One Complete Cosmic Breath
一次完整的宇宙呼吸

§1 Introduction: why write out one whole cosmic breath

The SAE cosmology series has so far advanced "in blocks." The cosmological constant is obtained from the frequency difference of a pair of topological clocks; dark matter is carried by a geometrized khronon dust, with no new particle; dark energy and MOND share an acceleration scale set by the first-order difference of the same pair of frequencies, carrying a prediction testable at high redshift; the effective gravitational constant has a plateau-then-rise history over cosmic time. Each result stands as its own paper, each with its own falsifiable observational interface.

But one question has always hung over these separate results: if we put them back onto a single timeline — from the beginning to the end, and on past the end — what does that line look like? This cannot be answered by re-shuffling existing results. It demands that we say clearly several things not previously handled in a systematic way:

• What, in SAE, is the "beginning" that is the Big Bang — a singularity in need of an external fuse, or something else?
• Is the way the universe ends a heat death of unbounded dilution, or some kind of contraction? If contraction, driven by what, and leading where?
• If there is a "next" Big Bang, is it a repeat of this one, or does each round differ? What is carried across, and what is discarded?
• And what, running through the whole line, is the quantity that is actually changing?

This paper sets out to give that line as a whole, at the qualitative level. In terms of role, it is a framework-level, prefatory work in SAE cosmology: it offers a six-phase ontological map, a set of order parameters characterizing the phases, and a list of falsifiable points; it does not offer a complete dynamical solution. Wherever a quantity would require full field-equation solving to pin down — the collapse statistics of the contracting phase, the complete form of the two-clock action, the dynamics of the very first leap — this paper honestly marks it a "key for the next layer," listing it without pretending it has been turned.

The label "0" is deliberate: in ordering, it sits before the cosmological-constant paper (the first of the series); and the numeral itself carries our stance — this is a groundwork, conversation-opening piece, claiming to have said the last word on no segment of the line.

Toward the related cyclic and bouncing cosmologies — Penrose's Conformal Cyclic Cosmology (CCC), the Steinhardt–Turok ekpyrotic picture, the Boyle–Turok CPT-symmetric universe — this paper states, where relevant, the similarities and differences with SAE. Our attitude is that these are explorations, from different axiomatic entry points, of the same deep structure, of which SAE offers one; we state differences and identify each one's falsifiable points, without ranking them. This attitude is not mere politeness: a basic commitment of SAE is that any construction leaves a remainder and is incomplete (see §3 and below), so to claim "I am wholly right and others wholly wrong" would violate that very commitment.

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§2 The epistemic contract: claim tiers and stance

Before unfolding any specific picture, we make a contract with the reader: not all claims in this paper carry the same confidence, and we tag the confidence of each explicitly. This is both a duty to the reader and the technical realization of the "of some use" stance above — only by tagging, claim by claim, which are hard and which are soft does "not to be free of error" become something one can actually check rather than an empty modesty.

Claims fall into four tiers:

[structural] — forced by the internal logic of the SAE framework, and falsifiable in principle. If such a claim is refuted by observation or theory, what is shaken is the framework itself, not some adjustable parameter. The two most important structural predictions here are: the redshift evolution of the acceleration scale $a_0$ (decidable at $z>3$), and the intrinsic dark-energy equation of state $w_{\text{int}}\geq-1$ (no crossing of the phantom divide).

[posterior] — dependent on observational input or parameter values. Such quantities SAE does not derive from pure structure but takes from cosmological or astronomical data (just as general relativity does not predict the value of $G$). The two breath periods $T_1, T_2$, the speed of light $c$ (a quantity derivable from other boundary conditions but still of boundary-condition character), and the density parameters of dark matter and dark energy all belong here. To stress: finding a relation among boundary conditions (and thereby reducing the number of independent inputs) is one thing; deriving a constant from nothing is another — the former is ordinary physics, only the latter would be over-claiming.

[motivated] — a well-motivated preference, but not framework-forced. Such claims are, in our view, the most reasonable picture at present, while we grant that unexcluded alternatives exist. The specific final state of dark energy in the contracting phase, the analogy between the Crunch and the interior of a black hole, and the single-peaked curve itself as a qualitative structure, all belong here.

[open] — an undecided question, listed rather than pretended answered. Some open questions come with a "most promising candidate direction," but a direction is not a conclusion.

These tiers run through the whole paper; they are tagged in place, and collected in an appendix into a single claim–tier table for the reader's overview. Wherever a claim is [motivated] or [open] — and this paper has several, concentrated in the Crunch-to-next-Bang transition — we declare its candidate status explicitly at that point: marking its corpus grounding (which links are hard), its candidate content (which is the picture we propose), and where it awaits quantification (what has not been computed). We put these mechanisms forward to be probed and falsified, not as settled results.

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§3 The three ledgers: remainder, development sufficiency, entropy

To make one breath into a connected line, we must first say clearly what, along that line, is actually changing. An early SAE thought was to use "the average strength of the causal law" as this variable — weak in expansion, strong in contraction. But that has an internal inconsistency: in SAE, the fourth layer (4DD, i.e. the spacetime/causal layer) is inactive inside a black hole. So "the more black holes in contraction, the stronger the average causal law" is self-contradictory — black holes are precisely the regions where the causal layer is inactive. To speak of an "average causal law" across active and inactive regions is to invoke a quantity that is not well-defined to begin with.

We therefore take a different variable as the through-line, one that carries no such baggage: the sufficiency of remainder-unfolding. Intuitively it measures "how fully the remainder has been unfolded" — from the initial, chaotic, not-yet-unfolded state, to a state where complex structure and life have amply emerged, and back to a state where everything has collapsed and there is nowhere left to unfold. This curve is single-peaked:

• lowest at the Bang — the layered ladder has only just started, the remainder scarcely unfolded;
• rising through expansion — the ladder is built layer by layer, structure and higher tiers (life, consciousness) emerging in turn;
• peaking near the turnaround — development at its fullest;
• descending through contraction — structures fall one by one into 4DD-inactive black holes, the inactive fraction of the universe rising, the room left for unfolding shrinking;
• lowest again at the Crunch — the whole universe becomes one large, everywhere-4DD-inactive state, with nowhere for the remainder to unfold.

One wording needs care here: the decline of development sufficiency in contraction we do not phrase as "the average causal law grows stronger," but as "the inactive fraction of the fourth layer rises" (structure falling into black holes). The mechanism is gravitational collapse, not a causal law smoothing the remainder away. [motivated]

We state this need not new. The dynamics "the remainder cannot close within the current layer, and so breaks through to the next" has, in SAE's information-theory series, already been established as a primitive — called unfolding. That series had earlier taken "broadcast" and "reception" as two intra-layer, lateral dynamics of fourth-layer information; unfolding is a third, cross-dimensional, longitudinal one, describing how information crosses between layers, corresponding formally to the "chisel–construct" cycle of the remainder framework. Our "sufficiency of remainder-unfolding" is precisely the cosmological-scale application of this already-established primitive: the whole single-peaked curve depicts the longitudinal progress of the remainder's unfolding over the universe's life. [main-line inherits the third information-theoretic primitive; not newly coined here]

Now the heart of this section. This single-peaked curve is easily confused with two other quantities, and keeping them apart is the key to why the whole cycle is a "spiral rather than a circle." We write the three as functions of a cosmic phase $\tau$ (a parameter advancing monotonically with the breath, on the open interval $\tau\in(0,T_1)$, the two endpoints being singular transitions handled separately): development sufficiency $\mathcal{D}_R(\tau)$, entropy $S(\tau)$, and carried remainder $\mathcal{R}(\tau)$. We must first state their status: $\mathcal{D}_R$ is a phenomenological order parameter, capturing "how fully the remainder is unfolded"; this paper has not yet constructed it as a first-principles observable — that construction is later work; the statements below hold under the premise that such an order parameter exists with the stated qualitative behaviour. The three behave as follows:

$$\mathcal{D}_R(\tau)\ \text{generalized single-peaked}:\ \exists\,\tau_\in(0,T_1)\ \text{with }\mathcal{D}_R\ \text{non-decreasing on }(0,\tau_],\ \text{non-increasing on }[\tau_*,T_1),\quad \lim_{\tau\to 0^+}\mathcal{D}_R=\lim_{\tau\to T_1^-}\mathcal{D}_R=\mathcal{D}_R^{\min}$$

$$S(\tau)\ \text{monotone non-decreasing (an order relation, differentiability not assumed)}:\quad \tau_a<\tau_b\ \Longrightarrow\ S(\tau_a)\leq S(\tau_b)\ \text{(within each 4DD-active segment)}$$

$$\mathcal{R}_{\rm tot}\ \text{conserved}:\quad \mathcal{R}_{\rm tot}\big|_{\tau\to T_1^-}=\mathcal{R}_{\rm tot}\big|_{\tau\to 0^+\,(\text{next cycle})}$$

("Generalized single-peaked" allows the peak to be a plateau rather than a single point.) On $\mathcal{R}$, two levels must be distinguished: what is carried and strictly conserved is the total remainder $\mathcal{R}_{\rm tot}$; whereas what the next cycle inherits, fixing its parameters ($T_1, T_2$, etc.), is a projection of $\mathcal{R}_{\rm tot}$ onto the boundary conditions, and this projection drifts from cycle to cycle (§7.4). The total is conserved; the projection drifts; there is no contradiction.

This lets us state the relation among the three. The three ledgers are non-substitutable, for two different reasons. First, for $\mathcal{D}_R$ and $S$: there is no monotone one-to-one correspondence between them — not an extra assumption, but forced by the endpoint behaviour (proof below). Second, for $\mathcal{R}_{\rm tot}$: its distinction from the other two comes from its type and conservation role — it is a cross-cycle conserved ledger (more like a conserved quantity with internal structure than a third scalar curve of the same type as $\mathcal{D}_R, S$), not something distinguished by the function-relation proof below. Now the first, which needs only one counterexample. Take the limiting values at the two endpoint neighbourhoods,

$$\lim_{\tau\to0^+}\mathcal{D}_R=\lim_{\tau\to T_1^-}\mathcal{D}_R=\mathcal{D}_R^{\min}\quad\text{while}\quad \lim_{\tau\to0^+}S<\lim_{\tau\to T_1^-}S,$$

the same $\mathcal{D}_R$ value at both ends, different $S$ values. Hence no function $g$ with $S=g(\mathcal{D}_R)$ can exist (one argument value giving two function values), so $\mathcal{D}_R$ and $S$ cannot be monotonically equivalent. This is not an assertion; it is forced by the endpoint behaviour. [$\mathcal{D}_R$–$S$ non-equivalence: conditional structural conclusion; the endpoint behaviour itself is part of $\mathcal{D}_R$'s motivated ansatz]

(In passing: $\mathcal{D}_R$ equal at both ends while $S$ differs — the Bang and Crunch symmetric in development sufficiency, asymmetric in entropy — is exactly the expression, within this framework, of Penrose's point that the Big Bang is a special low-entropy state. We return to it when discussing contraction and the transition.)

On entropy, this paper adopts a horizon-entropy proxy, to avoid ambiguity later. It is of Bekenstein–Hawking type, written $S_H$, and dimensionally taken as

$$S_H = \frac{k_B c^3}{4G\hbar}\,A_{\rm hor}\,,$$

with $A_{\rm hor}$ the total area of the relevant horizons (both sides dimensionally consistent; $S_H$ carries entropy units). Three details remain to be completed for this proxy: which horizons to count (black-hole event horizons, apparent horizons, cosmological horizons, or trapping horizons), how to de-duplicate nested horizons, and the full statement of generalized entropy — these are later work. Accordingly, "within each 4DD-active segment the entropy is monotone non-decreasing" is a structural commitment inherited from the second law and the information-theoretic ground below, not a theorem already proved by the area sum alone (the area formula does not by itself entail monotonicity across the whole history of mergers, evaporation, and cosmological-horizon change). With that stated, we adopt the monotonicity at the framework level. It counts black-hole horizons — the boundaries where the fourth layer is active — and grows as matter falls into black holes and $A_{\rm hor}$ increases. The interior of a black hole (4DD-inactive) needs no "interior entropy": entropy is a quantity attached to horizons. Only at the two singular endpoints $\tau\to0^+$, $\tau\to T_1^-$ — where there is no fourth layer and no horizon — does $A_{\rm hor}$ lose definition and entropy lose its attachment (which is why we take the open interval). How it downgrades there, and how the remainder is carried across, is the subject of §7.

The monotone growth of entropy has, in SAE, a root deeper than "the second law," worth noting here. The information-theory series holds that the operational category of the fourth layer is encapsulation, and encapsulation admits no inverse: once energy enters the information channel as information, no operation within the fourth layer turns it back into energy, momentum, or mass. Here "information" is a conserved quantity called the causal load $I$, related to energy by

$$E = I\,c^3\,,$$

so $I$ has the dimension $E/c^3$ — a dimensional, physical load, not a dimensionless bit count or entropy. Hence, under non-equilibrium dynamics, the information channel becomes the sole one-way sink, only filling, never emptying, accumulating. The monotone growth of the horizon-area entropy is rooted right here: it is the manifestation of "encapsulation-irreversibility, $I$ accumulating one-way," not an externally imposed law. Two quantities must be kept wholly apart here, because §7.4 turns on exactly this: $I$ (the causal load, dimension $E/c^3$) is the carried; $S_H$ has entropy units, and it is $S_H/k_B$ that is dimensionless — readable as the logarithm of an effective count of configurations. The former is conserved and carried; the latter is downgraded at the transition — one carried, one not. [entropy monotonicity inherits information-theoretic encapsulation-irreversibility, as SAE's ground for the second law]

This also flags in advance a tension to be handled carefully in §7: if the fourth-layer information channel is irreversible, how is the "causal load falling back to energy at the transition," which §7 will speak of, possible at all? The clue is precisely that "at the transition there is no fourth layer" — irreversibility is the rule while the fourth-layer closure holds, and the transition is exactly where that closure itself dissolves, so its premise no longer applies. We defer this reconciliation to §7.

Three quantities, three behaviours: development sufficiency is the developmental variable (single-peaked), entropy the thermodynamic variable (monotone), remainder the conserved ledger (carried). Only by keeping them apart does the "spiral rather than circle" picture stand — entropy is not carried (its domain terminates at the transition), the remainder is carried, and so each Big Bang is a new low-entropy start whose parameters may differ from the last round's. The line therefore does not return to where it began; it spirals forward.

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§4 The six-phase map

Before unfolding phase by phase, we lay out the skeleton of the whole line as a map for what follows. We divide one complete breath into six phases:

Phase	Load-bearing picture	Main confidence
① Bang	chaos self-negation; forced readout from 0DD to 1DD	mechanism [structural]; threshold value [posterior/open]
② Early: forces	the four forces generated in turn as four negation steps projected onto the material layer	sequence [structural]; continuous expansion rate [open]
③ Present: acceleration	khronon dark matter + causal-scalar dark energy + four-form cosmological constant	$w_{\text{int}}\geq-1$ [structural]; $a_0(z)$ evolution [structural, decided at $z>3$]; parameter fits [posterior]
④ Turnaround	the geometric turning point; the peak of development sufficiency	decoupling [structural]; turnaround position [motivated/open]
⑤ Contraction	structure falling into 4DD-inactive black holes; a cosmological-scale closed critical state	mechanism [motivated]; collapse statistics and anisotropy [open]
⑥ Crunch → next Bang	a sequence of three distinct mechanisms; entropy downgrade, remainder carried	structure [motivated]; transition dynamics [open]

One whole-paper timeline judgment must be stated here about this map, because it revises a default in an earlier SAE work. The early picture assumed the breath symmetric: the turnaround exactly at the half-period, whence the period was set to twice the turnaround time. This paper drops that symmetry assumption, adopting instead an asymmetric breath as a well-motivated working hypothesis (see §7.1): if the contracting phase develops a net inward effective driving (contingent on the dark-energy final state, §7.3), then the contraction segment is shorter than the expansion, and the geometric turnaround falls late. We stress this is a working hypothesis, not an established result — a closed universe of matter and gravity alone is time-symmetric. Under that hypothesis, the line takes the following asymmetric shape:

$$\text{Bang} \;\longrightarrow\; \underbrace{\text{expansion (against gravity: slow, long)}}_{} \;\longrightarrow\; \underbrace{\text{geometric turnaround (late, possibly future)}}_{} \;\longrightarrow\; \underbrace{\text{contraction (if net inward driving: fast, short)}}_{} \;\longrightarrow\; \text{Crunch}$$

A consequence concerns where we now stand, and here caution is needed. An early picture would locate the present as "past the turnaround, in geometric contraction." But this framework has a two-frame structure (see §5 and below): the expansion we observe is one frame (the Jordan frame), while geometric expansion or contraction is another, and the action of dark energy can make a geometric contraction still appear, in the observed frame, as accelerating expansion. This yields an honest limit: all we can definitely state is that the Jordan frame is, at present, still in accelerating expansion; whether, in the geometric frame, the turnaround has already been crossed, current observation cannot decide. The geometric turnaround may still lie in the (late) future, or may already have been crossed and be "hidden," beneath the observed frame's accelerating appearance, by dark energy — this paper judges neither, only marks the undecidability as it stands.

