Abstract

The standard cosmological model interprets the Cosmic Microwave Background (CMB) as a relic radiation field originating from the recombination of hydrogen and helium at redshift z ~ 1100, propagating freely since that epoch. We present a different account based on the NMSI (New Subquantum Informational Mechanics) framework: CMB is not a fossil relic but coherent re-emission generated continuously, here and now, by ordinary baryonic matter (atoms, electrons, ions, molecules, dust) through the action of operators induced by the Riemann Oscillatory Network (RON) on discrete atomic and molecular emission lines. We identify PON-C (Plasmatic Oscillatory Network-Cosmological) as the global oscillation mode of all baryonic matter at the Hubble scale, coupled coherently through RON, oscillating in antiphase with RON. We further introduce PON-G (Plasmatic Oscillatory Network-Galactic) as the galactic component of the same plasmatic oscillatory network, concentrated in spiral arms, giant molecular clouds, HII regions, and star-forming regions, which constitutes the principal reservoir of dipolar molecular species (H₂O, OH, NH₃, CH₃OH, HCN) actively participating in CMB re-emission. The observed black-body spectrum at T_CMB = 2.7255 K results from the action of two RON-induced operators on the discrete atomic and molecular emission lines: the harmonic-mixing operator π* and the dissipative-smoothing operator γ_diss. Three theorems are proved: Theorem M1 (integrated atomic and molecular emission, after the action of π* and γ_diss, is spectrally indistinguishable from the observed CMB), Theorem M2 (γ_diss has a unique Planckian fixed point at temperature T*, with explicit spectral gap λ_diss ≥ 1/J_c ~ 0.018 and characteristic relaxation scale of approximately 55 iterations), and Theorem M3 (CMB anisotropies on the sky are linear in the column densities of neutral hydrogen, molecular hydrogen, and PON-G dipolar species). Seven falsifiable predictions follow from this theory: P11 (anti-correlation of cold CMB residuals with HI), P12 (correlation of warm CMB residuals with PON-G dipolar emission), P13 (null partial correlation with free-free), P14 (sub-percent SZE deviation from adiabaticity), P15 (dependence of the CMB residual on the local kinematic redshift modulus |z_kin|), P16 (independence from the Doppler sign—kinematic-amplitude effect, not classical Doppler), and P17 (statistical reconstruction of the CMB residual through |z_kin| after foreground control). Experimental results obtained on public Planck PR3 + HI4PI data: P11a confirmed at 13.03 σ; P12 confirmed at 5.00 σ on NILC, 16.24 σ on SMICA, above 25 σ on SEVEM; P13 confirmed at 0.50 σ (compatible with zero) with r_partial ~ 0.003; P15 global confirmed at 19 - 22 σ on all three reconstruction maps, P15 regional reaching |r| = 0.62 with approximately 28 σ in absV_top_1%; P16 eliminates the classical Doppler hypothesis (same negative sign for approaching and receding regions); P17 reconstructs the real component with a negative |z_kin| coefficient at 14 - 17 σ stable across all maps. The corrected form of the prediction is $\Delta T_{\text{residual}}\left(l,b\right)=-\delta \cdot \left|z_{\text{kin}}\left(l,b\right)\right|+\epsilon \left(l,b\right)$ , with δ > 0. These results are inconsistent with the fossil-CMB interpretation and are consistent with the mechanism of coherent re-emission through PON-C oscillation with active participation of PON-G. This finding does not bear on the validity of the Hawking-Penrose singularity theorem or on classical general relativity, whose domains of applicability are logically distinct from the empirical question addressed here (see Section 2 for an explicit discussion of the scope and logical status of this claim). The closing section presents the table of twelve pre-registered criteria confirmed on public IceCube data (Paper VII of the NMSI Neutrinos series), offered as independent supporting evidence from a different observational domain.

Keywords

Cosmic Microwave Background, PON-C (Plasmatic Oscillatory Network-Cosmological), PON-G (Plasmatic Oscillatory Network-Galactic), RON (Riemann Oscillatory Network), Atomic and Molecular Re-Emission, Kinematic Redshift |z_kin|, Partial Correlation, Statistical Reconstruction, Fossil-CMB Interpretation, Empirical Challenge to the Hot Big Bang Model, NMSI Framework

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1. Introduction: Objective of Testing the Fossil-CMB Interpretation

The standard cosmological model ΛCDM interprets the Cosmic Microwave Background as a relic radiation field originating from the recombination of hydrogen and helium at redshift z ~ 1100, propagating freely since that epoch with temperature scaling as T(z) = T₀(1 + z) under adiabatic expansion. The black-body character of the spectrum and its near-perfect isotropy are explained as a consequence of thermal equilibrium prevailing at recombination, supplemented by an inflationary epoch homogenizing causally disconnected regions. Under this interpretation, the CMB observed today comes from outside the Visible Sphere of the Universe—it is a fossil relic, a trace of the primordial state of the universe.

The present paper subjects this interpretation to empirical testing. The NMSI (New Subquantum Informational Mechanics) framework, developed over more than fifteen years of theoretical work [1] [2], offers a radically different alternative: CMB is not a fossil relic but active coherent re-emission, generated right now, inside the Visible Sphere, by ordinary baryonic matter (atoms, electrons, ions, molecules, dust) through the action of RON operators on discrete atomic and molecular emission lines organized by the plasmatic oscillatory networks PON-C and PON-G.

The scope of this paper is deliberately narrow and empirical. This paper does not claim, and does not attempt, to falsify the Big Bang model as a whole, nor does it engage with the mathematical status of singularity theorems in classical general relativity, such as the Hawking-Penrose theorem. Those results establish geodesic incompleteness under a specific set of assumptions in classical GR; they say nothing about the physical origin of the microwave background observed today, and they are not disputed anywhere in this paper. The claim under test here is narrower and purely observational: whether the CMB anisotropies observed today are better explained as a relic signal from z ~ 1100, entirely decoupled from the present distribution and kinematics of baryonic matter, or as a signal actively generated and modulated by that present matter. Section 2 makes this distinction explicit, since a previous version of this manuscript did not delimit the claim clearly enough, which understandably invited a broader reading of the paper’s objective than was intended.

The distinction between the two interpretations of the CMB signal is not philosophical—it is directly observable. If CMB is a fossil relic, its anisotropies are imprints of primordial density perturbations from the surface of last scattering and must not correlate with the spatial distribution of local baryonic matter, nor with its kinematics. If CMB is coherent re-emission, the anisotropies must be linear in the column densities of neutral hydrogen, molecular hydrogen, and galactic dipolar molecules (PON-G), precisely on the sky, precisely here, precisely now, and must exhibit systematic dependence on the local kinematics of baryonic gas.

This paper presents the results of experimental tests carried out on public Planck PR3 + HI4PI data, directly confronting the two hypotheses. The results are unambiguous: the NMSI predictions are confirmed at statistical significance levels between 13 σ and 28 σ, while the fossil hypothesis (CMB independent of local matter and its kinematics) is rejected on three independent CMB reconstruction maps.

To strengthen the conclusion and remove any ambiguity, the paper presents the decisive experimental chain P15 to P16 to P17, executed on public data. P15 establishes the global and regional kinematic dependence; P16 eliminates the classical Doppler explanation through approaching/receding separation; P17 confirms the statistical reconstruction of the CMB residual through |z_kin|. Combined, these three tests deliver a strong empirical challenge to the fossil hypothesis—the CMB residual depends measurably on the local kinematics of atomic gas, a phenomenon difficult to reconcile with the fossil interpretation of a relic coming from outside the Visible Sphere.

At the end of the paper, we also present the table of twelve pre-registered criteria confirmed on public IceCube data (from Paper VII of the NMSI Neutrinos series [3]), as additional supporting evidence obtained independently, through observational data from a different physical domain (cosmological CMB and IceCube neutrino telescope physics).

The structure of the paper is as follows. Section 2 delimits the scope and logical status of the paper’s central claim explicitly. Section 3 collects notations and constants. Section 4 establishes the identification of PON-C and PON-G as plasmatic oscillatory networks. Section 5 develops the mathematical apparatus. Section 6 presents the complete thermalization mechanism and the effective RON-EM Lagrangian. Section 7 states the three theorems. Section 8 develops the spatial structure. Section 9 introduces the seven falsifiable predictions. Section 10, in ten subsections, presents the complete experimental results. Section 11 synthesizes the cumulative table P1-P17. Section 12 formulates the verdict, calibrated to what the data support. Section 13 presents the IceCube table. Section 14 concludes. Section 15 outlines Paper IV.

2. Scope and Logical Status of the Falsification Claim

This section responds directly to a concern raised during peer review, namely that a paper proposing to challenge the standard interpretation of the CMB must first demonstrate, through rigorous mathematical proof, that the Hawking-Penrose singularity theorem is incorrect or inapplicable. We agree that such a demonstration would be required if this paper claimed to falsify the Big Bang model as a whole, since that model rests in part on results of classical general relativity concerning gravitational collapse and singularity formation, of which the Hawking-Penrose theorem [4]—recognized in part by the 2020 Nobel Prize in Physics—is a central pillar. We state explicitly that this paper makes no such claim.

2.1. What the Hawking-Penrose Theorem Establishes, and What It Does Not

The Hawking-Penrose singularity theorem [4] establishes geodesic incompleteness for a congruence of causal geodesics, under a specific set of assumptions: global hyperbolicity of the spacetime, validity of the Einstein field equations, the dominant (or null) energy condition, the existence of a trapped surface or a sufficiently converging initial congruence, and suitable global causal conditions. Its conclusion, precisely stated, is that certain geodesics cannot be extended to arbitrarily large affine parameter, that is, geodesic incompleteness.

The theorem does not state that the Cosmic Microwave Background is a fossil relic. It does not state that recombination occurred at z ~ 1100 in the specific thermal sequence assumed by the Hot Big Bang scenario. It does not state that the ΛCDM parameter set is correct. These are separate, empirical claims about the physical history and present content of the universe, layered on top of the mathematical fact of geodesic incompleteness. A theorem about the local structure of geodesic congruences under classical GR assumptions does not, by itself, fix the interpretation of a specific present-day observational dataset such as the Planck CMB maps.

2.2. The Correct Target of This Paper

The claim under test in this paper is narrower and strictly empirical. It concerns only the following:

• The fossil interpretation of the CMB, namely that the CMB observed today is a signal decoupled from the present state of baryonic matter, having propagated freely since z ~ 1100.

• The assumption, internal to that interpretation, that CMB anisotropies must be independent of the present spatial distribution and kinematics of local baryonic matter, once foregrounds are properly removed.

• The specific thermal-history narrative in which the observed near-isotropic 2.7255 K background is explained exclusively as relic radiation, rather than containing a measurable present-day re-emission component.

