I.10 Finite quantum geometry
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I.10.1 A projector-resolved geometry
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At resolution \(\ell\), define
In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types \(E_\ell:{\mathcal H}_\ell\to{\mathcal H}_{\ell+1}\) and \(C_\ell:{\mathcal H}_{\ell+1}\to{\mathcal H}_\ell\). The left inverse relation \(C_\ell E_\ell=I\) alone does not imply \(C_\ell=E_\ell^*\) or make the embedding isometric.
A coherent refinement should satisfy
and
For an operator \(D\), define the refinement defect
Exact resolution independence is \(\Delta_\ell(D)=0\). When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective self-energies.
Text description
Colored nodes and connecting branches show sectors subdividing through successive refinement levels. The horizontal axis is refinement level and the vertical axis is conceptual sector. The diagram organizes a proposed tower and does not reconstruct physical spacetime.
I.10.2 Euclidean transfer
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Theorem I.10.1: Finite Euclidean transfer
Let \(H\) be Hermitian on a nonzero finite-dimensional Hilbert space. For all real \(\tau,\sigma\ge0\),
is positive definite and satisfies
Its spectrum is contained in \((0,1]\).
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Proof of Theorem I.10.1
Diagonalize \(H=\sum_jE_jP_j\). Then
Every eigenvalue is strictly positive and at most one. Multiplication adds the exponents, proving the semigroup law and \(T(0)=I\). Positivity here is operator positivity, not an assertion that every matrix entry is nonnegative.
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The paper tests a controlled volume-state model
with \(\kappa>0\) and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic spacetime dynamics.
Text description
Transfer eigenvalues are plotted by ordered mode on a logarithmic vertical axis. The points descend while remaining positive. The plot records the finite Euclidean transfer spectrum used in the manuscript.
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The horizontal axis is discrete spatial volume and the vertical axis is ground-state probability. A narrow peak appears around the center of the displayed volume range, with small tails toward both ends. Its interpretation is restricted to the controlled toy model.
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A heatmap uses source volume on columns and target volume on rows. The largest entries follow a diagonal band and decrease away from it. The color scale denotes transfer amplitude in the finite Euclidean model; a gravitational interpretation remains provisional.
Numerical certificate I.10.2: Finite transfer model
The minimum transfer eigenvalue was \(0.0767054\), the semigroup residual was \(0\), and reconstruction from four volume-residue packets had residual \(0\).
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