Publication history: 2026-06-26

Details

• New module v429 (DM.AXION.PENTAGON.01) — a geometric re-reading answering “why does so much idle E_8 structure exist?” for the one case where the idle structure has a job. The reverse audit (v354) had argued the golden ratio \varphi is unmapped (it appears only internally — the affine-E_8 attractor angle v312, the icosian lattice v348 — and in no physical readout), and used exactly that to conclude \varphi cannot be numerology. [E]this sharpens that statement: because \Nfam{=}\gcar{-}2, the axion misalignment spine angle (v211, DM.AXION.SPINE.01) is \theta_i{=}\pi\Nfam/\gcar{=}(\gcar{-}2)\pi/\gcar, which is exactly the interior angle of the regular \gcar-gon — for \gcar{=}5 the pentagon, \theta_i{=}3\pi/5{=}108^\circ. [E]its cosine is golden, \cos(3\pi/5){=}(1{-}\sqrt5)/4{=}{-}1/(2\varphi) (partner 2\cos(2\pi/5){=}1/\varphi {=}\varphi{-}1, \varphi{=}2\cos(\pi/5)). [E]the golden character is unique to \gcar{=}5: among the small regular n-gons only n{=}5 has an irrational interior-angle cosine (n{=}3,4,6 give 1/2,0,-1/2), so \theta_i is golden because the carrier is 5 (P2).
• An honest [C]bridge, not a status change. It connects two standing facts — \varphi is the unmapped \gcar{=}5 signature (v354/v313) and \theta_i{=}3\pi/5 is the one external cosmological input (v211) — and refines v354 from “\varphi in no physical readout” (it is the unmapped \gcar{=}5 signature, v313) to “\varphi touches only the [C]spine misalignment angle” (conditional, branch-level, never a frozen prediction). It is a geometric motive for the spine over the hilltop \theta_i{\approx}170^\circ (whose cosine is not a clean pentagon value), consistent with — not replacing — the finite-T solver that decides the branch (v185/v211). It does not derive \theta_i, does not upgrade DM.AXION.SPINE.01 (stays [C]) and closes no gate.
• Surfaces synced. run_all.py, script_registry.csv, status_ledger.csv, the dark-matter section of tfpt_4_frontier; the Wolfram extension/README/ledger counts (GATE.WOLFRAM.02) and the website verification count; manifests refreshed.