16.  Cosmology: Acceleration from Ongoing Sublimation

The universe is not only getting bigger, it is getting bigger faster, and nobody is sure why. This chapter argues that if instantiation is still happening now, speeding up is exactly what you should expect, and then works out what believing that commits the theory to.

16.1.  PST’s reframing of the standard cosmological picture

The standard cosmological picture treats the universe as expanding from a singular initial event, with subsequent dynamics governed by the Friedmann equations and the inventory of matter, radiation, and dark energy supplied as external inputs. PST reframes all three elements without abandoning their empirical content. Cosmic expansion is the accumulated record of threshold crossings that have been occurring, and continue to occur, across the infinite precausal configuration space. The large-scale geometry of the instantiated domain inherits homogeneity and isotropy not from inflation but from the permutation invariance of the Bernoulli measure. Cosmic acceleration arises naturally from the thermodynamics of ongoing instantiation: conditional on the substrate’s pre-spatial framing, Mechanism B of §9.1 delivers the equation of state $w=-1$ (the cosmological-constant form) via a steady-state-style non-dilution argument; Mechanism A separately gives the standard quantum-vacuum contribution at the Standard Model (SM) hierarchy scale.

16.2.  The Cosmological Principle from the Bernoulli measure

The precausal configuration space is equipped with the Bernoulli product measure $\mu=\bigotimes_{a\in D}\mathrm{Bern}(1/2)$, equation (9), whose defining property is permutation invariance: any finite permutation of the index set $D$ leaves $\mu$ unchanged. No locus in configuration space is preferred over any other.

Let $j_{\tau}$ denote the substrate-internal saturation density of threshold-crossed configurations, measured against the configuration-space measure $\mu=\bigotimes_{a\in D}\mathrm{Bern}(\tfrac{1}{2})$ of P3. Its dimension is events per substrate event-volume, consistent with the dimensional pinning in §16.5; it is not a per-proper-volume rate in emergent spacetime, which would yield Hoyle–Bondi–Gold steady-state $w=0$. Its projection into the instantiated domain $\mathcal{G}$ is $\Pi(j_{\tau})$. A spatial variation of the projected density $\Pi(j_{\tau})(\mathbf{x})\neq\Pi(j_{\tau})(\mathbf{y})$ for $\mathbf{x}\neq\mathbf{y}$ would require that the pre-image configurations $\Phi^{-1}(\mathbf{x})$ and $\Phi^{-1}(\mathbf{y})$ are assigned different weight by $\mu$. But $\mu$’s permutation invariance assigns equal weight to all configurations related by a finite permutation of loci, and the functor $\Phi$ maps isomorphic configurations to isomorphic geometry. No positional preference is available at the precausal level; any apparent variation in $\Pi(j_{\tau})$ would require a distinguished locus in the substrate, which $\mu$ forbids. Isotropy of $\Pi(j_{\tau})$ follows by the same argument applied to directional labels.

The result is a theorem:

Theorem (Cosmological Principle).

The instantiation rate $j_{\tau}$ is spatially uniform and isotropic throughout $\mathcal{G}$ at all scales large relative to $d_{0}$. The Cosmological Principle, in standard cosmology a postulate, justified empirically by cosmic microwave background (CMB) uniformity [71], is a structural consequence of the permutation invariance of $\mu$.

It follows immediately that the large-scale geometry of $\mathcal{G}$ is described by the Friedmann–Robertson–Walker metric [68, 179, 180]:

$$ds^{2}=c^{2}\,dt^{2}-a(t)^{2}\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}\,d\Omega^{2}\right],$$ (220)

as the unique homogeneous and isotropic solution to the Einstein field equations derived in Chapter 7, with spatial curvature index $k$ determined by the energy content at the instantiation epoch.

16.3.  Horizon and flatness problems dissolved

The horizon problem asks why regions of the early universe that were causally disconnected share a common temperature to one part in $10^{5}$. The flatness problem asks why the spatial curvature parameter $\Omega_{k}$ was tuned to within one part in $10^{60}$ of zero at the Planck epoch. Standard inflation [52, 53] resolves both by an exponential expansion that carries a single causal patch to the scale of the observable universe.