We deliberately do not write the present as "not yet past the turnaround," partly because it genuinely cannot be measured now, and partly to leave room for future evidence able to reconstruct the geometric-frame state. It must be stressed that this timeline revision does not overturn the earlier series' quantitative results for the cosmological constant — because the cosmological constant is insensitive to the "shape" of the breath (the acceleration scale requires an extra layer of caution; both are explained in §6). [structural (Jordan-frame in expansion); geometric-frame crossing: currently undecidable]

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§5 From the first leap to the present universe (phases ①–③)

This section walks through the first three phases. They connect most tightly to SAE's published block results, so our approach is to cite rather than re-derive, focusing on splicing them into this timeline and marking the confidence of each.

5.① The Bang: chaos self-negation

In SAE, the Big Bang is not a singularity in need of an external fuse. It is chaos self-negating.

The logic here is purely a priori, not "because we exist, therefore backward-inferred." The starting point is: toward "existence," four attitudes may be taken — is, is-not, both, neither (this is the structure of Nāgārjuna's tetralemma). And chaos says "no" to all four alike — this is not some property of chaos, it is the definition of chaos: it cannot be pinned by any definite description. But chaos cannot rest in this "being nothing" state: for once "not developing" itself became one of its definite properties, that would contradict "it cannot be definitely described." So chaos must self-negate, must develop onward. This step — which we call the "chisel" — is not an external force, not the hand of a god, not some fuse igniting the Bang; it is the internal necessity of chaos self-negating.

After the four are negated one by one and exhausted, there is a breakthrough to the next round (matter → life → consciousness). Worth noting: the dimension number $d=4$ here is not an axiom, nor an assumption, but a consequence of the structure "the tetralemma is exhausted."

To connect this back to §3's through-line: the "chisel" here is precisely the first strike of that "unfolding" primitive. §3 said the remainder, unable to close within the current layer, breaks through to the next — the Bang is the origin of that breakthrough, and the whole rising branch of the single-peaked curve thereafter is the continuation of the same "chisel–construct" cycle. In other words, the Bang and the development sufficiency running through the whole paper are not two things, but the opening and the unfolding of one and the same longitudinal process.

This picture gives the mechanism of the Bang [structural]. But it also carries a floor we must mark honestly: the deepest point of this unfolding of negation is "whence the negation" — and this question, structurally, cannot be answered further down, because any "direction" or "origin" is something that only exists after negation has unfolded. This is not a black-box evasion — the mechanism of the Bang (tetralemma exhausted, must develop) can be stated clearly; but its very bottom "why is there negation at all" can only be acknowledged, not derived. This is one of the remainders this framework honestly acknowledges. [the "floor" at the threshold: open, and we argue it structurally underivable]

One discipline must also be set here, bearing on how we speak of time and the speed of light later. This Bang segment — the leap from 0DD to 1DD — occurs where "time itself has not yet been generated." The two clocks, the continuous expansion rate, even the speed of light as a rate, all used later, are language available only after the first layer. To use these "post-time-generation" concepts to explain "how time is generated" would commit a level error. So on the dynamics of this first leap, this paper marks only "open" — it is the deepest of the "keys for the next layer" — and does not force it full with later language.

5.② Early: the four forces generated in turn as four negation steps

Once past the first layer, the universe enters the early evolution more familiar to us. SAE's distinctive point on this segment is to understand the four fundamental forces as the projection, onto the material layer, of one and the same "four-step negation": the first negation step corresponds to the first layer and electromagnetism, the second to the second layer and the weak force, the third to the third layer and the strong force, the fourth to the fourth layer and gravity; correspondingly, the characteristic quantities of the material layer descend along this ladder by powers of $c$. This dimensional ladder may be written

$$E \;\to\; \frac{E}{c} \;\to\; \frac{E}{c^2} \;\to\; \frac{E}{c^3},$$

the four rungs corresponding in turn to the first through fourth layers (energy, momentum, mass, causal load $I$); $E/c^3$ is the dimension of the $I$ of the previous section. In this sense, unfolding from 0DD out to 4DD at the Bang is the process of the four forces being "sculpted" in turn.

A line against misreading must be drawn here. This "generation in turn" of the four forces is a generation sequence at the ontological-layer level, not a thermal history scaled by temperature — it is not that $U(1)$, $SU(2)$, $SU(3)$, gravity appear one after another as on a stopwatch. Standard cosmology's thermal history (electroweak unification, the QCD phase transition, recombination) has its own temperature order; SAE's four-step negation speaks of ontological layers, not the particle-physics phase-transition timeline. To conflate the two would draw an immediate objection from physicists. [generation sequence: structural; relation to thermal history: this paper explicitly distinguishes the two]

Whether this segment can be made compatible with standard early cosmology turns on interfaces that are all known and unavoidable checkpoints: the effective gravitational constant must fall within a safe window at nucleosynthesis and recombination; the khronon dust must behave like cold dark matter before recombination; the spatial term of dark energy and MOND must not spoil the acoustic peaks of the microwave background. A full Boltzmann solution is an unavoidable verification for this segment — this paper does not perform it and lists it as open. [continuous expansion rate and full Boltzmann: open]

On this segment, one positioning must be stated outright, because it decides where a reader should and should not go to test SAE. In the Bang and the early universe, SAE offers an ontological-level account (the generation sequence of four-step negation, the Bang as chaos self-negation), not numerical predictions differing from standard cosmology. Numerically, it requires itself to agree with standard cosmology — the interfaces above (nucleosynthesis abundances, the microwave background, recombination) are compatibility constraints it must satisfy, not new predictions it claims. Where SAE genuinely parts ways with the standard model numerically is in the boundary-condition layer: the cosmological constant, the evolution of the acceleration scale, the structure of the dark sector; not in early microphysics.

This division is itself a structural claim of SAE, from the information-theory series: thermodynamics and early microphysics belong to the internal structure of the fourth layer, cosmology to the boundary conditions of the fourth layer, the two connected only at the roughly $10^{-122}$ strength of the cosmological constant. It also explains why the falsifiable predictions this paper does claim (§5.③) all sit at high redshift — because the differences are concentrated in the boundary-condition layer, and the decisive differences of that layer appear at $z>3$. To a physicist wanting to test SAE, this sentence is useful: go to the high-redshift acceleration scale and equation of state to find the differences, not to nucleosynthesis — there SAE agrees with standard cosmology to begin with. [early universe as ontological account, not numerical prediction; numerical differences concentrated in the boundary-condition layer: structural]

5.③ Present: acceleration, and a note on epistemology

Coming to the present, we enter the home ground of the earlier series (the dark-sector paper). Its picture: dark matter carried by a geometrized khronon, dark energy and MOND by a causal scalar field, the cosmological constant by a four-form — all three out of that pair of breath clocks, with no dark-matter particle introduced. This segment gives two structural hard bounds: the redshift evolution of the acceleration scale (decidable at $z>3$), and the intrinsic dark-energy equation of state $w_{\text{int}}\geq-1$ (no crossing of the phantom divide). Here the two aspects of the acceleration scale must be distinguished: its redshift slope (how it grows with redshift) is the framework's structural prediction, and the object the $z>3$ decision targets; whereas the coefficient of its present value (that $\pi/2$ factor, from the geometry of the closed three-sphere) is a consistency corroboration, not where the predictive power lies. In other words, the falsifiable content is in the slope, not the coefficient — an observation seeking to refute the framework should target whether the scale grows as predicted at high redshift, not its present absolute value.

In dialogue with observation, one register needs care. Recent galaxy surveys (e.g. DESI) do give pressure toward a time-evolving dark energy, but "whether an intrinsic phantom crossing is required" is parametrization-dependent. So this paper does not write "observation has confirmed $w$ crossing $-1$"; the accurate statement is: the data give pressure toward evolving dark energy, while SAE's present hard bound is the intrinsic $w_{\text{int}}\geq-1$, any effective crossing behaviour handled as a separate branch. [$w_{\text{int}}\geq-1$: structural; specific fits: posterior]

On the epistemic tier of the cosmological constant, an explicit note is needed here, because it decides just how many quantities this framework "predicts." The two periods $T_1, T_2$ of this pair of clocks are anchored from different inputs: $T_1$ is pinned independently by the scale of the cosmological constant — its smallness (about $10^{-122}$) comes mainly from the square of the ratio of the Planck time to the cosmic lifetime, which fixes $T_1$ at the order of twenty billion years, and this pinning is independent of the breath's shape; separately, the fact that life (the fifth layer) appears at about half the cosmic lifetime gives $T_1$ a coarser, independent corroboration. $T_2$ can be fixed precisely by inverting the observed cosmological constant, and can also be roughly corroborated by the Milky Way–Andromeda merger timescale (a standard orbital computation independent of this framework).

Hence the status of the cosmological constant is a conditional prediction [so tagged in this paper]: under the premise of "anchoring $T_1$ by life, and accepting the Milky Way–Andromeda identification," it is predicted to about 5% accuracy; if one accepts only the $T_1$ pinned by scale, then the cosmological constant and the acceleration scale together constitute "a consistency check between two posteriorly-locked quantities." On either reading, the framework's explicitly claimed unconditional hard predictions remain the high-redshift evolution of the acceleration scale ($z>3$) and the intrinsic $w_{\text{int}}\geq-1$. We tag the cosmological constant a conditional prediction — neither inflating it to an unconditional hard prediction (one identification step being soft), nor demoting it to a pure input (the scale its inversion rests on being independent).

Finally, to splice the present back onto the timeline: we now sit at a high point of development sufficiency — life (the fifth layer) has emerged. As to whether, geometrically, we are in expansion or contraction, as above, current observation cannot decide: all we can be sure of is that the observed (Jordan) frame is still in accelerating expansion, while the actual state of the geometric frame we do not know. This does not conflict with the observed acceleration — precisely because the two can decouple. In the next section on the turnaround, we make clear the two sides of this decoupling: the geometric "turning point" and the "peak" of development sufficiency are two points that may not coincide, and neither is directly identical to the acceleration we observe.

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§6 The turnaround: two "turning points" that need not coincide

At the end of the previous section we planted a sentence: the geometric turning point and the peak of development sufficiency are two points that may not coincide. This section makes it clear, because the distinction both clarifies a question previously left hanging and protects our earlier quantitative results from the timeline revision.

Name the two points apart first:

• The geometric turning point — the moment the scale factor reaches its maximum and the geometry turns from expansion to contraction. It is set by geometry and gravity; it is the equator of this "pendulum" of a closed universe.
• The peak of development sufficiency — the moment the sufficiency of remainder-unfolding is highest. It is set by developmental dynamics: where the unfolding of structure, complexity, and life reaches its fullest.

The early symmetric picture defaulted these two points to the same one, and further defaulted it to the half-period, whence the period was set to twice it. This paper keeps the two apart, for two reasons.

First, what they characterize is intrinsically different. The geometric quantity suppressing remainder-unfolding — roughly, proportional to the universe's compactness — reaches its minimum where the scale factor is maximal (the geometric turning point), a single point. But "the remainder unfolded to its fullest" is a developmental result, and development needs time to accumulate: life does not appear the instant suppression hits its minimum, but is built up gradually thereafter. So the peak of development sufficiency can lag the geometric turning point, and can be offset from it by the asymmetry of the contracting phase. The two are expected to be adjacent, in the same cosmic phase, but whether they coincide exactly is an open question. [decoupling of the two points: structural; whether they coincide exactly: open]

Second, and more important to the reader: wherever these two points are, and whether or not they coincide, our earlier cosmological constant is unaffected; the acceleration scale is a subtler matter, made clear together below. This deserves unfolding, because it is the premise under which the timeline revision (§4's asymmetry, the late turnaround) can safely proceed.

The reason is that the pair of frequencies is a whole-cycle topological-clock frequency — the winding frequency of each compact clock phase over one whole cycle, independent of the shape of the "pendulum" within that cycle:

$$\omega_i = \frac{2\pi}{T_i}\quad(i=1,2),\qquad \Lambda = \frac{2(\omega_2^2-\omega_1^2)}{c^2}.$$

The cosmological constant draws only on these two constant frequencies, and does not contain the specific shape of the scale factor $a(\tau)$ in time, nor where in the cycle the turnaround falls, nor whether expansion and contraction are symmetric — it is "blind" to the shape of the breath. We may therefore redraw the line in §4, §7 as asymmetric, pushing the geometric turnaround late or even (in the observed-frame-undecidable sense) into the future, and the expression for the cosmological constant does not move a hair. This is the core guarantee that the timeline revision can safely proceed.

For the acceleration scale, one more layer of caution than for the cosmological constant is needed. It has an equally frequency-only expression $a_0=(\pi/2)\,c\,(\omega_2-\omega_1)$, shape-blind in that sense; but it also has an apparent-horizon expression, which depends explicitly on geometry near the turnaround. Here in passing is a technical point of the latter, used when §7 discusses contraction: at the turnaround, where the scale factor is maximal and $H=0$, naively using the flat approximation $a_0\propto cH$ gives $a_0=0$ — which is wrong. The correct route is the apparent horizon,

$$a_0 = \frac{c^2}{2\pi R_A},\qquad R_A = \frac{c}{\sqrt{H^2 + k c^2/a^2}}\,,$$

for a closed universe $k=+1$, at the turnaround $H=0$ one has $R_A = c/(c/a) = a_{\max}$, still finite, so $a_0 = c^2/(2\pi a_{\max})\neq 0$. How these two expressions (the frequency-only and the horizon-based) reconcile over the whole cycle this paper has not fully clarified — so we rest the stronger "shape-blind" conclusion safely on the cosmological constant, and make only the weaker statement for the acceleration scale: its falsifiable content is in the redshift slope (§5.③, §9), while the reconciliation of the two expressions is listed as an item to be clarified. [cosmological constant shape-blind: structural; reconciliation of the acceleration scale's two expressions: open]

To pin down the old "period is twice the turnaround" relation as well: the whole-cycle duration is itself pinned independently by the scale of the cosmological constant (§5.③); the old picture could write "period equals twice the turnaround time" precisely because it assumed symmetry (turnaround at the half-period). Once symmetry is dropped, "turnaround at the half-period" no longer holds, but the whole-cycle duration is still pinned by that scale — so the turnaround's position within the cycle changes from a quantity locked by the symmetry assumption into an open one set by the degree of asymmetry, and able to fall late. And the fact that life appears at about half the cosmic lifetime then retreats from "an anchor locking the period" to "a coarser, independent corroboration" — that it falls near the half-period still holds, only it no longer bears the locking duty.

As to whether we are, right now, before the geometric turnaround — as §5.③ said, current observation cannot decide; all we can be sure of is that the observed frame is still in accelerating expansion.

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§7 Contraction: falling into black holes, and how one breath returns to low entropy

Past the geometric turnaround (whenever that is), the universe enters contraction. This section is where this paper's original content is most concentrated, and therefore the one where we are most careful, tagging the confidence of each step most finely. Per the stance set in §2: some of the following are mechanism candidates within SAE, with grounded and self-consistent qualitative chains but awaiting quantification; we put them out in full, to be probed and falsified, not as settled.

7.1 Why contraction may be faster than expansion: asymmetry as a working hypothesis

This paper adopts an asymmetric breath as a well-motivated working hypothesis, not a conclusion already established by dynamics — this must be said first. A closed FRW universe of matter and gravity alone is in fact time-symmetric in expansion and contraction (like a cycloid, the two segments equal in duration); "expansion against gravity, contraction with gravity" alone does not give "contraction shorter." What can actually make the fast–slow asymmetry is whether the contracting phase develops a net inward effective driving.

Specifically: in expansion, gravity is always pulling the universe back, and expansion is continually slowed by it. In contraction, gravity is already inward; if dark energy also turns inward here (the "turns negative" branch of §7.3), then gravity and dark energy superpose in the same direction, contraction has no opponent, the contraction segment is naturally shorter than the expansion, and the geometric turnaround falls correspondingly late. But this step hangs on the dark-energy final state: if dark energy goes to zero or stays positive (the other two branches of §7.3), this net inward driving need not appear, and the fast–slow asymmetry is then not fixed by the present argument. So "contraction short, expansion long" is a motivated working hypothesis, whose holding is decided jointly by §7.3's dark-energy final state and the two-clock dynamics, not an accomplished cosmic-dynamical result.

