P4
the redshift-isotropy floor, the empirical forcing of cosmic time and uniform expansion, and the measured resolution of the relativity of simultaneity
The standard cosmological model installs a global cosmic time, a preferred rest frame, and a spatially uniform expansion as kinematical assumptions, prior to and independent of the dynamical fit to data. We show that these assumptions are not conventions the data merely tolerate but structures the data force. The cosmological redshift is the path integral of the expansion rate, ln (1+z)=∫H dt, so the observed anisotropy of the cosmic microwave background (CMB) separates exactly into a source term, fixed at last scattering, and a cumulative term, set by the integrated expansion along each line of sight. Granted even perfect homogeneity at last scattering, the observed isotropy of the monopole is then a direct measurement of uniform expansion—so the common supposition that the isotropy is supplied by homogeneity at decoupling is a category error. We make the alternative quantitative: were the expansion rate to track the inhomogeneous matter distribution region by region, with no single global scale factor, the accumulated redshift would scatter by ∼10-3 across the sky, against an observed ≲3×10-6—an exclusion by three orders of magnitude. The effect is distinct from the Sachs–Wolfe anisotropies and not a re-derivation of them: when lumpiness clusters upon a single background, a photon's gravitational redshift descending into a potential well is undone climbing out, so the contributions telescope along the path and leave no √N accumulation, whereas a genuine differential expansion accumulates because there is nothing to cancel against—so the observed isotropy selects the telescoping picture over the accumulating one. Every choice in the estimate biases it downward, so the number is a floor. A differential programme can evade it only by requiring the local rates to average to a common value in every direction, which is itself a global uniform expansion in a cosmic time—the very structure such programmes set out to dispense with. What is excluded is an inhomogeneous expansion rate; the matter lumpiness is untouched, the density being the bend of the spatial cut, so uniform expansion and lumpy matter are consistent. The remaining, finely tuned, observer-centred escape is closed by the Copernican principle together with the independently measured isotropy of the expansion history. What the exclusion selects is a global, time-ordered expansion—the cosmic foliation made dynamical—which is logically prior to the notions of space, isotropy, and homogeneity the model otherwise assumes. The same datum thus establishes, from the bottom up, the cosmic time, the uniformity of its advance, and—within the observable region, under the Copernican principle—the maximal symmetry of its slices. Read ontologically, this measured foliation is the lapse of the 3+1 split—an objective rate of advance, the cosmic present—while the relativity of synchrony is the shift; the century-old reading of synchrony's relativity as the absence of any objective present is thereby a modal fallacy, which the measured isotropy falsifies outright. We prove that fixing the physical foliation and reading it so is both necessary and sufficient for a coherent formal description of an existing, evolving world—an augmentation of general relativity that changes none of its equations, its necessary half here measured. We also set down the history the measurement closes. The structure this datum forces was posited by Einstein in February 1917—explicitly “against the spirit of relativity”, and on the empirical ground that stellar proper motions are small compared with —countered a month later by de Sitter, who placed the choice among candidate universes outside physical argument altogether and pressed as his sharpest objection that in Einstein's solution “time has a separate position”; and defended in 1920 by Eddington on geometric grounds, in the declared absence of any experimental knowledge on cosmical scales. Einstein then went nearly silent on cosmology for the rest of his life, and never addressed how the cosmic time his own assumption distinguished stands to the relativity of simultaneity. The assumption was correct, and the objection was correct as a description of the structure though not as a reason to reject it. This is the empirical counterpart of the conceptual results of the companion papers [JanzenCRframework, JanzenBHcausality]: as stellar parallax converted the Earth's motion from an interpretive option into a measured fact, the isotropy of the cosmological redshift converts the cosmic foliation from a modelling convenience into a measured feature of the world.
The standard model of cosmology is assembled in a definite order [Robertson1933, Robertson1935, Walker1937, north1965]. One first posits that the average rest frame of matter—made manifest in the redshift–distance relation [Hubble1929]—defines a coherent cosmic time, the one direction in which the four-dimensional world is, in Eddington's phrase, not curved [Eddington1920]; then that space is the family of hypersurfaces orthogonal to it; then that this space is isotropic; then that it is homogeneous; and only last that its geometry evolves by a scale factor a(t) whose form the field equations fix through the energy content [Friedmann1922, Lemaitre1927]. The first four are kinematical posits; the dynamical fit comes fifth and last. The standard presentation treats the kinematical posits as idealizations vindicated after the fact by the success of the fit—a global cosmic time, a preferred rest frame, and a uniform expansion accepted because the model built on them works.