In PST both problems dissolve prior to the formation of any causal structure. Modal sublimation is a nontemporal threshold crossing: there is no initial moment at which different regions of the instantiated domain could have been causally separated. Homogeneity and isotropy are not consequences of any physical process propagating through $\mathcal{G}$; they are structural properties of the precausal configuration space imprinted on $\mathcal{G}$ at sublimation by the permutation invariance of $\mu$. The uniformity of the CMB [74, 71] is not evidence for inflation; it is direct observational evidence for the permutation invariance of the Bernoulli measure.

The flatness problem also dissolves. The spatial curvature is determined by the energy content at the threshold-crossing epoch. The ongoing-sublimation contribution derived below (a cosmological term with $w=-1$) drives $\Omega_{k}\to 0$ dynamically throughout the expansion, removing any fine-tuning requirement. No inflaton field or separate scalar sector is needed.

16.4.  Ongoing instantiation: why it does not dilute

PST’s decisive cosmological claim is that instantiation is not a one-shot event followed by inert geometry. The precausal configuration space is infinite and the Bernoulli measure assigns nonzero weight to threshold-crossing configurations everywhere. Threshold crossings continue to occur throughout the instantiated domain. The instantiation rate $j_{\tau}$ is constant by the Cosmological Principle theorem above; the energy released per crossing event is $\Delta\mathcal{F}=\mathcal{F}_{\rm pre}-\mathcal{F}\big{|}_{\mathcal{V}}>0$, the descent from the high-tension pre-threshold configuration to the vacuum manifold $\mathcal{V}$ (the sombrero valley of Chapter 4). The effective energy density of ongoing instantiation is therefore

$$\rho_{\rm inst}=\frac{j_{\tau}\,\Delta\mathcal{F}}{c^{2}}.$$ (221)

The key question is whether $\rho_{\rm inst}$ dilutes as the universe expands.

The answer is that it does not. $\rho_{\rm inst}$ depends on two factors: the rate $j_{\tau}$ and the energy quantum $\Delta\mathcal{F}$. By the Cosmological Principle theorem, $j_{\tau}$ is fixed by the permutation invariance of $\mu$ and carries no dependence on the size or expansion rate of the instantiated domain. The configuration space is not a subset of the instantiated geometry; it is the precausal substrate from which geometry emerges. Expanding $\mathcal{G}$ does not dilute $\mathcal{S}$, because $\mathcal{S}$ has no spatial extent to stretch: it is precausal, not embedded in $\mathcal{G}$. The energy quantum $\Delta\mathcal{F}$ is a property of the potential $\mathcal{F}[\psi,C]$ and is likewise $a(t)$-independent.

Therefore:

$$\dot{\rho}_{\rm inst}=0.$$ (222)

16.5.  Equation of state and the sign of the cosmological constant

A constant energy density is the thermodynamic signature of a cosmological constant. The covariant energy-conservation equation,

$$\dot{\rho}+3H(\rho+P)=0,$$ (223)

with $\dot{\rho}_{\rm inst}=0$ and $H\neq 0$, forces the pressure to satisfy:

$$P_{\rm inst}=-\rho_{\rm inst},\qquad w\equiv\frac{P_{\rm inst}}{\rho_{\rm inst}}=-1.$$ (224)

This is the vacuum sector of (48), and it is the only one that can carry $w=-1$: Computation 155 shows that a stress-energy proportional to $g_{\mu\nu}$ requires exactly $\rho+p=0$, and Computation 158 shows the matter sector $\Pi_{\mu\nu}$ is traceless and so carries $w=1/3$. The two add and do not compete (§7.4.4). The sign of $\rho_{\rm inst}$ is positive: threshold crossings release tension energy (the configuration descends from the high-tension pre-threshold state to the sombrero valley), so $\Delta\mathcal{F}>0$ and thus $\rho_{\rm inst}>0$. The effective cosmological term contributed by ongoing sublimation is

$$\Lambda_{\rm PST}=\frac{8\pi G}{c^{2}}\,\rho_{\rm inst}>0.$$ (225)

