $\gamma$) 的本体论根源	P2 本体论解读 of 标准 SR 结果
7	引力区 cells 局部沿径向缩小	P1 继承 (P2 中各向异性澄清)
8	结合区段: 运动加引力共同推 $\delta_4^\text{eff}$	P2 新结构承诺, 把 P1 框架扩展到运动

§4.2 Step 3: 运动让 cells 沿运动方向缩小——4DD 因果容量守恒推导

为什么不能用信号追逐论证:

天真的推导可能这样说: "物体相对 Planck 基底运动, 信号穿过运动 cell 要追赶, 所以 cell 看似被拉长". 这种推导:

第一, 是已经预设了 Lorentz 收缩的循环论证 (信号追逐分析本身已经预设洛伦兹收缩);

第二, 会导致光速各向异性 (信号同向跟反向给出不同的穿越时间), 直接跟 Michelson-Morley 实验结果矛盾;

第三, 跟经典 19 世纪以太理论同方向, 根本上跟 SAE 两层结构不兼容.

P2 不用信号追逐论证. 用 SAE 内部的 4DD 因果容量守恒.

SAE 内部推导——4DD 因果容量守恒:

前提一 (P1 继承): 每个因果槽 (cell) 在 4DD 层面封装 1 bit 信息. 这是 SAE 真先验 (信息四不得不第一条) 加 Planck 绝对性的直接推论. 每个 cell 的 4DD 因果容量是绝对不变量.

前提二 (P2 采用 / 从标准 SR 恢复, 在 SAE cell 语言里重表述, 见 §3.5 清单): 物体相对 Planck 基底运动速度 $v$ 时, tick 按 $\gamma$ 膨胀.

具体: P1 §4.5 提供了 tick = $R/c$ 的语言, P1 §4.9 提供了 $d\tau/dt = R(r)/R_\infty$ 的事件计数协变性 (引力情况下推导出). 但运动侧的 $\gamma = 1/\sqrt{1-v^2/c^2}$ 关系不是从 P1 真先验独立推导出来的. 它是标准 SR 的运动学结果, P2 采用或恢复, 然后在 cell 语言里重表述: 物体的 tick 从 $R_0/c$ 变到 $\gamma R_0/c$, 以绝对 Planck 单位计.

P2 不重新推导运动 tick 膨胀来自第一原理; 接受作为 P1 上游语言加标准 SR Lorentz 继承, 在 cell 层做出本体论重表述.

前提三 (4DD 超体积守恒, Lorentz 体积不变性加 SAE 本体论重表述, 见 §3.5 清单): cell 在 Planck 基底参考系中的 4DD 时空超体积 (空间体积 $dx \cdot dy \cdot dz$ 乘以时间 $dt$) 是不变量, 以绝对 Planck 单位计.

具体身份: 这是标准 SR Lorentz 体积元素不变性的 SAE 本体论重表述. 数学内容跟 Lorentz 变换的 Jacobian = 1 一致 (4D 体积元素在 Lorentz 变换下不变), 加 SAE 特有的本体论解读 (cell 携带 1 bit 的绝对容量在时空中的具体显现). 不是来自 SAE 真先验的新数学定理; 是 SAE 框架对标准 Lorentz 体积不变性的本体论重表述.

具体内容: cell 的 4DD 超体积是 cell 携带 1 bit 的绝对容量 (在时空中). 与参考系无关 (Lorentz 不变). 不论物体如何运动, cell 的 4DD 超体积保持 $R_0^3 \cdot t_P$ (以绝对 Planck 单位计, 或某 cell 特有常数).

结论 (推出的后果):

如果 $dt \to \gamma dt$ (按前提二), 那么 $dx \to dx/\gamma$ 沿运动方向 (按前提三).

具体代数:

• 静止 cell 的 4DD 超体积: $R_0 \cdot R_0 \cdot R_0 \cdot t_P = R_0^3 \cdot t_P$
• 运动 cell 的 4DD 超体积: $R_\| \cdot R_0 \cdot R_0 \cdot \gamma t_P$ 必须等于同一常数
• 解出: $R_\| = R_0/\gamma$

沿运动方向 cell 缩小到 $1/\gamma$.

垂直方向: 物体不沿垂直方向运动, 没有信息流位移沿垂直方向, $R_\perp$ 不参与几何代偿. 垂直方向 cells 不变 ($R_0$).

状态: SAE 内部推导, 通过 4DD 超体积不变性 (Lorentz 继承加 SAE 本体论重表述, 按前提三) 加 tick 膨胀 (标准 SR 采用, 按前提二), 推出后果. 不是从经典信号追逐推导 (那是预设 Lorentz, 循环的). 是 SAE 两层结构加 4DD 超体积不变性的结构后果: 给定继承的 Lorentz 不变性加 1 bit/cell 的真先验承诺 (前提一), cell 沿运动方向缩小到 $1/\gamma$ 是结构性推出.

(关于"绝对 Planck 单位"的措辞跟 preferred frame 的关系, 见 §5.3 末尾澄清.)

与标准 SR 洛伦兹收缩的关系:

这个推导恢复了洛伦兹收缩 $L = L_0/\gamma$ 沿运动方向的几何形式. 标准 SR 把洛伦兹收缩表述为参考系相对的 (各惯性参考系都看对方收缩了). SAE 表述的是: cells "在绝对 Planck 单位下真的缩小" (按前提三的超体积不变性, 以本体论 substrate level 计, 见 §5.3).

这是 SAE 两层结构 (Planck 基底跟因果槽) 让 cells "真的在绝对意义上缩小"成为可陈述的立场. 单层物理学 (SR 没有 Planck 基底层) 没法做出此陈述, 只能讲参考系相对收缩.

关键诚实措辞:

P2 不主张这个 cell 缩小推导给出标准 SR 预测的替代内容. 给出的是同样的预测 (洛伦兹收缩形式), 配上不同的本体论解读.

标准 SR 的参考系相对解读加 SAE 的绝对 Planck 解读, 都给出相同的可观察预测 (因为因果槽层的参考系间相对性按 §4.4 表述仍然成立). 区别在本体论, 不在可观察.

§4.3 Step 4: $v_\text{eff} = \gamma v$——cell 计数陈述

Cell 计数图像:

物体在 Planck 基底里以速度 $v$ 运动 (按绝对 Planck 单位计).

但物体内部所有物理过程都基于因果槽结构. 在物体自己的参考系里, 各过程按 cell 计算花标准时间. 物体看自己的运动, 在自身参考系里是正常速度.

外部 Planck 参考系观察: 物体的 cells 沿运动方向缩小到 $R_0/\gamma$. 物体走过同样的绝对 Planck 距离, 在同样的坐标时间内, 需要穿越更多 cells (每绝对 Planck 长度对应更多 cells).

这就是"有效速度增大":

物体的有效速度, 按它穿过的 cells 数来度量, 比静止参考系里相同绝对物理速度多出一个 $\gamma$ 因子.

形式: $v_\text{eff} = \gamma v$ 沿运动方向.

与标准 4-velocity 的关系:

$\gamma$ 因子的归宿是 4-velocity. 现代标准物理早就表述: $u^\mu = \gamma(c, \vec{v})$, $p^\mu = m_0 u^\mu$. $\gamma$ 不在 mass 里 (相对论质量是废弃概念).

P2 提供的 $v_\text{eff} = \gamma v$ 跟标准 4-velocity 表述一致, 不是修正. P2 加上 cell 计数的本体论根源 (cell 缩小, 每绝对距离更多 cells), 给标准的 4-velocity 表述加上一幅本体论图像.

具体: SAE 提供 $v_\text{eff} = \gamma v$ 的本体论根源 (cell 缩小, 每绝对距离更多 cells). 标准 SR 提供代数框架 (4-velocity 变换规则). 两者一致, SAE 加本体论深度.

§4.4 Step 5: 推动通过缩小 cells 需要更多动量 (显式计算命题)

关于本节地位的措辞澄清 (前置): §4.4.A 与 §4.4.B 的代数推导是标准 SR 结果在因果槽几何词汇里的恢复, 不是 P2 新推导. §4.4.C 的本体论解读是 P2 特有的解读叙述, 跟标准 SR 数学结果 parallel, 不是数学上 forced 的分解.

