Measured by the lab
0.25
Straight-edge shadows look sharp — but if light is a wave, what actually happens right at the edge of the shadow, and can a ray/corpuscular picture survive a close look? Can the famous I₀/4 at the geometric edge, the 1.37 overshoot, the 0.78 dip, the shadow leak and the √(λL) near-field scaling all be recovered from nothing but summed Huygens wavelets over a half-open aperture, with the Cornu spiral and every closed form banished from the recovery path?
Units: dimensionless intensity ratio I(edge)/I₀ at the geometrical shadow edge of a straight edge
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The ¼-lit shadow edge, from raw phasor sums: with an opaque half-plane coded ONLY as 'block y<0' and light as the plain Huygens sum of cylindrical wavelets K(χ)·e^{ikr}/√r over the open half — no Cornu spiral, no Fresnel-integral C(w)/S(w), no ¼ anywhere in the recovery — the intensity at the geometrical shadow edge comes back at exactly I₀/4 (0.250000, a mirror-symmetry consequence recovered to 1e-8; a 24-plate ±3% read lands 0.2494 ± 0.0009), the lit side OVERSHOOTS to 1.377 I₀ at w≈1.22 (txt 1.370) and dips to 0.7787 I₀ at w≈1.87 (txt 0.7783), light LEAKS into the geometric shadow (0.041 I₀ at w=−1, decaying monotonically), fringe positions scale as √(λL) (log-log slopes +0.5000 in both λ and L — the near-field signature), and geometric optics is falsified structurally: a ray step can never exceed I₀ (the 37% overshoot sits 123 SE above 1.0) nor light the shadow
A raw near-field Huygens–Fresnel phasor sum over the OPEN half-plane (y>0) onto a screen at distance L: cylindrical secondary wavelets K(χ)·e^{ikr}/√r with the exact Fresnel phase k(y−x)²/(2L) and obliquity K=½(1+cosχ), cosχ=L/r. The aperture is sampled at y=(j+½)·step (straddling the edge so no sample lands on y=0); the unobstructed reference I₀ is the SAME sum over the mirror half, so the edge ratio |open|²/|open+mirror|² = ¼ is a mirror-symmetry consequence, not an input. No ¼, no Cornu spiral, no Fresnel-integral C(w)/S(w), no closed-form intensity appears in the generator — λ enters only as the summed wavelength. Extrema are located by scan+parabolic refinement; the √(λL) scaling is a log-log fit of the first-fringe physical position x₁=w₁√(λL/2) across a λ×L sweep. A photographic plate (24 seeds, ±3% reading noise) reads the edge by linear interpolation at w=0 (an unbiased read on a steep, convex, asymmetric edge — a wide polynomial fit inherits the region's ~0.7% curvature bias) and the overshoot by a smoothed windowed-parabola peak (removing the max-of-noise upward bias). Rival: geometric optics (a sharp ray/corpuscular step) through the same coordinates. 8 physics gates + a 6-gate module-honesty certificate (I–N) that sha-pins and EXECUTES the shipped FresnelEdgeModule.ts headless, ~28 s.
Fresnel's 1818 near-field diffraction theory of a straight edge / opaque half-plane (Sommerfeld's rigorous half-plane solution, Math. Ann. 47, 317, 1896; Born & Wolf §8.7; Hecht §10.3.4): the intensity at the geometrical shadow boundary is I₀/4 (the Cornu-spiral endpoint at (½,½)), the first maximum is 1.37 I₀ at w≈1.22 and first minimum 0.78 I₀ at w≈1.87 in the Fresnel variable w=x√(2/λL). Non-circular in the same sense as the young/malus/airy oracles: the recovery path codes only the PRIMITIVE physics — Huygens cylindrical wavelets K(χ)·e^{ikr}/√r summed over the open half-plane — never a Cornu spiral, never the Fresnel integrals C(w)/S(w), never a closed-form intensity, and above all never the ¼, which is a consequence of the mirror symmetry between the open and blocked halves at the edge, not an input (tamper: known_value→0.30 ⇒ exit 1, recovery unchanged at 0.250000). The near-field OPENING of the lab's diffraction arc: where ?world=young/grating/airy work the Fraunhofer far field (patterns linear in λ), this is the Fresnel near field (√(λL) scaling), and its rival is the same one Young's slits faced — Newton's corpuscles, here falsified by an overshoot a ray can never make. ESTIMATOR NOTES: (1) the edge ¼ sits on a steep, strongly-convex, asymmetric part of the curve, so a wide polynomial fit inherits a ~0.7% curvature bias — linear interpolation at w=0 is the unbiased read; (2) the overshoot must be read from a smoothed windowed parabola, since a raw argmax of noisy pixels biases the peak upward (the same max-of-noise trap the estimator checklist warns of).
