exactShellGaugeUVStatus
plain-language theorem explainer
Canonical status record for the exact-shell Gaussian-UV path-sum module. It marks shell structure, entropy bound, UV summability, cutoff convergence, and non-vacuity as proved, and records regulator removal and continuum limit as open or unclaimed. Downstream grounding ties each true flag to a kernel theorem. Pure structure instance: five trues, two falses.
Claim. The status of the exact-shell Gaussian-UV module is: shell structure proved, shell entropy bound proved, UV shell-series summability proved, cutoff partial sums converge, and the regulated sum is non-vacuous; regulator removal ($\rho \to 0^+$) is not proved; no continuum (mesh-refinement) limit is claimed.
background
This module sits in the Seven Gaps gravity stack. It organizes the quotient-class path-sum configuration space into exact complexity shells (no size caps in the shell definition) and studies the shell-resummed path sum with an explicit Gaussian UV regulator $\exp(-\rho n^2)$.
Honesty constraints are binding: the regulator is a mathematical insertion, not derived physics; the action/phase is an arbitrary class-invariant parameter; regulator removal is a named open; nothing here is a physical continuum limit (complexity cutoff is not mesh refinement).
The status structure packages Stage 1 (relabeling-invariant complexity, exact setoid, Fintype shells, card bound $\mathrm{card}(\mathrm{ExactPathClass}, n)\le (n+1)^{12(n+1)}$, inhabited shells) and Stage 2 (class measure, modulus bounds, summability for every $\rho>0$, cutoff tendsto $Z_{\mathrm{RS,uv}}$) as boolean flags.
proof idea
Definitional structure instance, not a proof. Each field of ExactShellGaugeUVStatus is assigned a literal Boolean: five Stage-1/Stage-2 items set to true, regulator_removal_proved and continuum_limit_claimed set to false. No tactics, no lemmas applied at this site; the companion grounding theorem later witnesses that each true flag matches an actual kernel result.
why it matters
This record is the honest ledger entry for the exact-shell Gaussian-UV work. The grounding theorem exactShellGaugeUVStatus_grounded consumes it and ties every true flag to concrete theorems (inhabited shells, entropy bound, summability, cutoff limit, non-vacuity) while keeping the two open flags false. Non-vacuity is also used by the zero-phase positivity theorem for $Z_{\mathrm{RS,uv}}$.
In the Seven Gaps protocol it prevents silent overclaim: continuum-limit and FullTheoryLedger flags stay red; only the mathematical regulated shell sum is certified. It does not touch T0–T8 forcing, RCL, or the phi-ladder mass formula; it is infrastructure for a controlled path-sum measure under an explicit UV cutoff.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.