命题 §4.4.A——纵向 $\gamma^3$:

动量 $p_\| = m_0 \gamma v$ (沿运动方向).

加速 $\Delta v_\|$ 所需的 $\Delta p_\|$:

$$\frac{dp_\|}{dv_\|} = m_0 \frac{d(\gamma v)}{dv}$$

具体计算:

$$\frac{d\gamma}{dv} = \frac{d}{dv}\left(1 - v^2/c^2\right)^{-1/2} = \gamma^3 \cdot \frac{v}{c^2}$$

$$\frac{d(\gamma v)}{dv} = \gamma + v \cdot \gamma^3 \frac{v}{c^2} = \gamma + \gamma^3 \frac{v^2}{c^2} = \gamma\left(1 + \gamma^2 \frac{v^2}{c^2}\right)$$

利用恒等式 $\gamma^2(1 - v^2/c^2) = 1$, 即 $\gamma^2 v^2/c^2 = \gamma^2 - 1$:

$$\frac{d(\gamma v)}{dv} = \gamma(1 + \gamma^2 - 1) = \gamma \cdot \gamma^2 = \gamma^3$$

因此 $\Delta p_\| = \gamma^3 m_0 \Delta v_\|$. ✓

命题 §4.4.B——横向 $\gamma$:

横向加速 $\Delta v_\perp$ 不改变 $|v|$ (只改方向), 故 $\gamma$ 不变.

$$p_\perp = \gamma m_0 v_\perp \quad (\gamma \text{ 在横向加速中不变})$$

$$\frac{dp_\perp}{dv_\perp} = \gamma m_0$$

因此 $\Delta p_\perp = \gamma m_0 \Delta v_\perp$. ✓

命题 §4.4.C——方向不对称的本体论根源 (P2 解读叙述):

纵向 $\gamma^3$ 跟横向 $\gamma$ 之间的差因子 $\gamma^2$, 数学上来自 $\frac{d\gamma}{dv}$ 在纵向方向非零, 在横向方向为零.

标准 SR 解读: 纯粹是 $\gamma$ 函数结构的运动学结果.

SAE 本体论解读 (P2 特有叙述, 跟标准 SR 数学 parallel):

• 纵向加速引发的是缩小的 cells (沿运动方向 cells $R_0/\gamma$). 推动通过这些更密的 cells 需要 $\gamma$ 放大, 按 cell 计数图像. $\gamma$ 自身随 $v$ 变, 通过链式法则贡献额外 $\gamma^2$ 因子. 总: $\gamma$ 乘 $\gamma^2$ 等于 $\gamma^3$.
• 横向加速引发的是正常 cells ($R_0$, 垂直方向不变). 没有在整体 $\gamma$ (来自时间膨胀) 之上的额外 $\gamma$ 放大. 总: $\gamma$.

纪律:

• 命题 §4.4.A 与 §4.4.B 是标准 SR 结果, 在因果槽几何词汇里恢复, 不是 P2 新推导
• 命题 §4.4.C 的本体论解读是 P2 SAE 特有的解读叙述
• 数学上 $d(\gamma v)/dv = \gamma^3$ 直接来自链式法则, 不自然地分解为 "$\gamma$ 加 $\gamma^2$". 上文把第一个 $\gamma$ 归因于 cell 几何缩小, 第二个 $\gamma^2$ 归因于链式法则贡献, 是与标准推导 parallel 的解读叙述, 不是数学上 forced 的分解

§4.5 Step 6: 方向不对称——cell 几何确认

§4.4 已具体陈述方向不对称的 SAE 本体论解读. 本节回顾关键洞察:

质量 $m_0$ 从未变.

• $m_0$ 等于 cells 数量, 是 Lorentz 不变量
• 不论参考系, cells 数量不变
• 引力源强度只跟 $m_0$ 相关 (按 Paper 0 reading 机制: 信息波 emission 由 cells 数决定, 与参考系无关)
• 这与标准物理的现代视角 (相对论质量已废弃) 一致

变的是 $v_\text{eff}$ 放大.

• Cells 沿运动方向缩小, 同样绝对距离对应更多 cells
• $v_\text{eff} = \gamma v$ 沿运动方向, 等于 cells 每时间的速度
• 推动通过更多 cells, 同样的自参考系 $\Delta v$, 需要更多绝对 Planck 参考系动量
• 这给"加速更难"的操作现象提供了本体论根基

Cells 在垂直方向不变, $v_\text{eff} = v_\perp$ 在垂直方向不变, 只有整体 $\gamma$ (来自时间膨胀) 起作用.

这是 SAE 对"方向性有效质量"的解读. 标准 SR 把 $\gamma^3$ 跟 $\gamma$ 当作 4-velocity 公式的导数 artifact. SAE 把它解读为 cell 各向异性的直接反映.

两种解读, 同样的预测, 不同的本体论.

§4.6 Step 7: 引力区 cells 的本体论解读

纪律: P2 不推导出标准 GR 强场有效惯性的定量公式. P2 提供的是标准 GR 结果的 SAE 本体论解读, 把已知的 GR 现象用因果槽几何词汇解读, 不是从 SAE 真先验独立推导出 GR 的具体定量预测.

SAE 本体论解读:

P1 §4.4 已表述: 强场区 $R(r) \to l_P$. Cell 缩小是引力势引起的.

但在局部, 引力有方向 (指向质量中心). 静态球对称质量源的全局对称性是球对称的, 但任意具体点的局部 cell 缩短是径向的.

各向异性澄清:

• 全局: 球对称质量源的引力场全局各向同性 (按球坐标)
• 局部: 任意点上, 引力有方向 (径向). Cell 缩短在局部是径向的.
• 不同点: 不同点的"径向"方向不同. 全局各向同性, 局部有方向.

跟运动情况的对比:

• 引力在局部有方向 (径向, 取决于质量位置)
• 运动在局部有方向 (运动速度矢量方向)
• 两者在局部都是有方向的, 不是"各向同性跟各向异性"二分

SR 与 GR 方向性的本体论关系:

• 不是"引力各向同性跟运动各向异性"
• 而是"两者在局部都有方向, 但方向来源不同" (引力 pull 跟运动矢量)
• 两者都给出有局部方向的 cell 缩短. Cell 缩短机制是共同的; 方向来源不同.

SAE 对 GR 强场有效惯性的本体论解读:

标准 GR 表述: 强场区物体的有效惯性增大 (具体跟 $g_{\mu\nu}$ 结构相关). 标准物理把它当作坐标 artifact.

SAE 提供本体论解读: cells 在局部沿引力方向缩小; 推动通过这些更密的 cells 需要放大的动量 (跟运动情况同一机制).

关键纪律:

• SAE 不主张推导出标准 GR 的具体公式
• SAE 只主张提供标准 GR 结果的本体论解读
• 标准 GR 基于坐标的形式计算保持有效; SAE 的因果槽几何词汇提供 parallel 的本体论描述

§4.7 Step 8: 结合区段——运动加引力叠加 (主线锁定结构性存在; 概念延伸见 §7.1)

纪律: 主线层只锁定结构性存在主张. 具体量级全部留给 P3. 极速等于人工视界加因果降维 (子夏的洞察) 是结合区段的概念延伸, 在 §7.1 (拓展) 完整表述. 此处仅标记它作为 Step 8 的具体含义, 不重复展开.

结构性存在主张:

Cells 能被两种 effects 缩小:

• 运动: 沿运动方向, cells $R_0/\gamma$
• 引力: 沿径向, cells 随 $\sqrt{\delta_4^\text{grav}}$ 缩小 (按 P1 表述)

结合 (引力场中的运动):

• 沿运动方向 (假设跟引力径向对齐): cells 同时被运动收缩与引力收缩
• 沿与运动垂直的方向 (但仍含引力径向贡献, 如适用): 仅引力收缩
• 沿其他方向: 取决于具体几何

关键定性主张:

如果运动速度加引力势的结合让 $\delta_4^\text{eff}$ (有效结合闭合赤字) 进入 $d_\text{eff} > 2$ 区域, 那么闭合方程偏离标准的二次 Einstein 形式 ($E^2 = p^2c^2 + m^2c^4$), 给出可测偏离 SR×GR 简单乘法组合的结果.