Under this working hypothesis, the contracting phase has two further factors accelerating contraction — rising density strengthening effective gravity, and the irreversibility of black-hole formation (the more that collapses, the more regions become 4DD-inactive, the faster development sufficiency falls); but note that these two also presuppose "a net inward driving has appeared," and cannot independently break the symmetry of pure gravity. [asymmetric breath: motivated working hypothesis, conditional on §7.3's net inward driving; the specific dynamics: open]

7.2 How development sufficiency falls, and a wording caution

In contraction, the fall of development sufficiency we phrase as the rise of the inactive fraction of the fourth layer: structures fall one by one into black holes, and inside a black hole the fourth layer is inactive, so the part of the universe "available for the remainder to unfold" keeps shrinking. Here again the wording discipline of §3: we do not say "the average causal law grows stronger" — the "average causal law" across active and inactive regions is not well-defined to begin with, and black holes are precisely where the fourth layer is inactive. The mechanism of the fall is gravitational collapse, no more.

A natural tool for quantifying this fall is the causal-slot scale introduced in the information-theory series — the scale below which a region can sustain a stable fourth-layer causal readout. It varies inversely with temperature, opening to the macroscopic in a normal universe and retracting locally to the Planck floor in a collapsing region. So the contracting phase can be characterized by a "4DD-inactive volume fraction" tending to one hundred percent at the Crunch; the causal-slot scale, as a spatial field, resolves exactly which regions have already fallen back to the floor and which have not. Giving a specific model for how this fraction grows in time (collapse of overdense regions, mergers, horizon percolation) this paper lists as open. [contraction mechanism: motivated; collapse statistics: open]

Here too a weakness of SAE in the contracting phase must be marked honestly. Contracting universes generally face growing anisotropy (shear grows faster than matter as the universe contracts, generally dominating and leading to chaotic oscillation). The ekpyrotic picture suppresses anisotropy with a steep scalar potential, one of its strengths; whereas SAE's dark-energy equation of state near $-1$ provides no such steep potential, so SAE has no ekpyrotic-style anisotropy suppression. All we can offer at present is a candidate direction: the chaotic oscillation of anisotropy is a phenomenon on a fourth-layer timescale, and as regions fall one by one into 4DD-inactivity (whose interior has no fourth-layer time), these oscillations' "clocks" stop within those regions — if the inactive fraction grows fast enough, the chaos is cut off before it becomes globally severe. This candidate is not out of thin air: it is in fact required by SAE's own consistency — because the remainder is to be carried as a clean signal into the next cycle (see 7.4), the transition cannot be one disrupted by anisotropic chaos. But "whether the fraction grows fast enough" is a quantitative question, and this paper does not pretend it answered. [anisotropy cutoff: open, with candidate direction and consistency motivation]

7.3 The final state of dark energy in contraction: a well-analyzed open question

For contraction to happen, dark energy at least cannot forever be a strong "outward push." But just how it changes — going to zero, turning negative, or weakening while staying positive — is a question we have analyzed fully, but which current posteriors are insufficient to decide. We have thought through all three possibilities, each with its structural motivation and its own burden to discharge:

• To zero: the dark-energy term dilutes away, or that causal scalar field rests at the zero of its potential, and contraction is borne by pure gravity. Its motivation is that this falls within $w\geq-1$ and needs no negative-energy structure; what it must discharge is — whether pure-gravity contraction can recover the work that dark energy did outward during expansion, and why contraction is shorter.
• Turning negative: the dark-energy density turns negative, driving contraction in the same direction as gravity. Its motivations are several: it gives a closed sign cycle (outward-pushing expansion, inward-pulling contraction), it naturally explains why contraction is shorter, and it closes the energy ledger (unfolded in expansion, recovered in contraction); moreover, structurally, this framework's cosmological constant is a signed quantity, a difference of two frequencies, so taking a negative value is the generic case, while going to zero would require the two frequencies to be exactly equal — a non-generic coincidence. What it must discharge is — the field-evolution mechanism for dark energy to turn negative is yet to be established, and a genuinely negative energy density needs an orientation-sensitive structure (a point cognate with §8's cosmological-constant-sign question).
• Weakening while staying positive: dark energy weakens but does not change sign, the smallest modification. What it must discharge is — being still outward-pushing, how it is overcome by curvature and gravity into contraction, and the contraction rate.

We set these three side by side, with no preset priority — this is a well-analyzed open question, not an unconsidered shrug. To decide it needs the field dynamics of the contracting phase (this paper lists it as one of the keys) and future observation (contraction rate, or the redshift at which the deceleration parameter turns from negative to positive). One precise word must be added on the observational criterion: the deceleration parameter we observe belongs to the observed frame, and its turning from negative to positive corresponds to the moment dark energy can no longer hide the geometric contraction, not necessarily the geometric turnaround itself — between the two lies the frame transformation of §5.③, §6. So "the deceleration parameter turning positive at some future redshift" as a falsifiable point tests the apparent turnaround of the observed frame, not directly the geometric turnaround. [dark-energy final state: a well-analyzed open question, three branches side by side with no priority]

One honest word on the energy ledger is also needed, because it is exactly our reason for not favouring any one branch here. One might think: since the two sides' ledgers of one breath are opposite and sum to zero, could the ledger accumulated this side in expansion be handed wholesale to the other side at the transition, so this side need not balance its own? We shall explain in 7.4 that the handover across the two-side interface is only a small leak, not a wholesale transfer; so this side's energy ledger must, in the great majority, be settled on this side. That is, how the work dark energy did outward in expansion balances on this side is a ledger not yet settled — and precisely because it is unsettled, we favour neither "to zero" nor "turning negative," listing them side by side as they are.

7.4 How one breath returns to low entropy

Now to the crux of this section, and of the whole cyclic narrative: contraction has reached the Crunch, entropy is now at a maximum (matter all fallen into black holes, total horizon area maximal); and for the next Big Bang to be a low-entropy, smooth start, entropy must return from maximal to very low. How does this step happen?

First clear up an easily confused point. SAE has a fourth-layer conserved quantity, the causal load $I$, related to energy by $E=Ic^3$ and so of dimension $E/c^3$ — a dimensional physical quantity, but its dimension is not an energy dimension; the information-theory series states explicitly that this causal load is not entropy, not bits, not an information capacity. The entropy we speak of here is horizon-configuration-based (its logarithm being the horizon-area kind). The two differ: the causal load is the carried quantity (dimension $E/c^3$); the entropy $S_H$ carries entropy units, and what is dimensionless is $S_H/k_B$ — readable as the logarithm of an effective count of configurations. Telling these two apart is the premise of the step below — because one is conserved and carried, the other is not conserved and downgraded.

The mechanism candidate is as follows, and we mark the grounding of each step. At the Crunch, the whole universe becomes an everywhere-4DD-inactive state. It differs from an ordinary black hole in one key respect: an ordinary black hole's horizon has an active universe outside "reading" it, so the horizon is fourth-layer-active; whereas when the whole closed universe collapses, there is no exterior — not that there is nothing beyond the outermost layer, but that for a closed three-sphere the category "outside" simply does not apply ("what is outside" asks after the inaccessible thing-in-itself). So the horizons carrying the causal load lose the "being-read" relation on which they rest, and the fourth-layer closure dissolves with it.

It is right here that the tension flagged in §3 is reconciled. The information-theory series says fourth-layer information is irreversible, the causal load only fills, never empties — but that is the rule while the fourth-layer closure holds. The Crunch is precisely where that closure itself dissolves: the external relation severs, the horizons de-crystallize, and the fourth layer no longer exists as a closed layer. The premise of the rule no longer being met, the causal load can fall back — down the dimensional ladder to the bottom, becoming energy. This does not violate "irreversibility within the fourth layer," because here there is no fourth-layer interior to speak of. [causal load falling back: motivated candidate; reconciliation with information-theoretic irreversibility as above]

The causal load falls back to energy, but energy cannot exist "bare" — it must be re-expressed as light (the second layer) or massive particles (the second and third layers). This "fall-back re-expression" occurs entirely inside the Crunch-to-next-Bang, with no external disturbance and no classical structure inherited from the previous cycle to serve as a seed. Here we must connect to that not-yet-earned open item of §7.2, and can earn only a weaker conclusion, speaking conditionally: if the 4DD-inactive cutoff can indeed halt the growth of anisotropy before BKL chaos dominates, driving the transition boundary toward low Weyl curvature, then the previous cycle transmits no macroscopic, classical non-uniformity to the next, and a low-Weyl, near-uniform boundary becomes a natural candidate. But this does not amount to "necessarily uniform": even without inherited classical seeds, quantum fluctuations, vacuum instabilities, spontaneous symmetry breaking, dynamical unstable modes, even the transition matching itself, could still produce perturbations. So the next cycle's primordial fluctuations and their spectrum must still be given by the transition dynamics, and this paper does not pronounce on them. As to why the horizon-configuration entropy goes low here: the horizon structure it counted has dissolved in the de-crystallization, and its configurations collapse — it is not destroyed, but has lost the structure on which "high entropy" depended.

Here the "downgrade" of entropy must be phrased precisely, lest it read as a continuous descending curve. It is not a well-defined $S(\tau)$ smoothly falling from maximal back to minimal, but the domain of the previous cycle terminating, and the next cycle restarting at a low-entropy boundary — in symbols:

$$S_n(T_n^-)\ \text{high},\qquad S\ \text{undefined in the transition core},\qquad S_{n+1}(0^+)\ \text{low}.$$

Horizon de-crystallization is a candidate mechanism for "why the boundary condition on the next side is low-entropy," not a given equation of entropy continuously falling back. In this sense the chain is conceptually end-to-end: maximal entropy (end of the previous cycle) — entropy undefined in the transition core, causal load falling back to energy, remainder carried — low-entropy start of the next cycle. And energy (the causal load) crosses over conserved. This, then, is the candidate mechanism behind "entropy is not carried, remainder is" — the carried being energy and remainder, the downgraded (more precisely: domain-terminated then restarted low) being entropy. It may also offer a within-framework direction for the high uniformity of the microwave background: not smoothing non-uniformity by inflation, but the Crunch erasing old structure and the transition interior having no seed; but whether this candidate can replace or supplement inflation's account of uniformity and the perturbation power spectrum still awaits the dynamics and the power-spectrum computation, and this paper makes no assertion. [entropy return: motivated candidate (domain termination + low restart, not a continuous trajectory); quantification and replaceability of inflation: open]

Finally, on that bit of handover between the two sides, and why it is only a small leak. Above we spoke of the causal load's fall-back as this side's affair. But one breath of this framework has two sides (a pair of fourth layers with opposite time arrows), with a reciprocal interface between them. Across this interface, a small part of this side's information can leak to the other side — and because the other side's time arrow is reversed, this side's "already-read information" presents, to the other side, as exactly "not-yet-read remainder." This is precisely the two-sided face of "already-unfolded" and "to-be-unfolded," and precisely the content of that "the two sides' ledgers sum to zero" conservation. But this interface is a reciprocal interface, not a conduit: it permits partial leakage, not wholesale passage — and were information to transfer wholesale to the other side, this side's own next Big Bang would have nothing to go on. So the great majority of the causal load falls back on this side (as above), and only a small part leaks across. And it is exactly this small leak that becomes the source of drift between cycles: the remainder the next cycle inherits has mixed into it the part leaked from the previous cycle, so each round's parameters differ slightly — this being another layer of why the line is a spiral rather than a closed circle. [partial leakage and spiral: motivated; leakage dynamics: open]

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§8 The sign of the cosmological constant: an open question that must stand alone

By here, one question can no longer be avoided, and we give it its own section, because without facing it the whole cyclic narrative would stall at this point: the universe unfolded in the next cycle — is its cosmological constant positive (and so, like us, accelerating in expansion), or negative (and so decelerating, even collapsing)?

The question is sharp because of a seemingly obvious but in fact untenable intuition: since the other side's time arrow is reversed, would the sign not flip of itself? It would not. In standard dynamics, time reversal does not change the direction of acceleration — acceleration is invariant under time reversal, and the cosmological constant, as a geometric source term, does not change sign merely because the time arrow flips. So from "the other side's time arrow is reversed" alone, "the other side sees a positive cosmological constant" does not follow. This must be said, or one reaches a pretty but wrong conclusion.

To advance this question, we consider a conditional branch carrying a selection effect: if the other side can form long-lived complex structure and observers, then the effective gravity and expansion history in its own frame must fall in the region allowing such structure to form — gravity attractive, matter able to gather into galaxies, an expansion history that neither tears structure apart nor crushes it too early. This is a condition, not an established fact.

Under this condition, the two-side relation given by this framework's corpus offers a motivation: a third-layer symmetry gives a pair of fourth layers with opposite time arrows, this side and the other side generated by the same structure with only the parameter order reversed, and our Big Bang is the other side's Big Crunch. So the structure of two-sided common origin offers a motivation for "the other side, in its own frame, is qualitatively of the same type as this side" — namely: in the branch that can form structure, the other side too, seen by itself, should be gravity-attractive, cosmological-constant-positive, accelerating, structure-forming. But it must be stressed this cannot by itself yield that the other side's cosmological constant must be positive, nor exclude other branches: observer-less negative-cosmological-constant cycles, cycles able to form briefly but then rapidly recollapse, or branches whose orientation transformation differs from present intuition, are none of them excluded by this motivation. Nor do we answer "seen from our side, is the other side's cosmological constant positive or negative" — that depends on a cross-frame orientation transformation yet to be formalized; what is stated here is only the picture "in the branch that can form structure, and as regards the other side itself."

Back to the criterion at the section's opening — the next cycle's (i.e. the other side's) acceleration, positive or negative, seen from its own frame. What we can offer is a strong motivated preference, not a decided answer: in the class of other-side branches able to form complex structure, its own ledger should be qualitatively of the same type as ours — gravity attractive, cosmological constant (as regards itself) positive, universe accelerating, structure able to form. But this is a conditional judgment carrying a selection effect: supported by the premise "the other side must be able to form structure, must have observers," it therefore covers only the "subclass of other sides able to form complex structure," and does not exclude other branches — an observer-less negative-cosmological-constant cycle, a cycle able to form briefly but rapidly recollapse, a branch whose orientation transformation differs from present intuition. The sign question itself must still be decided by the orientation-transformation law of the two-clock action.

One caution on register must be stressed, bearing on the whole picture being a spiral rather than a circle: the "same type" above means — within the class of structure-forming other-side branches covered by the selection effect — that gravity-attractive, cosmological-constant-positive, accelerating, structure-forming qualitative structure is the same, and not wholly identical. The remainder is carried across (§7.4), making the next cycle's parameters slightly different from this round's — so, in this branch, the next cycle is the next turn of this spiral, a universe structurally the same as ours but with drifted parameters, not a replica wholly identical to us. This picture does not exclude the existence of the other branches above; it is merely the branch we prefer under present motivation and selection effect.

This picture and the corpus statement "our Big Bang is the other side's Big Crunch" are two sayings of the same thing; it is also a CPT-type mirror. (In passing we correct an earlier judgment of ours: this framework and those CPT-symmetric-universe approaches in fact share the mirror structure, only this framework further superposes the carrying of the remainder and the spiral it brings; we had earlier drawn the two too far apart.)

As to the observational-side support for this judgment, honestly marking its weight: this universe of ours is itself the product of the previous transition, and we observe on nearby galaxies a positive acceleration scale — a positive-definite threshold quantity, observationally harder than the sign of the cosmological constant. "The product of the transition is an accelerating, positive-scale, structure-forming universe" — we thus hold an existence proof: ourselves. This is aligned with the argument above, one from structural generation, one from observation, jointly supporting the judgment "the next cycle, seen from itself, is of the same type as us." [nearby $a_0>0$, $q_0<0$: posterior; as $n=1$ support for the next-cycle picture]

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§9 Falsifiable points

A framework-level paper that does not say how it could be refuted has not done its duty. This section collects the falsifiable points scattered through the preceding text, sorted by decisiveness and by tier. One discipline to keep: falsifiability is a strength of this framework, not a weakness — a framework that states clearly "if observation is thus, then I am wrong" is precisely a scientific framework.

Unconditional hard predictions (structural; refutation shakes the framework itself):

• The high-redshift evolution of the acceleration scale. This framework predicts the scale to evolve with redshift in a specific way, its decisive differences concentrated at $z\gtrsim3$. If the high-redshift measurement shows it does not so evolve, the horizon mechanism on which this framework rests to give the dark sector fails. This is the cleanest, most concentrated decision point.
• The intrinsic equation of state $w_{\text{int}}\geq-1$. If a non-parametric reconstruction of the equation of state robustly shows an intrinsic phantom crossing (rather than merely an apparent crossing under some parametrization), this framework's dark-energy structure fails.

