A Six-Parameter Cosmology with Fixed Matter and Baryon Fractions

1. Introduction

The microwave sky and the distribution of galaxies contain an acoustic pattern: preferred angular sizes in the cosmic microwave background (CMB), and a preferred separation in galaxy clustering, called baryon acoustic oscillations (BAO). A cosmological model must account for both the pattern and the distances through which it is observed. The question here is whether a proposed physical allocation of matter can do that with the same number of adjustable cosmological quantities as the conventional model.

Toroidal Scale Mechanics (TSM) proposes that space is a physical substrate whose organized flow and deformation produce particles, forces, and cosmic structure. Its cosmological picture is an ongoing universe without a Big Bang: matter forms, changes, and is recycled through continuing substrate processes [1]. The present study tests geometrically motivated density fractions within a conventional cosmological calculation, while interpreting its matter-like and tension-like terms through that material picture.

Two fixed density fractions can leave enough freedom to describe the acoustic microwave sky and late-time galaxy distances when the effective stress can vary with redshift. The family examined here fixes ${\mathsf{\Omega }}_m=1/\pi$ and ${\mathsf{\Omega }}_b=1/\left(2{\pi }^2\right)$ and fits two stress-profile coefficients alongside four common cosmological coordinates. It exchanges the two independently adjustable matter densities of flat $\mathsf{\Lambda }$CDM for two effective-stress coordinates while retaining six adjustable cosmological quantities.

The empirical question is how much likelihood fit survives this exchange, and whether it persists when supernova distances and shared high-redshift physics conventions are tested. This requires refitting complete parameter sets against observations. The cosmic microwave background (CMB) constrains physical densities such as ${\omega }_b={\mathsf{\Omega }}_b{h}^2$ and the acoustic angular scale. In the conventional calculation, its inference about a present density fraction depends on the transfer and distance relations linking recombination to the observer [2]. Agreement with a reference spectrum at one chosen Hubble constant therefore has a narrower meaning than agreement with a joint observational likelihood.

The geometry of the substrate’s structural support and mobile content is proposed to constrain the cosmological energy allocation [3,4,5]. This suggests a different order of inference: specify candidate fractions from the material model, then determine which effective stress profiles remain compatible with the data. Section 2 develops that connection from the conserved effective action and the geometric assignments to the observable density restriction.

The same numerical pair appears in an earlier, separate heuristic construction [6]. A related proposal fixes ${\mathsf{\Omega }}_m=1/\pi$ with a different expansion law and tests a compressed CMB acoustic-scale constraint alongside late-time distances [7]. Here the contribution is a matched likelihood test of the theory-motivated two-fraction restriction, including its cost relative to conventional free-density CPL dark energy and its response to additional distance data. This revision replaces the earlier reference-spectrum comparison with observed-likelihood tests; Appendix C preserves the scope of the earlier control results.

The effective stress profile uses the established Chevallier–Polarski–Linder (CPL) parametrization [8,9]. DESI has already reported improved fits in that family when the matter densities vary freely [10]. The relation between density assumptions, distance degeneracies, and CPL inference has also been examined explicitly [11]. The comparisons below quantify the cost of the imposed fractions, with an equal-count $\mathsf{\Lambda }$CDM control and a free-density CPL control. The comparison therefore asks whether the two geometric restrictions survive within a known, explicitly specified CPL family.

2. Theoretical Motivation and Connection to the Test

2.1. Substrate Interpretation and Observational Variables

The working architecture has four material support directions, one registered through scale, plus a progression degree experienced internally as time. Its tension-bearing and mobile-flow roles are called duron and pneumon [1]. These are effective response roles; the microscopic constitution remains open. TSM assigns the additional gravitating response and the effective cosmological tension to material behavior of this substrate. The conventional terms dark matter and dark energy identify the corresponding components in the observational model. In the TSM interpretation pursued here, the latter represents an effective substrate stress, and its variation is proposed to arise across the substrate’s scale structure.

Redshift z labels the observations. The solver uses $a={(1+z)}^{-1}$, which in standard cosmology is the expansion scale factor, evolving with cosmic time. TSM’s physical scale coordinate describes organization and motion through the substrate’s scale structure. The interpretation under investigation connects that structure to a redshift-dependent effective stress and hence to the measured distance–redshift relation. Establishing this connection requires a physical map between substrate scale, light propagation, clock response, and distance. Distance and substrate scale therefore have distinct roles in the proposed explanation.