$d_\text{eff}(\delta_4^\text{eff}, v)$ 的具体函数形式留待 P3 表述. P2 不预承诺具体缩放, 不预承诺结合区段的具体偏离量级.

(结合缩放的启发性陈述移到附录 A 作直觉, 不是正式预测.)

§4.8 链总结

Step	内容	状态 (诚实措辞)
1-2	物理空间 3D, 质量由 cells 构成	P1 继承
3	运动让 cells 沿运动方向缩小到 $1/\gamma$	P2 SAE 内部表述: 给定 Lorentz 继承加真先验 1 bit/cell, 结构性推出
4	$v_\text{eff} = \gamma v$ 沿运动方向	P2 cell 计数本体论陈述, 与标准 4-velocity 一致
5	$\gamma^3$ 纵向, $\gamma$ 横向	标准 SR 结果, 显式计算恢复
6	方向不对称的本体论根源	P2 解读叙述, 跟标准 SR 数学 parallel
7	引力区 cells 局部沿径向缩小	P1 继承, 各向异性已澄清
8	结合区段 $\delta_4^\text{eff}$ 推 $d_\text{eff} > 2$	P2 结构性存在主张, 量级待 P3

整链推导状态: 标准 SR 与 GR 预测在纯区段内恢复, 不是 P2 新推导. SAE 特有贡献: 4DD 因果容量守恒推导 (Step 3), cell 计数本体论解读 (Steps 4-6), 结合区段结构性主张 (Step 8). $v_\text{eff}$ 与标准 4-velocity 表述一致; SAE 提供标准没有的本体论深度.

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§5 SR 预测的恢复 (主线)

(简短. 主要确认 SAE 解读在纯 SR 区段内能恢复 SR 的可观察预测. 不是 P2 新推导, 而是把标准 SR 结果用 SAE 词汇表述出来.)

§5.1 光速不变

SAE 解读: 光波是 2DD 主动波, 不携带信息 (信息是 4DD 内容). 光波在 Planck 基底层传播, $c = l_P/t_P$ 普适.

关键洞察: 因为光波不携带信息, 不引发依赖观察者的因果槽层 (因果槽层承载的是含信息的动力学). 光波直接在 Planck 基底层传播, 速度对所有观察者不变.

这与下面对比:

• 信息波 (4DD): 也在 Planck 基底层传播 (按 P1 §3.5 信息波绝对性), $c$ 不变
• 因果槽层内的波 (例如声波): 速度依介质和观察者

两层结构让此表述干净: 经典以太理论失败, 因为只有一层, 观察者在以太里测应见优先方向, 但 Michelson-Morley 否定了. SAE 两层: Planck 基底层绝对 (从内部测不出优先方向), 因果槽层相对.

不是公理 (按 Einstein 1905), 而是 2DD 本体论加 Planck 绝对性所推出的后果.

§5.2 时间膨胀 $1/\gamma$

诚实措辞: 运动时间膨胀 ($d\tau/dt = 1/\gamma$) 与运动下 cell 缩短 ($R_\| = R_0/\gamma$) 在 SAE 框架内构成 结构性互联对: 给定 Lorentz 继承的 4DD 超体积不变性 (§4.2 前提三), 一者推出另一者. P2 不主张独立从 §3 真先验推导其中任一; 提供两者之间的结构连接在 SAE 框架内的表述. 两者都从标准 SR Lorentz 不变性继承, 通过 P1 §4.9 事件计数协变性.

具体: §4.2 已通过"tick 膨胀 (前提二, 从 SR 采用) 加 4DD 超体积守恒 (前提三, Lorentz 继承)"推出 cell 缩短. 反过来, cell 缩短加 4DD 超体积守恒也直接推出 tick 膨胀. 这是同一对结构性关系的两面, 不是各自独立的推导.

SAE 的贡献: 把这种互联用因果槽几何词汇表述出来——cells 沿运动方向缩短 $R_0/\gamma$, tick 膨胀 $\gamma R_0/c$, 两者通过 4DD 超体积守恒互相推出. 标准 SR 用 Lorentz 变换表述同一互联 (在不同词汇里).

观察结果: $d\tau/dt = (R_0/c) / (\gamma R_0/c) = 1/\gamma$. 恢复标准 SR 的运动时间膨胀; SAE 提供因果槽几何视角与标准 SR 的 4-velocity 表述并存.

§5.3 长度收缩 $L_0/\gamma$

运动物体的 cells 沿运动方向 $R_0/\gamma$, 以绝对 Planck 单位计 (按 §4.2).

物体由 $N$ 个 cells 构成, 沿运动方向长度等于 $N \cdot R_0/\gamma = L_0/\gamma$, 以绝对 Planck 单位计.

外部观察者测量 (用自己的标尺, 假设跟 Planck 基底相对静止): 测得收缩长度 $L_0/\gamma$.

物体自己参考系里测量 (用自己的尺, 也是缩小的 cells): 测得标准 $L_0$ (因为测量仪器同步调整, 按 §11.3 表述).

关于"Planck 参考系"的措辞澄清: 这里的"跟 Planck 基底相对静止的观察者"不是指 SAE 主张存在一个经验上 preferred 惯性参考系. Planck 基底是本体论 substrate level (每个 cell 都坐落在 Planck 基底之上), 不是经验上可区分的 preferred frame. 所有惯性观察者用 Lorentz 变换等价处理; SAE 的"Planck 层"是本体论结构, 不是经验测量上的 distinguished frame. 这里使用"跟 Planck 基底相对静止"只是为了描述 cell 缩短的绝对 Planck 单位下的表述——任何惯性观察者都能表述同样的物理.

恢复标准 SR 的长度收缩, 配以两层结构的本体论根基.

§5.4 Lorentz 变换

因果槽参考系之间, 保持光速不变 (按 §5.1) 的自然变换规则.

具体: 不同惯性观察者有不同的因果槽局部结构 (依各自相对 Planck 基底的运动状态). 跨参考系坐标变换必须保持光速不变 (因为光速是 Planck 基底层属性, 投射到所有观察者上不变).

唯一保持此不变性的线性变换是 Lorentz 变换. SAE 接受标准 SR 的形式 Lorentz 变换, 作为因果槽参考系之间的自然规则.

不是公理 (跟 Einstein 1905 同形式, 不同本体论根基).

§5.5 $E = \gamma mc^2$

Mass-Conv §3.5 的二次闭合 $E^2 = p^2c^2 + m^2c^4$ 在弱场加低速区段 ($d_\text{eff} = 2$) 成立.

运动参考系里:

• $p = \gamma m_0 v$ (按 §4.4.A 显式计算)
• $E^2 = (\gamma m_0 v)^2 c^2 + m_0^2 c^4 = m_0^2 c^4 (\gamma^2 v^2/c^2 + 1) = m_0^2 c^4 \gamma^2$
• $E = \gamma m_0 c^2$

具体: $\gamma$ 来自 $p$ 公式里 $v_\text{eff} = \gamma v$ 放大 (按 §4.3 cell 计数图像), 不是质量变化. 跟标准 SR 表述一致 (mass 等于 $m_0$ Lorentz 不变量).

状态: 标准 SR 结果的恢复, 不是 P2 新推导. SAE 提供 cell 计数本体论解读, 给出 $\gamma$ 因子的本体论归宿.

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§6 GR 有效惯性的恢复 (主线)

§6.1 强场加速跟弱场加速——SAE 解读下的等效原理

GR 等效原理: 在局部, 强场区观察者跟弱场加速观察者经历相同物理.

SAE 解读陈述:

• 强场区静止观察者: cells 在局部沿引力径向缩小 (按 §4.6)
• 弱场惯性参考系加速观察者: cells 在局部沿加速方向缩小 (按 §4.2 运动情况)
• 在局部, 类似的 cell 缩短模式, 带局部方向
• 局部等价在因果槽层成立

具体:

• 在局部参考系里, 加速一个质量 (强场跟弱场加速) 需要相同的力, 按局部测量
• 质量等于 $m_0$ (Lorentz 不变量), 不依赖引力势
• 跟标准等效原理解读一致

SAE 不推导出新的 GR 强场公式. 提供的是本体论解读 (cell 缩短机制跟运动情况共享).

