gap2_shell_balance_blocker_certified
plain-language theorem explainer
The shell-balance blocker certifies that an oscillatory continuum tail forces per-shell amplitude vanishing, so neither an eventually-zero phase nor a shell-constant phase can satisfy the remaining Z_q continuum obligation. Full-theory gravity auditors cite it as Pillar-2 item P2.4 in the machine-checked ledger. The proof is a one-line re-export of the certificate theorem from the shell-balance blocker module.
Claim. For every real phase assignment on exact shell path classes: (i) an oscillatory continuum tail implies that shell amplitudes vanish; (ii) an eventually-zero phase cannot produce an oscillatory continuum tail; (iii) a shell-constant phase cannot produce an oscillatory continuum tail. Equivalently, any phase that closes the tail obligation must rebalance every late shell.
background
The Full Theory Ledger records machine-checked status for the quantum-gravity campaign. Pillar 2 demands a well-defined quantum amplitude: a substrate-to-geometry bridge plus a path-sum measure with a proved continuum limit. Gap 2 tracks the remaining continuum/measure obligations on the $Z_q$ shell side.
An exact shell path class indexes admissible UV shell histories at shell index $n$. A phase is a real weight on those classes. The oscillatory-tail condition asks that late-shell contributions keep oscillating rather than dying or freezing. Shell-amplitude vanishing says those late contributions have zero amplitude; eventually-zero phase means the phase is finitely supported (a finite-cap repair); shell-constant phase means the weight is constant on each shell.
The ledger imports the campaign starting line and the dedicated shell-balance blocker module so that P2.4 flips only when the kernel-checked certificate is present. Related foundation material (eight-tick phases, recognition cost structure) sits upstream of the shell calculus but is not reopened here.
proof idea
One-line term wrapper: the theorem is definitionally the triple certificate already proved as p24_shell_balance_blocker_certificate in the shell-balance blocker module. No local case analysis or new algebra is performed; the ledger only re-exports the conjunction (tail $\Rightarrow$ amplitude vanish) $\wedge$ (eventually-zero $\Rightarrow$ not tail) $\wedge$ (shell-constant $\Rightarrow$ not tail) under the full-theory naming.
why it matters
P2.4 is the sharpness certificate for the remaining oscillatory-tail obligation on Pillar 2. It rules out two natural closing strategies: finite-cap (eventually zero) phase repairs and shell-constant phases. Any successful continuum phase must therefore rebalance every late shell, which tightens the design space for the path-sum measure and continuum limit demanded by the full-theory plan.
In the ledger architecture, flags flip only under kernel-checked, axiom-audited theorems; this declaration is the certified blocker for the shell-balance gap. It sits beside the measure-selection and metric-carrier blockers as part of the still-open Pillar-2 stack. The master claim that the full theory is not yet closed remains true until every pillar flag flips. Framework-wise this is continuum/path-sum work on the gravity side, not a T0–T8 forcing step, but it constrains how eight-tick shell structure can enter a quantum amplitude.
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