The claim of this paper is that observation forces those kinematical posits directly, and not through the success of the dynamical fit. The forcing is a falsification, not a confirmation: the one coherent alternative to the posits is given its observable consequence and rejected by the data. Where the standard argument runs only the affirmative half of the scientific loop—the model is consistent with uniform expansion—we run the half that decides, deriving what the sky would look like if the posits failed and finding that it does not look like that, by three orders of magnitude. The decisive observable is the isotropy of the CMB monopole redshift, read correctly: not as a property inherited from the emitting surface, but as the accumulated record of the expansion the radiation traversed. That this is a distinct effect from the Sachs–Wolfe anisotropies, and not a re-derivation of them, is the crux and is established in §3: lumpiness clustering upon a single background carries a metric potential whose contributions telescope along a photon's path—the redshift descending into a well undone climbing out—leaving no accumulation, whereas a differential expansion accumulates because there is nothing to cancel against. The discriminator is that the expansion rate is forced by the observation while the matter content is not, and the two must not be run together.
This is the modern counterpart of a familiar episode. Before Bessel, whether the Earth moved was, on a strict reading, an interpretive choice among empirically equivalent pictures; stellar parallax converted it into a measured fact [Bessel1838], and the heliocentric frame from an option into the one the measurement singled out [JanzenFortress]. A purely relational reading of relativity holds, by the same token, that a preferred cosmic rest frame could never be observable even if it existed. The CMB is the measurement that overturns that expectation: the universe furnished, through the physics that produced the background and the geometry of its uniform expansion, a direct measurement of our motion with respect to the cosmic rest frame, and with it of the cosmic time that frame carries [Janzen2025, JanzenWildGoose]. What follows makes that “modern parallax” quantitative.
A real, global cosmic time is a structure general relativity's formalism admits but does not supply, and a companion paper takes it as a commitment on other grounds [JanzenCRframework]. This paper shows that the commitment is not merely admissible but empirically forced. The two arguments are deliberately independent: a companion result on the causal occurrence of gravitational collapse stands on causal structure alone and needs none of the cosmological evidence below [JanzenBHcausality], and nothing in the present argument draws on it. That they meet is therefore evidence and not construction. We stress the register at the outset, as the companion papers do: nothing here modifies general relativity. The equations are unchanged. What is established is which of the formally available foliations is the physical one—and, read ontologically (§4, Theorem 1), that this physical foliation is the lapse, the objective cosmic present, while the relativity of synchrony is the shift: the measured isotropy thereby settles the century-old question the relativity of simultaneity left open—synchrony relative, simultaneity absolute—exposing the reading that denied an objective present as a modal fallacy, and fixing this reading as the necessary and sufficient augmentation of general relativity for a coherent description of an existing, evolving world.
The redshift along a line of sight is the total expansion the photon underwent, 1+z = aobs/aem, equivalently ln (1+z) = ∫H dt integrated along the ray: the cosmological redshift simply is the accumulated expansion in that direction, a path integral and not a property of the source. That path integral is a conserved quantity written out, which is why it is a property of the path. In conformal form ds2=a2(η)(-dη2+d x2) the vector ∂η is a conformal Killing vector, so its charge ξ·p is conserved along a geodesic only where p ·p=0; on the null cone that charge is aω, and its constancy is 1+z=aobs/aem, whose logarithm is ∫H dt. The restriction to the null cone is not incidental to the argument below but the reason it works: for a massive carrier p ·p=-m2, the charge is not conserved, and no such line-of-sight integral exists—so the redshift is a statement about light and could not have been obtained from any other messengerL14. The observed temperature is Tobs(n) = Tem(n)/(1+z(n)), so the observed anisotropy separates exactly into two physically distinct contributions,
a source term fixed by conditions on the last-scattering surface, and a cumulative term set by the directional variation of the integrated expansion rate over the whole path. The two are independent: the value of one constrains the other not at all. Observed isotropy therefore requires both to be small—uniformity at decoupling and uniformity of the subsequent expansion—and these are precisely the two kinematical assumptions the standard model installs by hand, homogeneity at decoupling zeroing the first term and a single direction-independent scale factor a(t) zeroing the second.