The implemented Cosmic Linear Anisotropy Solving System (CLASS) calculation retains the conventional background conservation, recombination, and perturbation equations. Its “expansion history” is the fitted background of that effective calculation. TSM’s proposed scale interpretation requires those observational relations to follow from its material dynamics; the present fit supplies quantitative targets for that derivation.

2.2. Conserved Material Content and an Effective Background

TSM describes the substrate through structural support and mobile content. Its minimal cosmological construction [3] represents these by two conserved currents, $J_d^{\mu }=b{u}_d^{\mu }$ and $J_p^{\mu }=n{u}_p^{\mu }$, where b is support-parcel density and n is mobile-content density. The symbol b here denotes a material variable, distinct from the observational baryon density. Three material-coordinate fields generate the identically conserved support current; a parcel constraint enforces ${\nabla }_{\mu }{J}_p^{\mu }=0$.

The proposed effective action in the four-dimensional observer description is

$$S_{\mathrm{eff}}=\int d^{4}x\sqrt{-g}\left[{\displaystyle \frac{M_{*}^2}{2}}R-V_T-b\epsilon (n/b,y,I_1,I_2)\right]+S_{\mathrm{vis}}+S_{\mathrm{cons}}.$$

(1)

Here g is the metric determinant, R is scalar spacetime curvature, and $d^{4}x\sqrt{-g}$ is the observer spacetime volume measure. The quantity $M_{*}$ is the effective gravitational scale, $V_T>0$ is a constant stored-tension energy density, $\epsilon$ is energy per support parcel, $y=-u_d·u_p$ measures relative motion, and $I_1,I_2$ describe support shape. The last two actions supply visible matter and the conservation constraints. This construction adopts Einstein metric coupling as the effective observer equation. On the homogeneous isotropic relative-rest branch, $y=1$, $I_1=I_2=3$, and $\partial \epsilon /\partial y=0$. For the effective scale factor a normalized to one at the observer reference, defining $r=n/b$ and ${\epsilon }_0\left(r\right)=\epsilon (r,1,3,3)$ gives

$$\begin{array}{cc}\hfill b& =b_0{a}^{-3},n=n_0{a}^{-3},r=r_0,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{sub}}& =V_T+b{\epsilon }_0\left(r\right),p_{\mathrm{sub}}=b{\partial }_b{\rho }_{\mathrm{sub}}+n{\partial }_n{\rho }_{\mathrm{sub}}-{\rho }_{\mathrm{sub}}=-V_T.\hfill \end{array}$$

(3)

Thus conservation and the specified parcel-energy form produce a pressureless structural density proportional to $a^{-3}$ and a constant negative-pressure component. Including all visible pressureless matter in ${\rho }_{\mathrm{cub},0}$, the flat branch with negligible radiation is

$$3M_{*}^2{H}^2={\rho }_{\mathrm{cub},0}{a}^{-3}+V_T,{\rho }_{\mathrm{cub},0}={\rho }_{\mathrm{vis},m,0}+b_0{\epsilon }_0\left(r_0\right).$$

(4)

This gives a mechanical basis for the two background components. The baryon–photon sector is separately conserved in this minimal construction, and its local acoustic equations retain their standard form. The full CMB calculation below includes radiation and specifies the perturbation model explicitly.

The common substrate field equations contain this background as a controlled limit and also supply coupled local acoustic and balanced-exchange equations [18]. Their material coefficients and observer energy functions remain open. The present six-parameter count refers to the fitted observational family; it is not a count of all constitutive freedom in the underlying material theory. Extending the constant-tension background to the fitted variable stress requires a material state and observer map that recover the same background and perturbation response together.

