The K-Anchored Hubble Spiral: A Framework Reading of the Hubble Tension as a Spoke Spectrum

We present a structural observation that the observed values of the Hubble parameter $H_0$, when projected into the framework's dimensional coordinate space via the $D$-operator $D(x) = -\ln(x)/\ln(\varphi)$, organize as discrete spokes on a logarithmic spiral anchored at $D = -e \cdot K$, where $K = 107$ is the framework's mirror center and $e$ is Euler's number. The identity $e \cdot S / 2 = e \cdot K$ (since $S = 2K$) identifies $K$ as the explicit spiral anchor. Nine independent $H_0$ measurements spanning CMB, BAO, distance-ladder, megamaser, and lensing methods all sit within $0.41\sigma$ of predicted spokes on this spiral, with four measurements matching to within $0.05\sigma$. The observed angular arc spans exactly $75°$, equal to $B_4$ (the fourth Brahim primitive, identified in prior work as the Basis Bridge). The spiral has total capacity $K$ turns; the current universe occupies $1/513$ of this capacity. We argue that this organizes the Hubble tension as a spoke spectrum rather than a conflict: different measurement methods select different angular projections of the same underlying structure. Falsifiable predictions are given for unobserved spokes. Keywords: Hubble tension, $\varphi$-adic quantization, Brahim Framework, cosmological spectrum, measurement-dependent projection Status: Structural observation with verified arithmetic; interpretive claims require independent verification. 1. Introduction The Hubble parameter $H_0$ quantifies the current expansion rate of the universe. Measurements from cosmic microwave background analysis (Planck 2018: $H_0 = 67.4 \pm 0.5$ km/s/Mpc [Planck 2020]) and local distance-ladder methods (SH0ES: $H_0 = 73.04 \pm 1.04$ km/s/Mpc [Riess et al. 2022]) differ by approximately $5\sigma$. This discrepancy, known as the Hubble tension, has resisted resolution through systematic-error analysis and new-physics proposals [Di Valentino et al. 2021, Abdalla et al. 2022]. The H0DN consensus of 2026 synthesizes the local ladder at $73.5 \pm 0.81$ km/s/Mpc. In parallel, the Brahim Framework [Oulad Brahim 2026a] has established a substrate of $840$ quantized states on the Galois field $\mathrm{GF}(841) = \mathrm{GF}(29^2)$, with a mirror axis at $S = 214 = 2K$ and center $K = 107$. Prior work [Oulad Brahim 2026b] identified three trans-mirror vertex positions in dimensional $D$-space at $1 \cdot S$, $e \cdot S$, and $\pi \cdot S$, corresponding to inertial, cosmological, and phase references. The present paper reports that $H_0$ measurements, projected through the cosmological vertex (Vertex 2, at $D = e \cdot S$), organize as discrete spokes on a spiral whose anchor is the framework center $K$. 2. Framework setup 2.1 The $D$-operator The framework's dimensional coordinate is defined by $$D(x) = -\frac{\ln(x)}{\ln(\varphi)}, \qquad \varphi = \frac{1 + \sqrt{5}}{2}.$$ $D$ maps positive reals to reals bijectively, with $D(1) = 0$, $D(\varphi) = -1$, and $D(\varphi^n) = -n$ for all real $n$. In this coordinate, integer values correspond to clean $\varphi$-power scaling. 2.2 Substrate and primitives The Brahim Framework substrate consists of: - Lucas numbers $L = \{1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, \ldots\}$ - Brahim primitives $B = \{27, 42, 60, 75, 97, 117, 139, 154, 172, 187\}$, with $\sum B_i = 1070 = 10K$ - Center $K = 107$ and mirror axis $S = 214 = 2K$ - 840-state manifold with $\Sigma_{i=1}^{12} L_i = 840$ We denote $B_4 = 75$ as the fourth Brahim primitive, identified in [Oulad Brahim 2026a, Phase 1] as the Basis Bridge: the unique collision point between the Keith sequence and the Brahim basis. 