Evidence for Dihedral D₃ Symmetry in the Planck CMB Temperature Anisotropy

1. Introduction

The temperature anisotropies of the cosmic microwave background (CMB), as measured by WMAP [4] and Planck [1], are broadly consistent with the predictions of the standard $\Lambda$CDM cosmology: a statistically isotropic, Gaussian random field with a nearly scale-invariant power spectrum. However, a persistent set of large-angle anomalies has resisted explanation within this framework [3,5].

These include the anomalous alignment of the quadrupole ($l=2$) and octupole ($l=3$) principal axes [6,7], the hemispherical power asymmetry [8], the low quadrupole amplitude, and the odd-parity preference at low l [9]. Individually, each anomaly has marginal significance ($\sim 2$–$3\sigma$); collectively, they suggest a preferred direction in the low-l CMB that the standard model does not predict.

Numerous explanations have been proposed, including non-trivial cosmic topology [11,12], anisotropic dark energy [13], and residual foregrounds [14]. Most address one anomaly at a time and leave the others unexplained.

In this work, we pursue a different approach: we test whether the dihedral group $D_3$—the symmetry group of the equilateral triangle, of order $|D_3|=6$—organizes the angular power spectrum at a single preferred axis. $D_3$ is the smallest non-abelian finite group and the Weyl group of the Lie algebra $\mathfrak{su}\left(3\right)$; it can simultaneously produce quadrupole–octupole alignment (through its $C_3$ rotational subgroup), hemispherical-type asymmetry (through its reflection planes), and a selection rule with period 3 in l (through its irreducible representation structure).

The analysis extends to $l_{\max }=150$ and is validated across all four Planck PR3 component-separation pipelines. This allows us to determine whether the signal persists to high multipoles or is confined to the largest angular scales, and to rule out pipeline-specific artifacts.

The paper is organized as follows. In Section 2 we derive the decomposition of spherical harmonics into $D_3$ irreducible representations. In Section 3 we describe the data and Monte Carlo methodology. In Section 4 we present the results: axis determination, per-multipole irrep fractions, combined significance, inter-irrep correlations, and the transition scale. Section 5 reports the cross-map validation. Section 6 presents null tests. Section 7 examines the cumulative transfer structure, including the parity-gated grammar and its factorized form. In Section 8 we discuss the physical interpretation and identify falsifiable predictions.

2. ${\mathbf{D}}_{\mathbf{3}}$ Irrep Decomposition of Spherical Harmonics

2.1. Action of $D_3$ on $S^2$

The irrep decomposition is not performed directly on the observed $a_{lm}$ in Galactic coordinates. Instead, a candidate $D_3$ axis is chosen and the coefficients are rotated into a frame where that axis is the pole; the group then acts in this rotated frame. The axis is later optimized from the data (Section 3.3).

Let the $D_3$ symmetry axis coincide with the z-axis of a coordinate system on $S^2$. The group $D_3$ is generated by a rotation $C_3$ through $2\pi /3$ about $\widehat{z}$ and a reflection ${\sigma }_v$ through a vertical plane containing $\widehat{z}$. Under $C_3$, a spherical harmonic $Y_{lm}$ acquires a phase:

$$C_3{Y}_{lm}=e^{2\pi im/3}{Y}_{lm}.$$

(1)

The harmonics therefore sort by $mmod3$:

• $m\equiv 0(mod3)$: invariant under $C_3$ (eigenvalue 1),
• $m\equiv 1(mod3)$: eigenvalue $\omega =e^{2\pi i/3}$,
• $m\equiv 2(mod3)$: eigenvalue ${\omega }^2=e^{4\pi i/3}$.

The $m\equiv \pm 1(mod3)$ pairs combine into two-dimensional E doublets. The $m\equiv 0(mod3)$ harmonics are further split by the reflection ${\sigma }_v$, which maps $m\to -m$:

$$\begin{array}{cc}\hfill A_1& :\frac{1}{\sqrt{2}}(Y_{l,m}+Y_{l,-m}),m=0,3,6,\dots \le l,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill A_2& :\frac{1}{\sqrt{2}}(Y_{l,m}-Y_{l,-m}),m=3,6,9,\dots \le l,\hfill \end{array}$$

(3)

where $Y_{l,0}$ is automatically symmetric and contributes only to $A_1$.

2.2. Subspace Dimensions

The dimensions of the three irrep subspaces at multipole l are:

$$\begin{array}{cc}\hfill d_{A_1}\left(l\right)& =\lfloor l/3\rfloor +1,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill d_{A_2}\left(l\right)& =\lfloor l/3\rfloor ,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill d_E\left(l\right)& =2l+1-2\lfloor l/3\rfloor -1=2\left(l-\lfloor l/3\rfloor \right),\hfill \end{array}$$

(6)

satisfying $d_{A_1}+d_{A_2}+d_E=2l+1$.

A critical structural property follows: $d_{A_2}$ increases by one unit precisely when l crosses a multiple of 3. This means that at $l\equiv 0(mod3)$, the $A_2$ subspace acquires a new degree of freedom—a new “antenna element” tuned to the reflection-plane boundary. At $l=1$, $d_{A_2}=0$: the dipole contains no $A_2$ subspace, and is structurally excluded from the present analysis by the representation theory itself.

2.3. Isotropic Baseline

For a statistically isotropic Gaussian random field, all $a_{lm}$ at fixed l are independent with equal variance. The expected power fraction in each irrep is therefore proportional to its subspace dimension:

$${\langle f_X\left(l\right)\rangle }_{\mathrm{iso}}={\displaystyle \frac{d_X\left(l\right)}{2l+1}},X\in \{A_1,A_2,E\}.$$

(7)

Table 1 lists the subspace dimensions and isotropic baselines for $l=2$–12.