2.3. Why These Two Fractions?

The geometric proposal uses round-sphere and angular normalizations [4,5]. With unit radius and the standard angular measure,

$${\kappa }_{W,\mathrm{cont}}={\displaystyle \frac{1}{Area\left(S^2\right)}}={\displaystyle \frac{1}{4\pi }},Vol\left(SO\left(2\right)\right)=2\pi .$$

(5)

The continuous yield proposal assigns four copies of ${\kappa }_{W,\mathrm{cont}}$ to the total matter fraction. A second representation counts the two effective substrate response roles against the angular period. Both give the candidate $4{\kappa }_{W,\mathrm{cont}}=2/\left(2\pi \right)=1/\pi$. The baryon proposal assigns the angular share $f_b\equiv {\mathsf{\Omega }}_b/{\mathsf{\Omega }}_m=1/\left(2\pi \right)$ [5]; multiplying by the proposed matter fraction gives ${\mathsf{\Omega }}_b=1/\left(2{\pi }^2\right)$. These are the material-model assignments that motivate the two restrictions. They concern, respectively, matter relative to critical density and baryons relative to total matter; the latter is not a second critical-density fraction until multiplied by ${\mathsf{\Omega }}_m$.

The exact geometric identities and the physical allocation have different derivations. For the positive flat two-component branch in Equation (4), normalization at $a=1$ gives the precise selection condition [4]

$${\mathsf{\Omega }}_{\mathrm{cub}}={\displaystyle \frac{{\rho }_{\mathrm{cub},0}}{{\rho }_{\mathrm{cub},0}+V_T}}={\displaystyle \frac{1}{\pi }}⟺V_T=(\pi -1){\rho }_{\mathrm{cub},0}.$$

(6)

The local conservation equations leave that ratio free. A global allocation law must relate the support and mobile inventories, the visible contribution, and the tension energy on the same physically specified reference slice. Likewise, the proposed baryon share requires a material partition law. The Global Yield Gate for the Fixed $1/\pi$ Profile [4] shows why the inherited channel counts and round-volume identities alone do not establish these energy relations. The fractions are therefore physically motivated restrictions to be tested, with their selection isolated as a definite constitutive problem. Supplementary Material B supplies the source derivations and the mapping used here.

2.4. Why Allow a Redshift-Dependent Effective Tension?

The minimal action gives constant effective tension. Its background corresponds to the constant-density control, which is costly in the matched likelihood calculation when both fractions are fixed. A scale-dependent substrate stress motivates considering a variable effective tension. CPL provides a specified redshift dependence for testing whether that freedom can retain the assigned fractions while recovering the observed acoustic and distance relations. Conventional conservation fixes the associated density profile, and the parametrized post-Friedmann (PPF) treatment supplies equations for the small disturbances around that background. The fitted $w_0,w_a$ characterize this effective test family. Connecting them to substrate scale is the material-to-observation problem described in Section 2.1.

This division gives the calculation a concrete purpose. The geometry supplies the density hypothesis; the conserved action explains the background components; the likelihood measures the observational room for a variable-tension extension. Comparing fixed-density CPL with free-density CPL measures the cost of the density hypothesis within one effective-stress family. Comparing it with six-coordinate $\mathsf{\Lambda }$CDM measures the consequence of assigning the same number of adjustable coordinates to different physical quantities.

3. Model and Parameter Allocation

Let $a={(1+z)}^{-1}$, with $a=1$ at $z=0$, and $H_0=100h\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$. The constrained family imposes

$${\mathsf{\Omega }}_m={\displaystyle \frac{1}{\pi }}=0.318309886\dots ,{\mathsf{\Omega }}_b={\displaystyle \frac{1}{2{\pi }^2}}=0.050660592\dots .$$

(7)

In the primary massless-neutrino convention, ordinary pressureless nonbaryonic matter has ${\mathsf{\Omega }}_c={\mathsf{\Omega }}_m-{\mathsf{\Omega }}_b$. Consequently,

$${\omega }_b={\displaystyle \frac{h^2}{2{\pi }^2}},{\omega }_c=h^2\left({\displaystyle \frac{1}{\pi }}-{\displaystyle \frac{1}{2{\pi }^2}}\right),{\displaystyle \frac{{\omega }_m}{{\omega }_b}}=2\pi .$$

(8)

The physical densities move together when h is fitted. The massive-neutrino sensitivity calculation includes the present massive-neutrino density in ${\mathsf{\Omega }}_m$. CLASS closes that budget using its Omega_m input, with ${\mathsf{\Omega }}_b$ fixed and ordinary nonbaryonic matter supplying the remainder.