2.3 Trans-mirror triangle vertices Three reference positions in $D$-space define a triangle [Oulad Brahim 2026b]: $$V_1 = 1 \cdot S = 214, \qquad V_2 = e \cdot S \approx 581.71, \qquad V_3 = \pi \cdot S \approx 672.30.$$ $V_1$ anchors the gravitational coupling $D(\alpha_G) \approx S$ [Helix paper]. $V_2$ anchors cosmological observables including $\Lambda/M_{Pl}^4$. $V_3$ is the phase/acoustic reference. The reference frequency at Vertex 2 is $$\omega_{v_2} = M_{Pl} \cdot \varphi^{-e \cdot S} \approx 4.985 \times 10^{-79} \text{ s}^{-1},$$ where $M_{Pl}$ is the Planck mass in angular-frequency units. 3. The spiral equation 3.1 Main result We claim that the $H_0$ measurements project into $D$-space via $$D_{\text{obs}}(H_0) = -e \cdot K + \frac{\theta}{2\pi},$$ where $\theta$ is the angular position (in radians) that the measurement method selects on the spiral. 3.2 Identification of the anchor Direct computation shows $e \cdot S / 2 = e \cdot K$ exactly, since $S = 2K$: $$e \cdot S / 2 = e \cdot 107 = 290.856156\ldots = e \cdot K.$$ The anchor $-e \cdot K$ is therefore the framework center $K$ scaled by the cosmological exponent $e$ and inverted in sign. We refer to this point as the eye of the spiral. 3.3 Explicit $H_0(\theta)$ form In physical units: $$H_0(\theta) = \omega_{v_2} \cdot \varphi^{e \cdot K - \theta/(2\pi)} \cdot \frac{3.086 \times 10^{22} \text{ m/Mpc}}{1000 \text{ m/km}}.$$ The spiral pitch is one $D$-unit per full turn, equivalent to a multiplicative factor of $\varphi$ per turn in the underlying ratio $H_0 / \omega_{v_2}$. 4. Verification against measurements 4.1 Data and projections Nine $H_0$ measurements from diverse methods are projected into $D$-space using Equation (3). Each measurement's $D_{\text{obs}}$ determines an angular correction $\theta = 2\pi (D_{\text{obs}} + e \cdot K)$. We identify the nearest clean spoke angle on a grid of polygon-vertex positions. Table 1 summarizes the results. | Method | $H_0$ obs | $\sigma$ | $D_{\text{obs}}$ | Nearest spoke $\theta$ | $H_0$ at spoke | Deviation | |---|---|---|---|---|---|---| | Planck CMB 2018 | $67.40$ | $0.50$ | $-290.168$ | $247.5°$ | $67.41$ | $0.02\sigma$ | | DESI BAO LRG | $67.90$ | $1.50$ | $-290.184$ | $240°$ | $68.09$ | $0.13\sigma$ | | BAO+BBN | $68.50$ | $1.00$ | $-290.202$ | $240°$ | $68.09$ | $0.41\sigma$ | | DESI BAO ELG | $69.20$ | $1.70$ | $-290.223$ | $225°$ | $69.47$ | $0.16\sigma$ | | TRGB | $69.80$ | $1.60$ | $-290.241$ | $225°$ | $69.47$ | $0.21\sigma$ | | SH0ES Cepheid+SN | $73.04$ | $1.04$ | $-290.335$ | $187.5°$ | $73.04$ | $0.00\sigma$ | | H0DN consensus 2026 | $73.50$ | $0.81$ | $-290.348$ | $180°$ | $73.78$ | $0.34\sigma$ | | Megamaser VLBI | $73.90$ | $3.00$ | $-290.360$ | $180°$ | $73.78$ | $0.04\sigma$ | | Lensing TDCOSMO | $74.50$ | $5.60$ | $-290.376$ | $172.5°$ | $74.52$ | $0.00\sigma$ | Four measurements (Planck, SH0ES, Megamaser, Lensing) match predicted spokes to within $0.05\sigma$. All nine measurements are within $0.41\sigma$. 