The decisive consequence is seen by granting the common reading its strongest form. Suppose the last-scattering surface were perfectly homogeneous, δTem = 0 exactly. Then (1) reads δTobs/Tobs = - δ∫H dt, with nothing left over: the entire observed isotropy is the statement that the integrated expansion was the same in every direction. Homogeneity at the source does not supply the isotropy; granted perfectly, it removes one term and leaves the observed isotropy as a direct measurement of uniform expansion. The widespread supposition that the monopole's isotropy is delivered by homogeneity at last scattering, full stop, is thus a category error—it credits the source for a fact about the observed radiation that resides entirely in the intervening expansion. What the remainder of this paper does is put a number to the cumulative term under the one coherent alternative to a global a(t), and read that number against the data.
The hypothesis the floor tests is that the large-scale expansion rate tracks the matter lumpiness region by region—that the expansion follows the local density, with no single global scale factor. This is the operative reading of the backreaction and timescape programmes [Buchert2000, Wiltshire2007]; we make it quantitative through the separate-universe correspondence, in which a spherical region of uniform overdensity evolves as a closed Friedmann patch.
To linear order the separate-universe correspondence assigns a region of fractional matter contrast δ≡δρ/ρ a fractional deficit in its local scale factor. Matter conservation gives ρ∝a-3, hence dρ/ρ= -3 da/a, so
An overdense region has expanded less; an underdense region more.
The observed CMB temperature in a given direction is Tobs = Tlss/(1+z) with 1+z = aobs/aem the total expansion factor along that line of sight, so to the order kept |δT/T| = |δa/a|: the temperature contrast between directions is the fractional contrast in accumulated expansion. A CMB photon traverses many regions of differing density; modelling its path as a tube binned into statistically independent cells of comoving size R = 8 h-1 Mpc, each carrying rms contrast σ8, the accumulated fractional redshift contrast is the path mean of the per-cell contrasts. By the central limit theorem its standard deviation is
the factor 1/3 from (2) and the 1/√N the averaging-down of independent contributions—a path mean, not a sum.
The effective contrast weights σ8 by its linear growth along the path, σ8,eff = ∫σ8(z) w(z) dz / ∫w(z) dz with σ8(z) = σ8(0) D(z) and w = H-1; for Ωm = 0.315, σ8(0) = 0.8 and zlss = 1089 [Aghanim2020] this gives σ8,eff ≈0.285. The comoving distance to last scattering is dlss = (c/H0)∫0zlss dz/E(z) ≈9.4 h-1 Gpc, so N = dlss/R ≈1174. Equation (3) then gives
Every choice in this estimate biases it downward, so the result is a genuine floor:
The monopole redshift is isotropic—after removal of the dipole, Galactic foregrounds, and the standard secondary anisotropies—to ≲3×10-6 [Aghanim2020].P4R1 The floor (4) exceeds this by some three orders of magnitude. The comparison is a lower bound against an upper limit, which is what makes it robust rather than close: every free choice in the estimate pushes the two further apart, so the exclusion cannot be weakened by a different estimate, only by a different physical claim. The margin is a factor of 923L25, and it is worth stating what would be required to close it—the observed isotropy would have to be 9× worse than measured for the exclusion to fall to two orders of magnitude, 92× worse for one, and 923× worse for it to vanish. The hypothesis that the expansion rate is locally set by the matter distribution is therefore excluded, and not marginally.
It is essential that this is a distinct effect from the ordinary Sachs–Wolfe and integrated Sachs–Wolfe anisotropies [Sachs1967], not a re-derivation of them. In the uniform-expansion picture the matter lumpiness is clustering upon a single background a(t), carried by a metric potential Φ; a photon's gravitational redshift descending into a potential well is undone climbing out, so the contributions telescope along the path, leaving the endpoint potential (∼10-5) and the small integrated term, with no √N accumulation. The floor computed here is the contrary case—a genuine differential expansion, a cosmological redshift that accumulates because there is nothing to cancel against. The observed anisotropy (∼10-5 in total, ≲3×10-6 once secondaries are removed) thus selects the telescoping, uniform-expansion picture over the accumulating, differential one. The isotropy of the monopole is not accounted for by large-scale statistical homogeneity—that homogeneity is fully consistent with the differential-expansion hypothesis, which would still produce the ∼10-3 scatter of (4)—but by uniform expansion in a cosmic time.
