Life and Death in a Branching Cosmos

Under the Many‑Worlds interpretation (Everettian quantum mechanics), every quantum event with multiple outcomes produces a superposition of decohered “branches.” The universal wavefunction evolves unitarily; it does not collapse. The squared magnitude of each branch’s amplitude—by the Born rule—acts as a measure that generalizes probability and determines how typical an experience is among all branches.

Schrödinger’s Cat, From the Cat’s Point of View

In the classic thought experiment, a sealed box contains a cat, a vial of poison, a Geiger counter, and a radioactive nucleus with a 50% chance to decay in a fixed interval (say, one hour). If the nucleus decays, the Geiger counter triggers, the vial breaks, and the cat dies; if it does not, the cat lives. Before an external observation, the composite system is modeled—mathematically—as a superposition of alive and dead branches that have decohered with respect to one another.

From the cat’s own point of view, the Many‑Worlds picture predicts that there are branches in which the cat experiences waking up alive after each interval, and branches in which experience ends. The Born rule says those branches are weighted by squared amplitudes—i.e., by probabilities like $p$ for survival and $1-p$ for death. If the experiment is fair each hour (the nucleus has a 50‑50 chance to decay), then $p=\frac{1}{2}$. After $n$ hours, the branch in which the cat has survived every single interval has measure ${\frac{1}{2}}^n$, and the aggregate measure of branches with at least $m$ survivals is a binomial tail. Subjective continuity is therefore possible but becomes measure‑atypical very quickly as $n$ grows.

This is exactly the scenario our calculator quantifies: it tells you how much of the total measure corresponds to the cat waking up alive after a sequence of risky intervals. You can interpret each “trial” as one hour in the box (or any repeated hazard), set $p$ to the survival amplitude‑squared per interval, and explore either the probability of perfect survival (survive every interval) or the probability of surviving at least $m$ intervals.

From the outside observer’s perspective, opening the box yields a single outcome, and classical language forces a choice: “the cat was alive” or “the cat was dead.” From the inside perspective, there is no mystical collapse—only a branching of futures weighted by measure. The cat’s continued experiences—if any—occur precisely in those branches whose measures this tool computes.

Thought experiments like quantum suicide ask whether a conscious observer who repeatedly faces lethal quantum tests should expect to keep awakening only in the branches where they survive. That notion (often labeled “quantum immortality”) depends crucially on measure: even if some branches contain continual survival, their combined measure can be astronomically small. This tool quantifies that measure precisely and does not encourage any risky behavior. See the ethical note below.

What This Calculator Computes

Suppose each trial independently yields “survive” with amplitude‑squared probability $p$, and you perform $n$ trials. Let $m$ be the minimum number of survivals you require. The measure of branches with exactly $k$ survivals is the binomial weight $\left(\frac{n!}{k!(n-k)!}\right)p^k{\left(}^{1-p}$. This calculator returns either the tail measure $M=\sum _{k=m}{n}^{}$ of branches with at least $m$ survivals, or the single‑point measure for exactly $m$, depending on the mode you choose.

Mapping to Schrödinger’s cat. Let one “trial” be a full hazard cycle in the box. Then $p$ is the cat’s per‑cycle survival measure (e.g., $\frac{1}{2}$ for a fair radioactive trigger), $n$ is the number of cycles, and choosing “exactly $m$” or “at least $m$” controls whether you count only one final‑alive pattern or all sufficiently alive patterns. Setting Perfect survival enforces $m=n$, matching “the cat survives every interval.”

We also show understandable diagnostics: percentage, “1 in X” odds, decimal‑log and bit surprise, an independent normal approximation (for cross‑checking), the expected number of surviving branches (≈ $2^n$ × $M$), and for comparison, the idealized “perfect survival” measure $p^n$.

Exponential Intuition

With “perfect survival” the measure decays as $p^n$. A useful diagnostic is the measure half‑life: the number of trials after which the perfect‑survival measure halves, calculated as $h=\frac{ln(1/2)}{ln\left(p\right)}$. For example, with $p=0.9$ the half‑life is about 6.6 trials, so every 6–7 trials the measure halves again.

Numerics You Can Trust

Direct factorials overflow quickly (even for n ≈ 200). We compute using log‑space arithmetic with a Lanczos approximation to the log‑Gamma function and a log‑sum‑exp accumulator. This stays stable for thousands of trials and tiny measures (e.g., 10⁻³⁰⁰ and below) and avoids the inaccuracies of naive summation.

Worked Examples

The table updates with your current settings for several presets (first row mirrors your inputs).

Assumptions, Limits & Ethics

• Trials are independent and identically distributed; environmental decoherence is assumed to isolate outcomes.
• “Measure” reflects amplitude‑squared weighting (Born rule). Many‑Worlds remains an interpretation, not a proved ontology.
• This tool is explanatory only. It must not be used to justify dangerous actions. If thoughts about self‑harm arise, seek immediate help from professionals and trusted people in your area.

Mathematical Appendix

Binomial point measure: $P(K=k)=\frac{n!}{k!(n-k)!}{p}^k{(}^1$. Tail ("at least m"): $M=\sum _{k=m}{n}^{}$ of the same term. Normal approximation with continuity correction uses $z=\frac{m-0.5-np}{\sqrt{np(1-p)}}$ and $M\approx 1-\Phi \left(z\right)$ when variance is not tiny.