Primary-Sector Width and Linked-Ruler Constraints in a Restricted Phase-Field Description of the Cosmic Microwave Background

The empirical CMB sector is tightly constrained. COBE/FIRAS fixed the monopole spectrum to an almost perfect blackbody and, together with later absolute thermometry, established a temperature $T_0=2.72548\pm0.00057\,\mathrm{K}$TeXT_0=2.72548\pm0.00057\,\mathrm{K} with very small allowed distortions [citation]. Planck measured the angular acoustic scale with $100\theta_*=1.0411\pm0.0003$TeX100\theta_*=1.0411\pm0.0003 [citation], and the final Planck PR4 likelihood is consistent with slightly smaller uncertainties [citation]. ACT DR6 maps cover 19,000 deg$^2$TeX^2 in three frequency bands, while the DR6 TT, TE, and EE power spectra use 10,000 deg$^2$TeX^2 measured to arcminute scales [citation]. The full 500-square-degree SPTpol dataset supplies additional precision measurements of the lensed damping tail in temperature and E-mode polarization [citation]. Planck PR4 lensing and ACT DR6 lensing provide independent late-transfer consistency checks [citation]. DESI DR2 measured BAO with more than 14 million galaxies and quasars and found distances consistent with the acoustic angle $\theta_*$TeX\theta_* measured by Planck [citation].

The analysis is restricted to the primary CMB sector. The homogeneous background expansion, the late angular-diameter distance $\DA(\eta_*)$TeX\DA(\eta_*), and the later transfer operator $\Tlate$TeX\Tlate are fixed external inputs throughout. No black-hole analytical input, horizon model, or black-hole-side inference enters the declared gate analysis. Two quantities are constrained: the admissible width of the last-accessibility layer compatible with a resolved acoustic peak sequence, and the residual freedom between the recombination ruler and the drag-era fossil ruler once the early-time background expansion, baryon loading, recombination kernel, and drag kernel are held fixed.

The source-level Gaussian identity derived below is exact only under a Gaussian last-accessibility kernel and a locally linear acoustic phase. The corresponding width bound is used only as a temperature-sector projection envelope on a chosen peak window. Temperature-sector structure is treated explicitly, while polarization and lensing enter only as external checks. Finite-thickness suppression and diffusion damping both contribute on small scales [citation], while polarization carries independent information about the duration of last scattering [citation]. Recent polarization-only analyses combining lensing reconstruction with delensed EE power spectra provide an independent precision check [citation]. Broader recombination-history freedom is constrained by combined early- and late-time probes [citation], remains directly testable in current parameter fits and recent specific recombination scenarios [citation], and ACT DR6 extended-model analyses find no evidence for modified recombination once lensing and BAO information is included [citation].

On the declared Gaussian/local-linear window, the analysis proves an exact source-level Gaussian identity and derives a temperature-sector width envelope on a representative peak window. It also studies, within a fixed-kernel one-parameter family, the first-order linked response of the recombination and drag rulers at fixed early-time background expansion, baryon loading, recombination kernel, and drag kernel. Any bridge between $\sigma_*$TeX\sigma_* and $\DeltaLSS$TeX\DeltaLSS, and any explicit coefficient set quoted for an illustrative benchmark, is family-conditional and used only as a fixed-convention consistency diagnostic.

The restricted primary-sector representation uses four inputs: (i) a scalar phase field $\HH$TeX\HH; (ii) a normalized last-scattering visibility kernel $\gstar(\eta)$TeX\gstar(\eta) (hereafter, the last-accessibility kernel); (iii) a source correction $\delta S_{\HH}$TeX\delta S_{\HH} to the standard primary source; and (iv) a sound-history correction $\delta c_{\HH}^2$TeX\delta c_{\HH}^2 to the standard baryon--photon response. No additional structure is introduced.

remark: Thermodynamic-temperature normalization. Write the cleaned specific intensity as

I_\nu(\hat{\mathbf n}) = B_\nu(T_0)+\delta I_\nu(\hat{\mathbf n}).

