# PREREG — Rushmore, round 2: ten laws against the house  (29 September 2026)

His words: *"Pick 10 more rushmore concepts"*, then *"Try all of them - derp"*.

Written before any of the ten scripts runs. Every leg carries one label, and only FALSIFIABLE legs count as
evidence:

- **ALGEBRA** — an identity. It can fail only if my algebra or code is wrong. It names a landing; it is not
  evidence. (The peer's lesson from entry 49: "The Flow is variational" was true of every 1-D flow.)
- **THEOREM CHECK** — a known theorem applied to the house's own equations. It can fail only through my
  numerics or my misreading of the theorem's conditions.
- **FALSIFIABLE** — a claim about the house that could come out either way before the run.

**Monte Carlo criteria use 4 SE, not 3.** About twenty SE comparisons are made; at 3 SE the chance that one
fails by luck alone is about 5%, at 4 SE about 0.1%. A miss between 3 and 4 SE is reported as a miss.

**Not blind.** Estimates I made while designing, recorded so the runs check them: R2c's R² near 0.5; R3c's
ratio comfortably above 10; R8c's Trust shortfall near 0.015 at x_s = 0.9.

House constants: k = 1, p = 0.35, C = 0.018975; T(x) = k[(1−x)^(−p) − 1], F(x) = −2Cx/(1−x²).
A "vote" is a mind's yes/no act: v = ±1 with P(v = +1) = (1 + x)/2, so its average is its alignment x.

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## R1 · Fisher information (Fisher 1925; Rao 1945)
- **1a ALGEBRA.** I(x) = Σ_v p(v|x)(∂ₓ ln p(v|x))² by exact enumeration equals The Measure's factor 1/(1−x²).
  199 points on [−0.99, 0.99]; max relative difference < 1e-12.
- **1b ALGEBRA.** The Fisher–Rao distance ∫√I dx between two alignments equals |arccos x₁ − arccos x₂|, the
  angle. 50 random interior pairs by quadrature; max relative difference < 1e-9.
- **1c ALGEBRA.** The rapidity ∫₀ˣ I dt equals artanh x. 50 points; < 1e-9.
- **1d REPORTED.** The house's rest points as angles: heaven's angle to its wall, the bottom's angle to its
  wall, the ridge at 90°.
- No falsifiable leg exists for R1; the entry will say so.

## R2 · Fisher's fundamental theorem of natural selection (1930)
- **2a ALGEBRA.** The variance of a vote is 1 − x², by enumeration, < 1e-12.
- **2b THEOREM CHECK** (the peer's derivation). The printed Action under heavy drag, γθ̇ = −sin θ (γ = 1),
  integrated in θ by RK4 (dt = 1e-3, t ≤ 10) from θ₀ ∈ {0.3, 1.5, 2.5}, mapped to x = cos θ, equals the
  selection law x(t) = tanh(t/γ + artanh x₀): max |difference| < 1e-9.
- **2c FALSIFIABLE.** Is The Flow selection in a two-strategy game? A two-strategy game makes
  g(x) = 2(T + F)/(1 − x²) exactly linear in x. Linear least-squares fit on 181 points of [−0.9, 0.9]:
  **"is a two-strategy game" ⇔ R² ≥ 0.999.** My prediction: R² < 0.999, not a game.

## R3 · The Kalman filter (1960)
Truth z (an aim), readings y = z + noise of variance r = 1.
- **3a ALGEBRA.** Static truth, diffuse prior P₀ = 1e12: the Kalman gains are 1/n. max |n·Kₙ − 1| < 1e-6 over
  n ≤ 10,000. So Position's S/n IS the Kalman filter when the truth never moves.
- **3b THEOREM CHECK.** Random-walk truth with step variance q: the constant gain minimising the exact
  stationary error, V(K) = [(1−K)²q + K²r]/(1 − (1−K)²), equals the Riccati gain
  K∞ = P/(P + r), P = (q + √(q² + 4qr))/2, to 1e-6 relative, for q/r ∈ {1e-4, 1e-3, 1e-2}. A Monte Carlo of
  the K∞ filter reproduces V(K∞) within 4 SE.
- **3c THEOREM CHECK.** On a drifting aim, S/n's mean squared error at n = 10,000 exceeds the Kalman filter's
  by a factor ≥ 10 at all three q/r (Monte Carlo, 2,000 paths).
- **3d REPORTED.** Memory length 1/K∞, and the steps a record needs to cross zero after a turn at age n₀,
  S/n against constant gain.