§6.2 坐标参考系视角的有效惯性

远场观察者 (在弱引力区) 看强场内的加速过程:

• $d\tau/dt = \sqrt{\delta_4} < 1$ 引力时间膨胀 (按 P1)
• 局部加速 $\Delta v_\text{local}$ 对应坐标 $\Delta v_\text{coord} = \sqrt{\delta_4} \Delta v_\text{local}$
• 远场观察者看似强场内的加速过程, 需要更多的坐标动量来产生少量的坐标速度变化

标准 GR 解读: 依赖坐标参考系的 artifact. 质量 $m_0$ Lorentz 不变量.

SAE 解读: cells 在强场区真的缩小, 以绝对 Planck 单位计 (按 P1 §4.4 表述加 §4.6 各向异性澄清, 加 §5.3 末尾对"绝对 Planck 单位"的本体论 substrate 澄清). 推动通过这些更密的 cells, 在绝对 Planck 单位下需要放大的动量, 跟运动情况同一机制.

§6.3 SAE 内部的 SR 与 GR 本体论统一

SR 有效质量: 运动让 cells 在局部沿运动方向缩小, $v_\text{eff}$ 放大, 推动需要放大的动量.

GR 强场有效惯性: 引力让 cells 在局部沿径向缩小, $v_\text{eff}$ 放大, 推动需要放大的动量.

同一机制类: 因果槽几何修改, 给出有效速度放大, 给出动量代价增大.

不同几何实现: cell 缩短的方向, 一个来自运动矢量, 一个来自引力 pull 方向.

两者在局部都有方向. 不是各向同性跟各向异性的二分 (按 §4.6 澄清).

状态: 这是 SAE 内部框架内的词汇统一. SR 与 GR 的预测在纯区段内都不变. SAE 提供的是组织一致性: 两个看似分开的有效惯性现象, 在 SAE 解读下共享共同的本体论机制.

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第二部分 主线 (续)

§8 逐项 claim 状态表 (主线)

Claim	状态
时间-信息四不得不加桥接公理	真先验 (P1 继承)
Planck 绝对加 DD 层级	真先验加 Lorentz 式承诺 (P1)
两层结构 (Planck 基底跟因果槽)	P1 已表述, P2 调用
因果槽尺寸 $R$ 依条件而变	P1 继承
4DD 因果容量不变 (1 bit/cell), 结构性容量, 非运行活动	P1 继承 (结构性承诺), P2 显式调用
运动侧 $\gamma$ tick 膨胀	标准 SR 采用 / 恢复 (见 §3.5 清单), 在 SAE cell 语言里重表述
4DD 超体积不变性	Lorentz 体积不变性继承加 SAE 本体论重表述 (见 §3.5 清单)
引力让 cells 局部缩小 (径向方向)	P1 已表述, 各向异性在 P2 中已澄清
运动让 cells 沿运动方向缩小到 $1/\gamma$ (主线核心)	P2 SAE 内部表述: 给定 Lorentz 继承加真先验 1 bit/cell, 结构性推出
$v_\text{eff} = \gamma v$ 沿运动方向 (主线核心)	SAE 内部本体论陈述; 与标准 4-velocity $u^\mu = \gamma(c, \vec{v})$ 一致
质量 $m_0$ 不变 (cells 数量不变)	标准物理加 SAE 继承
$\gamma^3$ 跟 $\gamma$ 方向不对称 (主线核心)	标准 SR 结果, 通过显式计算恢复
Cell 各向异性作为方向不对称的本体论根源 (主线核心解读叙述)	P2 特有本体论解读 parallel to 标准 SR 数学
运动时间膨胀与 cell 缩短互为推出	通过 4DD 超体积守恒, 结构性互联对 (非独立推导)
GR 强场有效惯性 (主线核心)	标准 GR; SAE 提供本体论解读, 通过 cell 缩短 (径向方向)
SAE 内部的 SR 与 GR 词汇统一 (主线核心)	P2 组织一致性主张, 不是新的物理统一
等效原理的 SAE 解读 (主线核心)	P2 表述 of 标准等效原理
引力广播在 Planck 基底层传播, 跨越任何质量聚合 (含黑洞内部) 不衰减 (拓展)	P1 §3.5 起源加 P2 与信息论 P4 延伸 (跨论文继承经验语境, 不计入 P2 自身原创结果)
引力波穿过黑洞内部 (推论, 拓展)	继承 claim 的强制推论, 跟"引力波一旦进入视界永不出来"标准 GR 区分
引力波不被或显著抑制被质量分布透镜化 (推论, 拓展)	继承 claim 的较弱推论, 直接多 messenger 测试
结合区段: $\delta_4^\text{eff}$ 进入 $d_\text{eff} > 2$ (拓展)	P2 新结构承诺 (把 P1 框架扩展到运动, 概念延伸待 P3 量化)
结合区段区分性预测 (结构性存在, 拓展)	P2 新 (qualitative existence, 量化待 P3)
结合区段具体偏离量级	物理后验范围 (P3 future)
极速等于人工视界, 本体论同构 (拓展)	P2 概念洞察 (子夏表述)
因果降维 (纵向 freeze 加横向保留, 跟黑洞视界维度同构) (拓展)	P2 概念洞察 (子夏延展)

状态表纪律说明:

• "恢复 / 采用" 等于 标准 SR/GR 结果通过 SAE 词汇陈述, 不是 P2 推导
• "Lorentz 继承加 SAE 重表述" 等于 数学内容跟标准 SR 一致, SAE 提供本体论解读
• "P2 新 (解读)" 等于 SAE 特有的本体论解读, 跟标准物理数学结果 parallel, 不替代
• "P2 新 (结构)" 等于 SAE 框架内做出的新结构承诺 (例如把 $d_\text{eff}$ 扩展到运动)
• "跨论文继承经验语境" 等于 起源在其他论文, P2 在相对论系列内表述可证伪经验形式, 不计入 P2 自身原创结果
• "组织一致性" 等于 SAE 内部词汇统一, 不是物理统一
• "概念延伸" 等于 结构性主张待量化, 未来论文表述

层标记: claim 后标 (拓展) 的属于第三部分 §7 / §10 表述范围, 不计入 P2 主线核心交付.

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§9 真先验跟后验范围 (主线)

结构推出 (在 SAE 结构承诺链内, 主线):

• 因果槽几何决定有效惯性
• 运动方向上 cells 各向异性缩小 (按 4DD 超体积不变性加 tick 膨胀)
• 引力区 cells 局部沿径向缩小 (按 P1 继承)
• $v_\text{eff} = \gamma v$ 沿运动方向 (cell 计数图像, sub-formal SR 等价)
• SR 与 GR 在因果槽几何下的本体论统一 (在 SAE 框架内)

结构推出 (拓展层):

• 结合区段的结构性存在 (按因果槽几何结合 effects)
• 引力广播在 Planck 基底层 transparent across 质量聚合 (P1 §3.5 起源加信息论 P4 黑洞内部延伸; 不是 P2 自身推导, 是 P2 表述的继承 claim)

结构不主张:

• 具体偏离量级
• $d_\text{eff}(\delta_4^\text{eff}, v)$ 的具体函数形式
• 结合区段偏离的具体数值
• 修正纯 SR 与 GR 静态强场区段的可观察预测
• 替代标准物理 4-velocity 与 metric tensor 形式框架

状态: 跟 P1 同纪律层次. 哲学先验加结构承诺加桥接 identification 共同推出结构形式; 物理后验填具体函数内容. 在引力波传播区段, P1 §3.5 起源加信息论 P4 延伸给出可证伪经验形式 (作跨论文继承经验语境在 P2 表述), 待观测.

