zRSUVShell
plain-language theorem explainer
Gaussian-regulated shell term at complexity n: exp(-ρ n²) times the finite sum over exact path classes of automorphism weight μ(c) times a unitary phase. Anyone proving UV summability or assembling the regulated recognition path sum cites it. Pure definition: multiplies the class-weighted unitary sum by the explicit Gaussian regulator.
Claim. For regulator strength $\rho\in\mathbb{R}$, a phase parameter $\theta$ assigning a real number to each exact complexity class, and shell index $n\in\mathbb{N}$, the regulated shell term is the complex number $e^{-\rho n^{2}}\sum_{c}\mu(c)\,e^{i\theta_{n}(c)}$, summed over all combinatorially distinct exact complexes $c$ of complexity exactly $n$, where $\mu(c)=1/|\mathrm{Aut}(c)|$ is the per-class measure.
background
This module organizes the quotient-class path-sum configuration space into exact complexity shells with no size caps, then studies the shell-resummed path sum under an explicit Gaussian UV regulator $\exp(-\rho n^{2})$.
The exact complexity shell at level $n$ is the disjoint union, over shell signatures, of the quotient of exact labeled complexes by global relabeling equivalence. It is finite. The per-class measure $\mu$ equals $1/|\mathrm{Aut}|$, is well-defined on classes, positive, and at most one.
Binding honesty disclosures: the regulator is a mathematical insertion, not derived physics; the phase is an arbitrary invariant function on classes (no physical action is derived); regulator removal $\rho\to 0^{+}$ is a named open and is never claimed. Nothing here is a continuum or mesh-refinement limit.
proof idea
Definition, not a proved claim. The body multiplies the real Gaussian factor $\exp(-\rho n^{2})$ by a finite complex sum ranging over the exact path class at level $n$. Each summand is the class measure cast to $\mathbb{C}$, times the unitary $e^{i\theta}$ built from the phase parameter at that class. No lemmas fire; the expression is the object later bounded and summed downstream.
why it matters
Stage 2b building block of the Gaussian-UV-regularized recognition path sum. Downstream, the modulus bound shows its complex norm is at most the regulator times shell cardinality; composing with the entropy bound yields comparison against $\exp(-\rho n^{2})(n+1)^{12(n+1)}$. That comparison feeds UV summability for every $\rho>0$, which legitimates the tsum definition of the full regulated path sum and the cutoff-convergence theorem.
The grounding theorem ties true status flags (shell structure, entropy bound, UV summability) to these kernel results. Regulator removal and continuum/mesh limits stay open or red in the ledger; this definition flips none of those flags. It supplies a mathematically controlled UV shell series inside the gravity seven-gaps path-sum program, not a step of the T0–T8 forcing chain.
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