A Rigorous Functional Measure for Spectral Quantum Gravity: Minlos Construction on the Substrate Krein Space, a Sharp Trace Bound, the Horizon KMS Property, and Wedge Duality

SQG Measure, version 12.

We construct a Gaussian functional measure dμ_SQG for Spectral Quantum Gravity on the substrate Krein space K_sub = H₊ ⊕ H₋, and establish the thermal and modular structure of the emergent de Sitter horizon. Four results are proved.

First, dμ_SQG exists as a Borel probability measure by a single application of the Bochner–Minlos theorem to the positive Hilbert structure of K_sub, for any ghost-block operator that is positive and bounded below; the physical indefinite two-point function is recovered by inserting the Krein symmetry into the observable (a Gupta–Bleuler prescription).
Second, the covariance is classified exactly: the mode operator reduces to Bessel form, K^(−1/2) is Hilbert–Schmidt with an exact norm in the limit of a wide window, but not trace class; in four dimensions the covariance lies in the Schatten class S_p only for p > 2, which the Minlos construction does not require.
Third, the vacuum of the emergent generalized free field is a KMS state on the static-patch algebra at the de Sitter temperature, with modular flow acting geometrically as the de Sitter boost.
Fourth, wedge duality follows from the same decomposition, with the modular conjugation implementing the antipodal reflection; the Borchers–Buchholz theorem gives an independent route and forces β to the Gibbons–Hawking value.

Beyond the Gaussian level, on a lattice at the Planck scale — the physical cutoff of the framework, so that no continuum limit is required — the interacting window measure with a quartic self-coupling exists, is reflection positive along the fiber, and its two-point functions have a limit to a non-compact fiber that does not depend on the order of the limits (Osterwalder–Seiler and Ginibre inequalities); a Monte Carlo certificate confirms reflection positivity at two couplings and detects a deliberately broken control.

The construction is situated within the literature on indefinite-metric quantization (Dirac, Pauli, Gupta–Bleuler, Lee–Wick, Gazeau–Renaud–Takook, Albeverio–Gottschalk). Claims are graded [proved], [derived], [computed], [argued], [imported] or [open].

Version 14 (30 October 2026). This version adds three remarks to Section 11 (the Planck-lattice measure) and leaves every theorem of v13 unchanged. (1) Discretisation of the scale axis: on the lattice itself, without a continuum limit, an ω-uniform scale axis reproduces the spectral dimension three, Weyl’s law and the Hilbert–Schmidt norm of the continuum window operator; a log-uniform axis gives spectral dimension two and one mode per site. Everything proved in Section 11 holds for either choice; what depends on the choice is stated. (2) The gauge field along the scale axis: a single plaquette coefficient makes the gauge field see the axis as a uniform chain in the site index, so a Kaluza–Klein tower along the axis exists only on the log-uniform discretisation. (3) The Lorentz-invariance remark of v13 is reformulated: the hypercubic requirement (ω-uniform axis, spectral dimension three) and the fermion-sector bandwidth condition (spacing a_ω ≥ (π/2)√κ a) meet on an ω-uniform axis of that spacing, at the price of a one-loop anisotropy of order 0.02 g² absorbed in one renormalisation of κ; the gauge coupling of the construction that keeps the chirality partner at the zone edge is, however, non-local (the lattice form of the Karsten–Smit obstruction), so the chiral problem is not settled by the choice of axis and remains open. Which discretisation is the Planck lattice of the theory is stated as open with its consequences. All three remarks are model calculations of the companion programme and are graded [computed]; the summary table and the closing section are updated accordingly. Author block unchanged.