In the solver’s dark-energy component, the effective stress-to-energy ratio, in units $c=1$, is

$$w\left(a\right)=w_0+w_a(1-a).$$

(9)

Equivalently, $w\left(z\right)=w_0+w_{a}z/(1+z)$: the two fitted coefficients set the reference value and its redshift dependence. In the TSM interpretation, this profile is a candidate observable projection of scale-dependent tension. Separate conservation in the conventional background, $d{\rho }_X/dlna=-3[1+w\left(a\right)]{\rho }_X$, integrates to

$$F\left(a\right)\equiv {\displaystyle \frac{{\rho }_X\left(a\right)}{{\rho }_{X0}}}=a^{-3(1+w_0+w_a)}exp\left[3w_a(a-1)\right].$$

(10)

For the primary radiation convention, spatial flatness gives

$${\displaystyle \frac{H^2\left(a\right)}{H_0^2}}={\mathsf{\Omega }}_r{a}^{-4}+{\mathsf{\Omega }}_m{a}^{-3}+(1-{\mathsf{\Omega }}_m-{\mathsf{\Omega }}_r)F\left(a\right).$$

(11)

The massive-neutrino calculation instead follows that species’ evolving distribution in CLASS. Setting $(w_0,w_a)=(-1,0)$ recovers a constant effective tension density in that component.

Perturbations use the standard parametrized post-Friedmann (PPF) completion [12], with cs2_fld=1 and c_gamma_over_c_fld=0.4. This gives a regular effective evolution through $w=-1$ and is part of the model tested. A microscopic material realization requires its own stability and matching calculation.

All new runs implement the nonbaryonic matter component as ordinary unsplit pressureless matter in CLASS. This specifies the effective perturbation response used to represent the additional gravitating contribution. The earlier TSM implementation divided the nonbaryonic inventory into support and mobile response channels, with vanishing acoustic entrainment in the retained fit. At that point, ordinary unsplit matter changes $Q_{\mathrm{CMB}+\mathrm{BAO}}$ by about $7\times {10}^{-6}$ [13]. The gain is therefore attributable to the restricted CPL cosmology and its standard completion.

The shared coordinates are ${\theta }_4=(h,n_s,\tau ,ln\left({10}^{10}{A}_s\right))$, describing the Hubble constant, scalar tilt, reionization optical depth, and primordial scalar amplitude. Table 1 defines the families. A common Planck calibration is an additional fitted nuisance in all cases. Joint supernova fits also profile a magnitude intercept. These nuisances enter the objective and are excluded from the cosmological count $N_{\cos }$.

4. Observations and Numerical Method

4.1. CMB and BAO

The primary objective combines Planck 2018 foreground-marginalized Plik-lite temperature (TT), temperature–polarization (TE), and E-mode polarization (EE) spectra with native low-multipole TT and EE likelihoods [14], and the full thirteen-observable Dark Energy Spectroscopic Instrument second data release (DESI DR2) BAO vector [10]:

$$Q_{\mathrm{CMB}+\mathrm{BAO}}=Q_{\mathrm{high}-\ell }+Q_{\mathrm{lowTT}}+Q_{\mathrm{lowEE}}+{\left({\displaystyle \frac{A_{\mathrm{Planck}}-1}{0.0025}}\right)}^2+{\mathbf{r}}_{\mathrm{BAO}}^T{C}_{\mathrm{BAO}}^{-1}{\mathbf{r}}_{\mathrm{BAO}}.$$

(12)

Native CMB likelihood normalizations are retained. Identical additive constants apply within each comparison. Lower Q indicates a better fit; absolute totals contain likelihood offsets.

CLASS v3.3.4 [15] supplies lensed spectra through $\ell =2508$, with nonlinear corrections disabled and primordial pivot $0.05{\mathrm{Mpc}}^{-1}$. Cobaya 3.6.2 supplies the native likelihoods [16]. The primary convention has $N_{\mathrm{ur}}=3.046$, no massive species, $T_{\mathrm{CMB}}=2.7255\mathrm{K}$, and $Y_{\mathrm{He}}=0.245$. Recombination uses the shared HyRec 2020 implementation. The sensitivity calculation changes both families to one $0.06\mathrm{eV}$ species, $N_{\mathrm{ur}}=2.0328$, and big-bang-nucleosynthesis (BBN) consistent helium. It tests the joint convention change, without separating its two contributions.