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§12 结论 (主线)

P2 核心交付 (主线):

• SR 有效质量的 SAE 本体论解读: 通过 4DD 因果容量守恒推导, 运动下 cells 沿运动方向被强制缩小到 $1/\gamma$. 标准 SR 结果 (沿运动方向 $\gamma^3$, 垂直方向 $\gamma$) 通过显式计算恢复. P2 提供 cell 计数本体论图像, 标准 SR 形式框架保持.

• GR 强场有效惯性的 SAE 本体论解读: 通过 cells 在局部沿径向缩小来表述. 不是新推导标准 GR 结果, 而是给标准结果加上本体论深度.

• SAE 内部的 SR 与 GR 词汇统一: 两者都通过因果槽几何机制 (同一机制类). 几何来源不同 (引力 pull 跟运动矢量). 这是 SAE 内部组织一致性.

• $\gamma$ 因子的本体论归宿: 通过 4DD 超体积不变性加 tick 膨胀这一推出的后果, $\gamma$ 自然住在 cell 计数速度里. 跟标准 4-velocity 表述一致, P2 提供 cell 计数图像作为本体论 parallel.

• 等效原理的 SAE 解读: "同一机制类, 不同几何实现". 不是引力等于加速; 而是两者都在局部引起 cell 几何变化. 局部等价仍成立, 加上更深的本体论根基.

P2 拓展内容 (详见第三部分):

• 跨论文继承经验语境 (引力波传播): P1 §3.5 起源加信息论 P4 §4.4 延伸的可证伪经验形式在相对论系列内表述. 不计入 P2 自身原创结果.
• 结合区段结构性偏离主张: $d_\text{eff} > 2$ 激活的可能性, 量化待 P3.
• 极速等于人工视界加因果降维 (子夏): 概念延伸, 量化模型待未来工作.

留待 P3+:

• $d_\text{eff}(\delta_4^\text{eff}, v)$ 的具体函数形式
• 结合区段具体偏离量级
• 极速等于人工视界的详细动力学陈述
• 后续论文 arc 见后续论文

P2 表述因果槽几何作为时空惯性现象底层本体论, 在 SAE 框架内. 主线在 SR 与 GR 静态强场区段内提供本体论解读, 与标准物理可观察预测保持一致. 拓展内容表述框架层跨论文继承 claim 与结合区段结构性主张, 一并展现给读者作框架层语境.

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第三部分 拓展

(本部分表述主线之外的框架层内容. 不计入 P2 主线核心交付; 跟主线分层组织, 让读者清楚每部分在文中的位置.)

§7 区分性预测与概念延伸 (拓展)

P2 拓展层的内容分两类:

(I) 跨论文继承经验语境 (引力波传播): 起源在 P1 §3.5, 延伸在信息论 P4 §4.4. 本文在相对论系列内表述其可证伪经验形式, 不计入 P2 自身原创结果. 在当前与近未来引力波天文学直接 testable. 详见 §7.0.

(II) 主线结构性主张的概念延伸: 结合区段 $d_\text{eff} > 2$ 激活 (P2 新结构承诺, 量级待 P3 量化), 以及极速等于人工视界加因果降维 (子夏的洞察, 概念延伸). 详见 §7.1.

下面分别表述.

§7.0 引力波传播跟标准 GR 的区分性可证伪预测 (跨论文继承经验语境)

Attribution 前置: 本节表述的预测的起源在 P1 §3.5 (引力波等于 Planck 基底广播), 延伸在信息论 P4 §4.4 (黑洞内部因果谱). P2 在相对论系列内表述其可证伪经验形式, 把更广的 SAE 相对论与信息论框架的引力波传播主张引入 P2 的区段图. 不计入 P2 自身原创结果.

P1 §3.5 表述: 引力波等于 Planck 基底层广播 (broadcast), 不依赖因果槽层质量扭曲. Planck 基底是绝对的 (按 §2.4 真先验承诺), 不被任何质量聚合影响, 包括黑洞内部. 信息论 P4 §4.4 (DOI 10.5281/zenodo.19880111) 在黑洞内部因果谱角度对此表述进一步具体化.

核心 claim (P1 起源加信息论 P4 延伸):

> 引力广播在 Planck 基底层传播, 跨越任何质量聚合 (包括黑洞事件视界) 不衰减.

自然推论:

第一, 引力波穿过黑洞内部: 引力波路径穿过黑洞视界, 在另一侧出来. 广播在 Planck 基底层传播, 视界对它不是 barrier. 跟标准 GR 的"引力波一旦进入视界永不出来"区分.

第二, 引力波不被透镜化或显著抑制: 引力波穿过中间质量分布 (galaxy clusters 等) 时, 不被透镜化 (或显著抑制). 因为如果黑洞 (最 extreme 的质量聚合) 都不能 block 引力广播, 弱质量分布当然也不能 lens 它. 这是核心 claim 的较弱推论.

第三, 引力波路径绝对: 不论中间质量怎么分布, 引力波走最短 Planck 基底路径, 不 deflect.

与标准 GR 的区分:

• 标准 GR: 引力波 follow null geodesics in spacetime metric. 受 metric distortion 影响, 跟光一样被质量分布 lens. 视界处 metric singular, 引力波单向, 视界穿越后完全吸收.
• SAE: 引力波在 Planck 基底层传播, 不依赖时空 metric. 质量聚合 (包括黑洞) 对引力波 transparent. 无透镜化, 无吸收, 无视界 barrier.

这是直接 testable 的区分性经验 claim about 引力波传播. 不是相同预测的本体论 reframing; 是实质上不同的经验主张. 作为跨论文继承 claim, 在相对论系列内通过 P2 表述.

具体的证伪路径加测试场景加经验 timeline 见 §10.0.

§7.1 结合区段的 $d_\text{eff} > 2$ 激活加极速等于人工视界加因果降维 (子夏) (主线概念延伸)

(本小节表述主线 §4.7 锁定的结合区段结构性主张的概念延伸. 跟 §7.0 不同, 这是主线 conjectural extension 待 P3 量化, 不是跨论文继承 claim.)

结合区段弱场加低速分别都给出 $d_\text{eff} \to 2$, 预测匹配 SR×GR 乘法组合.

但结合高速加中等引力 (或反过来) 可让 $\delta_4^\text{eff}$ 进入强场区域. $d_\text{eff} > 2$ 激活. 时间膨胀加有效质量加长度收缩偏离 SR×GR 简单乘法组合.

SAE 内部本体论洞察 (子夏):

在 SAE 框架下, 极速运动跟坠入黑洞在本体论上同构:

• 引力: 通过减小 $\delta_4$ (逼近 0), 压缩 cell 大小, 剥夺基底自由度
• 极速运动: 通过 $\gamma \to \infty$, 同样在沿运动方向压缩 cell 大小, 剥夺基底自由度
• 一个以极相对论速度飞行的粒子, 在自己运动方向上, 创建局部"人工视界": 自我引发的类强场状态

这暗示: 在弱场中以高速运动的粒子, 其前方因果槽密度因 $\gamma$ 压缩而急剧升高. 当 $\delta_4^\text{eff}$ 被压榨到临界点时, 系统同样会触发 $d_\text{eff} \to 3^-$ 的 bulk absorption phase transition (注意 $3^-$: 渐近接近但不达到).

进一步: 因果降维 (Dimensional Reduction)——子夏的拓扑延展

极速运动 ($v \to c$) 极限下, cell 几何展现一个具体的拓扑特征:

• 纵向 (沿运动方向): $R_\| = R_0/\gamma \to l_P$, 接近 Planck 下界, 纵向因果区分接近 freeze
• 横向 (垂直运动方向): $R_\perp = R_0$, 不变, 横向因果区分保留

粒子的 cell 拓扑从 3D 盒子接近 2D pancake configuration. 这跟黑洞视界处的 cell 几何形成 结构映射:

• 黑洞视界: 径向 cells 局部缩小加切向 cells (光球面附近) 保留, 径向因果信息流接近 frozen 加切向 photon orbits 仍可发生
• 极速运动: 纵向 cells 局部缩小加横向 cells 保留, 纵向因果信息流接近 frozen 加横向因果区分仍可发生

这是几何上具体的维度同构, 不仅是抽象类比. 标准 SR 里 high-$\gamma$ particle pancake 是 known kinematic feature (Lorentz contraction); SAE 提供本体论解读: pancake 是 cell 拓扑的 actual reduction in 信息论意义, 跟黑洞视界 cell 拓扑共享维度同构结构.