If the ratio

\frac{\delta I_\nu(\hat{\mathbf n})}{(\partial B_\nu/\partial T)_{T_0}}
=
T_0\,\Th(\hat{\mathbf n})

is independent of $\nu$TeX\nu on the chosen channels, then the thermodynamic temperature map $\Th(\hat{\mathbf n})$TeX\Th(\hat{\mathbf n}) is uniquely defined there. This is an observational normalization step and carries no dynamical load in the later theorems.

definition: Restricted primary sector. The restricted primary CMB sector is specified by:

- a normalized nonnegative last-scattering visibility kernel $\gstar(\eta)$TeX\gstar(\eta) centered at $\eta_*$TeX\eta_*; - a primary source equation (k,eta)=(k,eta)+delta S_(k,eta); equation - an effective sound history equation ^2(eta,k)=^2(eta)+delta c_^2(eta,k). equation

The temperature multipoles are represented by

\Th_{\ell}^{\mathrm{pri}}(k)
=
\int d\eta\,
\gstar(\eta)\,
\Seff(k,\eta)\,
j_\ell\!\big[k(\eta_0-\eta)\big].

All downstream primary-sector claims are read only relative to this declared restricted primary-sector object, its adopted last-accessibility kernel, and the fixed external inputs recorded in reference.

definition: Peak-window source class. Let $\Kpk$TeX\Kpk denote the acoustic wavenumber window projecting to the resolved CMB peak sequence. On $\Kpk$TeX\Kpk, admissible source deformations may alter only a smooth envelope of the standard acoustic source. They may not introduce an independent oscillatory $k$TeXk-dependence, a free phase function, or any other structure that restores peak coherence lost from an overbroad last-accessibility kernel. All downstream temperature-sector peak-window statements are read only on this declared peak-window source class and its chosen acoustic window $\Kpk$TeX\Kpk.

The observed spectrum is generated by a later transfer layer acting on the primary sector,

C_\ell^{\mathrm{obs}}=\Tlate\!\left[C_\ell^{\mathrm{pri}}\right].

remark: Later transfer. The operator $\Tlate$TeX\Tlate may lens, smooth, and reweight the primary spectra through standard late-time transfer. It may not reconstruct oscillatory power already excluded in the primary sector by scale-dependent retuning on $\Kpk$TeX\Kpk.

remark: Background and propagation inputs. The homogeneous background expansion, the late angular-diameter distance $\DA(\eta_*)$TeX\DA(\eta_*), and the later transfer operator $\Tlate$TeX\Tlate are taken as fixed external inputs and are not rederived here as new background or propagation laws.

Assume that on a narrow interval around $\eta_*$TeX\eta_* the oscillatory part of the primary source takes the form

\Seff(k,\eta)=A(k)\cos\!\big(k\,r(\eta)+\varphi(k)\big),

with acoustic phase

r(\eta)=\int_0^{\eta}\ceff(\eta',k)\,d\eta'.

Assume further that $\gstar$TeX\gstar is Gaussian,

\gstar(\eta)=\frac{1}{\sqrt{2\pi}\sigma_*}
\exp\!\left[-\frac{(\eta-\eta_*)^2}{2\sigma_*^2}\right],

and that the acoustic phase is locally linear across the support of $\gstar$TeX\gstar,

r(\eta)\approx r_*+\bar c_*(\eta-\eta_*),
\qquad
r_*:=r(\eta_*),
\qquad
\bar c_*:=\ceff(\eta_*,k).

The Gaussian kernel reference together with the local-linear phase window reference is the adopted kernel family for all downstream width diagnostics in the declared primary-sector analysis.

theorem: Gaussian damping of the kernel-smeared oscillatory source. Under reference, the kernel-smeared source integral obeys

\int d\eta\,\gstar(\eta)\,\Seff(k,\eta)
=
A(k)\exp\!\left[-\frac12 k^2 \bar c_*^{\,2}\sigma_*^2\right]
\cos\!\big(kr_*+\varphi(k)\big).

proof. Write the cosine as the real part of an exponential and shift $x=\eta-\eta_*$TeXx=\eta-\eta_*:

\int d\eta\,\gstar(\eta)\,\Seff(k,\eta)
=
A(k)\,\Re\!\left[
 e^{i(kr_*+\varphi)}
 \int \frac{dx}{\sqrt{2\pi}\sigma_*}
 \exp\!\left(-\frac{x^2}{2\sigma_*^2}+ik\bar c_*x\right)
\right].