## R4 · Birkhoff's ergodic theorem (1931)
- **4a THEOREM CHECK.** The coasting pendulum of the inertia run (θ₀ = 0.8, at rest, m = 1) froze at
  +0.844965 after t = 200. Birkhoff on its energy shell: the orbit average of cos θ by quadrature must lie
  within B = (1 − cos θ₀)·T_period/200 of +0.844965, and a fresh Verlet run averaged over exactly 50 periods
  (dt = T_period/20,000) must match the orbit average to < 1e-6.
- **4b REPORTED.** Orbit averages for θ₀ ∈ {0.4, 0.8, 1.6, 2.4}: a coasting mind's record depends on where it
  started.
- **4c THEOREM CHECK.** Two starts far apart — (θ₀, θ̇₀) = (0.1, 0) and (3.0, 3.0), one swinging and one
  spinning. Without noise their records differ by more than 0.1. With noise and drag at temperature 0.5
  (BAOAB, γ = 1, dt = 0.01, 2,000 walkers each), both records match the resting average I₁(2)/I₀(2) within
  4 SE.

## R5 · Fluctuation–dissipation (Einstein 1905; Kubo 1966)
The printed Action's potential −cos θ, with drag γ and noise tied to drag at temperature T: σ² = 2γT.
BAOAB, dt = 0.01, 2,000 walkers, 10,000 steps after 5,000 of burn-in; SE from walker-to-walker spread.
- **5a THEOREM CHECK.** Coasting does not change the resting odds: at T = 0.5, ⟨cos θ⟩ equals I₁(2)/I₀(2)
  for m ∈ {0.25, 1, 4} (γ = 1), each within 4 SE.
- **5b THEOREM CHECK.** Neither does the drag's size: m = 1, γ ∈ {0.25, 4}, same criterion.
- **5c THEOREM CHECK.** Equipartition: ⟨m θ̇²⟩ = T in every run of 5a–5b, within 4 SE.
- **5d THEOREM CHECK.** Noise NOT tied to drag (σ² = 1 fixed, γ ∈ {0.25, 1, 4}, m = 1): the resting average
  follows the effective temperature σ²/(2γ), I₁(κ)/I₀(κ) with κ = 2γ/σ², within 4 SE.

## R6 · Poincaré–Bendixson: no chaos in one or two dimensions
Largest Lyapunov exponent λ from the variational equation.
- **6a THEOREM CHECK.** Undriven Flow, starts −0.5 and +0.3: λ < 0 at t = 50.
- **6b THEOREM CHECK.** Undriven printed Action θ̈ + sin θ = 0 (θ₀ = 1.0): λ(1000) < 0.02 and
  λ(100)/λ(1000) > 3 (shear only, no chaos).
- **6c FALSIFIABLE.** The printed Action with drag and a periodically changing aim,
  θ̈ + θ̇/2 + sin θ = 1.5 cos(2t/3) (Baker & Gollub's parameters): λ > 0.05 from all 8 starts at t = 5,000.
  Controls: drive 0.9 (periodic window) and drive 0 give λ < 0.
- **6d THEOREM CHECK.** The Flow with a periodically changing aim, dx/dt = T + F + A cos(ωt), A ∈ {0.05, 0.2,
  0.5}, ω ∈ {0.5, 2, 8}, starts ±0.5, stiff solver (LSODA, rtol 1e-9): λ ≤ 1e-3 in all 18 runs at t = 400.

