The plate that lights a dark field

Two crossed polarizers pass nothing. Insert a THIRD polarizer between them — can adding an absorber brighten a dark field, and does the whole intensity chain really follow Malus's I = I₀cos²Δθ with nothing but 'a polarizer transmits the field component along its axis' as input?

Units: dimensionless (Malus exponent; companions: T₁ = 1/2, T₃ᵖᵉᵃᵏ = 1/8 at φ* = 45°)

▶ Run this simulationRead how it works

The finding

The plate that lights a dark field: with a polarizer coded ONLY as a projector E′=(â·E)â, Malus's exponent comes back as n = 1.99999 ± 0.0006, unpolarized light halves to 0.50011 ± 0.0001, and sliding a third plate between crossed polarizers lifts the output from exactly 0 to a peak of 0.124962 ± 0.00002 at φ* = 44.99° ± 0.02 (known 1/8 at 45°) — while the sieve rival that only removes light can never light the field at all

Method

Code the polarizer as a pure PROJECTOR and nothing else: E′ = (â·E)â, intensity = |E|². No cos² law, no ½, no ⅛ appears anywhere in the generator or the recovery — Malus's exponent, the unpolarized half, and the three-polarizer curve must all EMERGE from chained dot products over an ensemble of unit fields at uniform random polarization angle (stratified 4096 for ensemble means; genuine 2e5-realization Monte-Carlo for the ½ and the photon-counting twin). Recover (i) the exponent n from a log–log fit of the noisy analyzer sweep I vs (â₁·â₂) over 12–66°; (ii) the three-polarizer peak (φ*, T₃) from a ±1%-noise scan of φ on a 0.25° grid — located by a windowed least-squares quadratic around a smoothed argmax, because the ⅛sin²2φ peak is FLAT (curvature −1 rad⁻²; a naive 3-point parabolic refine wanders ±3° under 1% noise, which is exactly what the live module's on-screen argmax does); (iii) the same peak with the outer analyzer tilted to θ₃ ∈ {40,60,75,90}°. All knowns are loaded from scripts/oracles/malus.reference.json only to score; 24 deterministic LCG seeds. ?world=malus.

The law it recovers

I(Δθ) = I₀cos²Δθ (Malus 1809); unpolarized → ½; crossed + inserted plate → T₃(φ) = ⅛sin²2φ, peak 1/8 at 45° — a polarizer PROJECTS the surviving field onto its axis, it does not merely remove light

Measurements, controls & cross-checks

Recovered uncertainty

0.00061

Recovered se distance

0.013

Unpolarized half

Recovered

0.50011

Se

0.000142

Aligned second plate cost

0

Note

2e5 random-angle fields per seed through one plate: exactly half survives (⟨cos²⟩ emerges, never coded); a second ALIGNED plate then costs 0 to machine precision because the transmitted beam is fully polarized (|E_⊥| ≤ 1e-16)

Three polarizer paradox

Crossed pair T

1.9000e-33

Peak deg

44.992

Peak deg se

0.017

Peak T

0.124962

Peak T se

2.3600e-5

Peak T rel offset

-0.0003

Note

P1(0°)⊥P3(90°) pass 1.9e-33 of the source; inserting P2 lifts the output to 1/8 at 45° — the plate ADDS light because projection re-aligns the field, giving it a component the next plate can pass

Tilted analyzer law

Law

peak tracks φ* = θ₃/2 with height ½cos⁴(θ₃/2)

Worst peak deg offset

0.025

Worst peak T rel offset

0.00028

Note

θ₃ ∈ {40, 60, 75, 90}°: the recovered peak follows the half-angle law across the whole sweep — the 45°/⅛ point is one rung of a continuous law, not a coincidence

Rival falsified

Sieve crossed T

0

Sieve max T3 over all phi

0

Sieve malus curve max deviation

0.053

Note

the keep-angle SIEVE (a polarizer only removes photons; pass iff the photon's fixed angle is within 45° of the axis — the only keep-angle rule that reproduces the crossed dark field) predicts T₃ ≡ 0 for EVERY middle angle: no paradox, ever. Its two-polarizer curve is triangular in Δθ (max deviation 0.053 from the projection curve). The lit field at 45° falsifies filtering; light is re-aligned by measurement

Quantum cross check

Photon peak T

0.12508

Photon peak se

0.000121

Max curve deviation

0.00065

Note

single photons passing with Born probability (â·ê_ψ)² and COLLAPSING to the analyzer axis reproduce the classical ensemble point-by-point from raw counts — the same cos² is Born's rule |⟨θ|ψ⟩|², so Malus's 1809 bench is a quantum measurement chain run 10²⁰ photons at a time

Staircase

Law

N plates stepping evenly 0→90° transmit ½cos²ᴺ(90°/N) → ½

T 2

0.125

T 90

0.486478

Worst rel offset

9.5000e-9

Note

chaining more projections between the crossed pair transmits MORE: 1/8 at N=2 (the paradox), 48.6% at N=90 — the polarization is rotated 90° by pure projections with vanishing loss (the polarizer-staircase limit behind the quantum Zeno effect)