The Gaussian integral is elementary,

\int \frac{dx}{\sqrt{2\pi}\sigma_*}
\exp\!\left(-\frac{x^2}{2\sigma_*^2}+ik\bar c_*x\right)
=
\exp\!\left[-\frac12 k^2\bar c_*^{\,2}\sigma_*^2\right],

and taking the real part gives reference.

remark: Projection envelope. The identity in reference is exact for the kernel-smeared source under reference and reference. The exact factor is source-level only on the declared Gaussian/local-linear primary window. It is not promoted here to an exact statement for the full observed TT/TE/EE spectra after later transfer. Applying it to the projected multipoles $\Th_{\ell}^{\mathrm{pri}}$TeX\Th_{\ell}^{\mathrm{pri}} on a chosen peak window additionally requires that the line-of-sight factor $j_\ell[k(\eta_0-\eta)]$TeXj_\ell[k(\eta_0-\eta)] be slowly varying across the support of $\gstar$TeX\gstar, or be absorbed into the same narrow-support local approximation. The width bound derived below is therefore a controlled temperature-sector projection envelope rather than an exact full-multipole damping theorem.

corollary: Temperature-sector width envelope on a chosen primary peak window. If acoustic peaks remain resolved on a chosen primary window up to multipole $\ell_{\pk}$TeX\ell_{\pk} with minimum retained oscillatory fraction $\varepsilon_{\pk}$TeX\varepsilon_{\pk}, then

\frac{\sigma_*}{\rsstar}
\lesssim
\frac{\sqrt{2\ln(1/\varepsilon_{\pk})}}{\pi \bar c_*}
\frac{\ellA}{\ell_{\pk}},
\qquad
\ellA\simeq \pi \frac{\DA(\eta_*)}{\rsstar}.

proof. Within the narrow-support projection approximation of reference, set $k_{\pk}\simeq \ell_{\pk}/\DA(\eta_*)$TeXk_{\pk}\simeq \ell_{\pk}/\DA(\eta_*) and require the damping factor in reference to satisfy

\exp\!\left[-\frac12 k_{\pk}^2\bar c_*^{\,2}\sigma_*^2\right]\ge \varepsilon_{\pk}.

Using $k_{\pk}\simeq (\ell_{\pk}/\ellA)\pi/\rsstar$TeXk_{\pk}\simeq (\ell_{\pk}/\ellA)\pi/\rsstar yields reference.

Planck+ACT benchmark

Planck gives $100\theta_*=1.0411\pm0.0003$TeX100\theta_*=1.0411\pm0.0003, so

\ellA=\frac{\pi}{\theta_*}=301.75\pm0.09.

ACT DR6 provides power spectra estimated over 10,000 deg$^2$TeX^2 to arcminute scales in TT, TE, and EE [citation]. For a representative peak window with $\ell_{\pk}=2500$TeX\ell_{\pk}=2500, retained oscillatory fraction $\varepsilon_{\pk}=1/2$TeX\varepsilon_{\pk}=1/2, and $\bar c_*\in[0.45,0.47]$TeX\bar c_*\in[0.45,0.47], reference gives

\frac{\sigma_*}{\rsstar}\lesssim 0.096\text{--}0.10.

The interval $\bar c_*\in[0.45,0.47]$TeX\bar c_*\in[0.45,0.47] is used here as a representative baryon--photon sound-speed range across the narrow visibility window. The bound in reference is used only as a temperature-sector envelope and does not replace a full treatment of diffusion damping or polarization transfer. The representative peak window, retained oscillatory fraction, and the range $\bar c_*\in[0.45,0.47]$TeX\bar c_*\in[0.45,0.47] constitute the adopted benchmark coefficient set for the quantitative width envelope quoted here.