Figure 1. Left: Subspace dimensions $d_X\left(l\right)$ for the three $D_3$ irreps. $d_{A_2}=\lfloor l/3\rfloor$ increments (red dots) at $l\equiv 0(mod3)$; these dimension jumps drive the $lmod3$ selection rule. Right: Isotropic expected fractions $\langle f_X\rangle =d_X/(2l+1)$. All three approach their asymptotic values ($1/3$ for $A_1$ and $A_2$; $2/3$ for E) at large l, with sawtooth modulation at period 3.

Figure 1. Left: Subspace dimensions $d_X\left(l\right)$ for the three $D_3$ irreps. $d_{A_2}=\lfloor l/3\rfloor$ increments (red dots) at $l\equiv 0(mod3)$; these dimension jumps drive the $lmod3$ selection rule. Right: Isotropic expected fractions $\langle f_X\rangle =d_X/(2l+1)$. All three approach their asymptotic values ($1/3$ for $A_1$ and $A_2$; $2/3$ for E) at large l, with sawtooth modulation at period 3.

3. Data and Method

3.1. Planck Data

We analyze all four Planck PR3 (2018) component-separated temperature maps: SMICA, NILC, SEVEM, and Commander [2]. Each map is provided at HEALPix $N_{\mathrm{side}}=2048$ and internally downgraded to $N_{\mathrm{side}}=256$ (sufficient for $l_{\max }=150$). Spherical harmonic coefficients $a_{lm}$ are extracted using healpy.map2alm [15].

For the polarization null test (Section 6), E-mode $a_{lm}^E$ are extracted from the SMICA IQU polarization map at $N_{\mathrm{side}}=64$.

3.2. $D_3$ Decomposition

At each multipole l and a given axis $({\ell }_{\mathrm{gal}},b_{\mathrm{gal}})$, the $a_{lm}$ are rotated into the frame where the candidate $D_3$ axis is the pole. The rotation uses healpy.rotate_alm with Euler angles $(-\varphi ,-\theta ,0)$, where $(\theta ,\varphi )$ are the colatitude and longitude of the $D_3$ antipode. The power in each irrep is:

$$P_X\left(l\right)=\sum _{m\in {\mathcal{M}}_X\left(l\right)}{\left|a_{lm}\right|}^2,f_X\left(l\right)={\displaystyle \frac{P_X\left(l\right)}{C_l}},$$

(8)

where $C_l={\sum }_m{\left|a_{lm}\right|}^2$ and ${\mathcal{M}}_X\left(l\right)$ is the set of m-indices belonging to irrep X at multipole l. Note that $f_{A_1}+f_{A_2}+f_E=1$ by construction.

3.3. Axis Optimization

The preferred axis is determined by maximizing $f_{A_2}$ at $l=3$ (the octupole) over a HEALPix grid at $N_{\mathrm{side}}=32$ (12,288 directions; mean pixel spacing $1.8^{\circ }$), followed by a deterministic local refinement on a $0.2^{\circ }$-spaced sub-grid within $5^{\circ }$ of the coarse optimum. The optimal axis is then fixed for all subsequent analysis at $l\ne 3$. This protocol ensures that the results at all other multipoles are effectively pre-registered: the axis was determined from $l=3$ data alone.

3.4. Monte Carlo Simulations

Statistical significance is assessed against $N_{\mathrm{MC}}=10,000$ isotropic Gaussian random realizations. For isotropic fields, the $D_3$ fraction distribution is frame-invariant by rotational symmetry, so the MC simulations generate random $a_{l,m}$ directly in the $D_3$ frame without rotation: $a_{l,0}\sim \mathcal{N}(0,1)$ (real), $a_{l,m}\sim \mathcal{N}(0,1)+i\mathcal{N}(0,1)$ for $m>0$, with the Condon–Shortley convention $a_{l,-m}={(-1)}^m{\overline{a}}_{l,m}$. This avoids the $O\left(l^2\right)$ cost of Wigner-D rotation per simulation.

The probability to exceed (PTE) for each l is the fraction of simulations with $f_{A_2}\ge f_{A_2}^{\mathrm{obs}}$. PTEs of exactly zero (no simulation exceeded the observed value) are set to $0.5/N_{\mathrm{MC}}$ as a conservative upper bound.

Combined significance across multipoles uses Fisher’s method:

$${\chi }_{\mathrm{Fisher}}^2=-2\sum _{i}ln\left({\mathrm{PTE}}_i\right),$$

(9)

compared to a ${\chi }^2$ distribution with $2k$ degrees of freedom, where k is the number of contributing multipoles.

3.5. Statistical Protocol

To assist the reader in assessing trial factors, we summarize the complete analysis protocol:

1.

Axis selection. The axis is chosen by maximizing $f_{A_2}(l=3)$ on a HEALPix $N_{\mathrm{side}}=32$ grid ($1.8^{\circ }$ resolution) with local refinement. This is the sole optimization step and constitutes a single directional degree of freedom.

2.

Axis freezing. The same axis is then held fixed for all multipoles $l\ne 3$, all four component-separation maps, and all subsequent tests. The $l=3$ octupole should therefore be understood as the axis-defining (registration) statistic, not as an independently pre-registered detection.

3.

Significance assessment. Per-multipole PTEs are computed against $N_{\mathrm{MC}}=10,000$ isotropic Gaussian Monte Carlo realizations with the same decomposition.

4.

Primary significance claim. The Fisher combined PTE for $l=2$–15 across all four maps (Table 4).

5.

Replication criterion. Cross-map agreement is used as a replication standard, not as a tuning parameter: the axis is never re-optimized per pipeline.

6.

Secondary analyses. The top-9 peak residue-class test and the $lmod3$ permutation test are designated a posteriori supportive analyses; they are not part of the primary significance claim.