A differential-expansion programme can evade the floor only by requiring the local rates to average to a common value in every direction: a uniform large-scale Hubble flow. But that requirement is itself a global uniform expansion in a cosmic time, the very structure such programmes set out to dispense with; imposing it to recover the observed isotropy reinstates the cosmic rest frame and the objective simultaneity it carries. Either way, the observed isotropy of the monopole redshift forces a uniform cosmic expansion.
The floor constrains the expansion rate, not the matter content. CR's real space is locally curved and inhomogeneous—the matter density is the bend of the spatial cut off the de Sitter vacuum profile [JanzenOperator, JanzenRange]—and that lumpiness is untouched by the argument. What is excluded is an inhomogeneous expansion rate tracking the lumpiness: the layer advances uniformly in cosmic time while its intrinsic geometry carries the bend. Uniform expansion and lumpy matter are consistent, and only the former is at issue here.
The floor closes the statistical escape, and one further escape is logically available and must be closed separately. A finely tuned, spherically symmetric inhomogeneity centred on the observer would produce isotropic redshift about us with no global uniform expansion; being structured rather than statistical, it is not touched by the √N floor. It is closed instead by the Copernican principle together with the independent isotropy of the expansion history itself: standard rulers (baryon acoustic oscillations [Eisenstein2005]) and standard candles (Type Ia supernovae [Riess1998, Perlmutter1999]) reconstruct the expansion along many directions and agree across the sky, so the uniformity is not the vantage artefact of a privileged centre [Ellis2006, Barrett2000, Clarkson2010]. With the statistical alternative excluded by the floor and the centred one by the Copernican step, the disjunction is exhausted: the observed isotropy forces uniform cosmic expansion.
What that exclusion reaches is the point of the paper. The region the data reject is not a parameter corner but the entire class of histories lacking a single global scale factor on a single global time; to reject that class is to select a global, time-ordered expansion, which is the cosmic foliation made dynamical. And the foliation is logically prior to everything the standard model otherwise assumes—“space” is one of its slices, “expansion” an ordered family of them, “isotropy” and “homogeneity” properties of a slice—none of which is even statable until the foliation is in hand. So a single datum establishes, from the bottom up, the cosmic time on which the rest stands, the uniformity of its advance, and—within the observable region and under the Copernican principle—the maximal symmetry of its slices. The model's foundational kinematics are not idealizations the data tolerate; they are what the data force, the one coherent alternative excluded.
The structure this datum forces was posited, attacked, defended, and abandoned within four years, and the record is worth setting down because the abandonment shaped a century of practice.
Einstein's cosmological paper of February 1917 assumed, in his own words and marked as such, “against the spirit of relativity”, that “there is a system of reference relatively to which matter may be looked upon as being permanently at rest” [Einstein1917]. His ground was empirical: “the most important fact that we draw from experience as to the distribution of matter is that the relative velocities of the stars are very small as compared with the velocity of light”. In a special-relativistic world one would expect neighbouring bodies at velocities drawn uniformly from (0,c); the observed smallness of proper motions is evidence that the world lines are nearly parallel, and the three-space in which they are at rest is distinguished. That inference is the one this paper's floor now closes at three orders of magnitude, and it was drawn from stellar velocities a century before.
De Sitter's reply came one month later and countered every consideration in turn [deSitter1917]. The boundary conditions were dismissed as unobservable in principle and, given the new λ-term, achievable anyway; the world-matter Einstein thought essential was declared to “serve no other purpose than to enable us to suppose it not to exist”; and the choice among the candidate universes was placed outside physics altogether—it “cannot be decided by physical arguments, but must depend on metaphysical or philosophical considerations, in which of course also personal judgment or predilections will have the same influence”. And the criticism he pressed most was the one that matters here: that in Einstein's solution “time has a separate position” from space.