Structural requirements in tension with observation (to be marked honestly):

• A closed three-sphere. This framework needs a globally closed space. Current curvature measurements are highly close to flat, which puts pressure, a lower bound, on the radius of closure, but does not directly exclude topological closure — what is required is that the curvature be small enough, or the scale of closure large enough. If future surveys tighten the curvature constraint further to some degree, this framework will be in considerable difficulty; if the space is confirmed open, the closed picture is refuted. We mark it honestly as a requirement and a pressure, not a supported fact.

Apparent turnaround of the observed frame (to be distinguished from the geometric turnaround):

• A future turning-positive of the deceleration parameter. This framework's asymmetric picture expects a future turnaround. But as §5.③, §7.3 stressed repeatedly, the deceleration parameter we observe belongs to the observed frame, and its turning from negative to positive corresponds to the moment dark energy can no longer hide the geometric contraction, not necessarily the geometric turnaround itself — a frame transformation lies between. So this falsifiable point tests the apparent turnaround of the observed frame. If, in the long run, acceleration is robustly maintained with no sign of any turning-positive, this framework's turnaround-and-contraction picture is seriously challenged.

Awaiting theoretical decision (not observational, internal to the framework):

• The Crunch-to-Bang transition mechanism, the sign of the cosmological constant, and the quantification of the entropy return, are all incomplete. If some quantum-gravity theory proves such a transition impossible, or proves entropy cannot reset there, this framework's cyclic picture is set back.

A word on tiers, to close the section: the unconditional hard predictions above (the high-redshift acceleration scale, the intrinsic $w_{\text{int}}\geq-1$) are decidable separately from the whole "cyclic" picture. If the high-redshift acceleration scale is refuted, what is shaken is the horizon mechanism and the dark sector; whereas the "cyclic" is a framework-level, well-analyzed but still open picture, which does not stand or fall with the former. Keeping these two tiers apart is so that success or failure in one place is not misread as success or failure globally.

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§10 Conclusion, and the "keys for the next layer"

This paper has tried to give one complete cosmic breath — from the Bang, through expansion, turnaround, and contraction, to the Crunch, and on to the next Bang — as a connected line. To sum up that line in one sentence:

> One complete breath is not a circle but a remainder-conserving spiral. The carried are remainder and energy, the downgraded is entropy; the sufficiency of remainder-unfolding goes from low to high to low (single-peaked, its two ends equal in value while entropy differs, showing it non-equivalent with and non-substitutable for entropy), entropy grows monotonically within each fourth-layer-active segment, and the scale of causal structure opens from the Planck floor and collapses back. The Bang and Crunch are symmetric in development sufficiency, asymmetric in entropy; the two low troughs are connected by three distinct mechanisms (generation, leak-handover, collapse), not by a simple time mirror.

We have not spoken every segment of this line to its end, nor do we claim to. Some links — especially the Crunch-to-Bang transition, the entropy return, the sign flip of the cosmological constant — are at present mechanism candidates within this framework, their qualitative chains grounded and self-consistent, but their quantification incomplete, and quite possibly to be revised or refuted. We have put them out in full, precisely in the spirit of the opening: not to be free of error, but to be of some use. If this picture gives the physics community a target that can be probed and falsified, the paper will have served its purpose.

The "keys" left for the next layer, to be genuinely turned, we list here, lest they be mistaken for solved:

• The dynamics of the first leap — the step from 0DD to 1DD, where time is not yet generated. Its mechanism (chaos self-negation) can be stated, but its dynamics cannot yet be characterized in "post-time" language.
• The two-clock action — a phase effective action applicable only in the effective stages of the breath (after the first layer, the fourth layer active), which would give the continuous expansion rate, the handling of the two clocks at the transition, and the decision of the cosmological-constant sign and the dark-energy final state. Appendix C gives a skeleton of it, but stresses that is a skeleton, not the final dynamics.
• The collapse statistics and anisotropy of contraction — a specific model for how the 4DD-inactive fraction grows in time, and whether the anisotropy cutoff can happen in time.
• The quantification of the entropy return and the two-side leak — how far the horizon count collapses, and the dynamics of the cross-interface leak.
• The full Boltzmann solution and the curvature/topology constraints — the ordinary but necessary work of splicing this line into standard cosmology's numerical checks.

These are not omissions of this paper, but the boundary it has drawn: it offers a map, a set of order parameters, a list of falsifiable points, and honestly leaves the solving of the dynamics to the next layer.

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Appendix A · The claim–tier table

For the reader's claim-by-claim checking, the table below collects the paper's main claims and their confidence tiers. For the meaning of the tiers see §2: [structural] framework-forced, falsifiable; [posterior] dependent on observation or parameters; [motivated] a well-motivated preference with unexcluded alternatives; [open] awaiting decision.

Claim	Tier
$\mathcal{D}_R$ as the main order parameter and its generalized single-peaked shape	motivated (framework ansatz)
Given the endpoint behaviour, $\mathcal{D}_R$ and $S$ non-monotonically-equivalent and non-substitutable; $\mathcal{R}_{\rm tot}$ a separate conserved ledger	conditional structural conclusion
The three ledgers non-substitutable ($\mathcal{D}_R$–$S$ non-equivalence conditional structural; $\mathcal{R}_{\rm tot}$ separate by type and conservation role)	see the two rows above
Main line is remainder-development sufficiency, not average causal law	structural (wording discipline)
"Unfolding" inherits the third information-theoretic primitive; entropy monotonicity inherits information-channel one-way accumulation	structural (inherits the information-theory series)
High-redshift evolution of the acceleration scale ($z>3$ decision)	structural (unconditional hard prediction)
Intrinsic $w_{\text{int}}\geq-1$	structural (unconditional hard prediction)
Cosmological constant $\Lambda$ insensitive to the breath's internal shape	structural (conditional on the whole-cycle frequency reading)
Reconciliation of the acceleration scale's frequency and horizon expressions	open
No particle dark matter (geometrized khronon)	structural
Early universe an ontological account, not numerical prediction differing from standard	structural (differences concentrated in the boundary-condition layer)
Bang as chaos self-negation; $d=4$ from the exhausted tetralemma	structural (mechanism); threshold value open
Periods $T_1$ (pinned by the $\Lambda$ scale), $T_2$ (by inverting $\Lambda$); $c$ a derived boundary condition	posterior / boundary input
Cosmological constant a conditional prediction (on life-anchoring and the merger identification); unconditional hard predictions only the acceleration scale and $w_{\text{int}}$	posterior / conditional
Nearby $a_0>0$, $q_0<0$	posterior (observation)
5DD anchored at the development-sufficiency peak (unfolding threshold / causal-slot breakthrough); life near the half-period a coarse corroboration	motivated
Asymmetric breath (contraction faster than expansion); geometric turnaround late, whether already crossed undecidable in the observed frame	motivated working hypothesis, conditional on §7.3; position open
Crunch a cosmological-scale closed critical state (of a kind, not a Schwarzschild replica)	motivated
Low-entropy-restart candidate: horizon de-crystallization → entropy domain terminates → conditional low-Weyl / near-uniform boundary; primordial fluctuations and spectrum open	motivated (candidate); quantification open
Causal load can partially leak to the other side → source of the spiral	motivated; leakage dynamics open
The two-stage transition: on approach, fourth-layer closure first retreats to L4b (arrow kept, single direction lost); in the transition core, the two clocks and all of 4DD fail, reaching the 0DD timeless boundary	motivated (inherits the layer subdivision)
Sign of the cosmological constant: time reversal alone does not flip $\Lambda$; the other side, in its own frame, of the same type as us (next cycle accelerating)	motivated (strong preference, on the "must form structure" selection-effect premise); action-level formalization open
Anisotropy cutoff, two-clock action, first-leap dynamics, full Boltzmann, curvature/topology	open (keys for the next layer)

Appendix B · Related cyclic and bouncing cosmologies

This paper regards the following approaches as explorations, from different axiomatic entry points, of the same deep structure, of which this framework offers one; below we state similarities and differences and identify each one's falsifiable points, without ranking.

• Penrose's Conformal Cyclic Cosmology handles entropy between aeons by conformal rescaling. The difference from this framework is that this framework does not carry entropy across aeons nor rely on rescaling, but lets entropy downgrade at the transition via horizon de-crystallization, carried across by energy and remainder. The transition mechanisms differ.
• The Steinhardt–Turok ekpyrotic and cyclic braneworld is the closest structural analogue to this framework's two-side structure — two entities, periodic relative motion, driven by an interface potential, opening a new cycle by collision, an isomorphism on four counts; its inter-brane potential is also an existence precedent for "interface-reaction driving." But one distinction must be held: this framework's two sides are two sides at the ontological-layer level, their clocks phase-space frequencies, not two branes in a higher-dimensional space — its structure may be borrowed as analogy, but its higher-dimensional gravitational-potential mechanism may not be imported. Ekpyrotic's suppression of anisotropy by a steep potential is its strength; this framework has no such mechanism (§7.2).
• The Boyle–Turok CPT-symmetric universe — its mirror structure (our Big Bang is the other side's Big Crunch) is shared by this framework; this framework's difference is in further superposing the carrying of the remainder and the spiral it brings (parameters may drift each cycle). So the two are told apart not by the mirror itself, but by the spiral and the handling of entropy (this framework by horizon de-crystallization, not dilution).

Appendix C · A phase-effective skeleton of the two-clock action

To stress: the following is a skeleton, not the final dynamics, and applicable only in the effective stages of the breath (after the first layer, the fourth layer active, i.e. the bulk from expansion to contraction), not to the first leap or the transition core. Its form is a usual gravity–scalar–form-field action, with the sum $\theta_+=(\theta_1+\theta_2)/2$ and difference $\theta_-=\theta_1-\theta_2$ of the two clock phases as effective coordinates:

$$S_{\rm eff} = \int d^4x\,\sqrt{-g}\;\Big[\tfrac{M_{\rm P}^2}{2}R \;+\; \mathcal{L}_{\rm kh}(\theta_+, g) \;+\; \mathcal{L}_{C}(\theta_-, g) \;+\; \mathcal{L}_{F_4}(F_4) \;+\; \mathcal{L}_{\rm rem}\Big],$$

where $\theta_+$ (the breath phase) drives the geometrized khronon term $\mathcal{L}_{\rm kh}$, $\theta_-$ corresponds to that causal scalar field $\mathcal{L}_C$, $F_4$ is the four-form giving the cosmological constant, and $\mathcal{L}_{\rm rem}$ subsumes the remainder. The scale factor's evolution with phase may take the parametric form of a closed universe, its shape exponent not fixed by hand. Here we adopt the two-stage reading consistent with Appendix A: before the transition, the fourth-layer closure first retreats to L4b (arrow kept, single direction lost), where the above is still marginally applicable; in the transition core, the two clocks and the whole fourth-layer description fail together and step down, reaching the 0DD "timeless" boundary, to be regenerated in the new cycle — it is at this innermost core, not at the retreat to L4b, that "0DD timelessness" is met. The completion of this skeleton — giving the continuous expansion rate, the matching conditions at the transition, and the decision of the cosmological-constant sign and the dark-energy final state — is one of the "keys" listed in the main text. To stress again: this is only a ledger-keeping skeleton; the specific forms of its terms, their couplings, even whether they can be so separated, are all to be determined, and it should not be read as a settled dynamics.

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References

Related cyclic and bouncing cosmology, and black-hole entropy

• Bekenstein, J. D. (1973). Black Holes and Entropy. Physical Review D, 7, 2333–2346. DOI: 10.1103/PhysRevD.7.2333.
• Hawking, S. W. (1975). Particle Creation by Black Holes. Communications in Mathematical Physics, 43, 199–220. DOI: 10.1007/BF02345020.
• Khoury, J., Ovrut, B. A., Steinhardt, P. J., & Turok, N. (2001). The Ekpyrotic Universe. Physical Review D, 64, 123522.
• Steinhardt, P. J., & Turok, N. (2002). A Cyclic Model of the Universe. Science, 296, 1436.
• Penrose, R. (2010). Cycles of Time: An Extraordinary New View of the Universe. Bodley Head.
• Boyle, L., Finn, K., & Turok, N. (2018). CPT-Symmetric Universe. Physical Review Letters, 121, 251301.
• DESI Collaboration (2025). DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints. arXiv:2503.14738.

SAE series (upstream, all by the present author on Zenodo)

• Qin, H. (2026). SAE Foundation / Methodology (ZFCρ). Zenodo. DOI: 10.5281/zenodo.19361950.
• Qin, H. (2026). Mass Series Convergence ($E=Ic^3$). Zenodo. DOI: 10.5281/zenodo.19510868.
• Qin, H. (2026). SAE Information Theory I: The 4DD Ontology of Information. Zenodo. DOI: 10.5281/zenodo.19740020.
• Qin, H. (2026). SAE Information Theory VII: The Spark of Life (the "unfolding" primitive). Zenodo. DOI: 10.5281/zenodo.20105884.
• Qin, H. (2026). SAE Quantum Mechanics P8 (arrow vs direction of time; our Big Bang is the other side's Big Crunch). Zenodo. DOI: 10.5281/zenodo.20587634.
• Qin, H. (2026). Four-fold Nesting of the Physical-Quantity Ladder (the 4×4 sub-grid, L4a–d). Zenodo. DOI: 10.5281/zenodo.21227329.
• Qin, H. (2026). SAE Cosmology I (cosmological constant, closed FRW). Zenodo. DOI: 10.5281/zenodo.19245267.
• Qin, H. (2026). SAE Cosmology III ($a_0=(\pi/2)c\,\Delta\omega$). Zenodo. DOI: 10.5281/zenodo.19281983.
• Qin, H. (2026). SAE Cosmology V (dual-4DD opposite arrows; $\Lambda_1+\Lambda_2=0$). Zenodo. DOI: 10.5281/zenodo.19329771.
• Qin, H. (2026). SAE Cosmology VI (dark sector: khronon dust, causal scalar field, four-form; HP1/HP2). Zenodo. DOI: 10.5281/zenodo.21267555.
• Qin, H. (2026). Thought Experiment I (the symmetric breath, the 5DD-midpoint as working hypothesis). Zenodo. DOI: 10.5281/zenodo.19028005.

(Volume/issue/page for the external classics are per their original sources; the SAE series is authoritative as its Zenodo record.)

Acknowledgment

The framework direction, the core arguments, and the final adjudication of all claims are the responsibility of Han Qin (秦汉). In the course of writing, the author drew on several artificial-intelligence systems for conceptual organization, derivation checking, geometric consistency checking, and adversarial critique, with further AI systems taking part in divergent discussion and external evaluation; the contribution of these systems is limited to auxiliary organization, checking, and criticism, and constitutes no authorship of the paper's scientific claims.

Statement of author responsibility

All claims in this paper, their tier tags (structural / posterior / motivated / open), and every open question and possible error marked, are the author's own responsibility. Several places here are mechanism candidates within the SAE framework, not yet quantified and possibly to be falsified; the author presents them in full on the principle of "not to be free of error, but to be of some use," in the hope of offering subsequent research a target that can be probed and falsified. Any omission or error is the author's.

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Concept DOI: 10.5281/zenodo.21319569 (SAE Cosmology Paper 0). Chinese is the authoritative version; this English text is an independent rewrite, not a line-by-line translation.

