Moon Formation by Triple Phase Transition in the Differentiating Proto-Earth A dynamical endogeneity: a single mechanism that jointly sets lunar mass, rotation, and angular momentum

The Moon exists, with a set of measured, non-negotiable characteristics: near-perfect Earth–Moon isotopic identity, a small metallic core, crustal dichotomy, the angular momentum of the Earth–Moon system, and the residual ≈5° inclination of the lunar orbit relative to Earth's orbital plane. These facts impose a strong constraint: the causal chain that produced the Moon necessarily occurred, and its links belong to a narrowly constrained domain — the one compatible with this outcome. We therefore adopt a retrodictive approach: rather than tuning free parameters to reproduce a desired result, we determine the conditions of the Hadean proto-Earth that an already-certain result imposes. The number of links in such a chain does not measure an excess of degrees of freedom, but a degree of specification; the narrowness of the window of admissible conditions is not a weakness but the content of a prediction about the early Earth. The dominant scenarios, which derive the Moon from a body external to Earth, founder on these constraints taken together: they reproduce the isotopic identity only through fine-tuned reservoirs, leave the angular-momentum budget as an adjustable quantity, and offer no mechanical link between lunar formation, crustal asymmetry, and the delayed terrestrial dynamo. Each anomaly is handled separately, by a mechanism appended after the fact. We propose a change of framework: the Moon is endogenous, formed from the very material of the proto-Earth, extracted from its differentiating mantle. This resolves the isotopic identity at the outset — not a coincidence to be explained, but a consequence: the Moon is made of Earth. The framework rests on a thermodynamic necessity: any Earth-mass planet emerges from accretion in a state of near-total silicate-mantle melting, the accretion energy exceeding the melting energy by a factor of ≈121. The proto-Earth is thus a rapidly rotating magmatic body (T_rot ≈ 3.5 h), lacking a stabilizing satellite, whose axis undergoes large-amplitude free nutation within the interval [40°, 70°] in its own co-rotating frame. A single driver — the progressive segregation of metallic iron-nickel toward the forming core — triggers three coupled transitions. (1) Rheological: the magma acquires a Bingham-Herschel yield stress and organizes a Coherent Magmatic Torus (CMT) within the intertropical band |φ| < 30°. (2) Mechanical: the CMT undergoes N = 2–3 episodes of cohesive hypersonic ejection, arising from a deterministic instability — a parcel leaves the torus when the Solberg–Höiland coefficient λ(U) becomes positive. Each ejection shifts the rotation axis toward a new obliquity, and with it the active band, so that each successive CMT is geometrically distinct — a mechanical basis for the observed crustal dichotomy. (3) Magnetic: the dynamo ignites ≈350 Myr after the ejections end, consistent with the Jack Hills zircons. With an ejected mass of 2.2–4.0 × 10²² kg per episode and a motivated capture efficiency of ≈0.70, the lunar mass is reconstituted without an external impactor; the isotopic identity and the metallic-iron depletion follow, the ejected material being drawn off after iron segregation. The core of the argument is the angular-momentum budget, which we treat not as a free parameter but as a vector balance to be closed. The excess to be removed, ≈3 × 10³⁴ J·s, cannot be carried away by the ejecta or by direct tides; it is evacuated through a single physically identified channel — the crossing of the Laplace-plane transition, a resonant post-formation sink. The present-day angular momentum and the 5° lunar inclination jointly demand this crossing, whose activation condition — an inertial obliquity ε_I ≳ 60° — is precisely what the endogenous framework independently derives from the absence of a stabilizer. Two independent lines — one from the momentum balance, one from the obliquity — converge on the same Hadean configuration. The quantitative closure of the balance remains conditional on a rheological quantity (the global quality factor) whose determination we identify as the highest-priority numerical test. The central prediction — the linchpin of falsifiability — is one or more seismic interfaces between 200 and 530 km depth, with impedance contrast |R| ∈ [0.01, 0.04], imminently testable by Chang'e 7 (South Pole, August 2026), then by FSS, LEMS, and Artemis III (2028–2029). Preliminary support comes from the Chang'e-6 samples: norites dated at 4247 ± 5 Ma and high Fe/Mn ratios in deep olivines. These interfaces are the expected signature of a Moon built layer by layer from successive ejections. A animation is available : https://orion4622.github.io/moon-formation-triple-phase-transition/Animation_Formation_de_la_Lune_v2.html The retrodictive approach establishes the legitimacy of the reasoning; these predictions will decide whether this is the causal chain that built the Moon. If they are refuted, so too must the theory be.