Throw two fair dice. Listing every outcome gives 6 × 6 = 36 ordered pairs, and exactly 6 of them sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So the chance of a 7 is 6/36 = 1/6. Nothing was measured. The probability is a ratio of two counts. COMPUTED
Two counting rules do most of the work. The multiplication principle: if a process has n stages and stage i can go ki ways, there are k1×…×kn paths through it (four throws of a die: 64 = 1296). And choosing: the number of ways to pick r of n things, order ignored, is C(n, r) (which 2 of 5 checkers are the ones that catch an error: C(5,2) = 10). Weight each path by its chance and you have probability. COMPUTED
Suppose an argument has n steps, each independently right with chance p, and the conclusion is right only if every step is. There is one all-right path among the 2n right-or-wrong paths, and it carries weight pn. That is the multiplication principle again. At p = 0.95, a figure chosen for illustration, not the accuracy of any system: COMPUTED
| steps n | chance the whole chain holds |
|---|---|
| 1 | 0.9500 |
| 5 | 0.7738 |
| 10 | 0.5987 |
| 13 | 0.5133 |
| 14 | 0.4877 |
| 20 | 0.3585 |
| 50 | 0.0769 |
The chain is worse than a coin flip at 14 steps (0.5133 at 13, 0.4877 at 14). Ten steps hold about 60% of the time, and 50 steps hold under 8%. Turned around: to be right 90% of the time over 50 steps, each step has to be right 99.79% of the time. A seeded simulation of 200,000 ten-step chains gave 0.6009, against the exact 0.5987. COMPUTED
This is the arithmetic behind a remark of Wolfram's about language models: because the way they work is statistical, it becomes “progressively less likely that, as things get more complicated, they will work out correctly”. The chain law puts a number on it for a chosen p, and it applies to a human working through a long proof as well. CITED
Now put the checkers in parallel instead of in series. Each of n independent checkers (a script, a second source, a teacher) catches a given error with chance q. The error survives only if all of them miss it, with chance (1 − q)n, so it is caught with chance 1 − (1 − q)n. Counting explains the same formula: the chance that exactly r of n checkers catch it is C(n, r) qr(1 − q)n−r, the binomial counts, and they sum to 1. Three checkers at 70% each all miss with chance 0.027. COMPUTED
| detection | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| q = 0.5 | 0.5000 | 0.7500 | 0.8750 | 0.9375 | 0.9688 |
| q = 0.7 | 0.7000 | 0.9100 | 0.9730 | 0.9919 | 0.9976 |
| q = 0.9 | 0.9000 | 0.9900 | 0.9990 | 0.9999 | 1.0000 |
The two laws mirror each other. Series multiplies the chances of success and shrinks. Parallel multiplies the chances of failure and shrinks that instead. A weak checker repeated is strong, and a strong chain repeated is weak.
The catch is the word independent. Suppose 30% of errors are invisible to every checker, for example because they share a blind spot. Then detection is 0.7 × (1 − 0.3n), which is 0.49, 0.68, 0.70 and 0.70 for 1, 3, 10 and 100 checkers. It never passes 70%, however many are added. The 30% is an illustration, not a measurement. MODEL
Nogueira Grossi's The Swarm Simulator defines a swarm state Xt = (It, Ct, Mt, Ft): shared intent, coordination, type propagation and diffusion. Definition 3.1 gives It+1 = It · ftypes · fagents · (1 − η). Theorem 5.1 says the swarm map contracts when LI + LC + LM < 1, and Theorem 5.3 bounds the distance to the fixed point by Lt times the starting distance. The paper's closing sections credit Cucker–Smale, Olfati-Saber and others for the classical models. CITED
Shared intent is a chain. With the default parameters, ftypes × fagents × (1 − η) = 0.65 × 0.80 × 0.80 = 0.416, so It = I0 × 0.416t exactly: 1.55 × 10−4 after 10 steps and 2.41 × 10−8 after 20. It is the chain law of section 2 with p = 0.416, and the script confirms it at t = 1, 10, 20 and 60. The printed constants also check: LI = 0.4160, LC = 0.1664, LM = 0.2310, L = 0.8134 < 1. Running the recurrences, I, C and M all decay to zero, while the diffusion term Ft = 1 + αt keeps growing (3.0 at t = 100, 5.0 at t = 200), so on all four coordinates there is no fixed point to converge to. COMPUTED
A check on three starts is weak evidence. The bound of Theorem 5.3, taking the fixed point of (I, C, M) to be 0, holds for the three starting states the paper's convergence figure uses. From the start (100, 10, 1) it fails at every step from 1 to 5: the size of the state is 208.2, 1169.2, 3324.2, 3976.5 and 1981.4, against bounds of 90.3, 73.4, 59.7, 48.6 and 39.5. The reason is visible in the update Ct+1 = Ct · It+1/(1 + D): it multiplies two state variables, so the constant LC is only a local bound and grows with C. How often a random start breaks the bound depends on where you look. With seed 2026, starts drawn from [0,3]3 never break it, from [0,10]3 about 3% do, and from [0,100]3 about 88% do. This is the parallel law used backwards: if 10% of starts fail, three random tests all pass with chance 0.93 = 0.729. COMPUTED
None of this is hidden in the paper: its open-questions file already lists S1 (the full contraction proof) and S4 (the induction for the bound) as open. Version 3 of the deposit, prepared with this chapter, settles them by restating them. SwarmSimulatorV3.no_global_lipschitz proves that the map has no global Lipschitz constant, so Theorem 5.1 as printed cannot hold. step_lipschitz_ball proves the constant max(a(1 + cR), m) on the ball of radius R, which is below 1 for R < 3.51 with the default parameters. orbit_nrm_le proves the decay ρ(R)t‖X0‖, paper_bound proves the printed Lt bound for starts of size at most 1, and printed_bound_fails proves the failure from (100, 10, 1) exactly, in rational arithmetic. The same file shows that two clusters that both satisfy the ball condition share the fixed point 0, so the multi-orbit picture of distinct fixed points does not follow for this model, and diffuse_unbounded proves that F has no fixed point. KERNEL-CHECKED
The size of that ball matches the experiment: starts drawn from [0,3]3 never broke the bound, and breaking begins beyond it. The V2 Lean file of the deposit does not compile on the toolchain this series pins (nine errors), and several of its items restated their hypotheses, which is the failure this series' Lean audits look for. V3 replaces it. COMPUTED
A chain of 14 steps that are each 95% reliable is already worse than flipping a coin. It is also why a verify script that runs before the sentence is worth its cost: it turns a long series chain into many short ones, each of which can be checked on its own. MODEL
swarm_simulator.py for 60 steps). The Lean file was compiled in the cloud against the pinned toolchain, and its axiom report is in swarmsimulatorv3.axioms.txt; it should be re-run on the Mac with tools/leancheck.sh --audit. It proves statements about the map, not the paper's Theorems 5.1–5.3 as printed. The V3 deposit, with the revised paper (LaTeX and PDF), was published on Zenodo on 29 September 2026 as version 3.0, doi:10.5281/zenodo.23027566, and is kept in papers/swarm-simulator-v3/. Whether the model can be calibrated to data (S3) is open. OPENswarm_simulator.py and OPEN_QUESTIONS_SwarmSimulator.md, held; the V2 PDF is not held).