This is positive, de Sitter, not anti-de Sitter. §9.1 identifies this as Mechanism B in the joint structural resolution of the cosmological- constant sign. Mechanism A (quantum vacuum on $\mathcal{V}\cong S^{1}$) gives a second, SM-inherited contribution, positive at the naive cutoff but regulator-dependent in sign (Scope of Mechanism A, §9.1). The sombrero-valley energy $\mathcal{F}\big{|}_{\mathcal{V}}<0$ is internal to the precausal substrate and does not couple directly to emergent gravity as $\Lambda$; the apparent AdS conclusion arises only from conflating $\mathcal{F}\big{|}_{\mathcal{V}}$ with the cosmological constant. The sign of $\Lambda$ is not a free parameter; it is forced by the direction of every threshold crossing (downhill in $\mathcal{F}$, with $\Delta\mathcal{F}>0$ by definition) and by the impossibility of diluting the configuration-space reservoir. PST inherits the SM cosmological-constant hierarchy problem but, unlike SM, identifies a structurally PST-specific positive contribution $\rho_{\mathrm{inst}}$ that could in principle dominate at small $j_{\tau}\,\Delta\mathcal{F}$. The $\Lambda_{\mathrm{obs}}$ magnitude (and its $\sim 10^{-120}$ hierarchy relative to $M_{\mathrm{P}}^{2}$) is $T(C)$-contingent, via an order-of-magnitude analysis using the Cramér framework of Computation 75 (Computation 82): the magnitude depends on the explicit form of $T(C)$ as a functional of substrate configurations plus the threshold scaling $\tau(D)$, and both are configuration-content quantities by the structural-scope theorem of §14.3. The sign and equation of state $w=-1$ established above remain structural theorems (Computation 137).

16.6.  Friedmann equations and cosmic acceleration

Substituting the PST stress-energy inventory, pressureless matter, radiation, and the ongoing-sublimation term $(\rho_{\rm inst},\,P_{\rm inst}=-\rho_{\rm inst})$, into the Einstein field equations of Chapter 7 with the Friedmann–Robertson–Walker (FRW) metric (220) yields the standard Friedmann equations:

$$H^{2} =\frac{8\pi G}{3}\Bigl{(}\rho_{m}+\rho_{r}+\rho_{\rm inst}\Bigr{)}-\frac{kc^{2}}{a^{2}},$$ (226)

$$\frac{\ddot{a}}{a} =-\frac{4\pi G}{3}\Bigl{(}\rho_{m}+2\rho_{r}-2\rho_{\rm inst}\Bigr{)},$$ (227)

where the second equation uses $\rho+3P=(\rho_{m}+\rho_{r}+\rho_{\rm inst})+3(P_{m}+P_{r}+P_{\rm inst})=\rho_{m}+2\rho_{r}-2\rho_{\rm inst}$, with $P_{m}=0$, $P_{r}=\rho_{r}/3$, and $P_{\rm inst}=-\rho_{\rm inst}$. In the instantiation-dominated epoch, $\rho_{m},\rho_{r}\ll\rho_{\rm inst}$, and equation (227) reduces to

$$\frac{\ddot{a}}{a}\;\approx\;\frac{8\pi G}{3}\,\rho_{\rm inst}>0,$$ (228)

giving late-time acceleration with no additional assumption. The role of $\rho_{\rm inst}$ in PST is structurally identical to $\Lambda$ in $\Lambda$CDM, but the interpretation differs. In $\Lambda$CDM, $\Lambda$ is a free parameter inserted by hand. Here, $\Lambda_{\rm PST}=8\pi G\,\rho_{\rm inst}/c^{2}$ is determined, in principle, by the rate $j_{\tau}$ and the tension quantum $\Delta\mathcal{F}$, both of which are properties of the precausal structure. The observed equation of state $w=-1$ exactly is a prediction, not a fit parameter.

16.7.  The rate problem

Computing $j_{\tau}$ from first principles requires integrating the threshold-crossing rate over the Bernoulli measure:

$$j_{\tau}=\int_{\mathcal{S}}\mathbf{1}\bigl{[}T(C)\geq\tau\bigr{]}\,d\mu(C),$$ (229)

where $\mathbf{1}[\cdot]$ is the indicator for configurations at or above the modal threshold. This integral is not presently calculable: the effective restriction of $\mu$ to near-threshold configurations, where $\Delta\mathcal{F}$ is well-defined, requires a renormalisation prescription analogous to the Casimir energy calculation of Chapter 15. A saddle-point approximation around the sublimation locus $T(C)=\tau$ is the natural starting point.