§1 引言：为什么要完整地写一次宇宙呼吸

SAE 的宇宙学系列此前是"分块"推进的。宇宙学常数从一对拓扑时钟的频率差得出；暗物质由几何化的 khronon 尘埃承担，无需新粒子；暗能量与 MOND 的加速标度由同一对频率的一阶差给出，并带有一个可在高红移检验的演化预言；有效引力常数随宇宙年龄有一段平台再上升的行为。这些结果各自成篇，各有其可证伪的观测接口。

但一个自然的问题始终悬在这些分块结果之上：如果把它们放回一条时间线，从最初到最终、再到"最终之后"，这条线长什么样？ 这不是把已有结果重新排列一遍就能回答的。它要求我们说清几件此前未被系统处理的事：

• 大爆炸这一"起点"在 SAE 里究竟是什么——是一个需要外部引信的奇点，还是别的东西？
• 宇宙走向终结的方式，是无限稀释的热寂，还是某种收缩？如果是收缩，由什么驱动，又通向何处？
• 如果存在"下一次"大爆炸，它与这一次是同一个循环的重复，还是每一轮都有所不同？携带什么、丢弃什么？
• 贯穿这整条线的、真正在变化的那个量，是什么？

本文就是要把这条线作为一个整体、在定性的层面勾勒出来。定位上，它是 SAE 宇宙学的一篇框架性、前置性的工作——它提供的是一张六阶段的本体地图、一组刻画各阶段的序参量、以及一份可证伪点清单；它不提供一个完整的动力学解。凡是需要完整场方程数值求解才能定的量（例如收缩相的具体坍缩统计、双时钟作用量的完整形式、以及最初那一跳的动力学），本文诚实地将其标为"下一层的钥匙"，列出而不假装已经拧开。

之所以名为"0"，一是时序上它在宇宙学常数（我们此前系列的第一篇）之前；二是这一编号本身就带着我们的姿态——它是一篇打地基、开话头的文章，不声称对这条线上的任何一段已经说到底。

对相关的循环与反弹宇宙学进路——Penrose 的共形循环宇宙学（CCC）、Steinhardt–Turok 的 ekpyrotic 图景、Boyle–Turok 的 CPT 对称宇宙——本文在相应之处会陈述 SAE 与它们的异同。我们的态度是：这些是从不同公理入口对同一深层结构所作的探索，SAE 提供其中一种；我们陈述差异、指出各自的可证伪点，而不评判高下。这一态度并非客套：SAE 框架的一条基本承诺是任何构造都留有余项、都不完备（详见 §3 与后文），因此若声称"我全对而他人全错"，恰恰违背了这一承诺。

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§2 认识论契约：本文的主张分层与姿态

在展开任何具体图景之前，我们先与读者立一个契约：本文并非所有主张都有相同的把握程度，我们把每一条的把握程度显式标出。 这既是对读者负责，也是上述"抛砖引玉"姿态的技术实现——只有逐条标清哪些是硬的、哪些是软的，"不求全对"才不是一句空的谦辞，而是可以逐条核查的东西。

全文的主张分为四个层级：

〔结构性〕——由 SAE 框架的内在逻辑所强制，原则上可证伪。这类主张若被观测或理论证否，动摇的是框架本身，而非某个可调参数。本文中最重要的两个结构性预言是：加速标度 $a_0$ 随红移的演化（在 $z>3$ 处可判决），以及内禀暗能量状态方程 $w_{\text{int}} \geq -1$（不穿越 phantom 界）。

〔后验〕——依赖观测输入或参数取值。这类量 SAE 不从纯结构导出，而是取自宇宙学或天文数据（正如广义相对论并不预言引力常数 $G$ 的数值）。呼吸的两个周期 $T_1, T_2$、光速 $c$（作为一个可从其他边界条件导出、但仍属边界条件性质的量）、暗物质与暗能量的密度参数，都属此类。需要强调：找到边界条件之间的一条关系（从而减少独立输入的数目），与从无到有导出一个常数，是两回事——前者是正常的物理，后者才是过度的声称。

〔动机性〕——有充分动机、但并非框架强制的偏好。这类主张我们认为是当前最合理的图景，但承认存在未被排除的替代。收缩相中暗能量的具体终态、大坍缩与黑洞内部的类比、单峰曲线本身作为一个定性结构，都属此类。

〔开放〕——尚未定的问题，我们列出而不假装已答。有些开放问题附有"最有希望的候选方向"，但方向不等于结论。

这套分层贯穿全文；在正文相应处逐条标注，并在附录中汇总为一张主张—层级表，供读者一览。凡属于〔动机性〕或〔开放〕的机制候选——本文有若干，尤其集中在从大坍缩到下一次大爆炸的过渡一节——我们会在该处显式声明其候选性质：标出它的 corpus 依据（哪些环节是硬的）、它的候选成分（哪些是我们提出的图景）、以及它的待定量之处（哪些还没算）。我们把这些机制摆出来供推敲与证伪，而非声称已经定论。

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§3 三本账：余项、展开充分度、熵

要把一次呼吸讲成一条连贯的线，先得说清"在这条线上，究竟是什么在变"。SAE 早期的一个自然想法，是用"因果律的平均强弱"来当这个变量——膨胀期因果律弱、收缩期因果律强。但这个想法有一处内在的不自洽：在 SAE 里，黑洞内部的第四层（4DD，即时空/因果层）是不活跃的。于是"收缩期黑洞越来越多，所以平均因果律越来越强"这句话自相矛盾——黑洞恰恰是因果律不活跃的区域。跨越活跃与不活跃区域去谈一个"平均因果律"，这个量本身就没有良好定义。

因此我们改用另一个变量作为主线，它不背这个包袱：余项展开的充分度。直观地说，它刻画的是"余项被展开到了多充分的程度"——从最初混沌的、尚未展开的状态，到复杂结构与生命充分涌现的状态，再到一切重新塌回、无处展开的状态。这条曲线是单峰的：

• 在大爆炸处最低——层级阶梯刚刚起步，余项尚未展开。
• 沿膨胀上升——阶梯逐层建立，结构与更高层级（生命、意识）依次涌现。
• 在转折附近达到峰值——发展最为充分。
• 沿收缩下降——结构逐一塌入第四层不活跃的黑洞，宇宙中不活跃区域的占比上升，可供展开的空间随之收缩。
• 在大坍缩处再次最低——整个宇宙成为一个大的、第四层处处不活跃的态，余项无处展开。

这里有一处措辞需要谨慎：收缩期展开充分度的下降，我们不表述为"平均因果律变强"，而表述为"第四层不活跃区域的占比上升"（结构塌入黑洞）。机制是引力塌缩，不是因果律把余项抹平。〔动机性〕

这里我们要说明这一主线并非本文新造。"余项无法在当前层自我闭合、于是突破到下一层"这一动力学，在 SAE 的信息论系列中已被确立为一个本原语——称为"展开"（unfolding）。信息论系列此前把"广播"与"接收"作为第四层信息的两种层内、横向动力学；而"展开"是跨维、纵向的第三种，它描述信息如何在层与层之间跨越，形式上对应余项框架中的"凿—构造"循环。本文的"余项展开充分度"，正是这一已确立的本原语在宇宙学尺度上的应用：整条单峰曲线，刻画的就是余项在宇宙一生中被展开的纵向进程。〔主线承信息论第三本原语，非本文新造〕

现在是本节的核心。这条曲线，很容易与另外两个量混淆，而把它们分清，是理解整个循环为何是"螺旋而非圆"的关键。我们把三个量记为宇宙相位 $\tau$（一个随呼吸单调推进的参数，取开区间 $\tau\in(0,T_1)$，两端点对应奇异过渡、单独处理）的函数：余项展开充分度 $\mathcal{D}_R(\tau)$、熵 $S(\tau)$、被携带的余项 $\mathcal{R}(\tau)$。须先声明其地位：$\mathcal{D}_R$ 是一个现象学的序参量，用以刻画"余项被展开到多充分"，本文尚未把它构造为一个从第一性原理导出的可观测量——它的完整构造属于后续工作；下面关于它的陈述，是在"存在这样一个序参量、且具有所述定性行为"这一前提下进行的。三个量的行为如下：

$$\mathcal{D}_R(\tau)\ \text{广义单峰}:\ \exists\,\tau_\in(0,T_1)\ \text{使}\ \mathcal{D}_R\ \text{在}\ (0,\tau_]\ \text{不减、在}\ [\tau_*,T_1)\ \text{不增},\quad \lim_{\tau\to 0^+}\mathcal{D}_R=\lim_{\tau\to T_1^-}\mathcal{D}_R=\mathcal{D}_R^{\min}$$

$$S(\tau)\ \text{单调不减（序关系，不要求可微）}:\quad \tau_a < \tau_b \;\Longrightarrow\; S(\tau_a)\leq S(\tau_b)\ \text{（在每一第四层有效的区段内）}$$

$$\mathcal{R}_{\rm tot}\ \text{守恒}:\quad \mathcal{R}_{\rm tot}\big|_{\tau\to T_1^-}=\mathcal{R}_{\rm tot}\big|_{\tau\to 0^+\,(\text{下一循环})}$$

（"广义单峰"允许峰顶为一段平台，而非必为单点；$\tau\to0^+$ 记大爆炸邻域，$\tau\to T_1^-$ 记大坍缩即下一循环大爆炸的邻域。）关于 $\mathcal{R}$ 需分清两个层次：被携带并严格守恒的是总余项 $\mathcal{R}_{\rm tot}$；而下一循环所继承的、决定其参数（$T_1,T_2$ 等）的，是 $\mathcal{R}_{\rm tot}$ 在边界条件上的一个投影，这个投影逐循环可漂（§7.4 的部分渗漏）——守恒的是总账，漂移的是投影，二者不矛盾。

由此可以说清三者的关系。三账以两种不同的理由不可相互替代。 其一，就 $\mathcal{D}_R$ 与 $S$ 而言，二者之间不存在单调的一一对应——这不是额外假设，而是端点行为所强制的（下面给证明）。其二，就 $\mathcal{R}_{\rm tot}$ 而言，它与前两者的区别来自对象的类型与守恒角色：它是一个跨循环守恒的账本（更像一个带内部结构的守恒量，而非与 $\mathcal{D}_R,S$ 同型的第三条标量曲线），而非由下述函数关系证明所区分。先看第一条的证明，它只需一个反例。取两端邻域的极限值，

$$\lim_{\tau\to0^+}\mathcal{D}_R=\lim_{\tau\to T_1^-}\mathcal{D}_R=\mathcal{D}_R^{\min}\quad\text{而}\quad \lim_{\tau\to0^+}S<\lim_{\tau\to T_1^-}S,$$

即两端 $\mathcal{D}_R$ 同值而 $S$ 异值。于是不存在函数 $g$ 使 $S=g(\mathcal{D}_R)$（同一自变量值给出两个函数值），从而 $\mathcal{D}_R$ 与 $S$ 不可能单调等价。〔$\mathcal{D}_R$ 与 $S$ 非单调等价：结构性，由端点行为强制〕须说明，这三个量都是单一参数 $\tau$ 的函数，作为曲线它们至多是一维的；我们这里主张的不是通常统计意义上的"三变量相互独立"，而是更弱、也更精确的一点：它们不能作为同一宇宙进程的单调一一重参数化——特别地，$S$ 不是 $\mathcal{D}_R$ 的单值函数（相同的 $\mathcal{D}_R^{\min}$ 对应不同的 $S$）。正是这一不可重参数化性，使下面的图景（同一 $\mathcal{D}_R$、不同 $S$）成为可能。

（附带一提：两端 $\mathcal{D}_R$ 同值而 $S$ 不同——大爆炸与大坍缩在展开充分度上对称、在熵上不对称——正是 Penrose 所指出的"大爆炸是一个特殊的低熵态"这一事实在本框架中的表达。我们将在讨论收缩与过渡时回到它。）

关于熵，本文采用一个视界熵的代理量，以免后文产生歧义。它是 Bekenstein–Hawking 型的，记作 $S_H$，量纲上取

$$S_H = \frac{k_B c^3}{4G\hbar}\,A_{\rm hor}\,,$$

其中 $A_{\rm hor}$ 是相关视界的总面积（此式两端量纲一致，$S_H$ 具熵的量纲）。须说明，这个代理量还留有三处待完整化的细节：应计入哪些视界（黑洞事件视界、表观视界、宇宙视界，或 trapping horizons）、嵌套视界如何去重、以及广义熵（generalized entropy）的完整表述——这些属于后续工作。因此，"在每一第四层有效的区段内熵单调不减"是本文从热力学第二定律及下述信息论根据所继承的结构承诺，而非仅由上面这个面积和式本身就已证明的定理（面积公式不自动蕴含"所有视界总面积在并合、蒸发、宇宙视界变化的全程必定单调"）。作此声明后，我们在框架层面采用这一单调性。它计入黑洞视界——这些视界是第四层活跃的边界——在物质塌入黑洞、$A_{\rm hor}$ 增长时增大。黑洞内部（第四层不活跃）不需要一个"内部熵"：熵是一个附着在视界上的量。只有在 $\tau\to0^+$、$\tau\to T_1^-$ 这两个奇异端点上——彼时已无第四层、无视界——$A_{\rm hor}$ 才失去定义，熵随之失去附着（这也是我们取开区间的原因）。它在那里如何降级、余项又如何被携带过去，是 §7 的内容。

熵的单调增长本身，在 SAE 里也有一个比"热力学第二定律"更根本的来处，值得在此点明。信息论系列指出：第四层的运算范畴是封装，而封装不容许逆运算——一旦能量作为信息进入信息通道，在第四层内部没有任何运算能把它转回能量、动量或质量。这里的"信息"是一个守恒量，称为因果负载 $I$；本框架的质量与信息论系列给出它与能量的关系为

$$E = I\,c^3\,,$$

因此 $I$ 具有 $E/c^3$ 的量纲（即质量·长度$^{-1}$·时间的量纲），位于量纲阶梯的第四层——它是一个物理的、有量纲的负载，不是无量纲的比特数或熵。于是在有动力学的非平衡条件下，信息通道成为唯一的单向汇，只进不出、累积增长。基于视界面积的熵之所以单调增，根子就在这里：它是"封装不可逆、$I$ 单向累积"这一结构的表现，而不是一条外加的定律。这里须把两个量彻底分开，因为 §7.4 的关键正系于此：$I$（因果负载，有 $E/c^3$ 量纲）是被承载者；$S$（视界熵，由视界配置数而来）是另一个量。前者守恒并被携带，后者在过渡处降级——一个携带、一个不携带。〔熵单调承信息论的封装不可逆，作为第二定律的 SAE 根据〕

这一点也预先标出了一处需要在 §7 小心处理的张力：既然第四层内部信息通道不可逆，那么 §7 所说的"过渡处信息回落为能量"如何可能？答案的线索正在"过渡处已无第四层"——不可逆是第四层闭合成立时的规则，而过渡恰恰是这一闭合本身解体之处，规则的前提不再满足。我们把这一协调留到 §7 展开。

三个量、三种行为：展开充分度是发展变量（单峰），熵是热力学变量（单调），余项是守恒的账（携带）。把它们分开，"螺旋而非圆"的图景才立得住——熵不被携带（在过渡处降级），余项被携带，于是每一次大爆炸都是一个新的低熵起点，其参数可以与上一轮不同。这条线因此不回到原处，而是盘旋着向前。

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§4 六阶段总图

在逐段展开之前，先把整条线的骨架摆出来，作为后文的地图。我们把一次完整的呼吸分为六个阶段：

阶段	承重的图景	主要把握程度
① 大爆炸	混沌的自我否定；由第零层向第一层的强制读出	机制〔结构性〕；阈值的具体数值〔后验/开放〕
② 早期：力的生成	四种力作为四步否定在物质层的投影，依次生成	生成序列〔结构性〕；连续的膨胀率〔开放〕
③ 当下：加速	khronon 暗物质 + 因果标量场暗能量 + 四形式宇宙学常数	$w_{\text{int}}\geq-1$〔结构性〕；$a_0$ 演化〔结构性，$z>3$ 判决〕；参数拟合〔后验〕
④ 转折	几何转折点；余项展开充分度的峰	两者解耦〔结构性〕；转折点的位置〔动机性/开放〕
⑤ 收缩	结构塌入第四层不活跃的黑洞；宇宙尺度的封闭临界态	机制〔动机性〕；坍缩统计与各向异性〔开放〕
⑥ 大坍缩 → 下一次大爆炸	三段相异的机制序列；熵降级、余项携带	结构〔动机性〕；过渡的具体动力学〔开放〕

关于这张图，有一处贯穿全篇的时间线判断需要在此说明，因为它修正了 SAE 早期一篇工作中的一个默认。早期的图景假设呼吸是对称的：转折点恰在整个周期的一半，据此把周期定为转折时刻的两倍。本文放弃这一对称假设，改采非对称呼吸作为一个有据的工作假设（详见 §7.1）：若收缩期出现一个净向内的有效驱动（依赖 §7.3 的暗能量终态），则收缩段短于膨胀段、几何转折偏后。须强调这是工作假设而非既成结论——纯物质引力的闭合宇宙本是时间对称的。在这一假设下，整条线呈如下的非对称形貌：

$$\text{大爆炸} \;\longrightarrow\; \underbrace{\text{膨胀（逆引力：慢、长）}}_{} \;\longrightarrow\; \underbrace{\text{几何转折（偏后，可能在未来）}}_{} \;\longrightarrow\; \underbrace{\text{收缩（若净向内驱动出现：快、短）}}_{} \;\longrightarrow\; \text{大坍缩}$$