4. Results

4.1. Axis Determination

The axis maximizing $f_{A_2}(l=3)$ is found at Galactic coordinates

$$(\ell ,b)=(50.3^{\circ },-64.9^{\circ }),$$

(10)

with $f_{A_2}(l=3)=0.940$ (SMICA). Because this multipole was used to define the axis, its significance ($z>4.4$ against isotropic simulations) should be interpreted as a detection statistic used for registration, not as an independently pre-registered exceedance. The real evidence for $D_3$ symmetry lies in the replication of the same axis in independent multipoles ($l=7,9$) and across all four maps.

The optimal axis lies $5.8^{\circ }$ from the antipode of the Axis of Evil [10], indicating that the $D_3$ signal and the previously reported quadrupole–octupole alignment share a common preferred direction.

Figure 2 visualizes the $D_3$ structure on the sky. When the Planck SMICA map is rotated so the $D_3$ axis points to the pole, the $l=3$ octupole (panel a) displays the predicted six-lobe pattern: three hot spots and three cold spots alternating at ${120}^{\circ }$ intervals, with the $C_3$ rotation axes (solid gold lines) passing through the alternating lobes and the ${\sigma }_v$ reflection planes (dashed white) bisecting them. Combining the three driving multipoles $l=3,7,9$ (panel b) sharpens the structure rather than diluting it, confirming coherent $D_3$ alignment across angular scales. The full signal band $l=2$–15 (panel c) retains the pattern even with the addition of E-dominated modes, demonstrating that the $A_2$ signal survives superposition. Panel (d) shows that the axis is objectively determined by the data: an all-sky scan of $f_{A_2}(\ell =3)$ peaks sharply at the optimal direction, with no comparably strong secondary maxima.

4.2. Per-Multipole Irrep Fractions

Figure 3 shows $f_{A_2}\left(l\right)$ for $l=2$–150 at the fixed axis (SMICA), compared to the isotropic baseline and Monte Carlo $1\sigma$/$2\sigma$ bands. The signal exhibits a two-tier structure: a dense cluster of excess at $l\le 15$, and sporadic but individually significant peaks at higher multipoles.

Table 2 provides detailed numerical results for the low-l region. Table 3 lists the nine strongest cross-map-consistent peaks across $l=2$–150. Three multipoles drive the primary detection in all four maps:

1.

Octupole ($l=3$):$f_{A_2}=0.910$–$0.943$ across maps, $\mathrm{PTE}<{10}^{-4}$ ($z>+4.4$). The octupole exceeds $4\sigma$ in every pipeline.

2.

$l=7$:$f_{A_2}=0.481$–$0.496$, $\mathrm{PTE}=0.013$–$0.016$ ($z\approx +2.9$).

3.

$l=9$:$f_{A_2}=0.392$–$0.478$, $\mathrm{PTE}=0.014$–$0.044$ ($z=+2.0$ to $+2.7$).

Beyond the primary cluster, two high-l multipoles show cross-map-consistent $A_2$ excess: $l=34$ (significant at $\mathrm{PTE}<0.05$ in SMICA, NILC, and Commander; SEVEM marginal at $0.056$) and $l=63$ ($\mathrm{PTE}<0.05$ in SMICA, NILC, and Commander; SEVEM at $0.050$). These are not isolated noise fluctuations: they replicate across independent pipelines at the same fixed axis.

4.3. Combined Significance

Table 4 presents the Fisher combined PTE for different multipole ranges across all four maps. The low-l sector ($l=2$–15) shows a statistically unlikely concentration of $A_2$ excess in all four pipelines. Excluding the axis-defining octupole, the combined Fisher statistic is null (PTE $=0.87$–$0.99$), indicating that $l=7$ and $l=9$ corroborate the same axis-specific structure but do not by themselves constitute an independent aggregate detection. Their evidential value is replication at the registered axis, not a second standalone Fisher-level discovery.

The aggregate Fisher PTE for $l=16$–150 is consistent with isotropy in every map (PTE $>0.91$). However, this aggregate statistic masks the presence of individually significant, cross-map-validated outliers at $l=34$ and $l=63$ (Table 3). The high-l range is best characterized as an isotropic background punctuated by isolated resonances at the same $D_3$ axis.

4.4. Inter-Irrep Correlations

The deviations $\Delta f_X=f_X^{\mathrm{obs}}-{\langle f_X\rangle }_{\mathrm{iso}}$ across $l=2$–150 satisfy a strict conservation law:

$$\Delta f_{A_1}\left(l\right)+\Delta f_{A_2}\left(l\right)+\Delta f_E\left(l\right)=0$$

(11)

at every l, exact to machine precision ($\sim {10}^{-15}$). This confirms that the $D_3$ decomposition is a pure redistribution of power among irreps, not a net excess or deficit.

The Pearson correlation coefficient between $\Delta f_{A_2}$ and $\Delta f_E$ is strongly negative across all maps:

$$r(\Delta f_{A_2},\Delta f_E)=-0.81\pm 0.03(\mathrm{range}\mathrm{across}\mathrm{maps}),$$

(12)

with individual values SMICA: $-0.805$, NILC: $-0.809$, Commander: $-0.806$, SEVEM: $-0.743$. The $A_2$ excess draws power specifically from the E (doublet) irrep, not from $A_1$. Figure 4 displays this anti-correlation.

4.5. The $lmod3$ Selection Rule

A permutation test ($10,000$ random partitions) for whether $l\equiv 0(mod3)$ multipoles show preferential $A_2$ loading yields marginal results when applied to the full range $l=2$–150: Commander PTE $=0.055$, SMICA PTE $=0.074$, NILC PTE $=0.079$, SEVEM PTE $=0.092$.