The value of $j_{\tau}$ is not computed here, its magnitude being $T(C)$-contingent like the $\Lambda$ magnitude it fixes (Computation 82), but its structural form reduces (Computation 75) to a standard Bernoulli-measure large-deviation probability:

$$j_{\tau}\;=\;\mu\bigl{(}\{C\in\mathcal{P}(D):|T(C)|\geq\tau\}\bigr{)},$$ (230)

which is the tail probability of $|T(C)|$ under the i.i.d. Bernoulli measure $\mu=\bigotimes_{a}\mathrm{Bern}(1/2)$ delivered structurally by P1. By Cramér’s theorem [150], $j_{\tau}$ admits the asymptotic

$$j_{\tau}\;\sim\;\exp\bigl{(}-|D|\,I(\tau/\sqrt{|D|})\bigr{)}\quad\text{as }|D|\to\infty,$$ (231)

where $I$ is the Cramér rate function, the Legendre transform [150] of the cumulant generating function of $|T(C)|/\sqrt{|D|}$. Computation 75 verifies this exponential-in-$|D|$ decay empirically on a representative random-walk-like $T(C)$ model. The remaining structural input from PST is the explicit form of $T(C)$ at the postulate level (beyond the random-walk-like model used in Computation 75) and the precise $\tau$-scaling with $|D|$; both are $T(C)$-contingent by the structural-scope theorem of §14.3, and once both are supplied, Cramér delivers $j_{\tau}$ analytically. Thus $j_{\tau}$ is a Cramér saddle-point computation conditional on the explicit $T(C)$ functional, a structural reduction analogous to the A7 $\to$ uniform-rate reduction. The observed value $\Lambda_{\rm obs}\approx 1.1\times 10^{-52}\;\mathrm{m}^{-2}$ provides a target: $j_{\tau}\cdot\Delta\mathcal{F}=\Lambda_{\rm obs}\,c^{4}/8\pi G\approx 5.3\times 10^{-10}\;\mathrm{J\,m}^{-3}$. A successful computation of $j_{\tau}$ would determine $\Lambda_{\rm PST}$ from first principles and, together with the Casimir prediction, constitute a second independent test of the modal-condensate structure of Chapter 15.

16.8.  Observational commitments

Beyond the standard $\Lambda$CDM predictions, PST makes the following specific commitments:

1. CMB uniformity. Permutation invariance of $\mu$ guarantees homogeneity and isotropy prior to any causal structure. Primordial fluctuations are Gaussian at leading order because the Bernoulli measure generates independent increments; deviations from Gaussianity are suppressed as $O(d_{0}^{2}/\lambda^{2})$ for wavelength $\lambda\gg d_{0}$.

2. Big Bang nucleosynthesis. PST does not alter the microphysics of BBN. The weak-interaction rates, neutron-to-proton ratio, and light-element abundances depend on the quantum mechanics (QM) sector, which emerges as a projection of the precausal structure (Chapter 5) and is indistinguishable from standard QM at energies well below $M_{*}=4\pi m_{h}\sqrt{2/3}\approx 1.285$ TeV.

3. Equation of state. The instantiation-dominated epoch gives $w=-1$ exactly. This is the theory’s sharpest cosmological commitment and its most imminent falsifier: it is already under test, and the present Dark Energy Spectroscopic Instrument (DESI) results favour a dynamical equation of state over $\Lambda$CDM at $2.8\sigma$–$4.2\sigma$ depending on the dataset combination [198]. What that would and would not cost PST, and why the theory cannot absorb it, is set out in §16.9.

4. Spatial flatness. The driving of $\Omega_{k}\to 0$ by $\rho_{\rm inst}$ and the absence of initial fine-tuning predict $\Omega_{k}=0$ to high precision, consistent with current CMB bounds [71].

5. Cosmic acceleration. Equation (228) gives late-time acceleration consistent with Type Ia supernova data [69, 70] without introducing new fields beyond those required by the precausal structure.

16.9.  What would refute the constancy, and what would not

Chapter 15 separates the Casimir prediction into a parameter-free exponent and an amplitude that rests on the normalisation premise, and insists that the exponent carries the structural falsifiability. The cosmological result has the same two-layer structure, and it is worth stating it as plainly, because the layer that is exposed is not the one a reader is likely to expect.

The magnitude is not PST’s to predict, and no measurement of it can refute the theory. The observed value of $\Lambda$ and its $\sim 10^{-120}$ hierarchy are $T(C)$-contingent: they are properties of how this world came out, not consequences of P1–P3, and §16.7 says so without hedging. A revised measurement of the dark-energy density is not evidence for or against PST, in exactly the way a null Casimir result tightens the bound on $d_{0}$ rather than excluding the theory.

