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| Format: | Recurso digital |
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Zenodo
2026
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| Accès en ligne: | https://doi.org/10.5281/zenodo.20349491 |
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- <div>This paper establishes a conditionally rigorous closure theorem for</div> <div>the low-energy spatial spectral dimension dspace = 3 in Algebraic Quan</div> <div>tum Morphogenesis (AQM). Earlier AQM material proposed a dimensional</div> <div>emergence argument based on nested Dirac recursion and a free-convolution</div> <div>attractor. The present paper makes a rigorous correction to that argu</div> <div>ment: free probability is no longer used as the primary criterion that</div> <div>by itself derives d = 3. Instead, it is used as an auxiliary mechanism for</div> <div>the stability of nested spectral fluctuations and for the non-drift of di</div> <div>mension. The actual determination of spatial spectral dimension comes</div> <div>from three functionally independent low-energy spatial spectral shells</div> <div>produced by the three-step condensation path, the Clifford-type princi</div> <div>pal symbol of a Dirac operator in the continuous limit, and the Weyl law</div> <div>/ heat-kernel short-time asymptotics.</div> <div>We first recall previous AQM results: the minimal noncommutative</div> <div>neck algebra M2(C), the legal algebraic landscape, the K-flow three-step</div> <div>condensation path</div> <div>Fib −→ Toric(e) −→ Z3,</div> <div>and the total theorem of boundary center readout. We then state three</div> <div>core lemmas. First, the three-step condensation path is functionally in</div> <div>dependent in the recursive-scale, boundary-circular, and triality-transverse</div> <div>branches. If the low-energy spatial spectral shell has fewer than three</div> <div>dimensions, it cannot accommodate all three independent structures; if</div> <div>it has more than three, it introduces unobserved redundant low-energy</div> <div>degrees of freedom. Hence three is the minimal nondegenerate number</div> <div>of spatial spectral-shell directions under the legal AQM path. Second,</div> <div>when the low-energy continuous spectral limit exists and the geome</div> <div>try is described by a first-order self-adjoint elliptic Dirac-type operator,</div> <div>the three independent first-order derivations induce a Clifford principal</div> <div>symbol</div> <div>σ(D∞)(x, ξ) =</div> <div>3</div> <div>∑</div> <div>a=1</div> <div>Γ a ξa,</div> <div>{Γ a , Γ b } = 2δ abI.</div> <div>Therefore</div> <div>σ(D∞ 2 )(x, ξ) = |ξ| 2 I,</div> <div>1so D∞ 2 is a three-dimensional Laplace-type elliptic operator. Third, free</div> <div>probability fluctuations in nested spectral recursion do not change the</div> <div>Weyl leading term or the heat-kernel exponent if they are zero-order</div> <div>bounded perturbations, relatively compact perturbations, or random</div> <div>spectral fluctuations that do not change the principal symbol.</div> <div>Combining these lemmas gives the main theorem: under the legal</div> <div>AQM three-step condensation path, the Dirac-type continuous spec</div> <div>tral limit, and free-probability spectral stability, the low-energy spatial</div> <div>spectral dimension is strictly</div> <div>dspace = 3.</div> <div>The paper also distinguishes spatial spectral dimension from full 3 + 1</div> <div>Lorentzian spacetime. The time direction comes from a one-parameter</div> <div>modular flow or RG scale flow and is to be handled in later papers on</div> <div>the time field and Lorentzian spacetime. This paper does not claim that</div> <div>a complete smooth spacetime or Einstein equations follow uncondition</div> <div>ally from arbitrary finite matrix algebras; it closes the three-dimensional</div> <div>spatial spectral dimension inside the legal AQM path.</div>