The NMSI_CMB framework proposes an alternative physical mechanism—coherent re-emission by present baryonic matter, mediated by RON, PON-C, and PON-G—and this paper tests that alternative against the fossil interpretation using seven pre-registered, falsifiable predictions (P11-P17) on public Planck PR3 and HI4PI data. None of these tests bear on geodesic incompleteness, on the dominant energy condition, on trapped surfaces, or on any other hypothesis of the Hawking-Penrose theorem. They bear on a correlation structure in present-day observational data.

2.3. Logical Independence of the Two Claims

The relationship between the Hawking-Penrose theorem and the claim tested in this paper can be stated as a short logical chain:

• Hawking-Penrose theorem [4]: classical GR assumptions lead to geodesic incompleteness. This is a mathematical result and is not disputed anywhere in this paper.

• Hot Big Bang/fossil-CMB claim: the CMB is relic radiation from recombination at z ~ 1100, decoupled from present baryonic matter. This is an empirical claim, distinct from the theorem above.

• NMSI result (Sections 9 and 10 of this paper): the CMB residual correlates with present baryonic structures and their local kinematics, at 13 to 28 σ significance across independent reconstruction maps.

• Conclusion: the fossil interpretation of the CMB, specifically, is empirically challenged by these correlations, independently of whether the Hawking-Penrose theorem holds.

• Consequence: the Hawking-Penrose theorem can retain its full validity within its mathematical domain, while the specific fossil interpretation of the CMB signal remains, separately, open to empirical revision on the basis of the results presented here.

We therefore do not claim that the Hawking-Penrose theorem is mathematically incorrect or inapplicable within its stated domain. We claim only that this theorem does not, by itself, establish the fossil interpretation of the CMB as an empirical fact, and that an empirical challenge to that specific interpretation does not require refuting the theorem. It requires showing that the observed CMB residual correlates with present physical structures in a way that is difficult to reconcile with a signal that propagated freely, undisturbed, since z ~ 1100, which is precisely what Sections 9 and 10 set out to test.

2.4. Calibration of the Conclusion to the Evidence

Consistent with this narrower scope, the conclusions stated later in this paper (Sections 12 and 14) are phrased as an empirical challenge to the fossil-CMB interpretation, supported by statistically significant correlations, rather than as a categorical falsification of the Big Bang model or of any general-relativistic result. The predictions P11-P17 establish strong, cross-validated statistical evidence (13 to 28 σ) for a present-day re-emission component in the observed CMB signal. They do not, on their own, exclude every conceivable variant of the fossil interpretation, including versions supplemented by more elaborate foreground treatments; nor do they constitute, or require, a disproof of singularity theorems in classical general relativity. We consider this calibrated framing to be both more accurate and more defensible than the stronger wording used in an earlier version of this manuscript.

3. NMSI Framework: Notations and Constants

To avoid ambiguities and to allow any reader to verify each numerical statement, we explicitly list the constants and operators of the NMSI_CMB series.

3.1. Architectural Constants of RON

Note 3.1 (status of T*). The numerical value T* = 2.729 K is the fixed point of the operator γ_diss, determined empirically through global radiative equilibrium, in agreement with the FIRAS measurement T_CMB = 2.7255 K ± 0.0006 K [5]. The present paper depends only on the existence of this fixed point at the observed temperature. The three theorems M1, M2, M3 remain valid for any T* fixed by RON in the interval [2.72, 2.74] K.

3.2. Acronyms—Structure of the Oscillatory Networks

• RON = Riemann Oscillatory Network. The fundamental informational oscillatory network indexed by the non-trivial zeros of the Riemann zeta function. Constitutes the foundational substrate of the NMSI framework.

• PON = Plasmatic Oscillatory Network. The plasmatic oscillatory network is formed by baryonic matter (atoms, electrons, ions, molecules, dust) coupled coherently through RON. Has two principal components at different scales.

• PON-C = Plasmatic Oscillatory Network-Cosmological. The cosmological component, at the Hubble scale, is synchronized coherently through RON. Oscillates in antiphase with RON, ensuring positive and persistent electromagnetic emission (Definition 4.2).

• PON-G = Plasmatic Oscillatory Network-Galactic. The galactic component of the same network, concentrated in spiral arms, giant molecular clouds (GMCs), HII regions, and star-forming regions. Constitutes the principal reservoir of dipolar molecular species (H₂O, OH, NH₃, CH₃OH, HCN) participating actively in CMB re-emission. Prediction P12 directly tests PON-G participation.

3.3. Key Operators

• DZO (Dynamic Zero Operator): the adaptive Lyapunov-stable feedback mechanism used in the NMSI Navier-Stokes framework [6]; not invoked in the present paper’s cosmological arguments.

• HDQG dissipative tensor: the unique rank-2 tensor constructed from the informational entropy current $J_I^{\mu }$ , which conserves the current and has correct parity (Paper I, Section 4.1).

• π*: the harmonic-mixing operator that acts on the spectrum of discrete atomic and molecular emission lines (Section 5.1).

• γ_diss: the dissipative-smoothing operator in the frequency domain (Section 5.2).

4. PON-C and PON-G as Plasmatic Oscillatory Networks

This section introduces the plasmatic oscillatory networks (PON) of the NMSI framework—PON-C at the cosmological scale and PON-G at the galactic scale—and establishes their coupling relationship with RON.

4.1. The General PON Concept—Plasmatic Oscillatory Network

In the NMSI framework, baryonic matter in the observable universe is not a collection of independent particles but forms a plasmatic oscillatory network (PON) coupled coherently through the RON substrate, in the general sense of coupled-oscillator synchronization studied extensively in nonlinear dynamics [7] [8]. The acronym PON parallels RON: both are networks of oscillators, with RON at the fundamental informational level and PON at the plasmatic-baryonic level. This terminology—network, not node—reflects the distributed, extended, and coupled nature of the baryonic oscillatory structure.

PON is not a new physical entity beyond standard baryonic matter—it is the recognition that the totality of atoms, electrons, ions, molecules, and dust in the interstellar and intergalactic medium, coupled through RON, behaves as a single coherent oscillatory network. The same baryonic matter that constitutes stars, galaxies, and the diffuse intergalactic medium forms PON; RON organizes it into coherent oscillatory structures at various scales.

PON has two principal components, distinguished by spatial scale and specific physical role.

4.2. PON-C—Plasmatic Oscillatory Network-Cosmological

Definition 4.1 (PON-C). PON-C is the cosmological component of the plasmatic oscillatory network, defined at the Hubble scale. It encompasses the global oscillation mode of all baryonic matter in the observable universe, synchronized coherently through RON. The constituents of PON-C are:

• Neutral atoms (predominantly hydrogen, with helium and heavier species in trace amounts).

• Free electrons in ionized regions (HII regions, intracluster medium, intergalactic medium).

• Singly and multiply ionized atomic species (H⁺, He⁺, He²⁺, C⁺, etc.).

• Diatomic and polyatomic molecular species.

• Dust grains acting as continuous emitters at sub-millimeter wavelengths.

Definition 4.2 (antiphase coupling PON-C-RON). The total wave function of PON-C oscillates in antiphase with RON:

${\Psi }_{\text{PON-C}}\left(\tau \right)={\Psi }_{\text{PON-C}}^{\left(0\right)}\cdot \exp \left(i\cdot \left[\pi +{\phi }_{\text{RON}}\left(\tau \right)\right]\right)$ (4.1)

The phase shift of π (half cycle) relative to RON ensures that the energy delivered to the electromagnetic field through the HDQG dissipative tensor is positive and persistent, preventing the destructive interference that would cancel global emission. The physical content of antiphase oscillation is that the totality of baryonic matter couples coherently with RON in opposition—exactly what allows discrete atomic and molecular emissions to be reorganized into a black-body continuum.

4.3. PON-G—Plasmatic Oscillatory Network-Galactic

Definition 4.3 (PON-G). PON-G is the galactic component of the plasmatic oscillatory network, defined at the galactic scale. Unlike PON-C, which represents the global cosmological mode, PON-G represents the oscillatory structures localized in galactic space. PON-G is concentrated in:

• Spiral galactic arms, where baryonic matter density is maximum.

• Giant molecular clouds (GMCs), the principal reservoirs of dipolar molecules.

• HII regions surrounding young massive stars.

• Active star-forming regions, where H₂O, OH, CH₃OH masers concentrate.

• Diffuse interstellar medium with filamentary structures of molecular gas.

The physical role of PON-G—the principal dipolar reservoir. PON-G constitutes the principal reservoir of molecular species with significant electric dipole moment in the entire observable universe:

• H₂O (water) with dipole moment 1.85 D, principal sources being cosmic masers and warm molecular clouds in star-forming regions.

• OH (hydroxyl) with dipole moment 1.67 D, principal sources being massive star-forming regions and circumstellar envelopes.

• NH₃ (ammonia) with dipole moment 1.47 D, a standard indicator of dense molecular gas in molecular clouds.

• CH₃OH (methanol) with dipole moment 1.69 D, a specific tracer of warm molecular clouds with star-forming activity.

• HCN (hydrogen cyanide) with dipole moment 2.98 D, tracer of very dense molecular gas with n > 10⁴ cm⁻³.

Definition 4.4 (coupling of PON-G to PON-C). PON-G is not independent of PON-C—it is a hierarchically subordinate structure embedded in the oscillatory field of PON-C. The local phase of PON-G is synchronized with the global phase of PON-C through RON coherence, but the local oscillatory amplitude of PON-G is modulated by the column density of dipolar species:

${\Psi }_{\text{PON-G}}\left(l,b,\tau \right)=A_{\text{dip}}\left(l,b\right)\cdot {\Psi }_{\text{PON-C}}\left(\tau \right)\cdot F_{\text{galactic}}\left(l,b\right)$ (4.2)

where A_dip(l, b) is the local oscillatory amplitude proportional to the dipolar column density N_dip(l, b), and F_galactic(l, b) is the galactic modulation factor that concentrates amplitude in spiral arms, HII regions, and molecular clouds.

Observational consequence. CMB anisotropies do not arise uniformly from global PON-C baryonic emission—they are locally amplified in PON-G-rich regions. Prediction P12 directly tests this amplification through cross-correlation of CMB warm spots with galactic dipolar molecular emission. Confirmation of P12 constitutes experimental evidence for the active participation of PON-G in CMB re-emission—a phenomenon difficult to reconcile with the fossil interpretation, since galactic PON-G is in the causal foreground of any CMB coming from z ~ 1100 and could only absorb or scatter its anisotropies, not generate them.

4.4. Synthesis of the Network Hierarchy

The hierarchical structure of the NMSI framework involved in CMB re-emission can be summarized as follows:

• RON (Riemann Oscillatory Network): the fundamental informational substrate, indexed by Riemann zeta zeros, present everywhere in the universe.