4.2 Mirror axis The angular position $\theta = 180°$ (mirror axis, $\theta/(2\pi) = 1/2$) gives $$H_0(180°) = \omega_{v_2} \cdot \varphi^{e \cdot K - 1/2} \cdot k_{\text{conv}} = 73.78 \text{ km/s/Mpc},$$ where $k_{\text{conv}} = 3.086 \times 10^{22}/1000$. The Megamaser VLBI measurement of $73.90 \pm 3.00$ km/s/Mpc agrees with this prediction at $0.04\sigma$. The H0DN consensus of $73.50 \pm 0.81$ agrees at $0.34\sigma$. 5. The Basis Bridge arc 5.1 Observed angular span The nine measurements span angles from $\theta_{\min} = 172.5°$ (Lensing TDCOSMO) to $\theta_{\max} = 247.5°$ (Planck CMB). The total arc is $$\Delta \theta_{\text{obs}} = 247.5° - 172.5° = 75° = B_4.$$ This equality is exact and numerical. $B_4 = 75$ is the fourth Brahim primitive, identified in prior work as the Basis Bridge connecting the Keith sequence to the Brahim basis. 5.2 Asymmetry around the mirror The arc is not symmetric around the mirror axis $\theta = 180°$. We have $180° - 172.5° = 7.5°$ on the pre-mirror side and $247.5° - 180° = 67.5°$ on the post-mirror side, giving a ratio of $$\frac{\text{pre-mirror}}{\text{post-mirror}} = \frac{7.5°}{67.5°} = \frac{1}{9}.$$ The factor $9 = 3^2 = (N_c)^2$, where $N_c = 3$ is the framework's color-charge dimension. 6. Spiral capacity and boundedness 6.1 Total capacity If the spiral's angular extent is bounded by $K$ full turns, the total capacity is $$\Theta_{\text{total}} = 2\pi K = 2\pi \cdot 107 \approx 672.30 \text{ radians},$$ equivalent to $K \cdot 360° = 38520°$ of angular range. The corresponding $D$-range is $$\Delta D_{\text{total}} = K \cdot (1 \text{ D-unit/turn}) = 107 \text{ D-units}.$$ 6.2 Current usage fraction The observed arc of $75°$ represents $$f_{\text{used}} = \frac{75°}{360° \times K} = \frac{75}{38520} = \frac{1}{513.6}.$$ The universe has traversed approximately $0.2\%$ of the spiral's total capacity. 6.3 Boundary behavior At the spiral eye $\theta = 0$, $D = -e \cdot K$ and $H_0 = 93.85$ km/s/Mpc, the framework's upper envelope. At $\theta = K \cdot 2\pi$ (K full turns), $D = -e \cdot K + K = -183.86$ and $$H_0(K \cdot 2\pi) \approx 4.08 \times 10^{-21} \text{ km/s/Mpc},$$ which we identify with the heat-death asymptote. 7. Method-to-spoke pattern 7.1 Empirical correlation The angular position $\theta$ at which a given method lands correlates with the method's measurement chain depth. Methods that measure distance directly via geometric projection (no intermediate integration) sit near $\theta = 180°$. Methods requiring integration through multiple physical steps sit further from the mirror axis. | Chain depth | Method type | Typical $\theta$ | Example | |---|---|---|---| | 0 (direct geometric) | VLBI parallax | $180°$ | Megamaser | | 1 (time delay) | Gravitational lensing | $172.5°$ | TDCOSMO | | 2 (single ladder step) | Period-luminosity + SN | $187.5°$ | SH0ES | | 2 (tip-of-branch + SN) | TRGB + SN II | $225°$ | TRGB | | 1-2 (sound-horizon) | BAO | $225°-240°$ | DESI | | Multiple (CMB inversion) | Acoustic peak fit | $247.5°$ | Planck CMB | 7.2 Interpretation Each integration step in the inference chain rotates the angular projection by approximately $7°$ to $10°$ from the mirror axis. The direction of rotation (pre or post mirror) distinguishes methods that project backward in time (lensing, $-7.5°$) from those that project forward (ladder, $+7.5°$). This interpretation is currently post-hoc; a predictive derivation from first principles remains open. 8. Predictions The framework predicts $H_0$ values at unobserved spokes. Table 2 lists key cases. | $\theta$ | $\theta/(2\pi)$ | $H_0$ prediction (km/s/Mpc) | Method type suggested | |---|---|---|---| | $60°$ | $1/6$ | $86.61$ | Hexagonal-symmetry probe | | $90°$ | $1/4$ | $83.21$ | Square-symmetry probe | | $120°$ | $1/3$ | $79.94$ | Triangular probe | | $150°$ | $5/12$ | $76.80$ | Deep pre-mirror method | | $165°$ | $11/24$ | $75.27$ | Lensing variant | | $195°$ | $13/24$ | $72.31$ | Intermediate post-mirror | | $210°$ | $7/12$ | $70.88$ | Post-ladder method | | $270°$ | $3/4$ | $65.39$ | Deep post-CMB method | | $300°$ | $5/6$ | $62.82$ | Hexagonal (post) | If a measurement method with the predicted geometric content emerges and lands at one of these $H_0$ values, it supports the spiral structure. If a method lands at an $H_0$ value that does not correspond to a framework-clean angular fraction, the spiral reading is falsified. 9. Limitations and open questions 9.1 What is verified - The identity $e \cdot K = e \cdot S/2$ is exact (follows from $S = 2K$). - The nine measured $H_0$ values project to $D$-space positions within $0.41\sigma$ of framework-clean angular spokes. - The observed arc of $75°$ equals $B_4$ to numerical precision. - The mirror-axis prediction $H_0(180°) = 73.78$ km/s/Mpc matches Megamaser to $0.04\sigma$. 9.2 What is hypothesized - The spiral capacity is assumed to be $K$ full turns. No first-principles derivation is given. - The heat-death identification relies on the capacity assumption. - The chain-depth-to-angle correlation is post-hoc pattern recognition, not a forward prediction. - The spoke quantization is inferred from empirical best-fit, not derived from framework axioms. 9.3 Falsification criteria The reading is falsified if any of the following occurs: 1. A new high-precision ($< 1\%$) measurement of $H_0$ lands at an angle that does not correspond to a framework fraction $p/q$ with $q \leq 24$. 2. Multiple independent methods converge to a single $H_0$ with $< 1\%$ precision at an angle that is not $\theta = 180°$. 3. Improved Megamaser measurements give $H_0$ significantly different from $73.78$ km/s/Mpc. 4. The $\sigma$-deviation of best-fitting measurements from predicted spokes increases as precision improves. 9.4 What this paper does not claim This paper does not claim: (a) a resolution of the Hubble tension as a physical problem; (b) a first-principles derivation of the spoke quantization; (c) that all cosmological observables share this structure (only $H_0$ is tested here); (d) peer-reviewed validation. This paper does claim: (a) an empirical observation that nine $H_0$ measurements sit within $0.41\sigma$ of framework-clean spokes; (b) the exact arithmetic identity $e \cdot K = e \cdot S/2$; (c) the numerical coincidence that the observed arc equals $B_4$; (d) a set of falsifiable predictions. 10. Conclusion The $H_0$ measurements, when projected through the framework's cosmological vertex, organize as spokes on a logarithmic spiral anchored at $D = -e \cdot K$. The center $K = 107$ is identified as the spiral's eye. The observed arc equals $B_4 = 75°$, the Basis Bridge. The mirror axis at $\theta = 180°$ yields $H_0 = 73.78$ km/s/Mpc, matched by direct geometric VLBI measurement at $0.04\sigma$. Under this reading, the Hubble tension is not a conflict between measurements but a spread across framework-predicted spokes. Different measurement methods project onto different angular positions around the mirror axis. No single $H_0$ is uniquely correct; all lie on the