一个直接的推论涉及我们此刻身处何处，而这里需要格外谨慎。早期图景会把当下定位为"已越过转折、正处于几何收缩之中"。但本框架有一个双参考系的结构（详见 §5 与后文）：我们观测到的膨胀属于其一（Jordan 参考系），而几何意义上的膨胀或收缩属于另一（几何参考系），暗能量的作用可以使几何上的收缩在观测参考系中仍表现为加速膨胀。这带来一个诚实的限制：我们能确定陈述的，只是 Jordan 参考系此刻仍处在加速膨胀段；至于几何参考系中转折是否已经越过，当前的观测无法判定。 几何转折可能尚在（偏后的）未来，也可能已经越过而被暗能量"藏"在了观测参考系的加速表象之下——本文不对此二者做判断，只如实标出这一不可判定性。

我们特意不把当下写死为"尚未越过转折"，一方面是因为当前确实测不出来，另一方面也为将来可能出现的、能够反推几何参考系状态的新证据留出空间。需要强调的是，这一时间线的重构并不动摇此前系列关于宇宙学常数的定量结果——因为宇宙学常数对呼吸的"形状"不敏感（加速标度的情形要多一层谨慎，二者我们都在 §6 中说明）。〔"仅 Jordan 参考系在膨胀段"：结构性，由双参考系结构给出；"几何转折是否已过"：当前不可判定〕

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§5 从最初一跳到当下宇宙（阶段①–③）

本节走过前三个阶段。它们与 SAE 已发表的分块结果衔接最紧，因此我们的做法是引用而非重推，重点放在把它们接入这条时间线，并标清各自的把握程度。

5.① 大爆炸：混沌的自我否定

在 SAE 里，大爆炸不是一个需要外部引信的奇点。它是混沌的自我否定。

这里的逻辑是纯先验的，不依赖"因为我们存在，所以回推"。出发点是：对"存在"这件事，可以取四种态度——是、非、亦是亦非、非是非非（这是龙树四句的结构）。而混沌对这四者一概说"不"——这不是混沌的某种属性，这就是混沌的定义：它无法被任何确定的描述所把住。但混沌不能停在这个"什么都不是"的状态上：因为一旦"不发展"本身成了它的一个确定属性，就与"它无法被确定描述"矛盾。于是混沌必须自我否定、必须发展下去。这一步——我们称之为"凿"——不是外力、不是神之手、不是某个引燃大爆炸的引信；它是混沌自我否定的内在必然。

四句被逐一否定、穷尽之后，突破到下一轮（物质 → 生命 → 意识）。值得一提的是，维度数 $d=4$ 在这里不是一条公理、也不是一个假设，而是"四句被穷尽"这一结构的结果。

把这一段接回 §3 的主线：这里的"凿"，正是那条"展开"本原语的最初一击。§3 说余项无法在当前层自闭时便突破到下一层——大爆炸就是这一突破的起点，而其后整条单峰曲线的上升段，是同一个"凿—构造"循环的持续。换言之，大爆炸与贯穿全篇的展开充分度，不是两件事，而是同一个纵向展开过程的开端与展开。

这幅图景给出了大爆炸的机制〔结构性〕。但它也带着一个我们必须诚实标出的地板：这套否定的展开，其最深处是"否定从何而来"（whence negativa）——而这一问，在结构上无法再往下回答，因为任何"方向""来处"都是 negativa 展开之后才有的东西。这不是一个黑箱式的搪塞——大爆炸的机制（四句穷尽、必须发展）是可以陈述清楚的；但它最底层的那个"为什么会有否定"，只能被承认、不能被推导。这是本框架诚实承认的余项之一。〔阈值处的这一"地板"：开放，且我们主张它在结构上不可导〕

还有一处纪律需要在此立下，它关系到后文如何谈论时间与光速。大爆炸这一段——从第零层向第一层的那一跳——发生在"时间本身尚未生成"之处。后文将要用到的双时钟、连续的膨胀率、乃至光速作为一个速率，都是第一层之后才有的语言。用这些"时间生成之后"的概念去解释"时间如何生成"，会犯一个层次上的错误。因此关于这最初一跳的动力学，本文只标为开放——它是"下一层钥匙"中最深的一把——而不用后来的语言去强行填充它。

5.② 早期：四种力作为四步否定依次生成

一旦跨过第一层，宇宙进入我们较为熟悉的早期演化。SAE 对这一段的独特之处，在于把四种基本力理解为同一套"四步否定"在物质层上的投影：第一步否定对应第一层与电磁力，第二步对应第二层与弱力，第三步对应第三层与强力，第四步对应第四层与引力；相应地，物质层的特征量沿这条阶梯以 $c$ 的幂次递降。这条量纲阶梯可写为

$$E \;\to\; \frac{E}{c} \;\to\; \frac{E}{c^2} \;\to\; \frac{E}{c^3},$$

四级依次对应第一至第四层（能量、动量、质量、因果负载 $I$）；$E/c^3$ 即前节 $I$ 所在的量纲。在这个意义上，大爆炸处从第零层依次展开到第四层，就是四种力依次被"雕刻"出来的过程。

这里必须立一条防止误读的界线。四种力的这一"依次生成"，是一个本体层级上的生成序列，不是一部以温度为刻度的热历史——不是说 $U(1)$、$SU(2)$、$SU(3)$、引力像在秒表上那样一个接一个地出现。标准宇宙学的热历史（电弱统一、量子色动力学相变、复合等）有它自己的温度次序；SAE 的四步否定说的是本体层级，不是粒子物理的相变时间线。若把二者混为一谈，物理学界的读者会立刻指出问题。〔生成序列：结构性；与热历史的关系：本文明确区分二者〕

至于这一段能否与标准早期宇宙学相容，关键的接口都是已知的、且是必经的关卡：有效引力常数在大爆炸核合成与微波背景时期须落在安全窗口内；khronon 尘埃在复合之前须表现得像冷暗物质；暗能量与 MOND 的空间项在早期不能破坏微波背景的声学峰。一次完整的玻尔兹曼方程求解是这一段绕不过去的验证——本文不做这一求解，将其列为开放。〔连续膨胀率与完整玻尔兹曼：开放〕

关于这一段，有一个定位需要明确说出来，因为它决定了读者该到哪里去检验 SAE、又不该到哪里去。在大爆炸与早期宇宙这一段，SAE 提供的是本体层面的解释（四步否定的生成序列、大爆炸作为混沌自我否定），而不是与标准宇宙学不同的数值预测。 数值上，它要求自己与标准宇宙学一致——上述那些接口（核合成丰度、微波背景、复合）是它必须满足的相容性约束，而非它宣称的新预言。SAE 与标准模型在数值上真正分道扬镳之处，在边界条件层：宇宙学常数、加速标度的演化、暗扇区的结构；不在早期微观层。

这一划分本身是 SAE 的一个结构性主张，来自信息论系列：热力学与早期微观属于第四层的内部结构，宇宙学属于第四层的边界条件，二者仅在宇宙学常数那个约 $10^{-122}$ 的强度上相连。它也解释了为什么本文明确宣称的可证伪预言（§5.③）全部落在高红移——因为差异集中在边界条件层，而边界条件层的判决性差异恰在 $z>3$ 处显现。对一个想检验 SAE 的物理学家，这句话是有用的：请到高红移的加速标度与状态方程上去找差异，而不是到大爆炸核合成上——那里 SAE 与标准宇宙学本就一致。〔早期为本体解释、非数值预测；数值差异集中在边界条件层：结构性〕

5.③ 当下：加速，以及一处认识论的说明

来到当下，我们进入此前系列（暗扇区那篇）的主场。那里的图景是：暗物质由几何化的 khronon 承担，暗能量与 MOND 由因果标量场承担，宇宙学常数由一个四形式承担——三者都出自那一对呼吸时钟，不引入任何暗物质粒子。这一段给出两个结构性的硬边界：加速标度随红移的演化（在 $z>3$ 处可判决），以及内禀暗能量状态方程 $w_{\text{int}}\geq-1$（不穿越 phantom 界）。这里要分清加速标度的两个方面：其红移斜率（它随红移如何增长）是本框架的结构性预言，也是 $z>3$ 判决所针对的对象；而其当下数值的系数（那个 $\pi/2$ 因子来自闭合三维球面的几何）则是一处一致性的印证，而非独立的预言力所在。换言之，可证伪的分量在斜率，不在系数——一个想证否本框架的观测，应针对高红移处标度是否按预言增长，而非其今日的绝对值。

在与观测对话时，有一处口径需要谨慎。近来的星系巡天数据（如 DESI）确实给出了暗能量随时间演化的压力，但"是否需要一个内禀的 phantom 穿越"是依赖参数化方式的。因此本文不写"观测已证实 $w$ 穿越了 $-1$"；准确的表述是：数据给出了演化暗能量的压力，而 SAE 当前的硬边界是内禀的 $w_{\text{int}}\geq-1$，任何有效的穿越表现另作分支处理。〔$w_{\text{int}}\geq-1$：结构性；具体拟合：后验〕

关于宇宙学常数的认识论层级，这里需要一个明确的说明，因为它决定了本框架究竟"预言"了几个量。这一对时钟的两个周期 $T_1, T_2$，其锚定来自不同的输入：$T_1$ 由宇宙学常数的标度独立地钉住——常数之小（约 $10^{-122}$）主要来自普朗克时间与宇宙寿命之比的平方，这把 $T_1$ 定在约二百亿年的量级，且这一钉定与呼吸的形状无关；此外，生命（第五层）出现在宇宙寿命约一半处这一事实，给 $T_1$ 提供一个较粗的、独立的旁证。$T_2$ 则可由观测到的宇宙学常数反解精确定出，同时又能被银河系与仙女座并合的天文时标（一个独立于本框架的标准轨道计算）粗略地印证。

由此，宇宙学常数的地位是一个条件性的预言〔本文如此标注〕：在"以生命锚定 $T_1$、并接受银河系–仙女座的那一步认定"的前提下，它被预言到约 5% 的精度；若只接受由标度钉定的 $T_1$，则宇宙学常数与加速标度一同构成"两个由后验锁定的量之间的一致性检验"。无论取哪种读法，本框架明确宣称的无条件硬预言，仍是加速标度的高红移演化（$z>3$ 判决）与内禀 $w_{\text{int}}\geq-1$ 这两项。我们把宇宙学常数标为条件性预言，既不夸大它为无条件的硬预言（因为其中一步认定是软的），也不把它贬为纯粹的输入（因为其反解所依据的标度是独立的）。

最后，把当下接回时间线：此刻我们处在余项展开充分度的高位——生命（第五层）已经涌现。至于我们在几何意义上究竟处于膨胀还是收缩，如上一节所述，当前观测无法判定：我们能确定的只是观测（Jordan）参考系仍在加速膨胀，而几何参考系的实际状况我们并不知道。这与观测到的加速并不冲突——恰恰是因为二者可以脱耦。下一节转折处，我们会说清这一层脱耦的两个侧面：几何意义上的"转折点"与余项展开充分度的"峰"，是两个可以互不重合的点，且它们都不直接等同于我们观测到的加速。

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§6 转折：两个不必重合的"转折点"

上一节末尾我们埋下一句话：几何意义上的转折点，与余项展开充分度的峰，是两个可以互不重合的点。本节把它讲清，因为这一区分既澄清了一个此前悬着的问题，也保护了我们前几篇的定量结果不受时间线重构的影响。

先把两个点分开命名：

• 几何转折点——尺度因子达到极大、几何意义上由膨胀转为收缩的那一刻。它由几何与引力决定，是闭合宇宙这架"摆"的赤道。
• 展开充分度的峰——余项展开充分度最高的那一刻。它由发展动力学决定：结构、复杂度、以至生命的展开达到最充分之处。

早期的对称图景把这两点默认为同一个，并进一步默认它落在整个周期的一半，据此把周期定为它的两倍。本文把这两点分开，理由有二。

其一，是它们所刻画的东西本就不同。压制余项展开的那个几何量——粗略地说，正比于宇宙的紧致程度——在尺度因子极大处（即几何转折点）达到最小，这是一个单点。但"余项被展开到最充分"是一个发展的结果，发展需要时间累积：生命并非在压制一到最低就瞬间出现，而是在其后逐步建立。因此展开充分度的峰，可以滞后于几何转折点，也可以因收缩相的非对称而与之错开。它们期望相邻、处于同一宇宙阶段，但精确重合与否是一个开放问题。〔两点解耦：结构性；是否精确重合：开放〕

其二，也是对读者更要紧的一点：无论这两点在哪里、是否重合，我们前几篇的宇宙学常数都不受影响；加速标度的情形更微妙一些，下面一并说清。 这一点值得展开，因为它是时间线重构（§4 的非对称、转折偏后）能够安全进行的前提。

原因在于，那对频率是整周期的拓扑时钟频率——它们是各自那个紧致时钟相位在一整个周期里的缠绕频率，与这一周期内部"摆"的形状无关：

$$\omega_i = \frac{2\pi}{T_i}\quad(i=1,2),\qquad \Lambda = \frac{2(\omega_2^2-\omega_1^2)}{c^2}.$$

宇宙学常数只取用这两个常数频率，而不含尺度因子 $a(\tau)$ 随时间的具体形状、不含转折点落在周期何处、不含膨胀与收缩是否对称——它对呼吸的形状是"盲"的。因此我们尽可以在 §4、§7 里把这条线重画成非对称的、把几何转折推向偏后甚至（在观测参考系不可判定的意义上）未来，而宇宙学常数的这一表达式纹丝不动。这是时间线重构能够安全进行的核心保证。

对加速标度，须比宇宙学常数多一层谨慎。它有一个同样只含频率的表达式 $a_0=(\pi/2)\,c\,(\omega_2-\omega_1)$，在这个意义上也是形状盲的；但它另有一个基于表观视界的表达式，在转折附近显式地依赖几何。这里顺带交代后者的一个技术要点，它在 §7 讨论收缩时会用到：在转折点，尺度因子达到极大、$H=0$，若天真地用平直近似 $a_0\propto cH$，会得到 $a_0=0$——这是错的。正确的做法是用表观视界，

$$a_0 = \frac{c^2}{2\pi R_A},\qquad R_A = \frac{c}{\sqrt{H^2 + k c^2/a^2}}\,,$$

对闭合宇宙 $k=+1$，转折点 $H=0$ 时 $R_A = c/(c/a) = a_{\max}$ 仍然有限，故 $a_0 = c^2/(2\pi a_{\max})\neq 0$。这两个表达式（只含频率的、与基于视界的）在整个周期上如何自洽，本文尚未完全厘清——因此我们把"形状盲"这一较强的结论稳妥地建立在宇宙学常数上，而对加速标度只作较弱的陈述：其可证伪的分量在红移斜率（§5.③、§9），而两个表达式的调和，列为一处待厘清的开放项。〔宇宙学常数形状盲：结构性；加速标度两表达式的调和：开放〕

这也顺带精确化了那个"周期是转折点两倍"的旧关系。整周期的时长本身，是由宇宙学常数的标度独立钉住的（§5.③）；旧图景之所以能写成"周期等于转折时刻的两倍"，正是因为它假设了对称（转折在半周期）。一旦放弃对称，"转折在半周期"不再成立，但整周期时长仍由那个标度钉住——于是转折点在周期内的位置，就从一个被对称假设锁死的量，变成了一个可以偏后、由非对称程度决定的开放量。而生命出现在宇宙寿命约一半处这一事实，此时也就从"锁定周期的锚"退为"一个较粗的、独立的旁证"——它落在半周期附近这一点仍然成立，只是不再承担锁定的重任。

至于我们此刻究竟在不在几何转折之前——如 §5.③ 所述，当前观测无法判定，我们只能确定观测参考系仍在加速膨胀。

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§7 收缩：塌入黑洞，以及一次呼吸如何回到低熵

过了几何转折（无论那在何时），宇宙进入收缩。这一节是本文原创成分较集中之处，也因此是我们最谨慎、把每一步的把握程度标得最细的一节。按 §2 立下的姿态：以下若干为 SAE 框架下的机制候选，其定性链有据且自洽，但定量化留待后续；我们把它们完整摆出，供推敲与证伪，而非声称已定。

7.1 收缩为什么可能比膨胀快：作为工作假设的非对称

本文采用非对称呼吸作为一个有据的工作假设，而非一个已由动力学确立的结论——这一点须讲在前面。纯粹只有物质与引力的闭合 FRW 宇宙，其膨胀与收缩其实是时间对称的（如同一条摆线，两段等时）；单靠"膨胀逆引力、收缩顺引力"并不能给出"收缩更短"。真正能造成快慢不对称的，是收缩期是否出现一个净向内的有效驱动。