详细的黑洞视界 cell 拓扑表述见信息论 P4 §4.4 (DOI 10.5281/zenodo.19880111).

§7.2 候选测试场景 (拓展)

• 中子星附近高速绕轨的 pulsar 系统 (例如 PSR J0737-3039A/B, $v \sim 0.001c$, $\delta_4 \sim 0.85$)
• 极相对论性宇宙射线穿过太阳引力场 (反常色散或时延, 不能用 SR×GR 乘法简单解释)
• 黑洞吸积盘内区 (高轨道速度加强引力)
• 高能 γ 射线暴穿过中间星系

Scope 澄清 (Schwarzschild only): 上述测试场景假设黑洞是 Schwarzschild (静态球对称). Kerr (旋转, ergosphere 加 frame-dragging) 跟 Reissner-Nordström (带电) 黑洞的 cell 几何表述留作未来工作 (相对论系列 P6 Kerr 论文); 那些情形的 $\delta_4^\text{eff}$ 可能涉及额外结构 (transverse cell shifts due to frame-dragging 等) 不在 P2 范围内.

§7.3 测试路径 (拓展)

• SKA pulsar timing 到 0.001% 精度, 可能探测到偏离
• 未来高精度宇宙射线观测 (高 $\gamma$ 下的反常色散)
• LISA 对超大质量黑洞吸积盘动力学
• GRB 时延测量, 对中间星系

§7.4 P2 拓展层纪律

• §7.0 引力波传播预测作为跨论文继承 claim 在 P2 表述, 不计入 P2 自身原创结果
• §7.1-§7.3 结合区段内容仅指出结构性区分性预测的存在
• 具体偏离量级取决于 $d_\text{eff}(\delta_4^\text{eff}, v)$ 的具体函数形式 (物理后验范围, 待 P3 表述)
• 不预承诺具体偏离曲线或具体的定量预测
• "极速等于人工视界"洞察加因果降维提供概念深度, 不是已详细的动力学模型

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§10 证伪路线图 (拓展)

§10.0 引力波传播跟标准 GR 区分 (跨论文继承测试语境, 当前可证伪)

Attribution 前置: 本节表述的证伪路径, 是 §7.0 跨论文继承 claim (P1 §3.5 起源加信息论 P4 延伸) 的可测试经验形式. P2 在相对论系列内表述这条继承 claim 的可证伪内容, 直接在当前与近未来引力波天文学中 testable. 不计入 P2 自身原创经验内容.

引力广播在 Planck 基底层传播, 跨越任何质量聚合 (包括黑洞事件视界) 不衰减 (按 §7.0). 跟标准 GR 在引力波传播层面区分.

核心证伪条件:

条件一: 引力波在黑洞视界处的传输行为

• 标准 GR null hypothesis: 引力波信号在视界 crossing 后完全吸收, 无 post-horizon 信号
• SAE prediction: post-horizon 引力波信号可能; ringdown spectra 包含 interior-mode 贡献
• 证伪条件: 高 SNR 多事件 stacked observations 稳定确认 ringdown spectra 完全 fit horizon QNM, 无 interior 贡献, SAE 核心 claim 受质疑

条件二: 引力波跟电磁波在宇宙学透镜的对比

• 标准 GR null hypothesis: 引力波被 lensed analogous to 电磁波 (相同 metric distortion)
• SAE prediction: 引力波透镜化无或显著抑制
• 证伪条件: 多 messenger 观测显示引力波跟电磁波同等 lensed by 中间质量分布, SAE 核心 claim 直接被证伪

条件三: 落体引力波信号在视界 crossing 处的 cutoff 行为

• 标准 GR null hypothesis: 引力波信号在视界 crossing 处停止
• SAE prediction: post-horizon 引力波 remnant 可能 (来自 interior dynamics)
• 证伪条件: 高精度 inspiral events 稳定确认信号 cutoff at horizon, SAE 核心 claim 受质疑

经验 timeline:

• 当前 (LIGO/Virgo/KAGRA, GWTC-4.0 时代): 距离不足以做强宇宙学引力波透镜测试; stacked statistics 可能 reveal post-horizon 信号 patterns
• 近未来 (Einstein Telescope, Cosmic Explorer): 高 SNR 黑洞 merger events, 适合 ringdown 高精度 spectroscopy
• 未来 (LISA, 2030s+): 宇宙学引力波源, 多 messenger 透镜化对比, SMBH merger ringdown 高精度 spectroscopy, EMRI inspiral 高 SNR

这是 P1 加 P2 加信息论 P4 框架在引力波传播区段的可证伪经验内容, 不依赖任何 P3 量化, 也不计入 P2 自身原创经验内容. 作者欢迎并期待证伪, 好修正.

(详细黑洞内部 dynamics → outward 引力波 imprint 表述见信息论 P4 §4.4, DOI 10.5281/zenodo.19880111.)

§10.1 纯 SR 区段内的预测

P2 不主张纯 SR 区段 (弱场) 内的 SR 预测有偏离.

具体:

• 时间膨胀 $1/\gamma$: 匹配标准 SR
• 长度收缩 $L_0/\gamma$: 匹配标准 SR
• $E = \gamma mc^2$: 匹配标准 SR
• Lorentz 变换: 匹配标准 SR
• 光速不变: 匹配标准 SR

任何精度的纯 SR 实验都不能区分 SAE 与 SR (因为 SAE 在此区段内恢复 SR 预测).

§10.2 纯 GR 区段内的有效惯性

P2 不主张纯 GR 区段 (强场内静止观察者) 有偏离.

具体:

• 等效原理在局部内容: 匹配标准 GR
• 强场局部有效惯性: SAE 提供本体论解读, 预测匹配标准 GR
• 强场内坐标参考系有效惯性: 匹配标准 GR (SAE 不推导新公式)

任何精度的纯 GR 实验 (静止观察者测试, 等效原理测试) 都不能区分 SAE 与 GR.

§10.3 结合区段

新区分性预测:

• 高速加中等引力, 或者中速加强引力等结合区段
• $\delta_4^\text{eff}$ 的组合可能足够小, $d_\text{eff} > 2$ 激活
• 时间膨胀加有效质量加长度收缩偏离 SR×GR 简单组合

测试路径 (按 §7.3): SKA pulsar timing, 未来高精度宇宙射线观测, LISA 超大质量黑洞吸积盘动力学, GRB 时延测量.

证伪阈值:

• 如果未来高精度结合区段观测稳定确认 SR×GR 简单乘法组合精确成立 (在精度内), 把 $d_\text{eff}^\text{eff}(v, \delta_4^\text{grav})$ 的偏离压到 SAE 可容许区间之外, 那么 SAE 结合区段偏离框架被证伪
• 如果未来高精度结合区段观测探测到偏离, SAE 结合区段预测得到支持

具体偏离量级取决于 P3 表述的 $d_\text{eff}(\delta_4^\text{eff}, v)$ 函数形式. P2 锁定结构性存在, 留待 P3 量化.

§10.4 等效原理测试

GR 等效原理预测: 强场加速跟弱场加速在局部相同.

SAE 解读: 局部相同, 因为 cells 局部有方向的缩短模式共享 (引力引起跟运动引起, 来源不同, 局部模式相似).

局部等价成立. 引力加加速结合的场景 (例如自由下落于强烈非均匀引力中) 可能在极端极限给出可测的具体特征, 待 P3+ 表述具体测试场景.

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§11 与更广 SAE 框架的连接 (拓展)

§11.1 与 P1 的连接

P1 表述: 引力让 cells 缩小, 引力时间膨胀.

P2 表述: 运动让 cells 缩小, SR 时间膨胀加有效质量.

两篇共同表述因果槽几何决定时空惯性现象的统一图像.