A complementary (a posteriori) test examines the top-9 cross-map peaks (Table 3). Their residue class distribution is: $lmod3=0$ (five multipoles: $l=3,9,18,12,63$), $lmod3=1$ (four: $l=7,34,31,46$), and $lmod3=2$ (zero). Under uniform random selection, the probability of drawing $0/9$ from the $l\equiv 2(mod3)$ class is $\approx 0.02$. The complete absence of the residue class where $d_{A_2}$ does not increment is consistent with the $C_3$ selection rule: the $A_2$ excess requires the new basis vector ${\displaystyle \frac{1}{\sqrt{2}}}(Y_{l,3k}-Y_{l,-3k})$ that appears at $l\equiv 0(mod3)$, or the adjacent $l\equiv 1(mod3)$ class where the E-doublet dimension also steps.

Figure 5 displays the mean $\Delta f_{A_2}$ grouped by residue class.

4.6. Signal Morphology

The $D_3$ signal does not exhibit a sharp cutoff. Instead, the $A_2$ excess shows a two-tier structure:

1.

Dense cluster ($l\le 15$): Four of the thirteen multipoles ($31\%$) exceed $f_{A_2}>0.27$, compared to an isotropic expectation of $\sim 5\%$. The Fisher combined PTE for this range is $4.2$–$12\times {10}^{-3}$ across maps.

2.

Sporadic resonances ($l>15$): Four of 135 multipoles ($3\%$) exceed $f_{A_2}>0.27$, compared to an expected $\sim 1.5\%$ under isotropy. The aggregate Fisher PTE ($>0.91$) is null, but two individual multipoles ($l=34$, $l=63$) achieve $\mathrm{PTE}<0.05$ in three of four maps.

Figure 6 shows the running sum of $\Delta f_{A_2}$, which rises steeply for $l\le 15$, partially plateaus, then shows mild step-ups at the sporadic resonances—consistent with a boundary condition whose geometry is fully resolved at large angular scales but produces isolated overtones at specific higher harmonics.

4.7. Real-Space $D_3$ Pattern

The harmonic-space analysis can be visualized in real space by rotating the Planck SMICA map into the $D_3$ frame (where the $D_3$ axis is the pole) and examining the temperature field as a function of azimuth $\varphi$ at fixed colatitude $\theta$. Figure 7 shows this visualization for the $D_3$-active multipoles ($l=2$–15 and the three driving multipoles $l=3,7,9$).

The equatorial ring ($\theta ={90}^{\circ }$) displays a clear large-amplitude oscillation with three-fold structure (Figure 7, upper left). This pattern persists across colatitudes from ${45}^{\circ }$ to ${110}^{\circ }$ (upper right), confirming that the signal is not confined to the equatorial plane but pervades the hemisphere geometry.

The azimuthal Fourier spectrum of the equatorial ring (lower left) confirms the $C_3$ origin: for the $l=3,7,9$ band, the $m=3$ mode dominates, with $m\equiv 0(mod3)$ modes carrying the bulk of the power. This is the real-space manifestation of the $mmod3$ sorting that defines the $D_3$ irrep decomposition (Equation (1)).

Most directly, the azimuthal auto-correlation (lower right) reveals:

$$\begin{array}{cc}\hfill r(\Delta \varphi ={120}^{\circ })& =+0.222,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill r(\Delta \varphi ={60}^{\circ })& =-0.234.\hfill \end{array}$$

(14)

Points separated by ${120}^{\circ }$ around the $D_3$ axis see correlated temperatures ($C_3$ rotational symmetry), while points at ${60}^{\circ }$—straddling a reflection plane—see anti-correlated temperatures (${\sigma }_v$ anti-symmetry). This alternating sign pattern with period ${60}^{\circ }$ is the real-space fingerprint of the $A_2$ (reflection-antisymmetric) irrep that dominates the harmonic-space signal.

5. Cross-Map Validation

A critical test is whether the $D_3$ signal is consistent across independent component-separation methods. The four Planck PR3 pipelines (SMICA, NILC, SEVEM, Commander) use different algorithms and foreground models; agreement among them rules out pipeline-specific artifacts.

Table 5 presents the Pearson correlation of $f_{A_2}\left(l\right)$ between all map pairs over $l=2$–150. The correlations range from $r=0.932$ (NILC–SEVEM) to $r=0.997$ (SMICA–NILC). SMICA and NILC are nearly identical; SEVEM shows slightly more variation but remains highly correlated with all other maps.

Figure 8 overlays $f_{A_2}\left(l\right)$ from all four maps, demonstrating the remarkable agreement—particularly in the signal region ($l\le 15$), where the four curves are nearly indistinguishable.

6. Null Tests

6.1. Aggregate High-l Null

The Fisher PTE for $l=16$–150 is $>0.91$ in all four maps (Table 4). The aggregate high-l range does not show a sustained collective $A_2$ excess. All four pipelines agree on this. However, this aggregate null coexists with the isolated cross-map-validated peaks at $l=34$ and $l=63$ discussed in Section 4.6—highlighting the distinction between a diffuse persistent signal (absent) and discrete resonances (present).

6.2. E-Mode Polarization

The $D_3$ decomposition was applied to E-mode polarization $a_{lm}^E$ from the Planck SMICA IQU map at $N_{\mathrm{side}}=64$. No multipole shows $f_{A_2}^{EE}>0.50$. The Fisher combined PTE for the key multipoles ($l=3,6,7,9$) is $0.70$—fully consistent with isotropy. An independent axis optimization on the EE data yields an axis ${83}^{\circ }$ from the TT optimum, confirming the signals are unrelated.

This null is physically expected. Low-l E-mode polarization arises from reionization Thomson scattering at $z\sim 6$–10, well inside the last-scattering surface where the $D_3$ boundary condition operates on acoustic modes.

6.3. Axis Stability

The TT-optimal axis at $l=3$ is stable under variation of the input data: using the full-resolution ($N_{\mathrm{side}}=2048$) map versus downgraded versions yields axis shifts $<2^{\circ }$.