Reduction sup

3.9000e-16

Module cross check

Executed peak deg

45.771623

Executed peak T

0.12490934

Executed exponent

1.9941788

Note

no longer a replica — the derisk EXECUTES the sha256-pinned shipped MalusModule.ts (gates J–O): the executed init-time measurement is held === the independent replica BIT-FOR-BIT, and 600 fl(1/120) engine calls of the real fixedUpdate/render are held bit-exact against a statics-built replica at every call (triangle sweep _phi/_phase/_acc, all 4 beam placements with the ½→cos²φ→sin²φ brightness chain, the rotating P2 axis hand, the %4 HUD cadence with full innerHTML, the chart cadence with full SVG), with the display proven LIVE (badge walks dark→peak→dark, chart marker moves) and the shown strings cert-pinned yet EARNED (executed _peakDeg !== 45, _peakT !== 0.125, _nExp !== 2). The oracle's windowed LSQ quadratic pins the same peak to 44.99° ± 0.02

Seeds

24

What it reduces to

Malus's law (É.-L. Malus, 1809): the transmitted intensity of polarized light through an analyzer at Δθ is I₀cos²Δθ — found with calcite and reflected sunlight before the wave theory of light existed. It VALIDATES, not derives: the lab assumes only that a polarizer transmits the vector component of the field along its axis (E′=(â·E)â, Hecht §8.2) and shows (i) the exponent 2, the unpolarized ½, and the three-polarizer ⅛sin²2φ all EMERGE from chained dot products, never coded; (ii) the paradox is real and quantitative — crossed plates pass 0, inserting a third at φ lights the field on the measured curve peaking at 1/8 at 45°, tracking the tilted-analyzer law φ*=θ₃/2, ½cos⁴(θ₃/2) across θ₃∈[40°,90°]; (iii) the intuition 'an absorber can only dim' — formalized as the keep-angle sieve, the only keep-angle rule consistent with the dark field — is falsified by that same lit field, so a polarizer PROJECTS (re-aligns), it does not filter; and (iv) a single-photon Born-rule twin (pass w.p. |⟨θ|ψ⟩|², collapse on pass — Dirac §I.2) reproduces the classical curve from raw counts: the cos² of 1809 is the Born rule of 1926, and the polarizer staircase (→ ½ as N→∞) is the projective-measurement limit behind the quantum Zeno effect. It does NOT model real polarizer extinction ratios, absorption anisotropy (Polaroid dichroism), or partial polarization — ideal projectors only; the polarization-by-reflection route to the same physics is ?world=brewster, and the entangled two-analyzer sequel is ?world=bell, whose E(a,b)=−cos(a−b) correlations no keep-angle model can reach at all.

Module systematics

The on-screen peak angle is plateau-limited, and the earlier ESTIMATED '±~3°' wander is now MEASURED BY EXECUTION: re-running the module's own naive argmax+parabolic estimator across the oracle's 24 seeds at the same ±1% noise gives wander SD 0.70°, max |dev| 1.31°, both inside the ceiling DERIVED from the noise level alone (a noise realization can only relocate the argmax where T ≥ (1−2a)·Tmax ⇒ acos(√(1−2a))/2 = 4.07°; the 1%-plateau half-width is acos(√(1−a))/2 = 2.87°). The executed screen value 45.77° sits +0.77° from 45° (inside the 2.87° plateau) and +0.78° from the oracle's windowed-LSQ 44.992° ± 0.017 — a display-estimator systematic, not a physics error; the shown height 0.1249 is the law evaluated at the plateau-limited angle (−0.07% from 1/8) and the shown exponent 1.994 sits 2.1 per-seed SD from 2 (measured naive-fit spread 2.0000 ± 0.0027). One module defect was found and fixed by the execution cert: render() guarded the HUD with a post-increment `_frame++ % 4`, then checked `_frame % 8 === 0` for the chart — always odd there, so the chart's 'live' φ-marker had been frozen at its init position (0°) since the world landed; the cadence now fires on the post-increment value and gate L proves every chart write bit-equals the replica SVG and the marker moves (verified live in a headless browser: marker 306.6→131.4 px as φ swept 83.4°→29.4°).

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- malus (scripts/malus-derisk.mjs)

Oracle

scripts/oracles/malus.reference.json

Sources

É.-L. Malus, 'Sur une propriété de la lumière réfléchie', Mém. Phys. Chim. Soc. d'Arcueil 2, 143 (1809). E. Hecht, Optics (5th ed., Pearson 2017), §8.2 — Malus's law, natural light, the three-polarizer arrangement. P. A. M. Dirac, The Principles of Quantum Mechanics (4th ed., Oxford 1958), §I.2 — photon polarization, Born probability and the projection postulate.