The inherited primary BAO mapping extracts an acoustic dilation from theoretical linear matter spectra:

$$lnP\left(k\right)=p_4\left(k\right)+Bln{P}_{\mathrm{ref}}\left(sk\right),r_{\mathrm{eff}}=s\left(147.092899\mathrm{Mpc}\right).$$

(13)

Here $p_4$ is a fourth-degree smooth polynomial and k is in physical ${\mathrm{Mpc}}^{-1}$. The extraction uses 800 equally spaced wavenumbers between $0.03$ and $0.20{\mathrm{Mpc}}^{-1}$, profiles the polynomial and amplitude, and minimizes over dilation. These coefficients are determined from theoretical spectra and add no parameters fitted to observed BAO. The observables are $D_M/r_{\mathrm{eff}}$, $D_H/r_{\mathrm{eff}}$, and $D_V/r_{\mathrm{eff}}$, where $D_H=c/H$ and $D_V={\left(z{D}_H{D}_M^2\right)}^{1/3}$.

This mapping originated in a wider acoustic-model comparison. Ordinary equations also provide the comoving sound horizon $r_d$ accumulated up to the baryon-drag redshift. We report the frozen score change under $r_{\mathrm{eff}}\to r_d$ as a separate control; the principal comparisons retain one common inherited objective. Both mappings use the complete published covariance and the survey’s distance-compression assumptions.

4.2. Supernovae and Refitting

The Pantheon+ extension [17] selects $z_{\mathrm{HD}}>0.01$, retaining 1590 light curves from 1473 distinct object identifiers out of 1701 input light curves. It uses the full statistical-plus-systematic covariance. With distances in Mpc,

$$\begin{array}{cc}\hfill D_{L,i}& =(1+z_{\mathrm{HEL},i})D_M\left(z_{\mathrm{HD},i}\right),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill u_i& =m_{B,i}^{\mathrm{corr}}-5{log}_{10}{D}_{L,i},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \widehat{\mathcal{M}}& ={\displaystyle \frac{{\mathbf{1}}^T{C}_{\mathrm{SN}}^{-1}\mathbf{u}}{{\mathbf{1}}^T{C}_{\mathrm{SN}}^{-1}\mathbf{1}}},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill Q_{\mathrm{SN}}& ={(\mathbf{u}-\widehat{\mathcal{M}}\mathbf{1})}^T{C}_{\mathrm{SN}}^{-1}(\mathbf{u}-\widehat{\mathcal{M}}\mathbf{1}).\hfill \end{array}$$

(17)

The profiled intercept absorbs a common distance scale; no SH0ES absolute-magnitude anchor is used. The new joint fits adjust all cosmological coordinates against $Q_{\mathrm{CMB}+\mathrm{BAO}+\mathrm{SN}}=Q_{\mathrm{CMB}+\mathrm{BAO}}+Q_{\mathrm{SN}}$. A frozen SN score at a CMB+BAO solution tests transfer of that particular distance–redshift relation; it does not determine the ordering of refitted model families.

4.3. Local Optimization and Uncertainty

Finite differences of fresh CLASS spectra and distances define a local response. Bounded Powell minimization proposes steps on that response, and fresh full calculations determine acceptance. A linearly interpolated low-EE table guides the search; endpoint scores use the original native table with calibration reprofiled. The reported solutions are local fitted points.

The validation design fixes bounds, iteration limits, nearby starting-point displacements, and the endpoint selection rule before the recovery runs. The joint starts are coordinates from an earlier exploratory run; its raw evidence was lost after execution interruption, and every reported new endpoint is regenerated. The machine-readable design records that provenance. The primary convention-sensitivity starts are the retained CMB+BAO points.

Each run receives a second start displaced in every fitted coordinate. The lower terminal smooth objective selects the cosmological endpoint, which is checked with fine spectra and dense matter calculations. Appendix A gives controls and actual stopping records.

Central spectral and distance differences define a local linear response retaining the high-multipole Gaussian term, smooth low-multipole likelihood, and distance covariances. Its Hessian $H_Q$ gives the approximation $C_{\mathrm{loc}}=2H_Q^{-1}$. Calibration is included and the SN intercept profiled. Hessian displacements of 0.2 and 0.1 in normalized coordinates test differentiation-scale sensitivity. These are conditional quadratic diagnostics, distinct from sampled posterior intervals. The fixed-fraction model has 6 full CLASS checks along nominal unit-width profile directions for h, $w_0$, and $w_a$.