7. Cumulative Transfer Structure

The per-multipole $A_2$ excesses discussed in Section 4 are signed: each multipole either collects power into $A_2$ ($\Delta f_{A_2}>0$) or pays power out ($\Delta f_{A_2}<0$). We define the transfer ledger ${\tau }_l\equiv f_{A_2}^{\mathrm{obs}}\left(l\right)-{\langle f_{A_2}\rangle }_{\mathrm{iso}}\left(l\right)$ and study its cumulative behavior across the full multipole range $l=2$–150.

7.1. Normalized Transfer Invariant

Partition the cumulative transfer into residue classes,

$$T_r\left(L\right)=\sum _{\begin{array}{c}l\le L\\ lmod3=r\end{array}}{\tau }_l,r\in \{0,1,2\},$$

(15)

and define the normalized invariant

$$\mathrm{\Xi }\left(L\right)={\displaystyle \frac{T_0\left(L\right)-T_2\left(L\right)}{{\sigma }_{\mathrm{MC}}[T_0\left(L\right)-T_2\left(L\right)]}},$$

(16)

where ${\sigma }_{\mathrm{MC}}$ is the standard deviation of $T_0-T_2$ under $N_{\mathrm{MC}}=10,000$ isotropic simulations, evaluated at each L. This quantity measures the cumulative residue-class asymmetry of the $A_2$ transfer in units of the isotropic null.

Figure 9 shows $\mathrm{\Xi }\left(L\right)$ as a function of the upper summation boundary $L_{\max }$. Three features are noteworthy:

1.

Stable plateau.$\mathrm{\Xi }\left(L\right)$ rises to $\approx +3\sigma$ by $L=15$ and remains between $+2.5$ and $+3.1$ across all four component-separation maps out to $L=150$. The inter-map spread is less than $0.3\sigma$ at every checkpoint (Table 6).

2.

$A_2$ specificity. The analogous invariant computed for the $A_1$ irrep fluctuates around zero ($\langle {\mathrm{\Xi }}_{A_1}\rangle <0.6$ at all L), while the E invariant mirrors $A_2$ at ${\mathrm{\Xi }}_E\approx -3\sigma$—a direct consequence of the per-multipole conservation $\Delta f_{A_1}+\Delta f_{A_2}+\Delta f_E=0$. The transfer is thus an $E\to A_2$ funnel with $A_1$ as spectator, extending the correlation reported in Section 8.2.

3.

Jackknife stability. Removing $l=9$ alone reduces $\mathrm{\Xi }\left(150\right)$ from $+2.5$ to $+2.1$; removing $l=3$ alone reduces it to $+1.2$. Removing all three low-l anchors ($l=3,7,9$) collapses the invariant to $+0.7$, confirming that the signal is dominated by the dense cluster at $l\le 15$. This is consistent with the two-tier morphology identified in Section 4.6: the boundary condition is fully resolved at large angular scales.

The persistence of $\mathrm{\Xi }\left(L\right)$ to $L=150$ does not imply an independent high-l detection; it reflects the survival of the low-l transfer asymmetry as additional high-l null multipoles are cumulatively added (cf. the aggregate high-l null in Section 6.1).

Restricting the summation to the 94 cross-map sign-consistent multipoles (63% of the total) strengthens the invariant: $\langle \mathrm{\Xi }\rangle$ rises from $+2.53$ to $+2.88$ at $L=150$, indicating that the sign-inconsistent multipoles dilute the signal rather than contribute to it.

7.2. One-Way Collect Rule

Among all multipoles individually significant at $\mathrm{PTE}<0.05$ in two or more pipelines, the cross-map mean $\langle {\tau }_l\rangle$ is positive in every case. That is, every statistically robust transfer event collects power into $A_2$; none pays out. This one-way rule holds under all thresholds tested (1/4 to 4/4 maps significant, $\mathrm{PTE}$ thresholds of $0.05$ and $0.10$), producing a binomial probability $\le 2^{-8}$ under the null hypothesis of symmetric pay/collect.

7.3. Parity-Gated Transfer Grammar

The $D_3$ group contains rotations and reflections but no spatial inversion; it does not distinguish odd from even l. Nevertheless, the transfer ledger splits sharply by parity. The mean excess $\langle \Delta f_{A_2}\rangle$ is positive for odd l and negative for even l in all four pipelines (Table 7). Aggregated via Fisher’s method, the odd-l subset yields a combined $\mathrm{PTE}<2\times {10}^{-4}$ in every map, while the even-l subset is entirely null ($\mathrm{PTE}>0.97$).

This parity split is not an artifact of $A_2$ subspace dimensions: odd and even l contribute nearly equal total $A_2$ capacity ($\sum d_{A_2}=1850$ vs. 1875 for $l=2$–150).

Crucially, the split is not confined to $A_2$. The E irrep shows the mirror pattern (even > odd) at comparable or stronger significance (${\mathrm{PTE}}_{\mathrm{perm}}<0.015$ in SMICA and Commander), while $A_1$ is marginally patterned ($\mathrm{PTE}\sim 0.06$). Because $\Delta f_{A_1}+\Delta f_{A_2}+\Delta f_E=0$ at every l, the parity gate acts on the entire redistribution system, not on a single output channel.

The Six-Cell Grammar

Crossing parity with the residue class $lmod3$ produces a $2\times 3$ transfer grammar (Table 8). The structure is dominated by a single cell: odd multipoles in the $l\equiv 0(mod3)$ class carry a cumulative $A_2$ transfer of $+1.15$, while every even-l cell drains. This is not “odd good, even bad”: the odd $(r=1)$ and odd $(r=2)$ cells are near-neutral or draining. The gate selects the intersection of odd parity and the $D_3$-preferred residue class.

