Finite-Support Origin, Present-Branch Window and de-Sitter Endstate Univers in the Mittermeier Attractor Theory (MAT): A Closure-First UV/IR Entropy Frame from Big Bang Support to Future Horizon

The Mittermeier Attractor Theory (MAT) recasts the cosmological boundary problem as a single closure-first readout rather than as a sequence of independently tuned sectors. In the standard effective picture, the hot big bang, primordial fluctuations, radiation endpoint, baryon abundance, cold component, neutrino loading, late acceleration and asymptotic horizon are described with different theoretical languages and then constrained jointly by data. MAT reverses the explanatory direction: the relevant thermodynamic, geometric, primordial and late-time quantities are read from one arithmetic-spectral support-scar-residue spine. The big-bang boundary is therefore not a zero-radius point, a compressed ordinary plasma, or a minimum length inserted by hand. It is the first admissible finite-support readout of the active branch. Its radius, entropy, compactness, density, temperature and curvature audits are fixed by the same threshold pair that also appears in the vacuum transport law. The support entropy is the ultraviolet control variable of the paper: it blocks the zero-support operation, supplies a finite information ledger, fixes the density/temperature/curvature triad, and projects into the relativistic-radiation endpoint through the closed trace law Nₑff^ (MAT, T) = SₜhMAT · (1 - alphaM/40), (SₜhMAT - Nₑff^ (MAT, T) ) /SₜhMAT = alphaM/40. This trace projection is the radiation lane. The divisor forty is used as the reduced finite-temperature trace dimension of the minimal 32 + 7 + 1 thermal support stack; the global statement that the final DMAT* forces this stack uniquely remains the microscopic completion boundary. The scalar amplitude and baryon abundance are different power layers of the same branch: they are scar-square readouts rather than refits of the thermal endpoint. The exponentiated thermal count, the nearby 21-count information marker, and the conventional scalar-amplitude count form a sharp cross-sector audit of this architecture without collapsing the lanes into one identity. The present universe becomes the readable middle of the branch. Radiation is dynamically small but still visible as CMB thermal memory; matter is subdominant but still decomposes into baryons, hot neutrinos and a cold residual; the vacuum already dominates the expansion and fixes the future horizon. The same active readout now also gives the native Hubble scale in direct form, (H₀*) ² = (Lambda₀* + 3 XiR^ (th, *) ) / 3 · (1 - pi^ (-1) + (4/3) delta alpha_ (chart, pi) * + epsiloncl), so the square of the Hubble scale is the radiation-loaded active vacuum scale divided by the active non-matter budget. This is the observationally rich interval in which origin trace, matter composition, native expansion scale and infrared horizon control are simultaneously legible. In the same present-window readout, MAT no longer treats the laboratory electromagnetic coupling as an external metrological input: the active coupling alpha_ (0, MAT) is generated as the support-to-chart laboratory readout of the finite-support chain. The same finite trace object now provides the compact root-scar and support-to-chart hierarchy epsilonₜhMAT = alphaM/40, epsiloncl = (400/pi) · (epsilonₜhMAT) ², Omega_ (m, D) * = pi^ (-1) - (4/3) delta alpha_ (chart, pi) * - (400/pi) · (epsilonₜhMAT) ². Thus the first thermal trace scar is the root of the closure-scar family, while the finite chart residue remains the active vacuum-depth residue. The support-radius defect is transported into that residue by the one-lane support-to-chart normal form delta alpha_ (chart, pi) * = (-ln Rₜh) · 1 - 1/ (10pi) + 2 epsilonₜhMAT/pi - epsiloncl/ (2 sqrt (pi) ) - epsiloncl epsilonₜhMAT, and this same normal form gives alpha_ (0, MAT) = kappaM pi² - 1 - epsiloncl / kappaM · (11 + e^ (-2) - delta alpha_ (chart, pi) *). The reduced trace block is fixed by the active SSOT readout; the full unique Hilbert-space realization remains the microscopic completion target. The far future is the complementary infrared surface of the same active readout. MAT connects the finite support entropy and the de Sitter horizon entropy through the UV/IR entropy-frame law SdSMAT / SₜhMAT = (RdSMAT / Rₜh) ² = 3pi / (Lambda₀* SₜhMAT), and compresses the entire origin-to-endstate radius hierarchy into N_ (support->dS) = N_*MAT Delta uₚi* + lne rho sqrt (3) / (2 sqrt (pi) ). This is the Mittermeier origin-endstate horizon product law: finite support origin, primordial horizon depth, active vacuum depth and de Sitter endstate appear in one active-branch formula. In this precise sense MAT functions as a quantum-gravity and Theory-of-Everything candidate: it replaces sector-by-sector retuning by correlated algebraic overconstraint from the beginning of the universe to its final horizon.