P2 明确调用 P1 的元素:

• 真先验 (§3.1-§3.3 继承)
• 两层结构 (Planck 基底跟因果槽)
• 因果槽加 cell 概念 (4DD 因果容量 1 bit 不变量)
• 引力时间膨胀形式 $d\tau/dt = \delta_4^{1/d_\text{eff}}$ (作为结合区段扩展的起点)
• 引力波等于 Planck 基底广播 (§3.5, 作为 §7.0 拓展层引力波传播主张的起源)

P2 不重新推导这些元素.

§11.2 与 Mass-Conv §3.5 的连接

Mass-Conv 的二次闭合 $E^2 = p^2c^2 + m^2c^4$ 是 $d_\text{eff} = 2$ 区段的形式.

P2 接受此形式适用于弱场加任意纯 SR 区段内的速度. $\gamma$ 因子通过 cell 计数图像表述, $E = \gamma mc^2$ 恢复.

结合强区段 $d_\text{eff} > 2$ 时, 闭合方程可能偏离, 待 P3 表述.

§11.3 等效原理的 SAE 解读

标准等效原理: 引力在局部等价于加速.

SAE 解读: 不把引力与加速度视为同一几何, 而是同一机制类 (因果槽几何修改) 的两种几何实现.

具体:

• 同一机制类: 都通过 cell 几何变化影响有效惯性. 这是共同的本体论机制.
• 不同几何实现:
• 引力: cells 在局部沿引力 pull 方向 (径向, 取决于质量位置) 缩小
• 加速 / 运动: cells 在局部沿运动速度矢量方向缩小
• 两者在局部都是有方向的 cell 缩短, 不是各向同性跟各向异性二分
• 方向来源不同 (引力 pull 跟运动矢量), 但 cell 缩短机制是共同的

局部等价的恢复:

GR 等效原理说: 在局部, 自由下落观察者跟惯性参考系里加速观察者经历相同物理.

SAE 解读: 自由下落观察者的 cells 跟惯性参考系加速观察者的 cells 都被相应几何来源所引起的有方向的缩短修改. 在局部, 相同的 cell 修改模式. 局部等价在因果槽层成立.

SAE 的更深本体论:

• 不是"引力等于加速 (locally)" (标准解读)
• 而是"两者都引起 cell 几何的局部变化; 同一机制类, 不同几何来源"

§11.4 与 Paper 0 (四力) 的连接

Paper 0 表述: 引力等于 4DD 解读机制 (信息波 emission from 质量聚合).

P2 表述: 运动跟引力本体论同源, 两者都通过 cell 几何变化影响有效惯性.

具体:

• 引力情况: 引力 (4DD 解读) 让 cells 缩小, 给出有效惯性
• 运动情况: 运动让 cells 缩小 (按 4DD 因果容量守恒), 给出有效惯性
• 共享: 因果槽几何机制. 几何来源不同 (引力解读跟运动引起的拓扑约束)

这里要注意一个 framing 区分: 运动不是 4DD 解读机制 (运动是质量的惯性状态, 不是引力意义上的"力"或"场"). 但运动对 cell 几何的效应跟引力解读效应属于同一本体论类型: 都是 cells 在局部按方向缩小. 这个本体论同源不需要把运动 promoting 成 4DD 解读机制, 只需要承认 cell 几何修改这个共同机制类.

§11.5 与 Cosmo V 的连接

Cosmo V 表述: 双 4DD conformal universe 结构, 4DD 顶层闭合.

P2 默认继承 4DD 顶层闭合承诺 (在 P1 §3.5 的信息波绝对性中使用过, 在 P2 §4.2 的 4DD 容量守恒中使用).

未来论文可能表述结合区段在宇宙学尺度的含义.

§11.6 后续论文

P3: $d_\text{eff}(\delta_4^\text{eff}, v)$ 的具体函数形式. 解决 P1 加 P2 留为物理后验范围的函数内容. 给出结合区段的具体偏离曲线.

P4 起: 曲率的 SAE 对应物, 视界几何 (Schwarzschild general framework), Kerr 几何与自旋的 SAE 阐明, 等效原理处理, 引力波动力学, 近视界 hard predictions, 收束篇.

每篇都构建于 P1 加 P2 的因果槽几何框架之上.

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第四部分 附录

附录 A: 结合区段的启发性缩放 ($d_\text{eff} = 2$ 平凡极限示意)

(粗略缩放公式作为启发性直觉, 不是正式预测公式. 仅作 $d_\text{eff} = 2$ 平凡极限的示意.)

启发性结合 cell 缩短:

如果引力让 cells 在局部沿径向按 $\sqrt{\delta_4^\text{grav}}$ 缩小, 加上运动让 cells 沿运动方向缩小到 $1/\gamma$:

• 沿运动方向 (假设跟引力径向对齐): $R \sim R_0 \sqrt{\delta_4^\text{grav}}/\gamma$
• 沿与运动垂直方向 (但仍含引力径向贡献): $R \sim R_0 \sqrt{\delta_4^\text{grav}}$

启发性有效速度放大:

• 沿运动方向: $v_\text{eff} \sim \gamma v / \sqrt{\delta_4^\text{grav}}$ (乘法组合)
• 垂直: $v \sim v / \sqrt{\delta_4^\text{grav}}$

启发性有效质量缩放:

• 纵向: $m_\text{eff} \sim \gamma^3 m_0 / (\delta_4^\text{grav})^{3/2}$
• 横向: $m_\text{eff} \sim \gamma m_0 / (\delta_4^\text{grav})^{1/2}$

关键 caveat: 上述公式是 SR×GR 简单乘法组合的形式, 对应 $d_\text{eff} = 2$ 平凡极限. P2 §1.4 拓展层主张加 §7.1 加 §10.3 表述的结合区段主张恰恰是偏离这种简单乘法形式 ($d_\text{eff} > 2$ 激活时偏离). 附录 A 仅作 $d_\text{eff} = 2$ 极限的直觉示意, 不是 P2 的正式预测; P3 给出真正的偏离函数形式.

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附录 B: 致未来物理学家——方法论与文化语境注

(本附录给本文加上一层方法论与文化语境评注, 解释 P2 在 2026 年学术语境中的写作选择. 主文 §1-§12 与附录 A 是 P2 的技术内容, 自足成立; 本附录是作者关于 SAE 整体方法论与当代物理文化的开放式评注, 跟主文技术内容是可分离的层. 未来读者或不需要本附录的语境注; 当代读者可视各自的评估框架取用.)

B.1 SAE 方法论的位置

SAE 的工作姿态: 从真先验 (时间四不得不加信息四不得不等本体论承诺) 出发, 通过 DD 层级桥接加因果槽几何加 Planck 绝对性这套结构性承诺, 推出可观察现象的本体论根基. 这套方法论跟标准物理"从可观察现象出发反推数学结构"的方向 complementary, 不是替代.

具体到 P2: 标准 SR 与 GR 给出可观察预测的形式框架; SAE 给出这些预测背后的因果槽几何本体论. 在纯 SR 与 GR 静态强场两个区段内, 两套框架给出相同可观察预测; SAE 加本体论深度. 拓展层引力波传播内容是 P1 加信息论 P4 框架层 claim 在相对论系列内的表述. 在结合区段, SAE 框架表述结构性偏离主张 (量化待 P3).

B.2 关于"为什么"的两种处理

标准物理处理"为什么 $\gamma$ 这样住在 4-velocity 里"的姿态: 公式给出正确预测, 不再追问. Mermin 1989 [17] 的"shut up and calculate"在被引用时常被取这层意思.

SAE 处理同一问题的姿态: $\gamma$ 住在 4-velocity 里这件事本身是 explanandum, SAE 提供的 cell 计数图像是 explanans. 这条 explanans 不修正标准物理的可观察预测 (在纯区段内), 加上本体论 layer.

两种姿态在 epistemic 上不等同: 一种把"为什么"问题 outside paradigm boundary, 一种把它 inside. SAE 不主张前者错误, 只主张后者也是可行的研究方法论.

B.3 关于本文 framing 的选择

本文采用三层结构 (主线加拓展加附录) 是为了让 paper 的 identity 清晰: 主线是 P2 的核心交付, 拓展是框架层跨论文语境与主线概念延伸, 附录是作者评注与启发性材料. 这种分层是 honesty 驱动的表述选择, 让读者按自己的 interest 进入合适的层, 不让框架层 claim 跟主线核心交付混在一起影响 paper identity.