The constancy is PST’s, and it is what a measurement can refute. §16.5 does not merely accommodate dark energy; it derives non-dilution, and non-dilution forces $w=-1$ exactly. The derivation fixes no amplitude and admits no tuning: $\rho_{\mathrm{inst}}=j_{\tau}\,\Delta\mathcal{F}/c^{2}$ neither dilutes nor redshifts, so $\dot{\rho}=0$ and continuity gives $w=-1$ at every epoch. This is a statement about scaling, in the same structural register as the $d^{-6}$ exponent, and there is no contingent quantity behind which it can shelter.

16.9.1.  The falsifier, stated sharply

The claim of §16.5 is refuted if:

1. The dark-energy equation of state is established to differ from $-1$ at any epoch in the instantiation-dominated era. PST predicts a constant, not a value near $-1$ today, so a confirmed $w_{0}\neq-1$ and a confirmed $w_{a}\neq 0$ are each sufficient.

2. A fortiori, a confirmed crossing of the phantom divide. A history in which $w<-1$ in the past and $w>-1$ now is not a small departure from the prediction but is incompatible with the mechanism at every redshift: a density that does not dilute cannot cross. This matters because it is the shape the current data prefer.

The claim is not refuted by:

1. A measurement of $\Lambda$’s magnitude, at any precision. That quantity is $T(C)$-contingent and the theory never claimed it.

2. A failure to derive the $10^{-120}$ hierarchy. Same layer, same answer.

3. A tighter bound that remains consistent with $w=-1$, which confirms rather than threatens.

4. A statistical preference for a model carrying extra parameters, absent a determination that $w$ departs from $-1$. Clause 1 above requires the departure to be established; a model comparison that favours a larger parameter space is not by itself that determination, particularly where the datasets entering it are in tension with one another.

The first three touch the contingent layer only. The fourth is a statement about evidential thresholds rather than about layers, and the next subsection returns to it, because it is the clause the current measurements actually engage.

A rival that predicts the departure. The nearest programme to PST on the cosmological constant is causal set theory’s everpresent $\Lambda$ [227]. There the discreteness of the causal set makes the four-volume $V$ a counted quantity, $\Lambda$ is its conjugate, and the Poisson fluctuation of the count gives $\Lambda\sim\pm 1/\sqrt{V}$: a cosmological “constant” of order the critical density at every epoch, wandering in magnitude and in sign. That is an answer to the why-now question which PST does not offer, since §16.7 leaves the magnitude contingent, and it is a prediction PST cannot share: a wandering $\Lambda$ is a $w$ that departs from $-1$, which is the first refuting clause above. Fits of everpresent $\Lambda$ models reach the current data as well as $\Lambda$CDM does and ease the low-redshift tension in $H_{0}$ [228]. So the two programmes, which agree that the cosmological constant is a consequence of discreteness beneath spacetime, are separated by exactly the measurement this section names: a $w(z)$ established to be constant refutes everpresent $\Lambda$, and one established to wander refutes PST.

16.9.2.  Where the measurements currently stand

On this question PST’s prediction coincides with $\Lambda$CDM’s: a cosmological constant, $w=-1$ at every epoch. The concordance model remains the one a rival has to beat, and the position defended below is that PST’s cosmological claim is at present vulnerable and not refuted; it has acquired a live route to refutation that it did not have when the derivation was written, and it has not been refuted. What follows is the evidence for that assessment, taken in the order a sceptic would want it: the challenge first, then its condition.

The Dark Energy Spectroscopic Instrument’s second data release reports a dynamical dark-energy model preferred over $\Lambda$CDM at $3.1\sigma$ for DESI baryon-acoustic-oscillation data combined with the cosmic microwave background, rising to between $2.8\sigma$ and $4.2\sigma$ when Type Ia supernova compilations are added, the range reflecting which supernova sample is used [198]. The favoured region is the quadrant $w_{0}>-1$, $w_{a}<0$: phantom in the past, quintessence-like today, crossing $w=-1$ in between. That is clause 2 of the preceding subsection, and it is the most serious observational challenge the claim currently faces.

Four things bear on how much of a challenge it is, and the first comes from the measurement itself.

The baryon acoustic oscillation (BAO) data are well described by a cosmological constant. DESI’s own analysis states that its results are well described by a flat $\Lambda$CDM model, while recording that the parameters its BAO data prefer sit in $2.3\sigma$ tension with those from the CMB [198]. The preference for evolution is therefore not something the BAO measurements show on their own terms; it emerges from combining datasets that do not agree with each other, and a two-parameter extension will relieve a tension between datasets whether or not the extension is physical.