The onset of this grammar is sharp. At $l=1$, the $A_2$ subspace has dimension zero (a group-theory identity confirmed to machine precision in all maps). At $l=2$ (even), $\Delta f_{A_2}=0$ to four decimal places. At $l=3$—the first multipole that is both odd and carries $d_{A_2}>0$—the signal switches on at $z\approx +4.5$. The transition from silence to the strongest detection in the entire spectrum occurs at the first algebraically allowed mode of the parity-gated $D_3$ grammar.

This two-layer structure—$D_3$ residue routing overlaid by a binary parity gate—is consistent with a two-branch boundary selection law rather than a generic parity anomaly. A background parity asymmetry would modulate total power $C_l$ without preferentially acting on the irrep redistribution; the observed effect acts specifically on the transfer grammar. Indeed, a permutation test on raw $C_l$ confirms this: the odd/even parity split in total power is null ($\mathrm{PTE}>0.61$) in all four maps, while the same test applied to $\Delta f_{A_2}$ yields $\mathrm{PTE}$ as low as $0.03$ (SEVEM). The binary gate acts on routing, not on total power.

7.4. Factorized Transfer Law

The six-cell grammar of Table 8 can be viewed as a $2\times 3$ matrix

$$M\left(L\right)=\left(\begin{array}{ccc}{T}_{\mathrm{odd},0}\left(L\right)& T_{\mathrm{odd},1}\left(L\right)& T_{\mathrm{odd},2}\left(L\right)\\ T_{\mathrm{even},0}\left(L\right)& T_{\mathrm{even},1}\left(L\right)& T_{\mathrm{even},2}\left(L\right)\end{array}\right),$$

(17)

where the rows index parity and the columns index the residue class $lmod3$. A singular-value decomposition (SVD) of $M\left(150\right)$ reveals that the matrix is approximately rank 1 (Table 9):

$$M\left(L\right)\approx {\sigma }_1\mathbf{u}{\mathbf{v}}^{\top },$$

(18)

with ${\sigma }_1/{\sigma }_2>4$ in three of four maps. The dominant left vector $\mathbf{u}\in {\mathbb{R}}^2$ encodes a binary gate (odd positive, even negative), and the dominant right vector $\mathbf{v}\in {\mathbb{R}}^3$ encodes a $D_3$ routing pattern ($r=0$ dominant, $r=1$ inert, $r=2$ draining).

Cross-map stability is high: the minimum pairwise cosine similarity is $0.967$ for $\mathbf{u}$ and $0.988$ for $\mathbf{v}$. The same binary branch and same $D_3$ routing vector are recovered by all four independent component-separation pipelines. The rank-1 fraction exceeds $94\%$ at $L=50$–75 in all maps and remains above $90\%$ in three of four pipelines through $L=150$. The exception is SEVEM ($77\%$), whose template-fitting method introduces larger even-l transfer amplitudes (roughly 2–$3\times$ those of the ILC and Bayesian pipelines), inflating ${\sigma }_2$ and diluting the rank-1 fraction; however, the SEVEM singular vectors $\mathbf{u}$ and $\mathbf{v}$ still align with the other three maps (pairwise cosine $>0.94$), so the same grammar is recovered with a noisier gate. The factorization reflects the dominant low-l boundary phase identified in Section 8.1 and remains stable in the cumulative matrix as null high-l multipoles are added. The factorization should be understood as a property of the cumulative low-l-dominated transfer matrix, not as a claim that the same singular vectors characterize the high-l sector in isolation.

In summary, the entire $D_3$ transfer structure admits an approximate factorization into two primitive components: a binary parity selector and a three-fold residue routing grammar. This factorized form is more constrained than either the residue-class hierarchy or the parity gate taken separately, and suggests that the underlying boundary law has an internal $2\times 3$ product structure.

Figure 10. The $2\times 3$ transfer grammar. Left: Observed four-map mean transfer matrix $M\left(150\right)$, partitioned by parity (rows) and residue class $lmod3$ (columns); error bars show inter-map spread. Center: Best rank-1 reconstruction ${\sigma }_1\mathbf{u}{\mathbf{v}}^{\top }$. Right: Residual (observed − reconstruction), showing that the rank-1 approximation captures the dominant structure. The single dominant cell (odd, $r=0$) and the near-zero residuals confirm the factorized transfer law.

Figure 10. The $2\times 3$ transfer grammar. Left: Observed four-map mean transfer matrix $M\left(150\right)$, partitioned by parity (rows) and residue class $lmod3$ (columns); error bars show inter-map spread. Center: Best rank-1 reconstruction ${\sigma }_1\mathbf{u}{\mathbf{v}}^{\top }$. Right: Residual (observed − reconstruction), showing that the rank-1 approximation captures the dominant structure. The single dominant cell (odd, $r=0$) and the near-zero residuals confirm the factorized transfer law.

7.5. Empirical Signature Summary

Table 10 collects the identified empirical properties of the $D_3$ signal, their observables, quantitative results, and evidential status. Primary claims are those supported by pre-registered statistics or clear null controls; supportive observations are a posteriori structural patterns that sharpen the interpretation but do not constitute standalone detections.

8. Discussion

8.1. A Large-Angle Boundary Condition

The detected $D_3$ symmetry should not be confused with a claim of multiply-connected spatial topology. Topological models (e.g., the Poincaré dodecahedral space [11]) tile the spatial 3-manifold with a discrete group. The $D_3$ signal reported here is an angular spectral property: the irrep decomposition of $a_{lm}$ at each l reveals preferential excitation of the reflection-antisymmetric subspace at a fixed axis on $S^2$.

The two-tier morphology—dense cluster at $l\le 15$ plus sporadic resonances at $l=34$ and $l=63$—constrains the physical mechanism. A rigid geometric template would produce excess at all l; a foreground systematic would correlate with the mask geometry and would not replicate across independent component-separation pipelines. The observed pattern is instead consistent with a boundary condition on the acoustic eigenvalue problem: the boundary geometry is fully resolved by modes at $l\lesssim 15$ ($\theta \gtrsim {12}^{\circ }$), producing the dense cluster, while specific higher harmonics whose angular wavelengths happen to alias the boundary structure produce isolated resonances.