Polarization width check

Hadzhiyska and Spergel showed that current CMB data constrain the thickness of the surface of last scatter to

\DeltaLSS = 19\pm0.065\,\mathrm{Mpc},

from the small-scale E-mode to temperature ratio [citation]. To connect this observable to the Gaussian width parameter of reference, write

\sigma_* = \avis\DeltaLSS.

The coefficient $\avis$TeX\avis depends on the chosen bridge prescription between the Gaussian width $\sigma_*$TeX\sigma_* and the polarization-width observable $\DeltaLSS$TeX\DeltaLSS; it is not fixed by temperature data alone. When used, the bridge $\sigma_*=\avis\DeltaLSS$TeX\sigma_*=\avis\DeltaLSS is an adopted family-level diagnostic convention and not a constitutive identity between the last-accessibility width and the polarization width. When this bridge is invoked below, all downstream polarization-width claims are read only relative to this adopted visibility-width convention on the representative Planck+ACT peak window.

corollary: Conditional polarization bound on $\avis$TeX\avis under an adopted $\sigma_*=\avis\DeltaLSS$TeX\sigma_*=\avis\DeltaLSS bridge. If reference holds on a chosen family, then combining reference with the P-ACT value $\rsstar=144.53\pm0.29\,\mathrm{Mpc}$TeX\rsstar=144.53\pm0.29\,\mathrm{Mpc} reported in the ACT DR6 power-spectrum analysis [citation] and reference gives

\avis\lesssim 0.73\text{--}0.76

on the representative Planck+ACT peak window.

proof. Substitute $\sigma_* = \avis\DeltaLSS$TeX\sigma_* = \avis\DeltaLSS into reference and use the quoted values of $\DeltaLSS$TeX\DeltaLSS and $\rsstar$TeX\rsstar.

The recombination ruler $\rsstar$TeX\rsstar and the drag ruler $\rsd$TeX\rsd are treated as two kernel-weighted functionals of one sound history evaluated with different kernels. The distinction is required because the drag-era BAO ruler depends on transfer physics beyond the instantaneous recombination kernel, including baryon loading and drag effects familiar from the standard transfer-function analysis [citation]. Throughout this section, the late angular-diameter distance and the later transfer sector remain fixed external inputs; only the primary-sector ruler response is analyzed. Define

\rsstar :=
\int d\eta\,\gstar(\eta)\int_0^\eta \ceff(\eta',k_{\mathrm{ac}})\,d\eta',

\rsd :=
\int d\eta\,\gd(\eta)\int_0^\eta \ceff(\eta',k_{\mathrm{ac}})\,d\eta',

where $\gd(\eta)$TeX\gd(\eta) is a normalized drag kernel and $k_{\mathrm{ac}}$TeXk_{\mathrm{ac}} denotes the low-$k$TeXk acoustic support on which the rulers are defined. Here $\rsd$TeX\rsd denotes only the drag-kernel fossil ruler associated with the declared drag-kernel class. No claim is made to replace a full BAO transfer analysis or the full drag-epoch phenomenology; only the primary-sector ruler response within the declared drag-kernel class is addressed.

definition: One-parameter family on the acoustic sector. A one-parameter family is defined at fixed early-time background expansion, fixed baryon loading, fixed recombination kernel $\gstar$TeX\gstar, fixed drag kernel $\gd$TeX\gd, and fixed low-$k$TeXk acoustic support $u(k_{\mathrm{ac}})=1$TeXu(k_{\mathrm{ac}})=1, with

\delta c_{\HH}^2(\eta,k)=\epsH\cs^2(\eta)f(\eta)u(k),

and with source deformation on the peak window restricted to

\delta S_{\HH}(k,\eta)=\epsH\,[a_0(\eta)+a_1(\eta)(k-k_{\mathrm{ac}})]\Sstd(k,\eta)+O\!\big(\epsH (k-k_{\mathrm{ac}})^2\big).