Three years later the same structure was defended on geometric grounds, by the relativist best placed to do it. Eddington held that “the world taken as a whole has one direction in which it is not curved; that direction gives a kind of absolute time distinct from space”, that relativity “is not concerned to deny the possibility of an absolute time, but to deny that it is concerned in any experimental knowledge yet found”, and that “it need not perturb us if the conception of absolute time turns up in a new form in a theory of phenomena on a cosmical scale, as to which no experimental knowledge is yet available” [Eddington1920]. The time in question is that of “a being coextensive with the world”, which is the foliation. He names the absence of the discriminator, and selects anyway.
What followed was silence. The 1917 paper was the last in which Einstein published the details of his cosmological considerations. After Hubble's result he wrote some five pages on the problem in two short articles, the second co-authored with de Sitter, while de Sitter published volumes; and his later rejection of the cosmological term shaped the expansion paradigm for the rest of the century [JanzenEinsteinCosmology]. He never returned to the question his own assumption raised—how a cosmic time distinguished by the matter's rest frame stands to the relativity of simultaneity—and so far as the record shows never addressed it in print.
The measurement reported above closes that arc. What Einstein posited apologetically on stellar velocities, what de Sitter placed beyond physical argument, and what Eddington defended without evidence, is what the redshift isotropy now forces. The assumption marked “against the spirit of relativity” was correct, and the objection that it gave time “a separate position” was correct too—as a description of the structure, though not as a reason to reject it. We record the history because the reasoning that abandoned this structure is still in the field's hands, and because a selection made in 1920 in the declared absence of evidence, met by a measurement in the decades after, is the pattern by which theory-choice earns its standing [JanzenShadowExistence]. And the parallax of this paper's title should be read with the same care the historical case requires. Stellar parallax went undetected for two millennia and was read as telling against the Earth's motion, but before the cosmos was given depth the relevant parallax was not a conceivable measurement at all—the stars sat on a sphere and had no distances to be displaced against [JanzenShadowExistence]. The objection is post-Digges, and it is Tycho who presses it. The lesson is the one this paper's floor turns on: an absent discriminator is not an absent fact, and whether a discriminator is even available is a question about the picture in hand rather than about the world.
What the standard model installs as assumptions, the present framework derives. The cosmic-time variable was put in by hand at the outset—assumed in Einstein's 1917 cosmology [Einstein1917] and carried through Friedmann's expanding solutions [Friedmann1922] to the present model, where a single global scale factor on a single global time is the background fixed before any matter is placed upon it. That foundational structure is here forced by measurement. A second assumption entered in the same 1917 paper, and it runs in the opposite direction from what the present result establishes. Einstein's static solution fixes the cosmic scale from the content: with λ=κρ/2 the radius is R=1/√λ, so the density of the world-matter determines the size of the world [Einstein1917]—and Friedmann's equations carry that direction of determination forward, the scale factor's evolution being sourced by what the universe contains [Friedmann1922]. What the isotropy forces is the reverse ordering for the rate: it is fixed independently of how the content is distributed, since a rate tracking the content is precisely what the measurement excludes. Whether the same reversal extends from the rate to the scale is not settled here and is taken up in a companion [JanzenCRframework]. This is the empirical motivation the title names: not a modification of general relativity, whose equations are untouched, but the measured selection of the foliation the theory admits and does not otherwise single out.
What the measurement forces is the empirical half of a structural result, and the half in hand fixes the other, so it is worth stating whole. General relativity supplies a four-geometry and field equations but does not single out which of the foliations its formalism admits is the physical one—as the companion papers stress and as the present forcing makes vivid [JanzenCRframework]. The standard interpretation fills that silence by collapsing two distinctions at once: it reads the relativity of simultaneity as the absence of any objective present, and it reifies the four-manifold as a thing that exists. Each is a category error of the same family as the one corrected in §2—there, crediting the source for a fact that resides in the intervening expansion; here, crediting the manifold for what the evolving world does. Pried apart, the two are the parts a coherent formal description of an existing, evolving world requires; and they are exactly the lapse and the shift of the 3+1 split read ontologically—the Newtonian part, an absolute rate of advance, and the Einsteinian part, the relativity of synchrony, the formalism the standard reading had fused and a coherent description must hold apart.