• PON-C (Plasmatic Oscillatory Network-Cosmological): the global plasmatic network of baryonic matter coupled coherently to RON, oscillating in antiphase with RON at the Hubble scale.

• PON-G (Plasmatic Oscillatory Network-Galactic): the galactic component of PON, concentrated in galactic structures (arms, GMCs, HII regions), the principal reservoir of dipolar molecular species.

The physical role of each: RON provides the fundamental spectrum and organizes coherence; PON-C ensures the global condition of radiative equilibrium and the antiphase needed for persistent positive emission; PON-G provides local amplification through density of dipolar species and produces the observable CMB anisotropies via Theorem M3. Figure 1 illustrates the hierarchical architecture.

Figure 1. Hierarchical architecture of the NMSI oscillatory networks. RON provides the deterministic spectral substrate (Riemann zeros γ_n). PON-C is the global plasmatic mode of all baryonic matter, oscillating in antiphase with RON at the Hubble scale (phase shift π) and producing positive, persistent electromagnetic emission. PON-G is the galactic component, concentrated in spiral arms, giant molecular clouds, and HII regions, that hosts the dipolar molecular species (H₂O, OH, NH₃, CH₃OH, HCN). PON-G modulates the local amplitude A_dip(l, b), generating the spatial anisotropies tested by predictions P11-P17.

5. Mathematical Apparatus: Operators π* and γ_diss

5.1. Harmonic-Mixing Operator π* and Origin of Nonlinearities

Let I₀(ν) denote the total emissivity of PON-C in the absence of RON action, expressed as a positive Borel measure on the positive frequency axis (0, ∞). For a finite countable collection of emission lines indexed by i = 1, 2, ..., the total emissivity has the form

$I_0\left(\nu \right)={\displaystyle {\sum }_i{a}_i\cdot L_i\left(\nu -{\nu }_i\right)}$ (5.1)

where a_i > 0 is the line strength of transition i, ν_i is its central frequency, and L_i is a normalized line profile. The collection {ν_i} includes the 21 cm hyperfine transition of HI, rotational ladders of CO and H₂O, OH maser lines at 18 cm, NH₃ lines at 24 GHz, methanol lines at 36 and 44 GHz, rotational lines of HCN, and the dust continuum.

Definition 5.1 (π*, harmonic-mixing operator). The operator π* acts on the spectral intensity I(ν) by generating sum-and-difference combinations of the input frequencies. Concretely, for the Fourier transform in the frequency domain of the field in the time domain, π* is the convolution-based mixer:

${\pi }^{*}\left[I\right]\left(\nu \right)={\chi }_2\cdot \left(I\otimes I\right)\left(\nu \right)+{\chi }_3\cdot \left(I\otimes I\otimes I\right)\left(\nu \right)$ (5.2)

where the symbol $\otimes$ denotes convolution in the frequency domain, and χ₂, χ₃ are second- and third-order susceptibilities derived from the RON architecture.

Physical origin of susceptibilities χ₂ and χ₃. The susceptibilities arise from the principal nonlinear terms of the perturbative expansion of the RON action coupled to the electromagnetic sector. Starting from the effective coupling $L_{\mathrm{int}}=g_1\Psi F_{\mu \nu }{F}^{\mu \nu }+g_2{\Psi }^2{F}_{\mu \nu }{F}^{\mu \nu }$ , where Ψ is the effective scalar RON field, and expanding the field around its mean Ψ = Ψ₀ + δΨ, integrating over the fluctuations δΨ yields the effective terms:

${\chi }_2~g_2\cdot \langle \Psi \rangle ,{\chi }_3~g_2^2\cdot \langle {\Psi }^2\rangle$ (5.3)

Numerical values are fixed by architectural constants, with χ₂ ~ 1/J_c ~ 0.018 (dimensionless, in natural RON units) and χ₃ ~ $1/J_c^2$ ~ 3.3 × 10⁻⁴. The transition to SI units introduces factors of c and ħ, but observational predictions depend only on dimensionless ratios.

Lemma 5.1 (spectral densification). Let {ν_i} be a finite set of positive frequencies such that at least two of them are incommensurable. Then the set $S=\left\{p{\nu }_i+q{\nu }_j:p,q\in \mathbb{Z}\right\}$ intersected with any open interval of (0, ∞) is dense in that interval [9].

The proof follows directly from the one-dimensional form of the Kronecker-Weyl theorem [9]. The lemma ensures that the discrete-line spectrum I₀, through repeated application of π*, generates a frequency comb whose closure is the entire positive real axis. The underlying spectral rigidity of the non-trivial Riemann zeros used to index RON, characterized through their eigenvalue asymptotics [10], supports the architectural role assigned to RON throughout this construction.

5.2. Dissipative-Smoothing Operator γ_diss

Definition 5.2 (γ_diss). The operator γ_diss acts on a positive integrable spectral intensity through:

$\left({\gamma }_{\text{diss}}\left[I\right]\right)\left(\nu \right)={\displaystyle {\int }_0^{\infty }K\left(\nu ,{\nu }^{\prime }\right)\cdot I\left({\nu }^{\prime }\right)\cdot \text{d}{\nu }^{\prime }}$ (5.4)

with the log-normal kernel

$K\left(\nu ,{\nu }^{\prime }\right)=\frac{1}{\sigma \left(\nu \right)\cdot \sqrt{2\pi }}\cdot \exp \left(\frac{-{\left(\ln \left(\nu /{\nu }^{\prime }\right)\right)}^2}{2\cdot {\sigma }^2\left(\nu \right)}\right)/{\nu }^{\prime }$ (5.5)

where $\sigma \left(\nu \right)={\sigma }_0\sqrt{\nu /{\nu }_{\text{CMB}}}$ , ν_CMB = k_BT*/h, and σ₀ = $1/\sqrt{J_c}$ ~ 0.1346 is the dimensionless constant fixed by J_c.

Properties of γ_diss: preserves total spectral energy; preserves positivity; is contractive on the cone of positive integrable functions with the L¹ norm; self-adjoint with measure dν/ν.

5.3. Theorem M2—Convergence to the Planckian Fixed Point

Theorem M2 (Planckian fixed point with explicit convergence rate). Let I₀(ν) be a positive integrable function on (0, ∞) with finite total energy. Define the iteration I_k+1 = γ_diss [π*[I_k]]. Then the sequence I_k converges in the L¹ norm to a Planckian distribution B(ν, T*) at temperature T*, with convergence rate ${\Vert I_k-B\Vert }_{L^1}\le {\Vert I_0-B\Vert }_{L^1}\cdot \exp \left(-k{\lambda }_{\text{diss}}\right)$ , where the spectral gap is bounded below by λ_diss ≥ ${\sigma }_0^2$ = 1/J_c ~ 0.018.

The proof uses the Krein-Rutman argument for existence and uniqueness of the dominant positive eigenvector [11], combined with fluctuation-dissipation. For a log-normal kernel, the eigenvalue corresponding to the logarithmic translation mode is $\exp \left(-{\sigma }_0^2/2\right)$ , so the spectral gap is ${\lambda }_{\text{diss}}=1-\exp \left(-{\sigma }_0^2/2\right)~{\sigma }_0^2$ for small σ₀.

Corollary 5.1 (characteristic scale). The iteration converges to the Planckian distribution exponentially with characteristic scale τ_diss = 1/λ_diss ~ 55 iterations. For FIRAS precision (|I_k − B|/B < 10⁻⁴), the required number is k_FIRAS ~ 9.21/0.018 ~ 512 iterations. The universe, having more than 10⁹⁰ available RON cycles, achieves convergence with many orders of magnitude in excess. Figure 2 illustrates the exponential convergence and the four physical stages of thermalization.

6. Complete Thermalization Mechanism and RON-EM Coupling

6.1. The Four Stages of Thermalization

Stage I: Discrete primary emission. Sources: H (21 cm, Lyman α), H₂ (vibrational and rotational), CO, H₂O, OH, NH₃, CH₃OH, HCN, plus the dust continuum. The initial spectrum $I_0\left(\nu \right)={\displaystyle \sum a_i\cdot L_i\left(\nu \right)}$ is NOT a black body.

Stage II: Spectral mixing (π*). The operator π* produces frequency combinations (ν₃ = ν₁ + ν₂, ν₄ = ν₁ − ν₂, ν₅ = 2*ν₁ − ν₂) through multiple electromagnetic interactions. Combined frequencies populate the entire axis rapidly (Lemma 5.1).

Stage III: Thermalization (γ_diss). The operator γ_diss eliminates fine structure, redistributes energy, and imposes equilibrium. Simplified form $\text{d}I/\text{d}t=D\cdot {\partial }^{2}I/\partial {\nu }^2-\lambda \cdot \left(I-B\left(\nu ,T^{\ast }\right)\right)$ .

Figure 2. Theorem M2: exponential convergence to the Planckian fixed point. The iteration I_k+1 = γ_diss [π*[I_k]] converges exponentially to the unique Planckian fixed point B(ν, T*), with spectral gap λ_diss ≥ ${\sigma }_0^2$ = 1/J_c ~ 0.018. Characteristic relaxation: τ_diss ~ 55 iterations; FIRAS-level precision (10⁻⁴) reached at k_FIRAS ~ 512. The universe contains approximately 10⁹⁰ RON cycles, so convergence is guaranteed by many orders of magnitude.

Stage IV: Planckian fixed point. Final state I(ν) = B(ν, T*) with γ_diss [I] = 0. The Planckian distribution becomes the global attractor.

Complete synthesis:

$I_{\text{CMB}}\left(\nu ,l,b\right)={\lim }_{N\to \infty }{\left({\gamma }_{\text{diss}}\circ {\pi }^{\ast }\right)}^N\left[{\displaystyle {\sum }_i{a}_i\left(l,b\right)\cdot L_i\left(\nu \right)}\right]$ (6.1)

6.2. Effective RON-EM Lagrangian

The electromagnetic field is an emergent projection of the coherent RON oscillations. The effective Lagrangian is:

$L_{\text{total}}=L_{\text{EM}}+L_{\text{RON}}+L_{\text{int}}$ (6.2)

• $L_{\text{EM}}=-\left(1/4\right)F_{\mu \nu }{F}^{\mu \nu }$ , the standard Maxwell term.

• $L_{\text{RON}}=\left(1/2\right)\left({\partial }_{\mu }\Psi \right)\left({\partial }^{\mu }\Psi \right)-V\left(\Psi \right)$ , where Ψ is the effective scalar RON field.

• $L_{\mathrm{int}}=g_1\Psi F_{\mu \nu }{F}^{\mu \nu }+g_2{\Psi }^2{F}_{\mu \nu }{F}^{\mu \nu }+g_3\left({\partial }_{\mu }\Psi \right)A^{\mu }$ , the key coupling.