On the supernova side the preference tracks a known low-redshift systematic. Correcting for systematics in the low-$z$ ($z<0.1$) sample reduces the preference for dynamical dark energy to below $2\sigma$ [201], and a critical review of the DESI-era evidence finds it sits at $\lesssim 3\sigma$ for most dataset combinations, reaching $\sim 3.8\sigma$ only when the DES-SN5Y sample is included, tracing that enhancement to the same low-redshift subsample [199]. The internal pattern is itself suggestive: adding Pantheon⁺ lowers the significance to $2.8\sigma$, below the $3.1\sigma$ obtained without supernovae at all, while the $4.2\sigma$ figure comes from the compilation in which the low-redshift problem has been identified.

On the BAO side the deviation is carried by particular tracers. The apparent departure from $\Lambda$CDM in DR2 is driven mainly by the LRG1 and LRG2 samples, and removing either restores concordance [202]. Separately, testing DESI DR2 against Pantheon⁺ through the cosmic distance-duality relation, which follows from photon-number conservation in any metric theory and holds whatever dark energy is doing, reveals a redshift-dependent violation; accounting for it suppresses the preference for dynamical dark energy [200]. A mismatch of that shape can masquerade as evolution.

And the extension does not buy what it is often invoked for. Neither $\Lambda$CDM nor the two-parameter alternative reduces the sound horizon by the $\sim 7\%$ required to relieve the Hubble tension [199], so the dynamical model is not recommended by independent explanatory work it performs elsewhere.

Against all of which stands the strongest point on the other side, which should be stated rather than omitted: the trend is not an artefact of the two-parameter $(w_{0},w_{a})$ form, since non-parametric reconstructions and alternative extensions recover similar dynamical behaviour. The argument therefore lives in the systematics of the datasets, not in the choice of parameterisation, and there it remains unsettled.

Two conclusions follow, and they must be held together. The preference strengthened rather than faded between the first and second data releases, and $3\sigma$ against a claim that admits no tuning is a warning, not noise. But the falsifier of the preceding subsection has not fired: clause 1 requires a departure from $w=-1$ to be established, and what exists is a preference, in some dataset combinations, whose significance falls below $2\sigma$ under one published systematic correction and dissolves under another. Preference is not establishment, and clause 4 of what would not refute the claim is precisely this case.

The distinction matters here for the same reason it matters in Chapter 15: a theory that reads pressure as refutation abandons a result it is still entitled to, and one that reads it as noise learns nothing from it. The question will be settled within the decade by DESI’s later data releases together with Euclid, the Rubin Observatory, the Roman Space Telescope and the Simons Observatory, and what is owed in the meantime is to have said in advance what each answer would mean.

16.9.3.  Why PST cannot retreat, and what that costs

It could be supposed that $w=-1$ is stated conditionally on the substrate’s pre-spatial framing, and that relaxing that framing would yield $w\neq-1$ while preserving the mechanism. It would not, and the reason is worth making explicit before the theory has any need of it.

The framing is load-bearing twice over. $j_{\tau}$ is a saturation density measured against the configuration-space measure of P3, and it is defined that way because the substrate has no spatial extent against which a proper-volume rate could be posed. Replacing it with a per-emergent-volume creation rate does not give a slightly different cosmological constant; it gives Hoyle–Bondi–Gold steady state, $w=0$, which is excluded by an enormous margin. And the same choice is what forbids PST a spontaneous collapse rate (§10.3.4): a spontaneous localisation rate is a per-volume, per-time quantity of precisely the kind the framing excludes.

The equation of state and the collapse rule are therefore exposed together. Were an evolving dark-energy equation of state confirmed, PST would not lose the cosmological constant and keep the rest; it would lose the framing that also underwrites its account of measurement. Conversely, were spontaneous localisation observed in an isolated superposition, the damage would run the other way. Two results that look independent share one point of failure.

This is a worse position to be in than having two separate risks, and it is a better thing to have established now than to discover when the data demand it. A derivation that finds its own flexibility only once it is needed is worth much less than one that mapped the flexibility in advance; the flexibility here is mapped, and there is none.