The coefficients $a_0$TeXa_0 and $a_1$TeXa_1 are smooth in $\eta$TeX\eta, and reference keeps only envelope-like local deformations around $k_{\mathrm{ac}}$TeXk_{\mathrm{ac}}; it forbids independent oscillatory $k$TeXk-structure on $\Kpk$TeX\Kpk. Within the family, $\epsH$TeX\epsH is the only parameter controlling the sound-history correction on the ruler-support sector. All downstream acoustic-scale, acoustic-angle, and two-ruler response claims are read only relative to this declared fixed-kernel family and its associated acoustic-scale / two-ruler comparison convention.

For $k=k_{\mathrm{ac}}$TeXk=k_{\mathrm{ac}}, reference implies

\ceff(\eta,k_{\mathrm{ac}})
=
\cs(\eta)\left[1+\frac{\epsH}{2}f(\eta)\right]+O(\epsH^2).

Define the functionals

\Jstar[f]
:=
\int d\eta\,\gstar(\eta)\int_0^\eta \cs(\eta')f(\eta')\,d\eta',
\qquad
\Jd[f]
:=
\int d\eta\,\gd(\eta)\int_0^\eta \cs(\eta')f(\eta')\,d\eta'.

Then

\rsstar
=
\rstarzero+\frac{\epsH}{2}\Jstar[f]+O(\epsH^2),
\qquad
\rsd
=
\rdzero+\frac{\epsH}{2}\Jd[f]+O(\epsH^2).

proposition: First-order response of the acoustic-scale proxy and the two-ruler ratio at fixed late geometry in the declared narrow-support class. Hold the late angular-diameter distance $\DA(\eta_*)$TeX\DA(\eta_*) fixed. Within the declared narrow-support class, the acoustic-scale proxy and the ruler ratio obey

\frac{\delta \ellA}{\ellA}
\simeq
-\frac{\epsH}{2\rstarzero}\Jstar[f]+O(\epsH^2),

\delta\ln\!\left(\frac{\rsd}{\rsstar}\right)
=
\frac{\epsH}{2}\left(
\frac{\Jd[f]}{\rdzero}
-
\frac{\Jstar[f]}{\rstarzero}
\right)
+O(\epsH^2).

proof. Within the declared narrow-support class, the acoustic scale satisfies $\ellA\simeq \pi \DA(\eta_*)/\rsstar$TeX\ellA\simeq \pi \DA(\eta_*)/\rsstar, so $\delta\ellA/\ellA\simeq-\delta\rsstar/\rsstar$TeX\delta\ellA/\ellA\simeq-\delta\rsstar/\rsstar. Using reference gives reference. Subtracting $\delta\ln \rsstar$TeX\delta\ln \rsstar from $\delta\ln \rsd$TeX\delta\ln \rsd gives reference.

corollary: No independent drag-ruler retuning within one fixed-kernel family. Within a family with fixed early-time background expansion, fixed baryon loading, and fixed recombination and drag kernels, the shifts of $\ellA$TeX\ellA and $\rsd/\rsstar$TeX\rsd/\rsstar are both controlled by the same parameter $\epsH$TeX\epsH. A fit that changes the drag-kernel fossil ruler while leaving the recombination ruler untouched is therefore not a retuning within the family but a change of family. Here $\rsd$TeX\rsd denotes only the drag-kernel fossil ruler within the declared fixed-kernel class. The statement does not by itself constrain the full late-time BAO observable once later transfer, reconstruction, and geometry choices are allowed to vary. It does not by itself exclude multi-parameter or kernel-changing early-time families beyond the declared fixed-kernel class.

Acoustic-angle constraint

In the declared narrow-support class, the acoustic-angle proxy satisfies

\theta_*\simeq\frac{\rsstar}{\DA(\eta_*)}.

For fixed late geometry this gives

\delta\ln \theta_* \simeq \delta\ln \rsstar
= \frac{\epsH}{2\rstarzero}\Jstar[f]+O(\epsH^2).

Using the Planck 0.03% precision on $\theta_*$TeX\theta_* [citation] yields

\abs{\frac{\epsH}{2\rstarzero}\Jstar[f]}
\lesssim 2.9\times 10^{-4}.