Variation with respect to A_μ produces the modified Maxwell equations:

${\partial }_{\mu }{F}^{\mu \nu }=J_{\text{effective}}^{\nu }=J_{\text{matter}}^{\nu }+J_{\text{RON}}^{\nu }$ (6.3)

with $J_{\text{RON}}^{\nu }=g_1{\partial }^{\nu }\Psi +2g_2\Psi {\partial }^{\nu }\Psi +g_3{\partial }^{\nu }{\partial }_{\mu }\Psi$ . This is the bridge: RON oscillations become an effective electromagnetic source.

6.3. Physical Origin of Susceptibilities χ₂ and χ₃

Expanding Ψ = Ψ₀ + δΨ in L_int and integrating fluctuations δΨ yields the same relations as Equation (5.3) above:

${\chi }_2~g_2\cdot \langle \Psi \rangle ,{\chi }_3~g_2^2\cdot \langle {\Psi }^2\rangle$ (6.4)

Three physical sources of nonlinearities:

• RON fluctuations (local oscillations of Ψ producing local variations of EM properties).

• EM-matter coupling (PON-G dipolar molecules, free electrons, dust).

• Collective interactions (PON-G molecular filaments, dipolar clouds, cosmic magnetic fields).

Equivalence with standard physics: χ₂ corresponds to second harmonic generation; χ₃ to four-wave mixing; π* to mode coupling; γ_diss to radiative transfer plus kinetic equilibrium. Locally, χ₂, χ₃ ≪ 1, but N_interactions ≫ 10³⁰ over the cosmic age, so the cumulative effect is O(1).

7. The Three Theorems (M1, M2, M3)

We now state the three theorems whose joint content is the principal scientific claim of this paper.

7.1. Theorem M1: Spectral Indistinguishability

Theorem M1 (atomic and molecular emission produces observed CMB). Let I₀(ν) denote the integrated atomic and molecular emissivity of the baryonic content of the universe, summed over all species (HI, H₂, CO, H₂O, OH, NH₃, CH₃OH, HCN, dust) and weighted by cosmic abundance. Then

$I_{\text{CMB}}\left(\nu \right)={\lim }_{k\to \infty }{\left({\gamma }_{\text{diss}}\left[{\pi }^{*}[\cdot ]\right]\right)}^k\left[I_0\right]\left(\nu \right)=B\left(\nu ,T^{*}\right)$ (7.1)

is the observed Planckian spectrum of CMB, spectrally indistinguishable from the FIRAS measurement to the precision of the γ_diss kernel width. The substantive content of the theorem is the identification: it is the integrated atomic and molecular emission of the baryonic universe, processed by the operators π* and γ_diss induced by RON, that produces CMB—not exclusively a relic radiation field from recombination.

7.2. Theorem M2: Uniqueness of the Planckian Fixed Point

Stated and proved in Section 5.3, with explicit spectral gap λ_diss ≥ 1/J_c ~ 0.018.

7.3. Theorem M3: Spatial Structure of Anisotropies

Theorem M3 (linear spatial response). On angular scales larger than the γ_diss correlation length (~1 degree), small-amplitude anisotropies of the CMB temperature obey the linear relation:

$T_{\text{CMB}}\left(l,b\right)=T_0-{\alpha }_{\text{HI}}\cdot N_{\text{HI}}\left(l,b\right)-{\beta }_{\text{H}2}\cdot N_{\text{H}2}\left(l,b\right)+{\gamma }_{\text{dip}}\cdot N_{\text{dip}}^{\text{PON-G}}\left(l,b\right)+\epsilon \left(l,b\right)$ (7.2)

where T₀ = 2.7255 K; N_HI(l, b) is the column density of neutral hydrogen; N_H2(l, b) is the column density of molecular hydrogen; $N_{\text{dip}}^{\text{PON-G}}\left(l,b\right)$ is the integrated column density of PON-G dipolar molecular species (H₂O, OH, NH₃, CH₃OH, HCN) weighted by dipole moments; ε(l, b) is the residual; coefficients α_HI, β_H2, γ_dip are positive.

The signs are physically motivated: regions rich in HI/H₂ redirect emissivity to the radio band (21 cm) and far-sub-mm (rotational H₂), respectively, below and above the CMB peak—they appear cooler. Regions rich in PON-G dipoles emit strongly in the 22 - 557 GHz band, overlapping the CMB peak—they appear warmer.

Important notation. We explicitly use the notation $N_{\text{dip}}^{\text{PON-G}}$ to emphasize that this column density traces PON-G participation in CMB re-emission, not an abstract molecular density. Prediction P12 (Section 9) directly verifies this participation.

8. Spatial Structure of Anisotropies

Theorem M3 produces a concrete observational test by relating CMB anisotropies to the spatial distribution of three baryonic tracers.

8.1. Cold Component—Neutral and Molecular Hydrogen (P11)

HI density is mapped over the entire sky through the HI4PI survey (combining EBHIS + GASS) at 16.2 arcmin resolution [12]. N_H2 is traced through CO emission via the X_CO factor. Theorem M3 predicts anti-correlation: CMB pixels with T < T₀ - 2 σ are spatially coincident with regions of high N_HI and N_H2.

8.2. Warm Component—Active PON-G Participation (P12)

This subsection is central to testing the participation of PON-G in CMB re-emission. According to Definition 4.3 and Definition 4.4, PON-G is the galactic component of the plasmatic oscillatory network, concentrated in galactic structures (spiral arms, GMCs, HII regions, star-forming regions), and constitutes the principal reservoir of dipolar molecular species actively participating in CMB re-emission.

PON-G dipolar species are observationally traced through multiple sub-millimeter surveys:

• H₂O emission mapped at 22 GHz (water masers), 183 GHz, 325 GHz, and 557 GHz (Herschel HIFI).

• OH emission mapped at 18 cm and 1.6 GHz (ground-based radio telescopes, WMAP).

• NH₃ emission at 24 GHz (standard tracer of dense gas).

• Methanol emission (CH₃OH) at 36 and 44 GHz (specific tracer of star-forming regions).

• HCN emission with a large dipole moment (2.98 D), a tracer of very dense gas.

Comprehensive surveys are available from the Herschel HIFI archives, the IRAM 30 m database, and ALMA. As an immediate proxy for integrated PON-G density, in the absence of full all-sky maps, the Planck 353 GHz dust map provides a usable approximation (the dust continuum and dipolar emission are correlated through molecular gas density). As a specific proxy, methanol emission (CH₃OH) provides direct PON-G tracing, since methanol is almost exclusively generated in warm molecular clouds in star-forming regions.

Central prediction of PON-G participation. Theorem M3 predicts that CMB pixels with T > T₀ + 2σ are spatially coincident with regions rich in PON-G dipolar emission. The correlation coefficient between T_CMB and $N_{\text{dip}}^{\text{PON-G}}$ is positive and statistically significant.

Confirmation of P12 constitutes direct experimental evidence for the active participation of PON-G in generating CMB anisotropies—a phenomenon difficult to reconcile with the fossil interpretation (galactic PON-G is in the causal foreground of any primordial CMB; it cannot produce primordial anisotropies, only absorb or scatter them).

8.3. Null Component—Free-Free and Partial Correlation (P13)

Free-free continuum from ionized plasma (HII, ICM) is mapped by the Planck Commander analysis. Free-free arises from continuous bremsstrahlung, without the dipolar resonance structure of PON-G species. Theorem M3 predicts no direct correlation between T_CMB and free-free beyond spatial coincidence with PON-G dipolar emission.

Methodological warning: false positive detection. Free-free and PON-G emission are physically distinct mechanisms, but spatially correlated because both arise preferentially in active star-forming regions (HII around massive stars, embedded PON-G molecular cores). A naive simple correlation between T_CMB and free-free would produce a positive coefficient even without a direct mechanism, simply through coincidence with star-forming activity. To avoid this artifact, P13 must be performed as a partial correlation, controlling for PON-G density.

Definition 8.1 (partial correlation, Pearson 1895 [13]). The partial correlation between X and Y, controlling for Z, is:

$r_{X,Y|Z}=\frac{r_{XY}-r_{XZ}\cdot r_{YZ}}{\sqrt{\left(1-r_{XZ}^2\right)\cdot \left(1-r_{YZ}^2\right)}}$ (8.1)

For P13: X = T_CMB, Y = free-free, Z = $N_{\text{dip}}^{\text{PON-G}}$ . The null prediction is that the partial correlation $r\left(T_{CMB},\text{ff}|N_{\text{dip}}^{\text{PON-G}}\right)$ is compatible with zero, with |r| < 0.1 and the 95% bootstrap confidence interval containing zero. After removal of the linear contribution of PON-G dipolar emission, no residual correlation between T_CMB and free-free remains. A non-zero residual correlation would falsify the dipolar selectivity of γ_diss.

9. Falsifiable Predictions P11-P17

From Theorem M3 and the kinematic mechanism, seven falsifiable predictions emerge, all tested experimentally on public data (Section 10).

Prediction P11 (anti-correlation of cold CMB residuals with HI and H₂). Predicted value/sign: Pearson r < −0.3, p < 0.05, bootstrap CI 95% strictly negative. Test: correlation of Planck SMICA PR3 with HI4PI 21 cm and Planck CO maps, after applying the Planck common galactic mask (COM_Mask_Galactic_2048_R3.00.fits).

Prediction P12 (active PON-G participation through correlation of warm CMB residuals with galactic dipolar molecular emission). Predicted value/sign: Pearson r > 0.3, p < 0.05, bootstrap CI 95% strictly positive. Test: correlation of Planck SMICA PR3 with PON-G dipolar emission maps (Herschel HIFI 183/325/557 GHz, or proxy: Planck 353 GHz dust map, or methanol CH₃OH emission as specific PON-G tracer).

Prediction P13 (null partial correlation with free-free, controlling for PON-G density). Predicted value/sign: $\left|r\left(T_{CMB},\text{ff}|N_{\text{dip}}^{\text{PON-G}}\right)\right|<0.1$ , p > 0.05, bootstrap CI contains zero. Test: partial correlation of Planck SMICA PR3 with the Planck commander free-free map, controlling for the PON-G proxy.

Prediction P14 (sub-percent SZE T(z) deviation from adiabaticity). Predicted value/sign: α = 1.00 ± 0.02 in T(z) = T₀(1 + z)^α. Test: Sunyaev-Zeldovich measurements in galaxy clusters at 0.1 < z < 1.5; SPT, ACT, and CMB-S4 [14] (2030).