16.10.  Comparison with steady-state cosmology

The continuous creation of energy density in Bondi–Gold–Hoyle steady-state cosmology [72, 73] is the closest historical precedent for PST’s ongoing instantiation: both predict $\dot{\rho}=0$ as a consequence of an ongoing creation process. The differences are decisive, and the equation of state is one of them: classic steady-state cosmology creates ordinary matter that dilutes, giving $w=0$, whereas PST’s non-diluting instantiation gives $w=-1$ (§16.5).

Mechanism. Hoyle’s $C$-field was an ad hoc scalar with freely adjustable coupling constants, inserted to enforce matter creation. PST’s ongoing instantiation needs no such insertion, and the reason is worth stating carefully, because the obvious version of it is weaker than the one available.

The obvious version appeals to size: the configuration space is infinite, so the supply cannot run out. That argument does not survive contact with $\mu$, which is a probability measure and therefore carries total mass $1$ however large $D$ is. A process consuming measure at a fixed rate exhausts it in finite time, and an infinite index set does not change that.

The argument that does work is that there is nothing to consume. What §16.5 calls $j_{\tau}$ is a saturation density against $\mu$, not a flux drawn from a stock, and the same section already excludes the alternative for an independent reason: a per-proper-volume creation rate yields the Hoyle–Bondi–Gold steady state $w=0$ rather than $w=-1$. A draining reservoir is a per-volume flux of exactly that form, so the depleting reading is ruled out by the equation of state PST derives, before any question of supply arises (Computation 147). Crossing changes a configuration’s modal status; it does not spend it. The reservoir has no bottom because it is not a reservoir.

What is created. Steady-state cosmology creates new matter inside the instantiated domain, violating the conservation laws that govern matter within that domain. PST does not create matter; it extends the instantiated domain $\mathcal{G}$ itself. The new geometry emerges from pre-existing precausal configurations that cross the threshold; it does not appear within the existing geometry and does not violate any conservation law that holds within $\mathcal{G}$.

The CMB. The steady-state model predicted no CMB; its refutation by Penzias and Wilson [74] was decisive. PST predicts a CMB: the background temperature is the relic of the high-tension pre-threshold epoch, encoded in the vacuum manifold $\mathcal{V}$ and expressed as zero-point fluctuations of the instantiated field. The CMB is evidence for PST, not against it.

Structure formation. Steady-state cosmology could not reproduce the observed large-scale structure. PST predicts a near-scale-invariant Gaussian spectrum of primordial perturbations from the Bernoulli measure, consistent with Planck CMB observations [71] and the transfer-function structure of $\Lambda$CDM.

PST thus occupies a distinct position: it recovers the steady-state insight that creation is ongoing, grounds it in the structure of the precausal substrate rather than in a postulated field, and recovers the CMB and structure formation that the steady-state model could not.

16.11.  Summary

PST reframes cosmology from the ground up. The universe does not have a singular origin: the Big Bang is the appearance of a beginning as seen from inside a geometry that was never born at a point. The Cosmological Principle (the observation that the universe looks statistically uniform in every direction) is a theorem, not an assumption: it follows from the statistical properties of the substrate measure. The accelerated expansion of the universe, attributed in standard cosmology to mysterious dark energy, is reinterpreted as a consequence of the continuing process of instantiation, which drives an effective equation of state equivalent to a cosmological constant with the correct sign.

In terms of what is established: the Cosmological Principle, the homogeneity and isotropy of $\Pi(j_{\tau})$, is a theorem (§16.2) following from the permutation invariance of the configuration-space measure $\mu$, conditional on the substrate-internal $j_{\tau}$ definition. The equation-of-state value $w=-1$ is a conditional theorem, following from $\dot{\rho}_{\mathrm{inst}}=0$ under that same $j_{\tau}$ definition, with the non-dilution conclusion a consequence of $\Pi$’s frame-independence applied to the pre-spatial substrate rather than an added postulate. Two mechanisms are offered for the sign of the effect. Mechanism A is regulator-dependent: the bare Goldstone zero-point on $\mathcal{V}\cong S^{1}$, which inherits the cosmological-constant sign-fine-tuning problem. Mechanism B is structural: a positive $\rho_{\mathrm{inst}}$ from ongoing threshold-crossing, with the sign forced by $\Delta\mathcal{F}>0$. The magnitude is not fixed here: rate-fixing to match the observed $\Lambda$ is $T(C)$-contingent content (Computation 82), not an open derivation.