Illustrative fixed-family benchmark

For an explicit benchmark, adopt a narrow-kernel family with Gaussian kernels in conformal time,

\gstar(\eta)=\mathcal N(\eta_*,\sigma_*^{(\eta)}),
\qquad
\gd(\eta)=\mathcal N(\eta_{\mathrm d},\sigma_{\mathrm d}^{(\eta)}),

with

\eta_{\mathrm d}=1.05\,\eta_*,
\qquad
\sigma_*^{(\eta)}=0.015\,\eta_*,
\qquad
\sigma_{\mathrm d}^{(\eta)}=0.020\,\eta_*,

and a lag profile

f_{\mathrm{lag}}(\eta)=\frac12\left[1+\tanh\!\left(\frac{\eta-\eta_t}{\Delta_t}\right)\right],
\qquad
\eta_t=0.95\,\eta_*,
\qquad
\Delta_t=0.010\,\eta_*.

Across this narrow overlap window the standard sound speed is treated as constant to first approximation. In this benchmark the Gaussian kernels $\gstar$TeX\gstar and $\gd$TeX\gd, the lag profile $f_{\mathrm{lag}}$TeXf_{\mathrm{lag}}, and any bridge map such as $\sigma_*=\avis\DeltaLSS$TeX\sigma_*=\avis\DeltaLSS are held fixed. Changing any of them moves to a different benchmark choice rather than a retuning within the same benchmark. Define the overlap coefficients

\alphastar:=\frac{\Jstar[f_{\mathrm{lag}}]}{\rstarzero},
\qquad
\alphad:=\frac{\Jd[f_{\mathrm{lag}}]}{\rdzero},
\qquad
\dalpha:=\alphad-\alphastar.

Numerical evaluation of the benchmark family gives

\alphastar=0.0499,
\qquad
\alphad=0.0951,
\qquad
\dalpha=0.0452.

Therefore

\delta\ln \theta_* \simeq \frac{\alphastar}{2}\epsH + O(\epsH^2),
\qquad
\delta\ln\!\left(\frac{\rsd}{\rsstar}\right)=\frac{\dalpha}{2}\epsH + O(\epsH^2).

Using reference with reference yields

|\epsH|\lesssim1.2\times10^{-2},

and hence

\left|\delta\ln\!\left(\frac{\rsd}{\rsstar}\right)\right|
\lesssim 2.7\times10^{-4}

within this benchmark. These coefficients are specific to the illustrative benchmark introduced above. Their numerical values are recorded in reference. All downstream benchmark-only quantitative claims are read only relative to this fixed benchmark coefficient set.

CMB--VP0 supplies a bounded public diagnostic comparison for the restricted primary-sector diagnostics used above [citation]. The source surfaces are Planck PR4/NPIPE, ACT DR6, and DESI DR2 public products. This comparison is not a full CMB likelihood fit, a Boltzmann-pipeline replacement, a complete polarization or lensing transfer calculation, a recombination-history inference, a late-transfer theorem, or a DE/LBD/DM cosmology closure.

The combined label is

\mathrm{CMB\text{-}VP0\text{-}PRIMARY\text{-}DIAGNOSTIC\text{-}PARTIAL}.

The reported sub-gates are: center

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

center

The CMB--VP0 label supports only the declared restricted primary-sector gate surface. It does not assign full temperature/polarization/lensing closure, fit a full CMB likelihood, infer a new background expansion, replace recombination or drag-transfer modeling, or establish a late-transfer law. Passage of the public gates means only that the declared width, acoustic-angle, linked-ruler, and later-transfer diagnostics survive the named public-source comparison.