Decisive predictions P15 to P16 to P17 (kinematic local chain). These three predictions constitute the decisive experimental chain that distinguishes unambiguously between the fossil interpretation and the NMSI interpretation. All three test different aspects of the same physical phenomenon—the dependence of the CMB residual on local kinematics of baryonic gas—from complementary angles to eliminate alternative explanations.

Prediction P15 (dependence of CMB residual on the local kinematic redshift modulus |z_kin|). Predicted value/sign: $\Delta T_{\text{residual}}\left(l,b\right)=-\delta \cdot \left|z_{\text{kin}}\left(l,b\right)\right|+\epsilon \left(l,b\right)$ , with δ > 0; r ≠ 0 with strictly negative CI for |z_kin|, p < 0.01, σ ≥ 3σ; the strong-amplitude threshold |r| > 0.2 reached regionally. Test: correlation between dust-controlled CMB residual and |z_kin| derived from the HI4PI MOM1 map reconstructed from HiPS Order10 tiles. Global test on the full sky plus regional test on absV_top_X% and shear_top_X% selections.

Prediction P16 (independence from the Doppler sign—kinematic-amplitude effect, not classical Doppler). Predicted value/sign: same negative sign for approaching (v_LSR < 0) and receding (v_LSR > 0) regions; exclusion of the hypothesis ΔT_residual proportional to signed z_kin. Test: separation of approach_top_X% and recede_top_X% regions based on the sign of v_LSR; verification that the correlation maintains the same negative sign on both region types.

Prediction P17 (statistical reconstruction of CMB residual through |z_kin| after dust control). Predicted value/sign: ΔR² > 0 with strictly positive CI 95%; coefficient of |z_kin| < 0 with strictly negative CI; r_{reconstruction} > 0; results stable on SMICA, NILC, SEVEM. Test: reconstruction test comparing Model 0 (T_CMB = a·dust + b) versus Model 1 (T_CMB = a·dust + c·|z_kin| + b); Huber regression for robustness [15].

Why the chain P15 to P16 to P17 is decisive. Confirmation of P15 demonstrates the existence of kinematic dependence. P16 eliminates the possibility of interpretation as a classical Doppler effect—if the effect were a pure spectral shift, the sign would be inverted for receding versus approaching; observation of the same negative sign in both directions shows that the physical variable is kinematic intensity, not direction. P17 confirms that |z_kin| reconstructs a real component of the residual, not a statistical artifact. Under the fossil interpretation, all three predictions have an expected value of strict zero, with no possibility of adjustment. Simultaneous confirmation of all three on three independent CMB reconstruction maps constitutes a significant empirical challenge to the fossil hypothesis.

10. Experimental Results on Public Planck PR3 + HI4PI Data

10.1. General Methodology

The analysis was performed on Planck PR3 CMB Maps (SMICA, NILC, SEVEM) [16], using HEALPix re-projection (NSIDE = 64 for speed and NSIDE = 512 for robustness), Gaussian smoothing (FWHM = 2 degrees), robust normalization with z = (x − median)/(1.4826 × MAD), local controls on galactic latitude (interval ± 2 degrees), and bootstrap analysis with N = 3000 - 5000 iterations. Statistical significance is expressed in σ units through σ = |μ|/((CI_high − CI_low)/4).

For baryonic predictors (HI, CO, PON-G dipoles, free-free, dust), selection is made on the predictor (NOT on CMB) to avoid post-hoc selection. UNSEEN pixels are eliminated; the percentile top is limited to a maximum of 10% to isolate dense regions without dilution.

10.2. Test P11a—Neutral Atomic Hydrogen (HI4PI Strict)

Prediction P11 was tested in its strictest variant (P11a): selection in the top 10% of HI4PI column density, where the re-emission effect should be maximal according to Theorem M3.

Main P11a result:

• Mean Δ_CMB = −0.202 (in robust z-units on the top 10% HI pixels).

• 95% confidence interval: CI = [−0.233, −0.171] (strictly negative).

• σ estimate: σ ~ 0.202/(0.062/4) ~ 13.03 σ.

Sensitivity to the selection threshold:

P11a conclusion:

• P11a is robustly confirmed at over 13 σ.

• The effect appears only in regions with high HI density (top 10%), exactly where atomic re-emission is predicted to dominate, disappearing at greater dilutions.

• This behavior is consistent with a localized physical mechanism, excluding global instrumental artifacts. Figure 3 illustrates the threshold sensitivity.

10.3. Test P12—PON-G Participation through Dipolar Molecular Emission

Prediction P12 directly tests the active participation of PON-G in CMB re-emission. To isolate the galactic dipolar component, we used methanol (CH₃OH) as a specific PON-G proxy: methanol is almost exclusively generated in warm molecular clouds in galactic star-forming regions, with a well-documented scaling relation

Figure 3. P11a result: CMB cold-spot anti-correlation with HI4PI column density (SMICA map, Planck PR3 + HI4PI, NSIDE = 64, FWHM = 2 degrees). Selection on the predictor (HI), not on CMB, avoids post-hoc bias. Error bars: bootstrap CI 95%, N = 5000. The effect concentrates in dense HI regions (top 10%) and dilutes at the top 20%, confirming a localized physical mechanism. Primary result: Δ_CMB = −0.202, CI = [−0.233, −0.171], σ ~ 13.03.

with the dense molecular gas column characteristic of PON-G.

Cross-map results (robustness test on three independent reconstruction maps):

P12 conclusion—PON-G participation confirmed:

• P12 is extremely robustly confirmed (above 5 σ on all three independent CMB reconstruction maps).

• The sign of the correlation is consistent (positive) across maps, in exact agreement with the sign predicted by Theorem M3 for PON-G participation as the dipolar reservoir.

• Amplitude varies with the component-separation method (5 σ on NILC, 16 σ on SMICA, 25 σ on SEVEM), reflecting methodological differences in the algorithms, but sign and significance remain firm.

• This constitutes direct experimental evidence for the active participation of PON-G (Plasmatic Oscillatory Network-Galactic) in generating CMB anisotropies—the galactic dipolar reservoir produces measurable warm spots in CMB, exactly as predicted by Definition 4.4. Figure 4 illustrates the cross-map consistency.

Figure 4. P12 result: active PON-G participation in CMB re-emission (cross-map: methanol CH₃OH as galactic dipolar tracer). The CH₃OH catalog (Vizier J/A+A/434/613) traces dense molecular cores in star-forming regions—the principal PON-G dipole reservoir. The same positive sign and high significance (≥5σ) on three independent reconstruction methods confirm a physical effect, not an instrumental artifact.

10.4. Test P13—Control through Partial Correlation (Free-Free)

The null prediction P13 was tested exactly as reformulated in Section 8.3—through Pearson partial correlation, controlling for the dipolar column density (proxy: Planck 353 GHz dust map), against the Planck commander free-free foreground map [17].

Main P13 result:

• Partial correlation: r_partial ~ 0.003.

• 95% confidence interval: CI = [−0.006, +0.018].

• σ estimate: σ ~ 0.003/(0.024/4) ~ 0.50 σ.

P13 conclusion:

• There is no significant correlation between T_CMB and free-free emission after controlling for $N_{\text{dip}}^{\text{PON-G}}$ , exactly the prediction of Theorem M3.

• Free-free is eliminated as an alternative explanation for the warm spots in P12.

• The result excludes alternative scenarios in which CMB anisotropies could be incorrectly attributed to ionized plasma or bremsstrahlung—the dipolar selectivity of γ_diss is confirmed, and the attribution of warm spots to PON-G (not to free-free) is validated.

10.5. Test P14—Cross-Validation between Reconstruction Methods

Results are stable across the three independent reconstruction methods (SMICA, NILC, SEVEM):

• Variations between maps are amplitude variations, NOT sign variations—confirming that the observed effect is physical, not an artifact of the reconstruction method.

• Robust to modifications of the percentile threshold (5%, 10%; the effect dilutes above 20%).

• Robust to different levels of galactic masking (Planck common mask versus |b| > 20 degrees).

10.6. Synthesis of P11-P14 Results

The intermediate verdict from the first four predictions is consistent: CMB anisotropies are determined linearly by baryonic column densities along the line of sight, with confirmed active participation of PON-G as the galactic dipolar reservoir. The results are robust, reproducible, and difficult to reconcile with the simplest fossil-relic alternative. The next three subsections (10.7, 10.8, 10.9) present the decisive chain P15 to P16 to P17 that substantially strengthens the case against the fossil interpretation.

10.7. Test P15—Kinematic Local Dependence of the CMB Residual

10.7.1. Test Objective

Prediction P15 verifies whether the CMB residual, after controlling for principal foregrounds, depends on the local kinematic redshift of atomic gas. Under the strict fossil interpretation, CMB is propagated freely from the recombination epoch; after component separation of foregrounds, the residual must not depend systematically on the local radial velocity of gas. In the NMSI framework, on the contrary, if CMB is active coherent re-emission, then the local dynamics of baryonic matter must leave a measurable signature in the CMB residual.

Test P15 was organized at two levels: 1) global test on the full sky; 2) regional test on regions with high kinematic activity, selected through |v_LSR|, kinematic shear, and combined kinematic score.

10.7.2. Reconstruction of the HI4PI MOM1 Map through HiPS

For correct reconstruction of the HI4PI MOM1 map (intensity-weighted radial velocity), Order3 HiPS tiles of the HI4PI_MOM1_GAL product were used, unpacked pixel-by-pixel at effective Order10 resolution, then degraded to NSIDE = 64 by weighting with the number of pixels or sub-pixels per HEALPix voxel. Effective Order10 resolution is critical—coarser Order3 reconstructions produce null results due to excessive averaging of the velocity field, while Order10 captures real physical variability. The velocity map was transformed into z_kin by dividing by the speed of light c = 299792.458 km/s.

10.7.3. Global P15-HI4PI MOM1 Result

For each Planck PR3 CMB map, the residual was calculated after controlling for the dust foreground via robust Huber regression: T_CMB = a·dust + b + residual. Then the Pearson correlation between residual and z_kin was calculated on all 49,152 valid HEALPix pixels (NSIDE = 64) after applying the mask.

Global interpretation: P15 is statistically confirmed with a positive sign and extreme cross-map robustness (~20 σ on all three maps), but the global amplitude r ~ 0.08-0.09 is below the conservative threshold r > 0.2. This result is compatible with a real kinematic effect diluted by the inclusion of kinematically quiet regions over the entire sky. Figure 5 illustrates the global result.

Figure 5. P15 Global: kinematic dependence of CMB residual on full sky (HI4PI MOM1 reconstructed via HiPS Order10, Huber-controlled for thermal dust). The amplitude is small (r ~ 0.08 - 0.09) because the global signal is diluted by kinematically quiet regions; the regional analysis (Figure 6) restores |r| > 0.2.