The relativity of synchrony is general—a feature of relative motion, the light postulate fixing only its magnitude [JanzenWildGoose, JanzenMisconstrue]—and it is the shift, the freedom of a moving observer to tilt the slices off the comoving normal. Simultaneity is the separate question, and to read the relativity of synchrony as the denial of an objective simultaneity is a modal fallacy, the absence of a local test mistaken for the absence of a fact—the imperative's characteristic dual, established as such in the companion epistemic paper [JanzenShadowExistence, JanzenWildGoose]. The lapse is the other part: the rate of advance of the layer that exists, what the clock measures [JanzenTrope]. To deny it—to hold the four-manifold itself the existent—is to treat occurrences as existents, which on analysis smuggles a fifth, meta-temporal dimension along which the block is supposed to “exist,” and is incoherent as a basic description [JanzenTrope, JanzenCanonicalTime]. The two corrections are not optional glosses on the formalism; they are what makes “a world that exists and evolves” a statable object at all. With them restored the measured result binds: the redshift isotropy forces the lapse—a global, uniform rate of advance in a cosmic time—while the relativity of synchrony, untouched, is the shift, the existent the evolving layer the foliation orders and the manifold its record.
It remains to weigh, against this result, the one objection that the uniform expansion is justified rather than assumed. It is held that Green and Wald have shown small-scale inhomogeneities to backreact negligibly on the large-scale metric, so that the Friedmann model is self-consistent with the averaged matter as its source [GreenWald2014], against the contrary case that backreaction is dynamically significant [Buchert2000, Wiltshire2007]. Kept apart from the theorem and then set against it, the demonstration is seen to take as its input precisely what the floor forces: it assumes the metric already close to Friedmann–Lemaître–Robertson–Walker form—the cosmic time, the single global expansion—and proves the lumpiness does not disturb it. That is a statement about the stability of the assumed foliation, presupposing it and establishing its robustness, not deriving it—what is small in the demonstration is the metric perturbation (∼10-5), while the matter contrast it rides on is nonlinear, so the licence to treat the configuration as a perturbed Friedmann model is itself a statement about the assumed background. The floor owes none of this, forcing the cosmic time and the uniformity of its advance from the redshift isotropy directly, a datum prior to any slice, with no Friedmann structure assumed in its derivation. This settles the standing dispute structurally and from outside it: the uniform expansion that one side assumes and the other questions is forced by measurement whichever way the averaging argument is adjudicated, so the most rigorous defence of the standard treatment is found to presuppose the very structure the measurement supplies, and the backreaction question, however it resolves dynamically, cannot reach the foliation the floor has fixed. Were the backreaction case to carry genuine large-scale content, it would enter on that merit and against this result; it does not enter as the ground of the uniform expansion, which the measurement has removed from its reach.
The scope is worth marking. This contradicts none of the standard model's fitted predictions; with its assumptions in place it reproduces the expansion history observed, which is why it stands. What the measurement changes is the standing of those assumptions: the cosmic time and the uniformity of its advance are forced from the bottom up, prior to and independent of the dynamical fit, not conventions adopted for it. What the measurement does not do is install the replacement—the positive account, that the expansion is set by the cosmological constant alone with the matter density a reading of cosmic epoch [JanzenCRframework], is motivated by this forcing but not compelled by it, and is left to the companion framework rather than asserted here.
This is the empirical counterpart, made quantitative, of the conceptual point that the cosmic present is measured rather than posited. As stellar parallax once converted the Earth's motion from an interpretive option into a measured fact, the isotropy of the cosmological redshift—with our own velocity through the rest frame read directly from the dipole, (369.82±0.11) km/s toward Crater [Aghanim2020]—converts the cosmic foliation from a modelling convenience into a measured feature of the world. The general result that no exterior temporal slicing can record a finite-time completion of gravitational collapse [JanzenBHcausality] stands on causal structure alone and needs none of this evidence; what the evidence adds is that the slicing that result concerns is not one option among many but the one the universe has let us measure. Scope. The temporal claim—uniform expansion throughout cosmic history—is clean, being the integral from decoupling to the present along the observed rays; the extension to a globally maximally symmetric space rides on the Copernican principle and is an extrapolation beyond the directly constrained region. And the scope of what is claimed is worth stating against what is not. What the isotropy measures is the uniformity of the expansion rate along the observed rays; it does not measure, and this paper does not claim, anything about which geometry carries that expansion or how the matter within it is arranged. The matter lumpiness is untouched. What is excluded is an inhomogeneous expansion rate, and the reason the two are separable is the reason the floor exists at all.