The construction must contain the standard linear CMB sector as a controlled limit. Let

\delta c_{\HH}^2\to 0,
\qquad
\delta S_{\HH}\to 0,
\qquad
\gstar\to g_*^{\mathrm{std}},
\qquad
\gd\to g_{\mathrm d}^{\mathrm{std}}.

lemma: Recovery of the standard sector within the declared kernel class. Under reference,

- $\ceff\to\cs$TeX\ceff\to\cs and the primary oscillator reduces to the standard baryon--photon acoustic mode; - $\Th_\ell^{\mathrm{pri}}$TeX\Th_\ell^{\mathrm{pri}} reduces to the standard line-of-sight source integral of linear CMB theory [citation]; - $(\rsstar,\rsd)$TeX(\rsstar,\rsd) reduce to the standard-sector recombination and drag rulers associated with the kernels $g_*^{\mathrm{std}}$TeXg_*^{\mathrm{std}} and $g_{\mathrm d}^{\mathrm{std}}$TeXg_{\mathrm d}^{\mathrm{std}}. If those kernels are further identified with narrow representatives centered on the standard recombination and drag epochs, these recover the usual acoustic and drag sound horizons.

proof. Statements (i) and (ii) follow directly from reference, reference, and reference when the phase-field corrections and kernel deformations vanish. Statement (iii) gives the standard-sector kernel-weighted ruler functionals associated with reference and reference once the phase-field corrections vanish. Equality with the usual horizon-at-epoch expressions requires the additional narrow-kernel identification stated above.

A surviving family must satisfy the following conditions.

R1. Blackbody consistency..

The monopole spectrum must remain consistent with the FIRAS distortion limits $|\yy|<1.5\times10^{-5}$TeX|\yy|<1.5\times10^{-5} and $|\mup|<9\times10^{-5}$TeX|\mup|<9\times10^{-5} [citation]. Post-thermalization mechanisms that inject energy above those bounds are excluded.

R2. Temperature-sector width envelope..

The last-accessibility width must satisfy reference; on the representative Planck+ACT window this becomes reference. This condition is interpreted only on the Gaussian/local-linear approximation window of reference.

R3. Polarization width check..

If a family identifies $\sigma_*$TeX\sigma_* with a fixed multiple of the polarization width $\DeltaLSS$TeX\DeltaLSS, then it must satisfy reference. More generally, any family that preserves temperature peak coherence by broadening the last-accessibility layer in a way that conflicts with the small-scale polarization width diagnostic is excluded [citation].

R4. Acoustic-angle constraint..

For fixed late geometry the family must satisfy reference. Violation of the observed $\theta_*$TeX\theta_* precision excludes the family before any late-time sector is invoked.

R5. Linked-ruler coupling..

Within a fixed-kernel family, the same parameter $\epsH$TeX\epsH controls both $\ellA$TeX\ellA and $\rsd/\rsstar$TeX\rsd/\rsstar through reference and reference. If an external BAO comparison requires a change in the drag-kernel fossil ruler that cannot be produced without simultaneously changing $\rsstar$TeX\rsstar, the fit must move to a different family. DESI DR2 provides the current BAO anchor for this comparison [citation].

R6. Later-transfer consistency..

The later transfer operator $\Tlate$TeX\Tlate must admit a realization consistent with CMB lensing. Planck PR4 lensing and ACT DR6 lensing therefore remain mandatory consistency checks on any surviving family [citation]. Polarization-only lensing and delensed-EE analyses provide an additional consistency check [citation]. By reference, later transfer may smooth or lens the primary spectra, but it may not reconstruct oscillatory power already lost in the primary sector.

Under a Gaussian last-accessibility kernel and a locally linear acoustic phase, reference gives the exact source-level factor on the declared Gaussian/local-linear primary window,

\exp[-k^2\bar c_*^{\,2}\sigma_*^2/2].

On a representative Planck+ACT peak window with $\bar c_*\in[0.45,0.47]$TeX\bar c_*\in[0.45,0.47], this implies $\sigma_*/\rsstar\lesssim0.096\text{--}0.10$TeX\sigma_*/\rsstar\lesssim0.096\text{--}0.10. If $\sigma_*=\avis\DeltaLSS$TeX\sigma_*=\avis\DeltaLSS is adopted as a family-level diagnostic convention, the same benchmark gives $\avis\lesssim0.73\text{--}0.76$TeX\avis\lesssim0.73\text{--}0.76.