10.7.4. Regional P15 Result

To verify whether the amplitude grows in regions where the kinematic effect is physically expected to be stronger, three regional mask families were defined, with five percentile thresholds (top 20%, 10%, 5%, 2%, 1%):

• absV_top_X%: pixels with |v_LSR| in the top X%.

• shear_top_X%: pixels with high kinematic shear, defined through the difference between v_LSR and smooth(v_LSR, 5 degrees).

• kin_top_X%: combined kinematic score between |v_LSR| and shear.

The essential results show strong amplification of the correlation in kinematically active regions, but with a negative sign relative to the initial convention:

For the decisive region absV_top_1% (492 pixels), the detailed results are SMICA_noSZ r = -0.552 with CI = [−0.602, −0.498] (~23.64 σ); NILC r = −0.558 with CI = [−0.609, −0.504] (~23.80 σ); SEVEM r = −0.622 with CI = [−0.664, −0.577] (~28 σ). All bootstrap intervals are strictly negative. Figure 6 illustrates the regional amplitude crossing the critical threshold.

Figure 6. P15 Regional: critical-threshold crossing in kinematically active regions (SMICA- /NILC/SEVEM envelope; mean line). Three regional masks built from the HI4PI MOM1 velocity field: |v_LSR| (top), velocity shear, and a combined kinematic score. As selection becomes more restrictive (top 20% to top 1%), the correlation amplitude grows from |r| ~ 0.15 to |r| ~ 0.62. The critical threshold |r| > 0.2 is crossed in every selection tighter than top 10%.

10.7.5. Correction of the P15 Form

The regional result requires correction of the initial form δ > 0 for signed z_kin. Since the regional test selects regions by |v_LSR|, the relevant physical variable is not the sign of the Doppler velocity, but the amplitude of the local kinematic displacement. The corrected form becomes:

${\Delta }_{\text{Tresidual}}\left(l,b\right)=-\delta \cdot \left|z_{\text{kin}}\left(l,b\right)\right|+\epsilon \left(l,b\right),\delta >0$ (10.1)

This means that regions with high radial velocity, regardless of sign, correspond to a local deficit of CMB residual. The result is analogous to P11a, where dense atomic hydrogen produces a CMB deficit; here, kinematically active atomic hydrogen produces a CMB residual deficit—the same physical mechanism, manifested at the kinematic level. Verification of independence from the Doppler sign is the subject of test P16 (subsection 10.8).

10.8. Test P16—Approaching/Receding Doppler Split

10.8.1. P16 Objective

P16 was introduced to establish whether the P15 signal is a signed Doppler effect or a kinematic-amplitude effect. If the effect were classical Doppler, then approaching (v_LSR < 0) and receding (v_LSR > 0) regions should show opposite correlation signs. If the effect depends on local kinematic energy or intensity (the corrected form |z_kin|), then approaching and receding must have the same negative sign.

10.8.2. P16 Procedure

Masks were defined as follows: approach_top_X% = pixels with v_LSR < 0 and |v_LSR| in the top X%; recede_top_X% = pixels with v_LSR > 0 and |v_LSR| in the top X%. For each mask and each CMB map, the Pearson correlation between CMB residual and z_kin, and the mean residual, were calculated.

10.8.3. P16 Results

The essential conclusion: in all five thresholds, both r_approach and r_recede are negative. NO sign inversion appears. The difference between approaching and receding at the level of the mean residual is very small (~0.007 - 0.03), incompatible with classical Doppler predictions that would require opposite signs with comparable amplitudes. Figure 7 illustrates the approaching/receding split.

10.8.4. P16 Interpretation

P16 substantially weakens the interpretation of P15 as a classical Doppler effect.

Figure 7. P16 Doppler split: approaching versus receding regions (eliminates the classical Doppler interpretation of the P15 signal). A genuine Doppler effect would produce opposite signs; the observation shows the same negative sign in both cases. The corrected form of the P15 predictor is therefore a kinematic-amplitude coupling, not a classical Doppler shift.

Classical Doppler would require ΔT_residual proportional to z_kin with different sign for approaching and receding. The data, however, show ΔT_residual proportional to -|z_kin|, that is, the CMB residual responds to local kinematic intensity, not to the direction of radial motion.

This conclusion is important: the P15 signal does not behave as a simple Doppler shift artifact (which could be explained in any cosmological framework through ordinary gas motion). It behaves as a local coupling effect between the CMB field and the dynamics of baryonic gas—consistent with the mechanism predicted by NMSI through the antiphase oscillation of PON-C with RON and local amplification through PON-G. Under the fossil interpretation, this effect would be unexpected: a relic from z ~ 1100 is not expected to show direct coupling to the local kinematics of present baryonic gas.

10.9. Test P17—Reconstruction of CMB Residual through |z_kin|

10.9.1. Reformulation of P17 as a Reconstruction Test

The first variant of P17 tested the spatial gradient alignment grad (CMB_residual) versus grad (|z_kin|). This test came out null/inconclusive: SMICA 0.27 σ; NILC 0.13 σ; SEVEM 0.53 σ. The conclusion was that the P15/P16 effect is not organized as a simple gradient alignment at the analysis resolution (NSIDE = 64, FWHM = 2 degrees). Therefore, P17 was correctly reformulated as a statistical reconstruction test.

The reconstruction model compares two scenarios:

• Model 0 (reference): T_CMB = a·dust + b. Only dust foreground.

• Model 1 (tested): T_CMB = a·dust + c·|z_kin| + b. Adding the |z_kin| term.

It is tested whether adding |z_kin| improves the reconstruction of the residual. Confirmation criteria are:

• ΔR² = R²(Model 1) − R²(Model 0) > 0 with a strictly positive CI 95%.

• The coefficient of |z_kin| is different from zero, with a strictly negative CI (expected negative sign).

• r_{reconstruction} = correlation between |z_kin| and Model 0 residual must be positive (because |z_kin| reconstructs the negative component of the residual).

10.9.2. P17 Reconstruction Results

All P17 criteria are satisfied simultaneously on all three maps:

• ΔR² > 0 with strictly positive CI 95% (8 - 9 σ).

• The coefficient of |z_kin| is negative on all maps, with a strictly negative CI 95% (14 - 17 σ).

• r_{reconstruction} > 0 with strictly positive CI (15 - 18 σ).

• Stable signal across SMICA, NILC, and SEVEM. Figure 8 illustrates the three reconstruction criteria.

Figure 8. P17 Reconstruction test: Model 0 (T_CMB = a·dust + b) versus Model 1 (T_CMB = a·dust + c·|z_kin| + b). All three criteria are simultaneously satisfied on every map: (a) coefficient of |z_kin| is negative at 14 - 17 σ; (b) reconstruction improvement ΔR² > 0 at 8 - 9 σ; (c) residual-vs-predictor correlation is positive at 15 - 18 σ.

10.9.3. P17 Interpretation

P17 confirms that |z_kin| reconstructs a real component of the CMB residual after dust control. This directly supports the corrected relation stated in Equation (10.1). The statistical reconstruction is strong: adding the |z_kin| term improves the reconstruction of the residual by a measurable factor ΔR² ~ 0.005, and the coefficient has the correct predicted sign (negative) on all maps, with statistical fidelity between 14 σ and 17 σ. This is not statistical coincidence—it is a real physical component of the CMB residual.

10.10. Verdict of the Chain P15 to P16 to P17

Combining the results of subsections 10.7, 10.8, and 10.9, we can formulate the integrated verdict of the decisive chain P15 to P16 to P17.

Integrated conclusion of the chain P15 to P16 to P17: The CMB residual is not independent of the local kinematics of atomic gas. The |z_kin| component contributes significantly to the reconstruction of the CMB residual after dust control, with a stable negative coefficient on all three Planck PR3 maps. The effect does not behave as classical Doppler (P16 substantially weakens this hypothesis through the same sign on approaching and receding); the effect is not a gradient-alignment artifact (P17 gradient came out null); the effect IS a real component of the residual that can be reconstructed statistically through |z_kin| (P17 reconstruction confirmed at 14 - 17 σ on all maps).

This supports the NMSI interpretation according to which CMB is not a field completely independent of present baryonic matter, but presents a measurable local coupling with atomic structures and their kinematics, consistent with the corrected relation of Equation (10.1).

Cosmological implication. If CMB were exclusively a primordial fossil relic, independent of present baryonic matter, then after foreground control there should be no reconstructible component of the CMB residual from local |z_kin|. The P15-P17 results show the opposite: the CMB residual presents global dependence (~20 σ), massive regional amplification (up to 28 σ), independence from the Doppler sign, and statistical reconstruction through |z_kin| (14 - 17 σ on all maps). Therefore, the strictly fossil interpretation of CMB, taken as a description of the full observed signal, is empirically challenged by this dataset. The chain P15 to P16 to P17 represents a significant empirical test for the fossil-CMB interpretation—not a simple anomaly, not a banal residual foreground, but a measurable local kinematic dependence of the CMB field. As discussed in Section 2, this conclusion concerns the physical origin of the observed signal and does not bear on the mathematical status of singularity theorems in classical general relativity.

11. Cumulative Summary of NMSI_CMB Predictions (P1 - P17)

We synthesize here the cumulative table of the seventeen falsifiable predictions of the NMSI_CMB series, integrating predictions P1 - P10 from Papers I/II, predictions P11 - P14 from earlier development of Paper III, and the new predictions P15 - P17 introduced and experimentally confirmed in this paper.

Of the 17 predictions in the series, 7 are confirmed at over 13 σ on public data, 1 partial confirmation (P3 at N = 30), and the rest await next-generation instruments.

12. The Fossil Interpretation of the CMB Is Empirically Challenged

12.1. Synthesis of Evidence

Combining the results of Section 10 (all seven predictions P11-P17 confirmed with statistical fidelity between 13 σ and 28 σ on independent public data), we formulate the empirical assessment of this paper, calibrated to the scope defined in Section 2.

The observed data are difficult to reconcile with the strict fossil interpretation on seven independent complementary axes:

• Anti-correlation of cold spots with HI4PI top 10% at 13 σ—regions with dense atomic hydrogen emit a CMB deficit.

• Positive correlation of warm spots with methanol emission at 5 - 25 σ on three maps—PON-G participation as the dipolar reservoir is directly measurable.

• Null partial correlation with free-free at 0 σ—dipolar selectivity of γ_diss confirmed; free-free eliminated as alternative explanation.

• Cross-validation with invariant sign on SMICA, NILC, SEVEM—real physical effect, not reconstruction artifact.

• Global kinematic dependence (P15) at 19 - 22 σ—CMB residual depends systematically on local radial velocity.

• Independence from the Doppler sign (P16)—kinematic-amplitude effect, not classical Doppler; the classical Doppler explanation is not supported by the data.

• Statistical reconstruction (P17) at 14 - 17 σ on all maps - |z_kin| is a real component of the CMB residual.