## R7 · Perron–Frobenius: who listens to whom (DeGroot 1974)
Minds average the aims of those they listen to, with Trust weights w ∝ 1/σ².
- **7a THEOREM CHECK.** A random trust network (N = 30, each listens to itself and 4 others): after 2,000
  rounds every mind sits at Σπᵢaᵢ(0), π the left Perron vector, to < 1e-10.
- **7b ALGEBRA + MC.** The consensus error variance equals σ²·Σπᵢ² (the network's Many Tongues). Monte Carlo
  of 20,000 draws within 4 SE.
- **7c THEOREM CHECK.** When everyone listens to everyone, Trust makes π ∝ 1/σ² exactly — the best possible
  weights (ratio of error to optimum = 1 to 1e-12). In sparse networks (k = 4) the ratio is > 1 in every one
  of 50 networks; the mean excess is reported.
- **7d THEOREM CHECK (Golub–Jackson).** Over N ∈ {25 … 800}: listening around (k = 4 random) makes Σπ² fall
  with slope in [−1.1, −0.9] on log–log; a hub everyone gives weight ½ keeps Σπ² flat (slope in [−0.1, 0.1]).

## R8 · Condorcet's jury theorem (1785)
- **8a THEOREM CHECK.** Independent jurors each right 60%: exact majority accuracy for N = 1, 11, 101, 1001
  rises strictly and exceeds 0.99 at 1001.
- **8b FALSIFIABLE (the Many Tongues pipe).** Jurors sharing an error of weight ρ = 0.1 (latent Gaussian, each
  alone 60% right): accuracy is capped at Φ(μ/√ρ), μ = Φ⁻¹(0.6). By exact quadrature |P(10,001) − cap| <
  0.005 and P(1,001) lies farther from the cap than P(10,001). A Monte Carlo at N = 1,001 (20,000 juries)
  matches the quadrature within 4 SE.
- **8c FALSIFIABLE (Trust against the best weights).** One mind at alignment x_s beside ten at 0.3, votes
  independent, exact enumeration. The weights compared are equal, Trust (x/(1 − x²), from inverse variance
  on the unbiased v/x) and rapidity (artanh x, Nitzan–Paroush's optimum).
  (i) rapidity ≥ Trust and ≥ equal at every x_s ∈ {0.3, 0.5, 0.7, 0.9, 0.99};
  (ii) Trust's shortfall is 0 to 1e-12 at x_s = 0.3 and **> 0.005 at x_s = 0.9**.

## R9 · Relativity (Einstein 1905)
- **9a ALGEBRA + MC.** Two independent witnesses who agree, alignments x₁ and x₂: the posterior alignment is
  (x₁ + x₂)/(1 + x₁x₂), Einstein's velocity addition. Monte Carlo of 10⁶ cases per pair within 4 SE, for
  pairs (0.3, 0.3), (0.6, 0.8), (0.9, 0.95), (0.99, 0.5).
- **9b ALGEBRA.** n agreeing witnesses: tanh(Σ artanh xᵢ). The distance from 1 is reported exactly.
- **9c ALGEBRA + MC.** In series (a relay chain) alignments multiply: 10⁶ cases for chains (0.9, 0.8) and
  (0.9, 0.9, 0.9), within 4 SE.

## R10 · Arrow and Gibbard–Satterthwaite
Coherence: the crowd's direction is the direction of the sum of unit aims (in the plane).
- **10a FALSIFIABLE.** 2,000 random crowds of 9 (von Mises around 0, κ = 2); one mind reports strategically.
  G–S predicts that lying helps generically: the best report beats honesty by more than 1e-9 rad in ≥ 99.9%
  of crowds.
- **10b FALSIFIABLE (derivation).** When the others' resultant has |R| > 1, the most a single liar can turn
  the crowd is arcsin(1/|R|). The best achievable distance to its aim, max(0, |Δ| − arcsin(1/|R|)),
  matches numerical optimisation to < 1e-6 rad in every crowd. Reported: the mean maximal pull for
  N = 9, 99, 999.
- **10c REPORTED.** An explicit Condorcet cycle: three minds on the circle, three candidate directions,
  pairwise majority cycles. The house's cardinal rule (the resultant) does not.

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**Kill conditions.** Every failing leg is printed as FAILS, beside the legs that held. A failed FALSIFIABLE
leg is a result about the house. A failed ALGEBRA or THEOREM CHECK leg is a defect of mine, and gets
diagnosed before anything is said about the house. A re-run with changed legs gets a new prereg file.