The primary scale $l\sim 15$ corresponds to angular separations of order ${12}^{\circ }$, or comoving distances of approximately 800 Mpc at the last scattering surface—comparable to the largest observed structures in the cosmic web. The resonances at $l=34$ ($\theta \sim 5.3^{\circ }$) and $l=63$ ($\theta \sim 2.9^{\circ }$) may represent overtones of the same boundary geometry.

The rank-1 factorization of the transfer matrix (Section 7.4) further constrains the boundary’s internal structure. The approximate decomposition $M\approx {\sigma }_1\mathbf{u}{\mathbf{v}}^{\top }$ implies that the boundary law is not irreducibly six-fold but has an internal $2\times 3$ product form: a binary parity selector composed with a three-fold residue routing grammar. This factorized structure is stable across all four pipelines and reflects the dominant low-l boundary phase, suggesting two primitive components in the boundary condition rather than a single monolithic selection rule.

8.2. Representation-Theoretic Structure

The signal is not a generic anisotropy. Four features identify it specifically as $D_3$:

1.

$A_2$ specificity. The collecting excess concentrates in the reflection-antisymmetric irrep, with a mirrored depletion in the E (rotation-doublet) irrep and $A_1$ largely acting as spectator. An arbitrary dipolar or quadrupolar modulation would populate all irreps comparably.

2.

$E\to A_2$ funnel. The anti-correlation $r(\Delta f_{A_2},\Delta f_E)=-0.81$ with $A_1$ decoupled is consistent with the reflection planes being the geometrically active elements. The cumulative transfer invariant (Section 7) confirms this structure: ${\mathrm{\Xi }}_{A_2}$ and ${\mathrm{\Xi }}_E$ are mirror images while ${\mathrm{\Xi }}_{A_1}\approx 0$, and every statistically robust event collects into $A_2$. In the Weyl chamber of $SU\left(3\right)$, the reflection hyperplanes (walls) separate adjacent chambers; the observed power transfer from E (rotations) to $A_2$ (reflections) admits an interpretation in which these walls act as boundary surfaces.

3.

$lmod3$ selection rule. The aggregate permutation test is marginal ($\mathrm{PTE}\sim 0.06$–$0.09$). As a supportive (a posteriori) observation, the top-9 cross-map peaks show zero entries from $l\equiv 2(mod3)$ ($p\approx 0.02$); however, the threshold defining “top 9” was not pre-specified and this test should be regarded as suggestive rather than confirmatory. The period of 3 is consistent with the order of the rotational subgroup $C_3\subset D_3$ and with the Coxeter number $h=3$ of the root system $A_2$, distinguishing $D_3=\mathrm{Weyl}\left(A_2\right)$ from other dihedral groups $D_n$ with $n\ne 3$.

4.

Parity-gated transfer. Although $D_3$ contains no spatial inversion, the irrep redistribution is sharply gated by parity: the entire $A_2$ collecting signal resides in odd-l multipoles, while even-l multipoles are null (Section 7.3). The gate acts across all three irreps simultaneously, ruling out an $A_2$-specific artifact. The six-cell grammar (parity × residue class) isolates the signal to a single cell—odd, $l\equiv 0(mod3)$—and the onset at $l=3$ (the first algebraically allowed mode) is step-like rather than gradual. This two-branch selection law is an additional structural constraint beyond $D_3$ alone, consistent with a hidden binary boundary encoding.

8.3. Connection to the Vacuum Energy Sector

The spherical boundary $\Sigma \simeq S^2$ on which $D_3$ acts is also the surface on which the monopole ($l=0$) channel of the cosmological causal diamond projects. The $D_3$ analysis operates at $l\ge 2$; the vacuum energy density, being spatially uniform, projects entirely onto $l=0$. These are different harmonic channels of the same $S^2$ boundary.

A first-principles derivation connecting the Standard Model field content on this boundary to the observed cosmological constant, yielding ${\Omega }_{\Lambda }=0.6948$ with zero free parameters (deviation $0.4\sigma$ from observation), has been presented elsewhere [16]. The present detection of $D_3$ at $l\ge 2$ is consistent with the hypothesis that the boundary $\Sigma \simeq S^2$ carries physically observable structure, though the CMB result alone does not require this interpretation. The cosmological-constant connection is not part of the statistical claim of the present paper; it is noted here only to indicate a possible theoretical context.

8.4. Falsifiable Predictions

The $D_3$ boundary-condition interpretation makes specific predictions testable with forthcoming data:

1.

Resonance spectrum. If $l=34$ and $l=63$ are overtones of a boundary geometry, additional resonances should appear at predictable multipoles determined by the boundary’s angular structure. Extension to $l_{\max }=500$–1000 with CMB-S4 [17] data will test whether the sporadic peaks continue or are confined to $l\lesssim 100$.

2.

$l\equiv 2(mod3)$ exclusion. The current $0/9$ absence of this residue class among the strongest peaks ($p\approx 0.02$) should persist or strengthen with deeper data. A single strong $l\equiv 2(mod3)$ peak would disfavor the $C_3$ selection-rule mechanism.

3.

Cross-survey consistency. The $D_3$ signal should appear in future independent CMB measurements (LiteBIRD [18], Simons Observatory [19]) at the same axis and with the same two-tier morphology.

4.

Lensing B-modes. If the boundary condition imprints on the matter field, gravitational-lensing B-modes should show a residual $D_3$ signal at low l.

5.

Parity-gate persistence. The odd-only concentration of the $D_3$ transfer should persist at higher l and in independent surveys. If the gate is a genuine two-branch boundary law, no even-l multipole should enter the dominant collecting cell at any $l_{\max }$.