5. Results

5.1. Retained CMB+BAO Comparison

Table 2 reproduces the retained matched comparison. Fixed-fraction CPL improves on six-coordinate $\mathsf{\Lambda }$CDM by $4.794421$ score units: $1.615568$ in CMB plus calibration and $3.178853$ in DESI. Freeing the two densities within CPL improves the score by a further $2.265689$.

The fixed-fraction endpoint has

$$\begin{array}{cc}\hfill (h,n_s,\tau ,ln\left({10}^{10}{A}_s\right))& =(0.666111802,0.967261275,0.055397015,3.043463932),\hfill \\ \hfill (w_0,w_a)& =(-0.757268654,-0.683799813).\hfill \end{array}$$

It implies ${\omega }_b=0.02247835$, ${\omega }_m=0.14123567$, and $H_0=66.61118\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$. These digits identify reproducible inputs. The fitted effective stress profile crosses $w=-1$ at $z\simeq 0.550$ and has an effective dark-energy fraction of approximately $9.25\times {10}^{-13}$ at $z=1090$.

Constant-density controls locate the necessary freedom. With $w=-1$ and only ${\mathsf{\Omega }}_m$ fixed, the five-coordinate fit gives $Q_{\mathrm{CMB}+\mathrm{BAO}}=1040.340$, a cost of $24.360$ relative to the conventional benchmark. Fixing both fractions leaves four coordinates and gives $1113.960$, a cost of $97.979$. A five-coordinate fixed-fraction constant-w model gives $1026.049$. The second CPL coefficient improves that result by $14.863$.

5.2. Joint Pantheon+ Comparison

At the frozen CMB+BAO points, the SN scores are $1406.2912$ for $\mathsf{\Lambda }$CDM, $1408.8446$ for fixed-fraction CPL, and $1457.0263$ for free-density CPL. The first two frozen sums differ by $-2.2409$. The large frozen penalty for the free-density point motivates the complete joint refit. The free-density search domain contains the reported fixed-fraction joint point, so its exact minimum cannot exceed that point’s score on the same objective.

Table 3 gives the joint scores. Fixed CPL differs from $\mathsf{\Lambda }$CDM by $-3.655797$ and costs $3.917164$ relative to free CPL. Its endpoint has $H_0=66.6176\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, $w_0=-0.80805$, and $w_a=-0.49002$. Free CPL gives $H_0=67.6798$ in the same units, ${\mathsf{\Omega }}_m=0.309584$, $w_0=-0.85552$, and $w_a=-0.49058$.

Figure 1 compares the retained and joint score differences.

5.3. Numerical and Convention Checks

The largest fine-minus-main score change is $0.022371$ and the largest native nearby-start spread is $0.021075$. Both pass the declared $0.2$-unit gates. These checks probe precision and nearby starting-point dependence within the recorded local basins.

The neutrino/helium comparison gives $\mathsf{\Delta }Q=-6.297538$ (Table 4). The primary advantage retains its sign. These are CMB+BAO fits; the joint SN fits retain the primary convention.

Frozen substitution of the standard CLASS drag ruler changes individual new totals by at most $0.107577$. Fixed-minus-$\mathsf{\Lambda }$CDM differences become $-3.628086$ for the joint SN comparison and $-6.196597$ for the convention-sensitivity comparison. This check makes no further cosmological adjustment.

Table 5 gives local fixed-CPL widths. Halving the Hessian displacement changes cosmological widths by up to about $0.6\%$. The 6 full-model unit-width displacements give $\mathsf{\Delta }{Q}_{\mathrm{smooth}}$ from $0.983$ to $1.099$. These diagnostics quantify the local quadratic approximation.

6. Interpretation

The imposed fractions tie the physical matter and baryon densities together through $h^2$. Two CPL coordinates then adjust the effective stress and distance–redshift profiles. This retains the familiar acoustic sensitivity to baryon loading and physical matter density. The equal-count comparison measures how well the stress-profile coordinates compensate for removing two independent densities.

The restricted family improves the primary fitted score by $4.794$ over $\mathsf{\Lambda }$CDM while costing $2.266$ relative to free-density CPL. Equal continuous parameter counts make the first comparison useful, but do not exhaust model assessment. The fractions and stress-profile form arose within a broader programme that explored alternative acoustic and tension responses. The numerical validation supplies neither a trials-corrected discovery significance nor an integrated model probability. The exact mathematical forms are hypotheses under test; a successful fit does not uniquely select them over nearby values.