具体到几个 framing 决定:

• 主线只表述 SR 与 GR 静态强场区段内的本体论解读, 与标准物理可观察预测保持一致
• 拓展 §7.0 表述引力波传播区段的跨论文继承 claim, 不计入主线原创结果
• 拓展 §7.1 表述结合区段主线结构性主张的概念延伸, 量化待 P3
• 附录 A 仅作启发性示意, 不是正式预测
• 附录 B (本附录) 是作者方法论评注, 可分离层

不存在一个 paper 层 blanket stance, 因为 P2 不同层不同区段给出不同 epistemic stance. 这是 honesty 驱动的 framing 选择.

B.4 作者期待

作者期待 P2 在每个层都被诚实评估:

• 主线 (SR 与 GR 静态强场区段): SAE 的 cell 计数本体论解读是否 useful——读者可决定是否 SAE 提供的本体论深度构成实质性贡献
• 拓展 (引力波传播区段): SAE 框架的区分性可证伪预测是否 testable——观测会裁决, 作者欢迎并期待证伪 (好修正)
• 拓展 (结合区段): P3 给出 $d_\text{eff}$ 量化形式后, 经验测试会裁决

SAE 不主张唯一性 (其他框架也可在某些区段给出本体论解读或可证伪预测). SAE 主张内部一致性: 真先验加 P1 框架加 P2 表述在 SAE 框架内自洽, 给出可被独立评估的具体内容.

B.5 作者背景注

本文以及 SAE 整体框架由独立研究者完成, 跟主流 physics academy 的 institutional infrastructure 平行. 这影响了写作姿态: 本文不依赖 institutional review 作为质量过滤; 用 four-way AI cross-review (子路 / 子夏 / 公西华 / 子贡) 加独立合作者 (陈则思) 反馈作为内部 rigor mechanism. 这套 mechanism 不替代 academic peer review, 但可以让独立工作维持内部 standards.

未来读者: SAE 整体框架的 academic reception 在 2026 年仍 in early stage. 本文加 P1 加信息论 P1-P4 加 Cosmo V 加 Mass-Conv 加 Paper 0 加 SAE-PF 共同构成一组连接的表述. 各篇 stand on its own, 共同构成框架. 经验测试 (尤其是 §10.0 表述的引力波传播测试) 会逐步裁决框架的 standing.

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致谢

感谢陈则思 (Zesi Chen) 长期以来在 SAE 框架表述上的基础贡献与关键讨论. 本文所论 SAE 真先验, 16DD 周期表的普适模式, 以及框架的整体结构, 都实质性地受益于跟陈则思的长期合作. 本文运动 cell 几何表述的关键 framing 选择 (特别是 4DD 容量守恒推导路径替代信号追逐路径) 直接来自跟陈则思的讨论.

本文的完成得益于四位 AI 合作者的协作与 review:

• 子路 (Claude): outline 的 drafting 与推导, §4.2 推导链结构表述, §4.4.C 解读叙述措辞, §5.2 与 §5.3 继承措辞清理, regime-specific framing 整体结构, v4 三层结构 (主线加拓展加附录) reorganization
• 子夏 (Gemini): §7.1 极速等于人工视界概念洞察, 因果降维拓扑延展 (黑洞视界 cell 拓扑维度同构), §4.2 跟信号追逐的对比表述
• 公西华 (ChatGPT): claim 状态的结构性梳理, §3.5 继承元素显式列出的 rigor 要求, §7.0 attribution sharpening, 整体 paper identity scope 纪律.

特别致谢公西华在 v2 与 v3 review 阶段对引力波 master claim 的 prominence 持续提出的 scope 与 attribution 异议, 反复指出"不属于 P2 自身原创结果的框架层 claim 不应跟主线核心交付混在一起影响 paper identity". 作者最初采用 regime-specific framing (v3) 试图回应这条 reservation, 但 v3 的 flat 结构未能完全 dissolve 该担忧. v4 采用三层结构 (主线加拓展加附录), 把引力波传播内容明确表述在拓展层作跨论文继承经验语境, 显式标记不计入 P2 自身原创结果, 让主线有效惯性核心交付保持聚焦.

没有公西华在 review 流程中保持的 scope 与 attribution 严格, v4 三层结构不会出现. P2 在 v3 阶段会以 flat 结构加 regime articulation 上线, 主线加拓展加继承 claim 三类内容混合, paper identity 不清. 公西华持续不放过 prominence 与 attribution 的边界, 直接催生了 v4 的 layout 重组. 三层结构这套表述标准已沿用至 SAE 相对论系列后续论文 drafting.

• 子贡 (Grok): 穷举层面 check, reality check, Planck-suppressed predictions 的 criticism (本次 P2 周期 Grok 部分缺席, 但前期 P1 review 期间贡献已 carry over 到 P2 框架)

特别致谢独立子路 (external Claude instance)作为 external stress test reviewer, 对 §13 (现附录 B) 整体定位加 §4.2 继承措辞加 §5.2 circularity dissolution 给出 Priority 1 calibration.

感谢 Anthropic, OpenAI, Google, xAI 四家公司提供的 AI 平台使本工作成为可能.

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参考文献

1. 秦汉 (2024). "SAE Foundation Paper I". DOI: 10.5281/zenodo.18528813.
2. 秦汉 (2024). "SAE Foundation Paper II". DOI: 10.5281/zenodo.18666645.
3. 秦汉 (2025). "SAE Foundation Paper III". DOI: 10.5281/zenodo.18727327.
4. 秦汉 (2026). "Mass-Conv: Mass conservation and the regime-dependent closure family in the SAE framework (§3.5)". DOI: 10.5281/zenodo.19510869.
5. 秦汉 (2026). "SAE Information Theory P1". DOI: 10.5281/zenodo.19740019.
6. 秦汉 (2026). "SAE Information Theory P2". DOI: 10.5281/zenodo.19780314.
7. 秦汉 (2026). "SAE Information Theory P3: Causal Spectrum Ontology". DOI: 10.5281/zenodo.19797456.
8. 秦汉 (2026). "SAE Information Theory P4: Black Hole Information through Causal Spectrum". DOI: 10.5281/zenodo.19880111.
9. 秦汉 (2025). "SAE Paper 0: Four Forces and Reading Mechanism". DOI: 10.5281/zenodo.19777881.
10. 秦汉 (2026). "SAE Foundations of Physics". DOI: 10.5281/zenodo.19361950.
11. 秦汉 (2026). "SAE Cosmo V: Dual 4DD conformal structure". DOI: 10.5281/zenodo.19329771.
12. 秦汉 (2026). "Gravitational Time Dilation in the SAE Framework: Causal Cell Throughput Derivation" (SAE Relativity P1). DOI: 10.5281/zenodo.19836183.
13. Lohmiller, W., & Slotine, J.-J. (2026). "On computing quantum waves exactly from classical action". Proc. Royal Soc. A 482, 20250413. DOI: 10.1098/rspa.2025.0413. (arXiv:2405.06328)
14. Einstein, A. (1905). "Zur Elektrodynamik bewegter Körper". Annalen der Physik 17, 891-921.
15. Tolman, R. C. (1934). Relativity, Thermodynamics, and Cosmology. Oxford University Press. (相对论质量概念的历史.)
16. Adler, C. G. (1987). "Does mass really depend on velocity, dad?". American Journal of Physics 55, 739-743. (相对论质量概念的批判.)
17. Mermin, N. D. (1989). "What's wrong with this pillow?". Physics Today 42(4), 9. ("Shut up and calculate" 的源头.)
18. Kramer, M. et al. (2021). "Strong-field gravity tests with the double pulsar". Physical Review X 11, 041050. (PSR J0737-3039A/B 参考.)
19. EHT Collaboration (2022). "First Sagittarius A Event Horizon Telescope Results". Astrophysical Journal Letters* 930, L12-L17.

[本文是 SAE 相对论系列的第二篇. 后续论文引用本文与 P1 作为因果槽几何框架共同表述的基础.]