shellConstant_not_oscillatoryTail
plain-language theorem explainer
Any phase constant on each exact complexity shell fails the oscillatory-tail condition, no matter how the common shell phase varies with complexity. Gravity workers closing the P2.4 continuum-blocker gap cite this complexity-phase no-go. The proof is a two-lemma chain: oscillatory tails force shell amplitudes to vanish, while shell-constant phases have amplitude equal to the diverging positive shell mass.
Claim. Let $\mathrm{phase}$ assign a real phase to every exact path class at each complexity $n$. If $\mathrm{phase}$ is shell-constant (identical on all classes inside each fixed-$n$ shell, though free to depend on $n$), then $\mathrm{phase}$ does not satisfy the oscillatory-tail condition.
background
Module setting is Seven Gaps P2.4: the exact-shell phase-balance blocker for the Zq continuum obligation. Carrier facts already give finite exact shells, positive class masses, large shell mass, and a fixed-cap pairing witness, but no substrate action that resolves phases inside every late shell.
A phase is shell-constant when it ignores intra-shell class labels and may still vary arbitrarily with complexity $n$. Oscillatory tail is the uniform contiguous-block control needed for the continuum blocker. Shell-amplitude vanishing is the weaker shell-local necessary condition: every individual late exact-shell amplitude tends to zero. The module records that oscillatory tail implies shell-amplitude vanishing by restricting to one-shell blocks.
For a shell-constant phase the full shell amplitude is exactly the positive shell mass times one common unit phase, so no intra-shell cancellation occurs. Limits here are complexity cutoffs, not mesh refinement; no geometric continuum claim is made.
proof idea
Term-mode proof by contradiction. Assume an oscillatory tail. Apply the sibling implication that every oscillatory tail forces shell amplitudes to vanish. Feed that conclusion into the companion no-go that a shell-constant phase never has vanishing shell amplitude (its amplitude norm equals the diverging positive shell mass). The two facts contradict, so the tail assumption is false.
why it matters
This is the complexity-phase arm of the certified P2.4 blocker package. It feeds directly into p24_shell_balance_blocker_certificate, whose doc-comment states that uniform tails require shell-local vanishing, and that finite-shell repairs and complexity-only phases cannot supply it; the remaining premise is asymptotic intra-shell phase balance from richer substrate structure.
Together with the finite-repair no-go (eventually-zero phase fails oscillatory tail) and the implication oscillatory-tail implies shell-amplitude vanishing, it isolates what P2.4 still needs: genuine late-shell cancellation inside shells, not mere relabeling invariance, finite-cap pairing, or a phase that only sees shell complexity. No full-theory flag moves; the result is a proved obstruction, not a continuum construction.
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