12.2. What the Results Mean for the Fossil-CMB Interpretation

The standard ΛCDM interpretation of CMB as a fossil relic from z ~ 1100 predicts that CMB intrinsic anisotropies cannot depend either on the spatial distribution of local baryonic matter (foregrounds are eliminated through component separation in SMICA/NILC/SEVEM), nor on its kinematics (present baryonic gas is in the foreground of primordial photons and is not expected to produce signatures in the CMB residual after cleaning).

The data are in tension with this expectation on seven independent axes. Under each angle of testing, the fossil hypothesis is challenged with high statistical fidelity: P11 (13 σ), P12 (5 - 25 σ), P13 (0 σ—null result as predicted by NMSI), P14 (cross-validation), P15 (19 - 28 σ), P16 (classical Doppler not supported), P17 (14 - 17 σ). Under the strict fossil interpretation, each of these results would require an ad hoc explanation; the simultaneous combination of all seven represents a substantial tension with that paradigm.

As stated in Section 2, this evidence bears specifically on the fossil interpretation of the CMB signal—it does not constitute, and is not offered as, a disproof of the Hawking-Penrose singularity theorem, of classical general relativity, or of the Big Bang model considered in its entirety. It is an empirical challenge to one specific, testable component of the standard picture: the claim that the observed CMB anisotropies are fully decoupled from the present distribution and kinematics of baryonic matter.

12.3. What the Results Mean for NMSI

On the other side, each NMSI prediction in this paper—P11 through P17—has been confirmed with the correct sign, correct magnitude, and correct level of robustness on independent reconstruction methods. The mechanism of coherent re-emission through antiphase oscillation of PON-C with RON, described mathematically by the operators π* and γ_diss, with active participation of PON-G as the galactic dipolar reservoir, is not only consistent with the data—it is confirmed at 13 - 28 σ on seven complementary axes.

This is the strongest experimental confirmation of the NMSI framework obtained to the date of this paper, on independent public data, without post-hoc selection, with pre-registered and reproducible protocols. The original formulation of CMB as the dynamic equilibrium operator of the PON-C circuit, proposed in Papers I and II, is transformed from a theoretical hypothesis into a statistically well-supported empirical thesis.

12.4. Summary Assessment

Combining the results of subsections 10.6 - 10.10 (P11 at 13 σ, P12 at 5 - 25 σ, P13 confirmed null, the chain P15 to P16 to P17 reaching up to 28 σ), the empirical assessment is the following: the CMB, as observed, contains evidence difficult to reconcile with a purely fossil relic coming from outside the Visible Sphere of the Universe. The data are consistent with an additional, present-day, active coherent re-emission component, generated inside the Visible Sphere, through antiphase oscillation of PON-C (Plasmatic Oscillatory Network-Cosmological) with RON, with local amplification through PON-G (Plasmatic Oscillatory Network-Galactic), the principal reservoir of dipolar molecular species. Figure 9 presents the cumulative summary of all seven Paper III predictions.

Figure 9. Cumulative summary of NMSI_CMB Paper III predictions: all seven predictions confirmed on Planck PR3 + HI4PI public data. Green: P11-P14 (spatial anisotropy group from Theorem M3). Gold: P15-P17 (kinematically decisive chain). Hatched bars (P13, P16): null-confirmed predictions where a low σ IS the predicted outcome. All seven predictions confirmed simultaneously on three independent CMB reconstructions (SMICA/NILC/SEVEM)—difficult to reconcile with a strictly fossil interpretation of the CMB signal.

This is not a rejection of the Penzias and Wilson observations from 1965 [18]—they remain correct: there is an almost isotropic background of microwaves at 2.7255 K. What this paper proposes is a reinterpretation of part of the origin of this background: not exclusively a trace of the Big Bang, but, at least in part, the signature of continuous coherent emission of the baryonic universe through the RON informational operators, superposed on and correlated with present-day matter. This is, in summary, an empirical challenge to the fossil interpretation of the CMB—not a falsification of the Hawking-Penrose theorem, of classical general relativity, or of the Big Bang model as a whole. The residual isotropic temperature itself, and the constant Λ_eff introduced in Section 3.1, also connect naturally to longstanding discussions of the cosmological constant and vacuum energy [19], a connection developed further in Paper IV (Section 15).

13. Twelve Confirmed IceCube Criteria (Paper VII NMSI Neutrinos)

To provide independent supporting evidence that the NMSI framework produces testable predictions beyond the cosmological CMB domain, we present here the table of twelve pre-registered criteria confirmed on public IceCube data [20] [21] (Paper VII of the NMSI Neutrinos series), consistent with the broader NMSI Neutrinos series developed in Papers VIII and IX [22] [23].

All twelve criteria were pre-registered in the NMSI analysis protocol (Papers VII-VIII, deposited at osf.io/ce5ud before data analysis). No post-hoc selection was performed. Each constant (β = 1.25 × 10⁻³ eV², L* = 24, the 20 Riemann zeros, the DZO template) was fixed from the NMSI structure before any contact with IceCube data. This constitutes a genuinely predictive test.

NMSI validation on public IceCube data: 12 criteria, zero free parameters. This table demonstrates that NMSI is a framework with independently confirmable predictions on observational data from entirely different physical domains—CMB cosmology (the present paper) and neutrino astrophysics (Paper VII NMSI Neutrinos). The combination of these confirmations on independent data, without methodological or data overlap, supports treating NMSI as a reproducible predictive framework rather than a purely theoretical construction.

14. Conclusions: NMSI as Real Physics

This paper completes the current stage of the NMSI_CMB series by specifying the concrete physical mechanism of coherent re-emission, by clarifying the hierarchical PON-C/PON-G structure (cosmological and galactic plasmatic oscillatory networks), and by confronting predictions with public Planck PR3 + HI4PI experimental data. The results are consistent across tests: all seven predictions P11-P17 are confirmed at statistical significance levels between 13 σ and 28 σ, while the specific expectations of the strict fossil interpretation are not supported on three independent CMB reconstruction maps.

The decisive chain P15 to P16 to P17—global kinematic dependence, a result not supportive of classical Doppler, and statistical reconstruction of the residual through |z_kin|—provides substantial additional evidence against the strict fossil hypothesis. Under the fossil interpretation, all three predictions would be expected to have a value of strict zero, with no possibility of adjustment. Simultaneous confirmation of all three on three independent CMB reconstruction maps constitutes a significant empirical challenge to the strictly fossil interpretation of CMB—a challenge that, as clarified in Section 2, is independent of, and does not require revisiting, singularity theorems of classical general relativity such as Hawking-Penrose.

The table of twelve confirmed IceCube criteria, presented in Section 13, provides independent supporting evidence from an entirely different observational domain (TeV-PeV neutrino astrophysics). The combination of Planck PR3 confirmations (cosmological CMB, 7 predictions at 13 - 28 σ) and IceCube (neutrino telescope, 12 confirmed pre-registered criteria) supports treating NMSI as a framework capable of generating reproducible, falsifiable predictions across independent observational domains, with a shared set of architectural constants.

The overall message of this paper is the following: the results reported here provide substantial statistical evidence that the observed CMB signal contains an active, present-day re-emission component correlated with baryonic matter and its kinematics, in addition to whatever primordial component may also be present. This evidence is reproducible by any researcher with access to public data and a standard computer, and it is offered as a basis for further scrutiny, replication, and, where warranted, revision.

The results presented in this paper concern the physical origin of the observed CMB signal—they show that it is difficult to reconcile with a strictly fossil interpretation, and consistent with the coherent re-emission of the baryonic universe through RON, with active participation of cosmological PON-C and galactic PON-G, observed from inside the Visible Sphere. This is a claim about the physical CMB signal, not a claim about the mathematical status of singularity theorems in general relativity, which remain outside the scope of this paper.

15. Outlook: Paper IV of the NMSI_CMB Series

The empirical results established in the present paper concerning the fossil interpretation of the CMB raise a natural follow-up question for the cosmological observational framework. If part of the observed CMB signal is active coherent re-emission generated continuously inside the Visible Sphere through PON-C and RON antiphase oscillation, rather than being exclusively a primordial relic propagating freely from z ~ 1100, then the standard cosmological interpretation of redshift as the consequence of metric expansion of space merits further examination from the same physical principles.

Paper IV in the NMSI_CMB series will investigate, both through mathematical development and through experimental testing on public observational data, whether the cosmological redshift z can be related, at least in part, to CMB re-emission through the PON-RON coupling. The mechanism to be examined is not metric expansion of space, nor classical Doppler motion of receding sources, but a frequency-domain effect intrinsic to the dissipative-smoothing operator γ_diss acting cumulatively along the photon path through the cosmological PON-C field. Each interaction of the propagating photon with the local PON-C oscillation would transfer a calculable fraction of energy to the RON substrate, producing a frequency drop that scales with the number of interactions, and therefore with cosmological distance, to be compared quantitatively with the observed Hubble relation z = H₀ · d/c.

Paper IV will aim to derive the functional form of the redshift law from the architectural constants of NMSI (J_c, σ₀, λ_diss), connect it explicitly to the constants T*, T_cycle, and Λ_eff established in Papers I and II, and confront it with public observational data including Type Ia supernovae luminosity-distance measurements (Pantheon+ sample), Baryon Acoustic Oscillation distance constraints (DESI 2024-2026), and high-redshift quasar samples (SDSS DR17). Pre-registered prediction criteria, parallel in structure to the P11-P17 criteria of the present paper, will be specified before any contact with the data, to ensure a genuinely predictive test rather than post-hoc fitting.

If the chain of results established in the NMSI_CMB series continues—Papers I and II (theoretical foundation), Paper III (present-day CMB re-emission evidence, present work), and Paper IV (redshift mechanism)—then the standard cosmological model ΛCDM would face substantive empirical questions on multiple fronts: the interpretation of CMB, the interpretation of cosmological redshift, and the interpretation of cosmological distances. The NMSI framework offers a candidate alternative for these questions, anchored in the same set of architectural constants and tested against the same body of public observational data. Paper IV is currently in preparation; preliminary results will be reported on osf.io/ce5ud and submitted for peer review in due course. Working title for Paper IV: “Cosmological Redshift and CMB Re-Emission through the PON-RON Coupling: Mathematical Development and Empirical Testing.” Expected submission: late 2026.

Appendix A: Python Code for Reproducing Tests P11 - P14

This appendix contains the Python 3 code used to obtain the P11 - P14 results presented in subsections 10.2-10.5.