Adding Pantheon+ and refitting changes the equal-count difference from $-4.794$ to $-3.656$. The fixed fractions retain a lower fitted score at the same cosmological parameter count. Free CPL improves the joint score by $3.917$, measuring the imposed fractions’ cost after including the SN distance shape.

The conserved material construction in Section 2 explains the origin of the background components and identifies the energy-allocation condition behind the matter-fraction proposal. The likelihood result adds an observational constraint on its extension: the tested constant-density limit is costly, while the conserved CPL–PPF extension improves the matched six-parameter fit. A material completion must reproduce an admissible scale-dependent stress, its observable projection, and the associated perturbation response together with the proposed inventory allocation. The fitted endpoints specify quantitative targets for that calculation.

The tested data boundary is also explicit. This analysis combines the stated Planck 2018 likelihoods, DESI DR2 compressed distances, and one uncalibrated SN compilation. Planck lensing reconstruction, overlapping BOSS distances, other SN samples, and newer CMB likelihoods are outside the objective. Sampled posteriors and model evidence address additional inferential questions.

7. Conclusions

Imposing ${\mathsf{\Omega }}_m=1/\pi$ and ${\mathsf{\Omega }}_b=1/\left(2{\pi }^2\right)$ within conserved CPL–PPF cosmology yields a six-coordinate family whose retained Planck-plus-DESI score improves on matched six-coordinate flat $\mathsf{\Lambda }$CDM by $4.794$. Its restriction costs $2.266$ relative to an eight-coordinate CPL fit. The constant-density controls show why both stress-profile coordinates matter.

The new joint refits give an equal-count difference of $-3.656$ and CPL restriction cost of $3.917$. The separate massive-neutrino/BBN-helium CMB+BAO check gives $-6.298$. Numerical and local-response receipts delimit the comparisons’ precision and inferential scope.

The result connects the geometric density proposal to a reproducible observational test: the two assigned fractions support a better fit than the matched conventional six-parameter model within the specified CPL family. TSM’s proposed explanation is an ongoing universe with scale-dependent substrate behavior. The material action, allocation condition, and mapping to redshift-dependent observables identify the work needed to connect that explanation to the fitted effective model.

8. Glossary

BAO. 

Baryon acoustic oscillations: the preferred separation in galaxy clustering used here as a distance probe.

Baryon fraction (${\mathsf{\Omega }}_b$). 

The present density of ordinary baryonic matter divided by the critical density; $f_b={\mathsf{\Omega }}_b/{\mathsf{\Omega }}_m$ is instead its share of total matter.

CLASS. 

The numerical solver used to calculate the conventional background, perturbations, and radiation-transfer observables.

CMB. 

Cosmic microwave background: the measured microwave sky and its temperature and polarization variations.

CPL. 

Chevallier–Polarski–Linder parametrization, $w\left(a\right)=w_0+w_a(1-a)$; w is effective pressure divided by energy density in units $c=1$.

Critical density. 

The conventional density $3H_0^2/\left(8\pi G\right)$ defining the present fractions, with G Newton’s gravitational constant.

Effective model. 

The specified observational equations used to represent candidate substrate behavior and compare it with data.

Likelihood score (Q). 

The fitted objective used in a declared comparison. Lower values indicate better fit; shared additive likelihood offsets cancel in differences.

Local width. 

An uncertainty diagnostic from curvature near a fitted point, conditional on the stated model and numerical approximation.

Matter fraction (${\mathsf{\Omega }}_m$). 

The present total matter density divided by the critical density. The physical density coordinate is ${\omega }_m={\mathsf{\Omega }}_m{h}^2$.

Nuisance parameter. 

A calibration or observational offset fitted alongside the cosmological coordinates.

PPF. 

Parametrized post-Friedmann treatment: the specified equations completing the effective stress perturbations through $w=-1$.

Redshift (z). 

The measured fractional increase of wavelength; the conventional solver uses $a={(1+z)}^{-1}$.

Substrate. 

TSM’s physical medium, supporting tension and mobile flow. Its physical scale coordinate needs a derived map to the solver’s observable redshift.