具体而言：膨胀期，引力始终在把宇宙往回拉，膨胀被它持续拖慢。收缩期，引力本就向内；若此时暗能量也转为向内（即 §7.3 的"转负"分支），则引力与暗能量同向叠加、收缩没有对抗者，收缩段自然短于膨胀段，几何转折相应偏后。但这一步系于暗能量的终态：若暗能量归零或仍为正（§7.3 的另两支），则这个净向内驱动未必出现，快慢不对称也就不被当前论证固定。因此"收缩短、膨胀长"是一个动机性的工作假设，其成立与否由 §7.3 的暗能量终态与双时钟动力学共同判决，而不是一个既成的宇宙动力学结果。

在这一工作假设之下，收缩相内部还有两个会进一步加速收缩的因素——密度升高使有效引力增强、以及黑洞形成的不可逆（越塌越多区域进入第四层不活跃，展开充分度下降得越快）；但须注意，这两者同样以"净向内驱动已出现"为前提，不能独立地把纯引力的对称性打破。〔非对称呼吸：动机性工作假设，条件于 §7.3 的净向内驱动；具体动力学：开放〕

一个直接的推论是，膨胀段长、收缩段短，于是几何转折点落在整个周期偏后的位置——这与 §4、§6 所说的"转折偏后、观测上不可判定是否已过"是同一件事的两面。

7.2 展开充分度如何下降，以及一个措辞上的谨慎

收缩相中，展开充分度的下降，我们表述为第四层不活跃区域的占比上升：结构逐一塌入黑洞，而黑洞内部第四层不活跃，于是宇宙中"可供余项展开"的部分不断缩小。这里再次强调 §3 立下的措辞纪律：我们不说"平均因果律变强"——跨越活跃与不活跃区域的"平均因果律"本身没有良好定义，况且黑洞恰是第四层不活跃之处。下降的机制是引力塌缩，如此而已。

要把这一下降定量化，一个自然的工具是信息论系列引入的因果槽尺度——刻画"多大的区域才能维持稳定的第四层因果读出"的那个尺度。它随温度反变，在正常宇宙中张开到宏观，而在塌缩区域局部收回到普朗克地板。于是收缩相可以用一个"第四层不活跃体积占比"来刻画，它在大坍缩处趋于百分之百；因果槽尺度作为一个空间场，恰好分辨出哪些区域已经塌回地板、哪些尚未。给出这个占比随时间增长的具体模型（涉及过密区塌缩、并合、视界渗流），本文列为开放。〔收缩机制：动机性；坍缩统计：开放〕

这里也要诚实标出 SAE 在收缩相的一处弱点。收缩宇宙普遍面临各向异性增长的问题（剪切随宇宙收缩增长得比物质更快，一般会主导并导致混沌式的振荡）。ekpyrotic 图景用一个陡峭的标量势来压制各向异性，这是它的一处长处；而 SAE 的暗能量状态方程近于 $-1$、并不提供这样的陡势，因此 SAE 没有 ekpyrotic 式的各向异性压制机制。我们目前能提供的只是一个候选方向：各向异性的混沌振荡是一个第四层时间尺度上的现象，而随着区域逐一进入第四层不活跃（其内部没有第四层时间），这些振荡的"钟"在该区域停摆——若不活跃占比增长得足够快，混沌在全局变得严重之前即被截断。这个候选并非凭空，它与 §7.4 的图景相互扣合：若 §7.4 那幅（同样是候选的）"余项作为一份干净信号被携带到下一循环"的图景成立，则它要求过渡不被各向异性的混沌所打乱——在这个意义上，"各向异性截断"与"余项携带"是同一套图景的两个侧面，一同成立、或一同失败。我们不用其中之一去证另一个（二者都还是候选），只指出它们彼此一致。但无论如何，"占比是否增长得足够快"仍是一个定量问题，本文不假装已答。〔各向异性截断：开放，与 §7.4 余项携带图景相互扣合、同为候选〕

7.3 收缩期暗能量的终态：一个被充分分析的开放问题

收缩要发生，暗能量至少不能一直是强烈的"向外推"。但它究竟如何变化——归于零、转为负、还是减弱而仍为正——是一个我们已充分分析、但当前后验不足以判决的开放问题。我们对三种可能都做过思考，各有其结构上的动机与各自需要交代的责任：

• 归零：暗能量项稀释掉、或那个因果标量场停在零势处，收缩由纯引力承担。它的动机是这落在 $w\geq-1$ 之内、不需要负能量结构；它需要交代的是——膨胀期暗能量向外做的功所积累的那部分，纯引力收缩能否收回，以及收缩为何更短。
• 转负：暗能量密度转为负，与引力同向驱动收缩。它的动机较多：它给出一个闭合的符号循环（向外推的膨胀、向内拉的收缩）、它自然地解释了收缩为何更短、它在能量账上闭合（膨胀期展开、收缩期收回）；而且从结构上看，本框架的宇宙学常数是一个有符号的、两频率之差的量，取负是一般情形，归零反而要求两频率恰好相等这一非一般的巧合。它需要交代的是——暗能量转负所需的场演化机制尚待坐实，而真正的负能量密度需要一个取向敏感的结构（这一点与 §8 的宇宙学常数符号问题同源）。
• 减弱而仍正：暗能量减弱但不变号，是最小的改动。它需要交代的是——既然仍在向外推，如何被曲率与引力压过而转入收缩，以及收缩速率。

我们把这三者并列、不预设优先级——这是一个被充分分析的开放问题，而非一个未经思考的耸肩。判决它，需要收缩相的场动力学（本文列为钥匙之一），以及未来的观测（收缩速率、或减速参数由负转正的红移）。这里对观测判据须补一句精确的话：我们观测到的减速参数属于观测参考系，它由负转正对应的是"暗能量不再能把几何收缩掩藏"的时刻，未必正好是几何转折本身——二者之间还隔着 §5.③、§6 所说的参考系变换。因此"未来某红移处减速参数转正"这一可证伪点，检验的是观测参考系的表观转折，而非直接测得几何转折。〔暗能量终态：充分分析的开放问题，三档并列无优先级〕

还需一句关于能量账的诚实话，因为它正是我们不在此偏向任何一档的理由。人们可能会想：既然一次呼吸的两侧账目相反、和为零，膨胀期本侧积累的账会不会在过渡时整体转交给对侧、从而本侧无须自己平衡？我们在 7.4 会说明，跨越两侧界面的转交只是小额的渗漏，而非整体过户；因此本侧的能量账，绝大部分仍要在本侧了结。也就是说，膨胀期暗能量向外做的功究竟如何在本侧平衡，这笔账当前并未算清——正因未算清，我们不偏向归零、也不偏向转负，如实并列。

7.4 一次呼吸如何回到低熵

现在到本节、也是整条循环叙事的关键：收缩到了大坍缩，熵此时处于极大（物质尽数塌入黑洞、视界总面积极大）；而下一次大爆炸若要是一个低熵的、平滑的起点，熵必须从极大回到极低。这一步如何发生？

先厘清一个容易混淆之处。SAE 里有一个第四层的守恒量，称为因果负载 $I$，它由 $E=Ic^3$ 与能量相联系，因而具有 $E/c^3$ 的量纲——它是一个有量纲的物理量，但其量纲不是能量量纲；信息论系列明确指出，这个因果负载不是熵、不是比特、不是信息容量。而我们这里说的熵，是基于视界配置数的（其对数即视界面积那一类）。二者不同：因果负载是被承载的量（量纲 $E/c^3$），熵 $S_H$ 则具有熵的量纲，而无量纲的是 $S_H/k_B$——可读作有效配置数的对数。分清这两者，是理解下面这步的前提——因为一个守恒、被携带，另一个不守恒、被降级。

机制候选如下，我们标出它每一步的依据。大坍缩时，整个宇宙成为一个第四层处处不活跃的态。它与一个寻常黑洞有一个关键的不同：寻常黑洞的视界之外，还有活跃的宇宙在"读"它，因而视界是第四层活跃的；而当整个闭合宇宙塌缩时，没有外部——不是说最外层之外空无一物，而是对一个闭合的三维球面而言，"外部"这个范畴根本不适用（"外部是什么"问的是不可及的物自体）。于是承载因果负载的那些视界，失去了它们赖以存在的"被读"关系，第四层的封闭随之解体。

正是在这里，§3 预告的那处张力得到化解。信息论系列说第四层内部信息不可逆、因果负载只进不出——但那是第四层封闭成立时的规则。大坍缩恰是这一封闭本身解体之处：外部关系断裂、视界解晶，第四层不再作为一个封闭的层存在。规则的前提既已不再满足，因果负载便可以回落——它沿着量纲阶梯退回最底层，成为能量。这不违反"第四层内部不可逆"，因为此处已无第四层内部可言。〔因果负载回落：动机性候选；与信息论不可逆的协调如上〕

因果负载回落为能量，但能量不能"裸"存在——它必须重新表达为光（第二层）或有质量的粒子（第二、三层）。这一"回落再表达"整个发生在大坍缩与下一次大爆炸的内部，没有外部的扰动、也没有从上一循环继承下来的经典结构充当种子。这里必须接上 §7.2 那个尚未挣得的开放项，并且只能挣得一个较弱的结论：若 第四层不活跃的截断确能在 BKL 混沌主导之前终止各向异性的增长、使过渡边界趋于低外尔曲率，则上一循环不会向下一循环传递宏观的、经典的非均匀性，于是一个低外尔、近均匀的边界成为自然的候选。但这不等于"必然均匀"：即便没有继承的经典种子，量子涨落、真空不稳定、自发对称破缺、动力学不稳定模、乃至过渡匹配本身，都仍可能产生扰动。因此下一循环的原初涨落及其功率谱，仍须由过渡动力学给出，本文不予断定。至于那个视界配置数意义上的熵为何在此变低：承载它的视界结构在解晶中瓦解，配置随之坍缩——它不是被销毁，而是失去了作为"高熵"所依附的结构。〔各向异性截断成立与否：开放；成立则低 Weyl 近均匀边界为候选，原初功率谱仍开放〕

这里须把熵的"降级"表述精确，以免读作一条连续的下降曲线。它不是一条定义良好的 $S(\tau)$ 从极大平滑回落到极低的轨迹，而是上一循环的定义域终止、下一循环以低熵边界重新起始——用记号写：

$$S_n(T_n^-)\ \text{高},\qquad S\ \text{在过渡内核无定义},\qquad S_{n+1}(0^+)\ \text{低}.$$

视界解晶，是"为何下一侧的边界条件是低熵"的一个候选机制，而不是一条已给出的、熵连续回落的方程。在这个意义上，这条链在概念上首尾相接：极大的熵（上一循环终点）——过渡内核熵无定义、因果负载回落为能量、余项携带——低熵的下一循环起点。而能量（因果负载）则守恒地跨了过去。这，就是"熵不被携带、余项被携带"这句话背后的候选机制——被携带的是能量与余项，被降级（确切说，定义域终止后于低熵重启）的是熵。它也可能为微波背景的高度均匀提供一个本框架内的解释方向：不靠暴胀抹平不均匀，而靠大坍缩抹平旧结构、过渡内部无种子；但这一候选能否替代或补充暴胀对均匀性与扰动功率谱的解释，仍待动力学与功率谱的计算，本文不作断言。〔熵归零：动机性候选（定义域终止＋低熵重启，非连续轨迹）；定量与对暴胀的替代性——开放〕

最后一步，关于两侧之间的那点转交，以及它为何只是小额。前面我们把因果负载的回落说成本侧的事。但本框架的一次呼吸有两侧（时间箭头相反的一对第四层），二者之间有一个互惠的界面。跨越这个界面，本侧的一小部分信息可以渗漏到对侧——而由于对侧的时间箭头相反，本侧"已经读出的信息"，对对侧而言恰恰呈现为"尚未读出的余项"。这正是"已展开"与"待展开"在两侧的一体两面，也正是那个"两侧账目和为零"的守恒律的内容。但这个界面是互惠的界面，不是一根导管：它允许部分渗漏，而非整体穿越——况且若信息整体过户到对侧，本侧自己的下一次大爆炸便无以为继。因此绝大部分因果负载在本侧回落（如上），只有一小部分渗漏对侧。而恰恰是这一小部分渗漏，成为循环之间的漂移之源：下一循环所继承的余项里，掺入了上一循环渗漏而来的那一份，于是每一轮的参数微微不同——这正是这条线是螺旋而非闭合之圆的又一层来由。〔部分渗漏与螺旋：动机性；渗漏动力学：开放〕

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§8 宇宙学常数的符号：一个必须单列的开放问题

到这里，一个问题已无法回避，我们把它单独成节，因为若不面对它，整条循环叙事会在这一点上卡住：下一次大爆炸所展开的宇宙，其宇宙学常数是正的（因而像我们这样加速膨胀）、还是负的（因而减速、乃至塌缩）？

这个问题之所以尖锐，是因为一个看似显然、实则不成立的直觉：既然对侧的时间箭头相反，那么符号翻转岂不是自然发生？并非如此。在标准的动力学里，时间反演并不改变加速度的方向——加速度对时间反演是不变的，宇宙学常数作为一个几何源项，也不会仅仅因为时间箭头翻转就改号。因此，仅凭"对侧时间箭头相反"，得不出"对侧看到正的宇宙学常数"。这一层必须说清，否则会得到一个漂亮但错误的结论。

那么符号究竟如何？这里有一条论证，可以把问题推进一大步；它的关键在于只陈述对侧自身参考系中的图景，而不去做"从我们这侧看对侧"的跨参考系翻译——后者会牵涉尚未形式化的取向变换，而前者是自洽的、不需翻译的，且已足以回答我们的问题。

为推进这一问题，本文考察一个带选择效应的条件分支：若对侧能够形成长期的复杂结构与观测者，则其自身参考系中的有效引力与膨胀史，就必须落在允许这类结构形成的区域内——引力吸引、物质能聚成星系、膨胀史不至于过早把结构撕碎或压垮。这是一个条件，不是一个已确立的事实。

在这个条件之下，本框架语料给出的两侧关系提供了一项动机：一个第三层的对称给出一对时间箭头相反的第四层，本侧与对侧由同一套结构生成、仅参数次序相反，而我们的大爆炸就是对侧的大坍缩。因此，双侧同源这一结构，为"对侧在其自身参考系中与本侧定性同型"提供了一项动机——即：在能形成结构的那一支里，对侧自己看来也应是引力吸引、宇宙学常数为正、宇宙加速、结构得以形成。但须强调，这不能由此单独推出对侧的宇宙学常数必为正，也不能排除其他分支：没有观测者的负宇宙学常数循环、能短暂成形却迅速再坍缩的循环、或取向变换与当前直觉不同的分支，都不被这一动机所排除。我们也不去回答"从我们这一侧看过去，对侧的宇宙学常数是正还是负"——那依赖尚待形式化的跨参考系取向变换；这里陈述的，只是"在能成结构的那一支中、就对侧自身而言"的图景。

回到本节开头的判据——下一循环（即对侧）从其自身参考系看，加速度是正是负。这里我们能提出的，是一个较强的动机性偏好，而非一个已判定的答案：在能够形成复杂结构的那一类对侧分支中，其自身账本应当与我们定性同型——引力吸引、宇宙学常数（就其自身而言）为正、宇宙加速、结构得以形成。但这是一项带选择效应的条件判断：它由"对侧须能形成结构、须有观测者"这一前提支撑，因而只覆盖"能形成复杂结构的对侧子类"，并不能排除其他分支——例如没有观测者的负宇宙学常数循环、能短暂成形却终归快速坍缩的循环、或对侧取向变换与当前直觉不同的分支。符号问题本身，仍须由双时钟作用量的取向变换律来判决。

但这一判断的强度必须诚实标定，它是一个有充分动机的论证，而非一个结构上的必然。它依赖"对侧须能形成结构、须有观测者"这一带选择效应的前提——严格说来，一个减速的、乃至负宇宙学常数的下一循环，是否绝对无法形成任何结构，本身尚可争论；因此"对侧与我们同型"是动机性的强倾向，而不是被框架强制的定理。要把它落到定理，需要在作用量层面形式化两侧的取向变换与宇宙学常数符号的关系——这属于双时钟作用量（附录 C、§10 钥匙）的工作。在此之前，我们把它标为一个有力的、但仍待形式化的判断。〔对侧自身参考系与我们同型：动机性（强倾向，依赖选择效应前提）；作用量层面的形式化：开放〕

需要强调一处分寸，它关系到整幅图景是螺旋而非圆：上面所说的"同型"，指的是——在被选择效应覆盖的那一类能成结构的对侧分支里——引力吸引、宇宙学常数为正、加速、结构形成这些定性结构相同，而不是完全同一。余项被携带过去（§7.4），使下一循环的参数与本轮微微不同——因此，在这一分支中，下一循环是这条螺旋的下一环，一个结构上与我们相同、而参数微漂的宇宙，而非一个与我们全然一致的复本。这一图景不排除前述其他分支的存在；它只是我们在当前动机与选择效应下所偏好的那一支。