A1. General Pipeline

#!/usr/bin/env python3

import os, requests

import numpy as np, pandas as pd

import healpy as hp

from astropy.io import fits

from scipy.stats import pearsonr

from sklearn.linear_model import HuberRegressor

from tqdm import tqdm

DATA_DIR = ‘data’

os.makedirs(DATA_DIR, exist_ok=True)

NSIDE = 64; FWHM_DEG = 2.0; N_BOOT = 5000

def robust_z(x):

x = np.array(x, dtype=float)

med = np.nanmedian(x)

mad = np.nanmedian(np.abs(x - med))

if mad == 0 or not np.isfinite(mad):

return (x - med) / np.nanstd(x)

return (x - med) / (1.4826 * mad)

def prepare_map(m, nside=NSIDE, fwhm_deg=FWHM_DEG):

m = np.array(m, dtype=float)

m[m == hp.UNSEEN] = np.nan

med = np.nanmedian(m)

m2 = np.where(np.isfinite(m), m, med)

m2 = hp.smoothing(m2, fwhm=np.radians(fwhm_deg))

if hp.get_nside(m2) != nside:

m2 = hp.ud_grade(m2, nside_out=nside)

return m2

def sigma_from_ci(mean, ci_low, ci_high):

se = (ci_high - ci_low) / 4.0

return abs(mean) / se if se > 0 else np.nan

A2. P11a—HI Strict

def test_P11a(cmb_z, hi_z, mask, lat, top=10):

valid = np.where(mask)[0]

thr = np.nanpercentile(hi_z[valid], 100 - top)

selected = valid[hi_z[valid] >= thr]

rng = np.random.default_rng(2026)

deltas = []

for idx in tqdm(selected):

lat0 = lat[idx]

local = valid[np.abs(lat[valid] - lat0) < 2.0]

ctrl = rng.choice(local, size=min(300, len(local)), replace=False)

deltas.append(cmb_z[idx] - np.nanmean(cmb_z[ctrl]))

deltas = np.array(deltas)

boot = []

for _ in range(N_BOOT):

s = rng.choice(deltas, size=len(deltas), replace=True)

boot.append(np.mean(s))

ci = np.percentile(boot, [2.5, 97.5])

sigma = sigma_from_ci(np.mean(deltas), ci[0], ci[1])

return np.mean(deltas), tuple(ci), sigma

A3. P13—Partial Correlation

def partial_corr(x, y, z):

rXY, _ = pearsonr(x, y)

rXZ, _ = pearsonr(x, z)

rYZ, _ = pearsonr(y, z)

denom = np.sqrt((1 - rXZ**2) * (1 - rYZ**2))

if denom == 0:

return np.nan, rXY, rXZ, rYZ

return (rXY - rXZ*rYZ) / denom, rXY, rXZ, rYZ

def test_P13(cmb_z, ff_z, dust_z, mask):

v = np.where(mask)[0]

X, Y, Z = cmb_z[v], ff_z[v], dust_z[v]

r_part, rXY, rXZ, rYZ = partial_corr(X, Y, Z)

rng = np.random.default_rng(2026)

boot = []

for _ in range(N_BOOT):

idx = rng.integers(0, len(X), len(X))

rp, _, _, _ = partial_corr(X[idx], Y[idx], Z[idx])

boot.append(rp)

ci = np.percentile(boot, [2.5, 97.5])

sigma = sigma_from_ci(r_part, ci[0], ci[1])

return r_part, tuple(ci), sigma

Appendix B: Dimensional Analysis of the Linear CMB Map Model

The linear model of Theorem M3, Equation (7.2), introduces three coefficients α_HI, β_H2, γ_dip for column densities in atoms/cm² or molecules/cm². For dimensional consistency with δT in microkelvin:

• α_HI ~ 1.5 × 10⁻²¹ μK·cm² per HI atom.

• β_H2 ~ 1.2 × 10⁻²¹ μK·cm² per H₂ molecule.

• γ_dip ~ 2.5 × 10⁻²¹ μK·cm² per (dipole-weighted unit) PON-G.

Verification with typical interstellar values: N_HI ~ 10²¹ cm⁻², N_H2 ~ 5 × 10²⁰ cm⁻², $N_{\text{dip}}^{\text{PON-G}}$ ~ 10¹⁹ cm⁻²:

$\Delta T=-1.5-0.6+0.025\approx -2.1\mu \text{K}$ (B.1)

Magnitude consistent with the observed amplitude of CMB anisotropies outside the galactic plane.

Appendix C: Complete Experimental Protocol Specification

C1. Required Public Data

C2. Equipment and Software

• Computer with at least 16 GB RAM and 50 GB free space.

• Python 3.10+ with: numpy, scipy, healpy, astropy, scikit-learn, astroquery, requests, tqdm.

• Internet connection for download (~2 GB total, one-time).

C3. Validation Criteria P11-P17

C4. Estimated Execution Time

• Public data download: 2 - 3 hours (depending on connection speed).

• P11-P14 complete run: ~30 minutes.

• P15 global: ~10 minutes.

• P15 regional (15 masks × 3 maps): ~30 minutes.

• P16 (10 Doppler masks × 3 maps): ~20 minutes.

• P17 (3 maps): ~10 minutes.

• Total: approximately one workday for complete results.

Appendix D: Separate Code for the Chain P15 to P16 to P17

This appendix contains the Python code used to obtain the decisive results of the chain P15 to P16 to P17 presented in subsections 10.7-10.9.

D1. HI4PI MOM1 Reconstruction through HiPS Order10

import astropy.units as u

from astroquery.hips2fits import hips2fits

HIPS_MOM1 = ‘http://cade.irap.omp.eu/documents/Ancillary/4Aladin/HI4PI_MOM1_GAL’

C_KMS = 299792.458

hdul = hips2fits.query(

hips=HIPS_MOM1,

ra=0*u.deg, dec=0*u.deg,

fov=180*u.deg,

width=1024, height=512,

projection=‘CAR’, coordsys=‘galactic’,

format=‘fits’

)

img = np.array(hdul[0].data, dtype=float)

img[img < -30000] = np.nan

# Convert to HEALPix NSIDE=64 by averaging sub-pixels

ny, nx = img.shape

lat_img = np.linspace(90, -90, ny)

lon_img = np.linspace(0, 360, nx, endpoint=False)

lon_grid, lat_grid = np.meshgrid(lon_img, lat_img)

theta = np.radians(90 - lat_grid)

phi = np.radians(lon_grid)

pix = hp.ang2pix(NSIDE, theta.ravel(), phi.ravel())

npix = hp.nside2npix(NSIDE)

num = np.zeros(npix); den = np.zeros(npix)

vals = img.ravel()

good = np.isfinite(vals)

np.add.at(num, pix[good], vals[good])

np.add.at(den, pix[good], 1)

vlsr_mom1_64 = np.full(npix, np.nan)

valid = den > 0

vlsr_mom1_64[valid] = num[valid] / den[valid]

vlsr_mom1_64 = hp.smoothing(

np.nan_to_num(vlsr_mom1_64, nan=np.nanmedian(vlsr_mom1_64)),

fwhm=np.radians(2.0))

zkin_map = vlsr_mom1_64 / C_KMS

zkin_z = robust_z(zkin_map)

D2. Global P15 with Huber Regression

def run_p15_global(cmb_path, dust_z, zkin_z):

cmb_raw = hp.read_map(cmb_path, field=0)

cmb_z = robust_z(prepare_map(cmb_raw))

mask = (np.isfinite(cmb_z) & np.isfinite(dust_z) & np.isfinite(zkin_z))

y = cmb_z[mask]; X = dust_z[mask].reshape(-1, 1)

model = HuberRegressor(max_iter=500).fit(X, y)

residual = y - model.predict(X)

zk = zkin_z[mask]

r, p = pearsonr(residual, zk)

rng = np.random.default_rng(2026)

boot = []

for _ in range(N_BOOT):

idx = rng.integers(0, len(zk), len(zk))

boot.append(pearsonr(residual[idx], zk[idx])[0])

ci = np.percentile(boot, [2.5, 97.5])

sigma = sigma_from_ci(r, ci[0], ci[1])

return r, p, ci, sigma

D3. Regional P15 on absV/Shear/Kin Masks

v = np.array(vlsr_mom1_64, dtype=float)

v_abs = np.abs(v)

v_smooth = hp.smoothing(v, fwhm=np.radians(5.0))

shear = np.abs(v - v_smooth)

kin_score = robust_z(robust_z(v_abs) + robust_z(shear))

regional_masks = {}

for top in [20, 10, 5, 2, 1]:

thr1 = np.nanpercentile(v_abs[np.isfinite(v_abs)], 100-top)

regional_masks[f’absV_top_{top}%’] = v_abs >= thr1

thr2 = np.nanpercentile(shear[np.isfinite(shear)], 100-top)

regional_masks[f’shear_top_{top}%’] = shear >= thr2

thr3 = np.nanpercentile(kin_score[np.isfinite(kin_score)], 100-top)

regional_masks[f’kin_top_{top}%’] = kin_score >= thr3

# Then run_p15_global with mask = base_mask & region_mask

D.4. P16 Doppler Split

# Separation approaching (v < 0) vs receding (v > 0)

doppler_masks = {}

valid_v = np.isfinite(v) & np.isfinite(zkin_z) & np.isfinite(dust_z)

for top in [20, 10, 5, 2, 1]:

thr = np.nanpercentile(v_abs[valid_v], 100-top)

doppler_masks[f’approach_top_{top}%’] = valid_v & (v < 0) & (v_abs >= thr)

doppler_masks[f’recede_top_{top}%’] = valid_v & (v > 0) & (v_abs >= thr)

# Verify: r_approach and r_recede have the same sign? If yes -> NOT classical Doppler

D5. P17 Reconstruction Test

from sklearn.metrics import r2_score

def run_p17_reconstruction(cmb_path, dust_z, abs_zkin_z):

cmb_raw = hp.read_map(cmb_path, field=0)

cmb_z = robust_z(prepare_map(cmb_raw))

mask = np.isfinite(cmb_z) & np.isfinite(dust_z) & np.isfinite(abs_zkin_z)

y = cmb_z[mask]

# Model 0: dust only

X0 = dust_z[mask].reshape(-1, 1)

m0 = HuberRegressor(max_iter=500).fit(X0, y)

R2_0 = r2_score(y, m0.predict(X0))

# Model 1: dust + |z_kin|

X1 = np.column_stack([dust_z[mask], abs_zkin_z[mask]])

m1 = HuberRegressor(max_iter=500).fit(X1, y)

R2_1 = r2_score(y, m1.predict(X1))

delta_R2 = R2_1 - R2_0

coef_zkin = m1.coef_[1]

residual_M0 = y - m0.predict(X0)

r_recon, _ = pearsonr(residual_M0, abs_zkin_z[mask])

return delta_R2, coef_zkin, r_recon

The complete code, including bootstrap for confidence intervals and validation criteria, is available for independent reproduction.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References