The live module (src/modules/FresnelEdgeModule.ts) is HONEST as physics: its on-screen numbers come from the SAME phasor sum as the oracle (no closed form, no ¼ coded), run at a lighter aperture (150 zones × 320 samples/zone vs the oracle's 240 × 500). As of the honest-module certificate (derisk gates I–N: the shipped module sha256-pinned, TS-stripped with 27 exactly-once pairs, and EXECUTED via new Function with Babylon/DOM recorder stubs) this is PROVEN BY EXECUTION, zero module edits: (1) the executed init measurement equals a statics-built replica BIT-FOR-BIT — edge, overshoot, dip, shadow leak, √(λL) slope, and all 150 display-pattern bars; (2) 600 real fixedUpdate/render engine calls at fl(1/120) match a replica at EVERY call (shimmer accumulator, all 150 bar emissive triples, 150 %4 HUD writes each bit-equal, bar-0 changed 300×); (3) the displayed edge is EXACTLY 0.25 in floating point — not a pasted constant but a float-commuting mirror symmetry (the mirror half-sum is bit-identical to the open half, so I₀ = 4·|open|² and division by 4 commutes with IEEE rounding), witnessed live by symmetry-breaking probes (I(w=0.05)=0.2767, I(w=0.2)=0.3719 ≠ ¼); (4) the shown overshoot 1.380 vs textbook 1.3704 is DECOMPOSED with every stage deterministic and ZERO noise term — the module has NO RNG at all, the first world in the honest-module streak with a completely noise-free display: +0.00616 disclosed cylindrical-taper physics (present at oracle resolution), +0.00361 aperture-resolution (150×320 vs 240×500, gated at 5e-3), −4.0e-5 scan-grid (0.002-pitch argmax, ≤ derived curvature ceiling 1.7e-3), stages summing EXACTLY to the shown offset; dip 0.779 = res +0.00029 + grid 2.5e-5; leak res Δ −4.7e-4; (5) the executed √(λL) exponent is EXACTLY 1 (the grid argmax lands on the same w-cell in all four λ×L configs — a quantization identity, displayed as ½ after the /2); (6) the measurement block is answer-free (no 0.25/1.37/0.778/1.2172/1.8725/0.041 literals; exactly six disclosed M.LAMBDA + six M.L_SCREEN instrument refs; planted-violation self-test proves the scanner live). KNOWN COSMETIC DEFECT, disclosed not fixed (6th world with the malus/young/newton/grating/airy cadence bug): render()'s chart branch tests _frame % 8 === 0 AFTER the %4 post-increment guard where _frame ≡ 1 mod 4, so the branch is DEAD — proven HARMLESS because the chart is statics-only (written exactly once at init; frozen SVG === replica === a fresh executed _buildChart() bit-for-bit). Tamper self-tests are surgical: known_value→0.4 fails only scoring gates A/F with the recovery unchanged; module-sha tamper fails only gate I; a 1-ulp edit to a pinned exec value fails only gate L. The transverse fringe scale is stretched for legibility (disclosed on-screen); the displayed intensities are the true recovered ratios. The chart overlays the measured wave pattern against the geometric-optics step (the falsified rival) so the overshoot and shadow leak are visible directly.
npm run derisk -- fresnel (scripts/fresnel-derisk.mjs)scripts/oracles/fresnel.reference.jsonA. Fresnel, 'Mémoire sur la diffraction de la lumière' (1818, crowned 1819 by the Académie des Sciences) — the wave theory of diffraction, over Poisson's corpuscular objection and Arago's confirming bright spot. A. Sommerfeld, 'Mathematische Theorie der Diffraction', Math. Ann. 47, 317 (1896). E. Hecht, Optics, 5th ed., §10.3.4 (straight-edge Fresnel diffraction, the Cornu spiral): edge I₀/4, first maximum 1.37 I₀ at w≈1.22, first minimum 0.78 I₀ at w≈1.87. M. Born & E. Wolf, Principles of Optics, 7th ed., §8.7.