classMu_le_one

This module organizes the quotient-class path-sum configuration space into exact complexity shells (no size caps in the shell definition) and proves that the shell-resummed path sum with an explicit Gaussian UV regulator $\exp(-\rho n^2)$ converges for every $\rho>0$.

An exact complexity shell is the disjoint union, over shell signatures of total complexity $n$, of the quotient of exact labeled complexes by global relabeling equivalence. The per-class measure is the reciprocal of the automorphism-group order of a representative; it is well-defined on classes by the congruence of the labeled measure under global equivalence.

Module honesty disclosures bind the reading: the regulator is inserted by hand (not derived physics), the phase is an arbitrary class function, and regulator removal ($\rho\to 0^+$) remains a named open. Nothing here is a continuum or mesh-refinement limit.

Destructure the shell class as a shell signature paired with a quotient element. Apply quotient induction on that element, reducing the claim to the labeled statement that the reciprocal automorphism order of any exact complex is at most one. The rest is elementary: a nonempty finite automorphism group has cardinality at least 1, so its reciprocal is at most 1.

Stage 2 of the Seven Gaps exact-shell program records that the per-class measure is well-defined, positive, and at most one. This unit upper bound is the ingredient named in the modulus bound for the regulated shell term: with unit-modulus phases, each shell contribution is dominated by the Gaussian regulator times the shell cardinality. That modulus bound feeds summability of the shell series for every $\rho>0$ and convergence of the cutoff partial sums to the regulated path sum. The result does not touch continuum limits, ledger flags, or the open regulator-removal question.