9. Conclusions

We have presented evidence for dihedral $D_3$ symmetry in the Planck CMB temperature anisotropy, validated across all four independent component-separation pipelines at a single preferred axis. The signal has a two-tier morphology: a dense cluster at $l\le 15$ (Fisher PTE $=4.2\times {10}^{-3}$ to $1.2\times {10}^{-2}$), driven by three multipoles ($l=3,7,9$) significant in every map, plus cross-map-validated resonances at $l=34$ and $l=63$ (each significant in 3/4 pipelines).

Four features specifically identify $D_3$ rather than a generic anisotropy: concentration in the $A_2$ (reflection-antisymmetric) irrep, an $E\to A_2$ power funnel with $r=-0.81$, supportive evidence for a $C_3$ selection rule from the absence of $l\equiv 2(mod3)$ multipoles among the strongest peaks, and a parity-gated transfer grammar in which the entire signal resides in odd-l multipoles while even-l is null across all irreps. The cumulative transfer invariant ${\mathrm{\Xi }}_{A_2}\left(L\right)\approx +3\sigma$ remains stable from $L=15$ to 150 across all four pipelines, with a one-way collect rule (every statistically robust event adds power to $A_2$) and the expected $A_1$/E conservation-law mirror. The six-cell parity × residue-class grammar isolates the collecting signal to a single cell (odd, $l\equiv 0(mod3)$), with step-function onset at $l=3$. Singular-value decomposition of the $2\times 3$ transfer matrix reveals an approximate rank-1 factorization into a binary parity selector $\mathbf{u}\approx (+,-)$ and a $D_3$ routing vector $\mathbf{v}\approx (+,0,-)$, stable across pipelines (minimum pairwise cosine $>0.96$). The boundary law thus has an internal $2\times 3$ product structure rather than a single monolithic selection rule. Cross-map correlations exceeding $r=0.93$ for all pipeline pairs rule out component-separation artifacts, and a null result in E-mode polarization (Fisher PTE $=0.70$) confirms the signal is confined to the temperature channel.

The primary empirical claims are the fixed-axis low-l$A_2$ concentration, cross-map replication, EE null, parity-gated transfer, cumulative transfer invariant, and the rank-1 factorization. The residue-class top-9 test and overtone interpretation are supportive structural observations rather than standalone confirmatory statistics (Table 10).

The group $D_3\cong \mathrm{Weyl}\left(A_2\right)$ is the Weyl group of the $\mathfrak{su}\left(3\right)$ root system. The period-3 selection rule matches the order of the rotational subgroup $C_3\subset D_3$ and the Coxeter number $h=3$ of $A_2$, suggesting a connection between the angular structure of the microwave sky and the gauge group of quantum chromodynamics through the shared boundary $S^2$.

Appendix A.1. Multipole Window

Table A1 shows the Fisher combined PTE for the $A_2$ excess (SMICA) as a function of the upper multipole boundary $l_{\max }$, with $l_{\min }=2$ fixed. The significance is strongest at $l\le 10$ and degrades gradually as null high-l multipoles are added, but remains below $0.05$ for all windows up to $l_{\max }=30$. This confirms that the signal is not an artifact of the particular choice $l_{\max }=15$.

Table A1. Fisher combined PTE (SMICA) for varying $l_{\max }$, with $l_{\min }=2$.

l range	Fisher PTE
2–10	$8.5\times {10}^{-4}$
2–15	$4.6\times {10}^{-3}$
2–20	$9.2\times {10}^{-3}$
2–25	$1.9\times {10}^{-2}$
2–30	$3.5\times {10}^{-2}$

Table A1. Fisher combined PTE (SMICA) for varying $l_{\max }$, with $l_{\min }=2$.

l range	Fisher PTE
2–10	$8.5\times {10}^{-4}$
2–15	$4.6\times {10}^{-3}$
2–20	$9.2\times {10}^{-3}$
2–25	$1.9\times {10}^{-2}$
2–30	$3.5\times {10}^{-2}$

Table A2 varies the lower boundary $l_{\min }$ with $l_{\max }=15$ fixed. Removing $l=2$ (which is E-dominated and contributes no $A_2$ signal) improves the Fisher PTE from $4.6\times {10}^{-3}$ to $2.2\times {10}^{-3}$, demonstrating that the quadrupole dilutes rather than drives the result. Removing both $l=2$ and $l=3$ (the axis-defining multipole) yields PTE $=0.14$: the signal from $l=4$–15 alone is not independently significant, confirming that $l=3$ anchors the detection.

Table A2. Fisher combined PTE (SMICA) for varying $l_{\min }$, with $l_{\max }=15$.

l range	Fisher PTE
2–15	$4.6\times {10}^{-3}$
3–15	$2.2\times {10}^{-3}$
4–15	$0.14$

Table A2. Fisher combined PTE (SMICA) for varying $l_{\min }$, with $l_{\max }=15$.

l range	Fisher PTE
2–15	$4.6\times {10}^{-3}$
3–15	$2.2\times {10}^{-3}$
4–15	$0.14$

Appendix A.2. Axis Seeding

The axis optimization uses HEALPix $N_{\mathrm{side}}=32$ as the initial grid. Repeating the search at $N_{\mathrm{side}}=16$ (3,072 directions; $3.7^{\circ }$ resolution) and $N_{\mathrm{side}}=64$ (49,152 directions; $0.9^{\circ }$ resolution) yields axis shifts $<1.5^{\circ }$ and $f_{A_2}(l=3)$ values within $0.002$ of the baseline, confirming that the optimum is a broad, well-resolved feature of the sky.

Appendix A.3. Resolution

Using the full-resolution ($N_{\mathrm{side}}=2048$) SMICA map versus downgraded versions ($N_{\mathrm{side}}=128$, 256, 512) yields axis shifts $<2^{\circ }$ and identical per-multipole PTEs to within Monte Carlo noise, indicating that the signal is insensitive to the pixel resolution.