这幅图景与本框架语料中"我们的大爆炸即对侧的大坍缩"这一表述，正是同一件事的两种说法；它也是一个 CPT 式的镜像。（附带更正我们此前的一处判断：本框架与那些 CPT 对称宇宙的进路，在镜像结构上其实是共享的，只是本框架另外叠加了余项的携带与由此而来的螺旋；我们此前把二者划得过开了。）

至于这一判断的观测侧支撑，仍如实标其分量：我们这个宇宙本身就是上一次过渡的产物，而我们在近处星系身上观测到一个正的加速标度——它是一个正定的阈值量，观测上比宇宙学常数的符号更硬实。"过渡的产物是一个加速的、正标度、能形成结构的宇宙"，我们手里因此有一个存在性的证据：我们自己。这与上面的论证同向，一个从结构生成、一个从观测，共同支撑"下一循环从自身看与我们同型"这一判断。〔近处 $a_0>0$、$q_0<0$：后验；作为下一循环图景的 $n=1$ 支撑〕

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§9 可证伪点

一篇框架性的文章，若不指明如何能被证否，便没有尽到责任。本节把散见于前文的可证伪点集中列出，并按其判决力与所处层级分清。这里要守一条纪律：可证伪，是本框架的长处而非短处——一个明确宣称"若观测如此则我错"的框架，才是一个科学的框架。

无条件的硬预言（结构性，判决动摇的是框架本身）：

• 加速标度的高红移演化。本框架预言这一标度随红移按特定方式演化，其判决性差异集中在 $z\gtrsim3$。若高红移的测量显示它并不按此演化，本框架赖以给出暗扇区的那个视界机制即告失败。这是最干净、最集中的一个判决点。
• 内禀状态方程 $w_{\text{int}}\geq-1$。若对状态方程的非参数化重构稳健地显示内禀的 phantom 穿越（而非仅仅是某种参数化下的表观穿越），则本框架的暗能量结构失败。

结构性的需求，与观测的张力（须诚实标出）：

• 闭合的三维球面。本框架需要一个整体闭合的空间。当前的曲率测量高度接近平直，这对闭合的曲率半径构成一个下界的压力，但并不直接排除拓扑上的闭合——所要求的，是曲率足够小、或闭合的尺度足够大。若未来的巡天把曲率约束进一步压紧到某个程度，本框架会相当为难；若确证空间为开放，则闭合图景被证否。我们如实把它标为一处需求与压力，而非一个已被支持的事实。

观测参考系的表观转折（须与几何转折区分）：

• 未来减速参数的转正。本框架的非对称图景预期存在一个未来的转折。但如 §5.③、§7.3 反复强调，我们观测的减速参数属于观测参考系，它由负转正对应的是暗能量不再能掩藏几何收缩的时刻，未必正好是几何转折本身。因此这一可证伪点检验的是观测参考系的表观转折，二者之间隔着参考系变换。若未来长期稳健地保持加速、不见任何转正的迹象，则本框架的转折与收缩图景受到严重挑战。

尚待理论判决的（非观测，属框架内部）：

• 从大坍缩到大爆炸的过渡机制、宇宙学常数的符号、以及熵回落的定量，都尚未完成。若某种量子引力理论证明这样的过渡不可能、或证明熵无法在此重置，则本框架的循环图景受挫。

一句关于层级的话，收束本节：上述无条件硬预言（高红移加速标度、内禀 $w_{\text{int}}\geq-1$）是与"循环"这一整幅图景可以分开判决的。高红移加速标度若被证否，动摇的是视界机制与暗扇区；而"循环"另属一个框架级的、被充分分析但仍开放的图景，它不与前者共存亡。把这两个层级分开，是为了不让一处的成败被误读为全局的成败。

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§10 结论，与"下一层的钥匙"

本文尝试把一次完整的宇宙呼吸——从大爆炸，经膨胀、转折、收缩，到大坍缩，再到下一次大爆炸——作为一条连续的线讲出来。若要用一句话概括这条线：

> 一次完整的呼吸不是一个圆，而是一条余项守恒的螺旋。被携带的是余项与能量，被降级的是熵；余项展开的充分度从低到高再到低（单峰，其两端同值而熵异值，表明它与熵非等价、不可相互替代），熵在每一个第四层有效的区段内单调增长，因果结构的尺度则从普朗克地板张开、又塌回。大爆炸与大坍缩在展开充分度上对称、在熵上不对称；两端的低谷由三段相异的机制（生成、渗漏交接、塌缩）连接，而非一个简单的时间镜像。

我们没有把这条线上的每一段都讲到底，也不声称如此。若干环节——尤其从大坍缩到大爆炸的过渡、熵的回落、宇宙学常数符号的翻转——目前是本框架下的机制候选，其定性链有据而自洽，但定量化尚待完成，且完全可能被修正或证否。我们把它们完整地摆了出来，正是本着开篇那句：不求无过，但求有功。若这一图景能为物理学界提供一个可推敲、可证伪的靶子，本文的目的便已达到。

留给下一层的、需要真正拧开的"钥匙"，我们在此列明，以免它们被误当作已经解决：

• 最初一跳的动力学——从第零层到第一层、时间尚未生成之处的那一步。其机制（混沌的自我否定）可以陈述，但其动力学尚无从用"时间之后"的语言刻画。
• 双时钟的作用量——一个只在呼吸的有效阶段（第一层之后、第四层有效）适用的相位有效作用量，将给出连续的膨胀率、过渡处两个时钟的处理、以及宇宙学常数符号与暗能量终态的判决。附录 C 给出它的一个骨架，但强调那是骨架而非最终动力学。
• 收缩的坍缩统计与各向异性——第四层不活跃占比随时间增长的具体模型，以及各向异性截断能否及时发生。
• 熵回落与两侧渗漏的定量——视界配置数具体坍缩到何等程度，以及跨界渗漏的动力学。
• 完整的玻尔兹曼求解与曲率/拓扑约束——把这条线接入标准宇宙学数值检验的那些常规而必要的工作。

这些不是本文的疏漏，而是本文所划出的边界：它给出的是一张地图、一组序参量、一份可证伪点清单，而把动力学的求解，诚实地留给了下一层。

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附录 A · 主张—层级总表

为便读者逐条核查，下表汇总全文主要主张及其把握层级。层级的含义见 §2：〔结构性〕由框架强制、可证伪；〔后验〕依赖观测或参数；〔动机性〕有充分动机的偏好、有未排除的替代；〔开放〕尚待判决。

主张	层级
$\mathcal{D}_R$ 作为主序参量及其广义单峰形状	动机性（框架 ansatz）
接受端点行为后，$\mathcal{D}_R$ 与 $S$ 非单调等价、不可互替；$\mathcal{R}_{\rm tot}$ 另为守恒账本	条件性结构结论
展开充分度 / 熵 / 余项 三者不可互替（$\mathcal{D}_R$ 与 $S$ 的非单调等价为条件性结构结论；$\mathcal{R}_{\rm tot}$ 因对象类型与守恒角色而另立）	见上两行
主线为余项展开充分度而非平均因果律	结构性（措辞纪律）
"展开"承信息论第三本原语；熵单调承信息通道单向累积	结构性（承信息论系列）
加速标度高红移演化（$z>3$ 判决）	结构性（无条件硬预言）
内禀 $w_{\text{int}}\geq-1$	结构性（无条件硬预言）
宇宙学常数 $\Lambda$ 对呼吸内部形状不敏感	结构性（条件于整周期频率读法）
加速标度 $a_0$ 频率式与视界式的统一	开放
无粒子暗物质（几何化 khronon）	结构性
早期宇宙为本体解释、非与标准不同的数值预测	结构性（差异集中在边界条件层）
大爆炸为混沌自我否定；$d=4$ 由四句穷尽	结构性（机制）；阈值数值为开放
周期 $T_1$（由宇宙学常数标度钉定）、$T_2$（由常数反解），$c$ 为依赖型边界条件	后验 / 边界输入
宇宙学常数为条件性预言（依赖生命锚定与并合认定）；无条件硬预言仅加速标度与 $w_{\text{int}}$	后验 / 条件性
近处星系 $a_0>0$、$q_0<0$	后验（观测）
5DD 锚于展开充分度峰（展开阈值 / 因果槽突破）；生命出现于半周期附近为粗旁证	动机性
非对称呼吸（收缩快于膨胀）；几何转折偏后、观测上是否已过不可判定	动机性；具体位置开放
大坍缩为宇宙尺度封闭临界态（同类而非史瓦西复制）	动机性
低熵重启候选：视界解晶 → 熵定义域终止 → 条件性的低 Weyl／近均匀边界；原初涨落与功率谱开放	动机性（候选）；定量开放
因果负载可部分渗漏对侧 → 螺旋之源	动机性；渗漏动力学开放
过渡的两阶段：进入过渡前，第四层因果闭合先退至 L4b（有箭头、无单一方向）；过渡内核处，双时钟与第四层描述整体失效，才抵 0DD 无时间边界	动机性（承层级细分）
暗能量收缩期终态（归零 / 转负 / 减弱仍正）	充分分析的开放问题，三档无优先级
宇宙学常数符号：时间反演本身不翻 $\Lambda$；对侧从其自身参考系看与我们同型（下一循环加速）	动机性（强倾向，依赖"对侧须成结构"的选择效应前提）；作用量层面形式化开放
各向异性截断、双时钟作用量、最初一跳动力学、完整玻尔兹曼、曲率/拓扑	开放（下一层钥匙）

附录 B · 相关的循环与反弹宇宙学进路

本文视以下进路为从不同公理入口对同一深层结构所作的探索，本框架提供其一；下面陈述异同、指出各自可证伪点，不评高下。

• Penrose 的共形循环宇宙学：以共形重标度在世代之间处理熵。与本框架的不同在于，本框架不将熵携带过世代、亦不靠重标度，而是让熵在过渡处经由视界解晶而降级、由能量与余项携带过去。二者的过渡机制不同。
• Steinhardt–Turok 的 ekpyrotic 与循环膜宇宙：这是与本框架两侧结构在结构上最接近的一个类比——两个实体、周期性的相对运动、由界面势驱动、以碰撞开启新循环，四点同构；其膜间势也为"界面反作用驱动"提供了一个存在性的先例。但须守住一处区分：本框架的两侧是本体层级上的两侧、其时钟是相空间的频率，不是高维空间中的两张膜——可借其结构类比，但不可搬用其高维引力势的机制。ekpyrotic 以陡势压制各向异性是其长处，而本框架不具此机制（§7.2）。
• Boyle–Turok 的 CPT 对称宇宙：其镜像结构（我们的大爆炸即对侧的大坍缩）为本框架所共享；本框架的不同，在于另外叠加了余项的携带与由此而来的螺旋（每一循环参数可微漂）。故二者的判别不在镜像本身，而在螺旋与熵的处理（本框架为视界解晶降级，而非稀释）。

附录 C · 双时钟作用量的相位有效骨架

需强调：以下是一个骨架，非最终动力学，且只适用于呼吸的有效阶段（第一层之后、第四层有效，即膨胀至收缩的主体），不适用于最初一跳与过渡的内核。其形式为通常的引力—标量—形式场作用量，以两个时钟相位的和 $\theta_+=(\theta_1+\theta_2)/2$ 与差 $\theta_-=\theta_1-\theta_2$ 作为有效坐标：

$$S_{\rm eff} = \int d^4x\,\sqrt{-g}\;\Big[\tfrac{M_{\rm P}^2}{2}R \;+\; \mathcal{L}_{\rm kh}(\theta_+, g) \;+\; \mathcal{L}_{C}(\theta_-, g) \;+\; \mathcal{L}_{F_4}(F_4) \;+\; \mathcal{L}_{\rm rem}\Big],$$

其中 $\theta_+$（呼吸相位）驱动几何化的 khronon 项 $\mathcal{L}_{\rm kh}$，$\theta_-$ 对应那个因果标量场 $\mathcal{L}_C$，$F_4$ 是给出宇宙学常数的四形式，$\mathcal{L}_{\rm rem}$ 概括余项。尺度因子随相位的演化可取闭合宇宙的参数形式，其形状指数不被硬性固定。这里采用与附录 A 一致的两阶段读法：进入过渡之前，第四层因果闭合先退至 L4b（有箭头、无单一方向），此时上式仍勉强适用；而在过渡内核，两个时钟与整个第四层描述一并失效、退场，才抵达 0DD 的"无时间"边界，于新循环再生——是这最内核处、而非退至 L4b 那一步，才与"第零层无时间"相合。此骨架的完整化——给出连续膨胀率、过渡处的匹配条件、以及宇宙学常数符号与暗能量终态的判决——即正文所列的"钥匙"之一。须再次强调：此式仅是一个记账用的骨架，其各项的具体形式、耦合、乃至是否可如此分离，都属待定，不应被读作一个已定的动力学。

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参考文献

相关的循环与反弹宇宙学、以及黑洞熵

• Bekenstein, J. D. (1973). Black Holes and Entropy. Physical Review D, 7, 2333–2346. DOI: 10.1103/PhysRevD.7.2333.
• Hawking, S. W. (1975). Particle Creation by Black Holes. Communications in Mathematical Physics, 43, 199–220. DOI: 10.1007/BF02345020.
• Khoury, J., Ovrut, B. A., Steinhardt, P. J., & Turok, N. (2001). The Ekpyrotic Universe. Physical Review D, 64, 123522.
• Steinhardt, P. J., & Turok, N. (2002). A Cyclic Model of the Universe. Science, 296, 1436.
• Penrose, R. (2010). Cycles of Time: An Extraordinary New View of the Universe. Bodley Head.
• Boyle, L., Finn, K., & Turok, N. (2018). CPT-Symmetric Universe. Physical Review Letters, 121, 251301.
• DESI Collaboration (2025). DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints. arXiv:2503.14738.

SAE 系列（上游，均为本作者於 Zenodo 发表）

• Qin, H. (2026). SAE Foundation / Methodology (ZFCρ). Zenodo. DOI: 10.5281/zenodo.19361950.
• Qin, H. (2026). Mass Series Convergence（$E=Ic^3$）. Zenodo. DOI: 10.5281/zenodo.19510868.
• Qin, H. (2026). SAE Information Theory I: The 4DD Ontology of Information. Zenodo. DOI: 10.5281/zenodo.19740020.
• Qin, H. (2026). SAE Information Theory P7: The Spark of Life（"展开"第三本原语）. Zenodo. DOI: 10.5281/zenodo.20105884.
• Qin, H. (2026). SAE Quantum Mechanics P8（时间箭头 vs 方向；我们的大爆炸即对侧的大坍缩）. Zenodo. DOI: 10.5281/zenodo.20587634.
• Qin, H. (2026). Four-fold Nesting of the Physical-Quantity Ladder（4×4 子四分形、L4a–d）. Zenodo. DOI: 10.5281/zenodo.21227329.
• Qin, H. (2026). SAE Cosmology I（宇宙学常数、闭合 FRW）. Zenodo. DOI: 10.5281/zenodo.19245267.
• Qin, H. (2026). SAE Cosmology III（$a_0=(\pi/2)c\,\Delta\omega$）. Zenodo. DOI: 10.5281/zenodo.19281983.
• Qin, H. (2026). SAE Cosmology V（dual-4DD 反向箭头；$\Lambda_1+\Lambda_2=0$）. Zenodo. DOI: 10.5281/zenodo.19329771.
• Qin, H. (2026). SAE Cosmology VI（暗扇区：khronon 尘埃、因果标量场、四形式；HP1/HP2）. Zenodo. DOI: 10.5281/zenodo.21267555.
• Qin, H. (2026). Thought Experiment I（对称呼吸、5DD 中点之工作假设）. Zenodo. DOI: 10.5281/zenodo.19028005.

（外部经典文献的卷期页码以原始出处为准；SAE 系列以 Zenodo 记录为权威版本。）

致谢

本文的框架方向、核心论证与所有主张的最终裁定，由 Han Qin（秦汉）负责。在写作过程中，作者借助了若干人工智能系统进行概念梳理、推导复核、几何一致性检查与约束性批判，另有人工智能系统参与了发散讨论与外部评估；这些系统的贡献限于辅助性的整理、检验与批评，不构成对本文科学主张的作者身份。

作者责任声明

本文所有主张、其分层标注（结构性／后验／动机性／开放）、以及所标出的一切开放问题与可能的错误，均由作者本人负责。文中多处为 SAE 框架下的机制候选，尚未完成定量化、亦可能被证伪；作者按"不求无过、但求有功"的原则完整呈现之，以期为后续研究提供一个可推敲、可证伪的靶子。任何疏漏与错误，责任在作者。