At fixed early-time background expansion, baryon loading, recombination kernel, and drag kernel, the same family parameter $\epsH$TeX\epsH controls both $\rsstar$TeX\rsstar and $\rsd$TeX\rsd. In the worked lag benchmark, Planck's acoustic-angle precision implies $|\epsH|\lesssim1.2\times10^{-2}$TeX|\epsH|\lesssim1.2\times10^{-2}, which limits the induced linked-ruler shift to $|\delta\ln(\rsd/\rsstar)|\lesssim2.7\times10^{-4}$TeX|\delta\ln(\rsd/\rsstar)|\lesssim2.7\times10^{-4}. Within a fixed-kernel family, independent drag-ruler retuning is therefore excluded at first order.

The admissible primary sector is therefore constrained by blackbody consistency, the temperature-sector width envelope, the polarization width check, the acoustic-angle constraint, linked-ruler coupling, and later-transfer consistency. These results apply on the declared Gaussian/local-linear window, under the adopted $\sigma_*=\avis\DeltaLSS$TeX\sigma_*=\avis\DeltaLSS bridge when that bridge is used, and within the first-order fixed-kernel one-parameter family. The later transfer operator remains imported throughout.

The CMB--VP0 public-data gate supplies bounded external comparison for this restricted primary-sector surface. Its combined label is $\mathrm{CMB\text{-}VP0\text{-}PRIMARY\text{-}DIAGNOSTIC\text{-}PARTIAL}$TeX\mathrm{CMB\text{-}VP0\text{-}PRIMARY\text{-}DIAGNOSTIC\text{-}PARTIAL}, with width, acoustic-angle, linked-ruler, and later-transfer sub-gates passing only under their declared diagnostic meanings. This status is not a full CMB likelihood closure, Boltzmann-pipeline replacement, complete polarization/lensing closure, drag-transfer inference, late-transfer theorem, or new DE/LBD/DM cosmology closure. No black-hole analytical input is used, and full TT/TE/EE+lensing Boltzmann-hierarchy closure and full drag-transfer inference are not attempted here. The illustrative lag benchmark records one explicit admissible point within the declared class and is not transferred beyond that benchmark family.

The Gaussian identity used in reference is

\int_{-\infty}^{\infty}\frac{dx}{\sqrt{2\pi}\sigma}
\exp\!\left(-\frac{x^2}{2\sigma^2}+iqx\right)
=
\exp\!\left(-\frac12 q^2\sigma^2\right).

Setting $q=k\bar c_*$TeXq=k\bar c_* gives the damping factor in reference.

For $|\epsH|\ll1$TeX|\epsH|\ll1, reference gives

\ceff=\cs\sqrt{1+\epsH f}=
\cs\left(1+\frac{\epsH}{2}f\right)+O(\epsH^2)

on the ruler-support sector $u(k_{\mathrm{ac}})=1$TeXu(k_{\mathrm{ac}})=1. Substituting into reference and reference yields reference, from which

\delta\ln \rsstar=\frac{\epsH}{2\rstarzero}\Jstar[f]+O(\epsH^2),
\qquad
\delta\ln \rsd=\frac{\epsH}{2\rdzero}\Jd[f]+O(\epsH^2).

Equations reference and reference follow immediately.

For the benchmark of reference, let the narrow kernels and lag profile be given by reference and reference and treat $\cs$TeX\cs as constant across the overlap window. Then

\rstarzero \propto \int d\eta\,\gstar(\eta)\eta,
\qquad
\rdzero \propto \int d\eta\,\gd(\eta)\eta,

\Jstar[f_{\mathrm{lag}}] \propto \int d\eta\,\gstar(\eta)\int_0^\eta f_{\mathrm{lag}}(\eta')\,d\eta',
\qquad
\Jd[f_{\mathrm{lag}}] \propto \int d\eta\,\gd(\eta)\int_0^\eta f_{\mathrm{lag}}(\eta')\,d\eta'.

Numerical evaluation gives the coefficient set quoted in reference.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

These coefficients fix the first-order laws reference and hence the bounds reference and reference.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.