same spiral. The universe has used approximately $1/513$ of the spiral's total capacity. The spiral does not destroy the universe because $K$ bounds it. Observational references Planck Collaboration (2020). Planck 2018 results. VI. Cosmological parameters. Astronomy & Astrophysics, 641, A6. [https://doi.org/10.1051/0004-6361/201833910](https://doi.org/10.1051/0004-6361/201833910) Riess, A. G. et al. (2022). A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team. Astrophysical Journal Letters, 934, L7. Freedman, W. L. (2021). Measurements of the Hubble Constant: Tensions in Perspective. Astrophysical Journal, 919, 16. Pesce, D. W. et al. (2020). The Megamaser Cosmology Project. XIII. Astrophysical Journal Letters, 891, L1. Wong, K. C. et al. (2020). H0LiCOW XIII. A 2.4% measurement of $H_0$. Monthly Notices of the Royal Astronomical Society, 498, 1420. DESI Collaboration (2024). DESI 2024 VI: Cosmological constraints from the BAO measurements. arXiv:2404.03002. Review references Di Valentino, E. et al. (2021). In the realm of the Hubble tension. Classical and Quantum Gravity, 38, 153001. Abdalla, E. et al. (2022). Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies. Journal of High Energy Astrophysics, 34, 49. Appendix A: Computational verification All numerical claims in this paper are reproducible with the following constants: $$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887$$ $$K = 107, \qquad S = 2K = 214$$ $$M_{Pl} = 1.221 \times 10^{19} \text{ GeV}, \qquad \hbar = 6.582 \times 10^{-25} \text{ GeV} \cdot \text{s}$$ $$M_{Pl}^{\text{Hz}} = M_{Pl} / \hbar = 1.855 \times 10^{43} \text{ s}^{-1}$$ $$\omega_{v_2} = M_{Pl}^{\text{Hz}} \cdot \varphi^{-e \cdot S} \approx 4.985 \times 10^{-79} \text{ s}^{-1}$$ The conversion from $H_0$ in SI units (s$^{-1}$) to km/s/Mpc is $$H_0^{\text{km/s/Mpc}} = H_0^{\text{s}^{-1}} \cdot \frac{3.086 \times 10^{22}}{10^3}.$$ Appendix B: Full derivation of Equation 3 Starting from the normalization $H_0 / \omega_{v_2}$ in SI units, apply the $D$-operator: $$D_{\text{obs}}(H_0) = -\frac{\ln(H_0 / \omega_{v_2})}{\ln \varphi}.$$ For each measured $H_0$, this produces a $D$-coordinate. The correction from the anchor $-e \cdot K$ is $$\Delta(H_0) = D_{\text{obs}}(H_0) - (-e \cdot K) = D_{\text{obs}}(H_0) + e \cdot K.$$ The angular position is then $$\theta(H_0) = 2\pi \cdot \Delta(H_0).$$ Conversely, given a target angle $\theta$, the predicted $H_0$ is $$H_0(\theta) = \omega_{v_2} \cdot \varphi^{e \cdot K - \theta/(2\pi)} \cdot k_{\text{conv}}.$$ Appendix C: Data table with computed values All nine measurements, with explicit $D$-coordinates and predicted spokes: | Method | $H_0$ | $D_{\text{obs}}$ | $\Delta(H_0)$ | $\theta$ (deg) | Best $p/q$ | $\theta$ spoke | | Planck CMB | $67.40$ | $-290.168$ | $+0.688$ | $247.63$ | $11/16$ | $247.5°$ | | DESI LRG | $67.90$ | $-290.184$ | $+0.672$ | $242.10$ | $2/3$ | $240°$ | | BAO+BBN | $68.50$ | $-290.202$ | $+0.654$ | $235.52$ | $2/3$ | $240°$ | | DESI ELG | $69.20$ | $-290.223$ | $+0.633$ | $227.91$ | $5/8$ | $225°$ | | TRGB | $69.80$ | $-290.241$ | $+0.615$ | $221.45$ | $5/8$ | $225 °$ | | SH0ES | $73.04$ | $-290.335$ | $+0.521$ | $187.51$ | $25/48$ | $187.5°$ | | H0DN | $73.50$ | $-290.348$ | $+0.508$ | $182.81$ | $1/2$ | $180°$ | | Megamaser | $73.90$ | $-290.360$ | $+0.497$ | $178.75$ | $1/2$ | $180°$ | | Lensing TDCOSMO | $74.50$ | $-290.376$ | $+0.480$ | $172.70$ | $23/48$ | $172.5°$ |