Wave speed c = 0.499920 ± 0.000051 cells/step (known 0.5, 1.6e-4) recovered from the emergent two-source interference field of a…

Drive two in-phase point sources on a membrane that knows ONLY the local rule uⁿ⁺¹ = 2uⁿ − uⁿ⁻¹ + C²∇²uⁿ. Does the classic ripple-tank physics — a definite emergent wavelength λ = c/f, deep nodal lines on Young's hyperbolae |r₁−r₂| = (m+½)λ, λ/2 node spacing on the axis, one phase speed for every frequency — all come out of the raw stencil, quantitatively, including the computable lattice correction?

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The finding

Wave speed c = 0.499920 ± 0.000051 cells/step (known 0.5, 1.6e-4) recovered from the emergent two-source interference field of a raw leapfrog wave equation — the medium retunes λ to every drive per von Neumann's discrete dispersion (5-point sweep ≤ 3e-4), nodes sit on Young's half-integer hyperbolae, and incoherent intensity addition is rejected

Method

Replicate the module's stencil headless (hard in-phase sinusoidal sources, quadratic sponge) on a 600×400 oracle grid whose 100-cell adiabatic sponge cuts boundary reflections to R² ≥ 0.99 fit residuals (the module's own 18-cell sponge reflects a few percent and was measured to bias λ̂ up to 1.3% at λ=17 — an instrument lesson, not physics). λ̂ per snapshot from a least-squares fit of the source-row field to a two-circular-wave model a·cos(kr₁)/√r₁ + … (k the only nonlinear parameter; 6·MAD robust aggregation because ~1 snapshot phase in 8 has a near-degenerate secondary SSE minimum); uncertainty from 12 hardcoded noise seeds at 1%-of-max measurement noise. ĉ = sin(ω/2)/sin(k̂/2) inverts the discrete dispersion on the SCORER side only. Independent geometry gates: envelope-minima node spacing λ/2 on the axis, off-axis minima at half-integer path difference. Perturbation: drive λ ∈ {9,11,13,15,17} — λ̂ must track the von Neumann prediction pointwise and the continuum deficit of λ̂f from c must be negative, match the formula, and shrink monotonically. Rival: each source run ALONE through the same generator, intensities added.

Measurements, controls & cross-checks

Recovered sigma

5.1000e-5

Lambda measured cells

Value

12.9012

Se

0.0014

Von neumann prediction

12.90332

Rel

0.000163

Fit quality

Median r2

0.9938

Points per row

282

Noisy fits surviving mad

12/12

Lattice dispersion

Continuum c lambda f

0.496201

Bias measured pct

-0.76

Bias predicted pct

-0.744

Note

the deficit of λ̂f from c IS von Neumann's sin(ω/2)=C·sin(k/2), matched to 1.6e-4 and removed exactly by the scorer-side inversion

Axis nodes

Lambda from spacing

12.8445

Rel vs fit

0.0044

Count

6

Hyperbolae

Mean half integer residual

0.0165

Minima used

47

Perturbation

Lambda src

• 9
• 11
• 13
• 15
• 17

Lambda hat

• 8.856
• 10.885
• 12.902
• 14.919
• 16.926

C rec

• 0.4999
• 0.5
• 0.5
• 0.5001
• 0.5

Bias pct

• -1.6
• -1.04
• -0.75
• -0.54
• -0.43

Monotone to zero

true

Control

Name

incoherent intensity addition I = A₁² + A₂²

Coherent nodes

6

Coherent median depth

0.644

Incoherent max depth

0.072

Note

rejected — the deep λ/2 banding requires coherent AMPLITUDE addition; energies alone can never produce dark lines where both sources deliver energy

Module systematic

Onscreen lambda cycle mean cells

13.5385

Onscreen lambda flicker range cells

• 13.285
• 13.949

Module grid clean fit cells

13.219

Oracle lambda cells

12.9012

Note

EXECUTING the shipped module (not a re-typed replica) exposed what the pooled-median replica had hidden: the 660-step measurement stride strobes the 26-step drive period (660 = 10 mod 26), so the crest-median sequence cycles with period 13, and at 2 of the 13 strobe phases a crest dipped under the 0.015 detection floor, doubling its gap and dragging that median to ~2 lambda - the HUD's EMA flickered over 13.7-17.8 cells (c = lambda*f up to 0.73 vs model 0.50, +46%). Fix (this run's one module edit): _measure now folds aliased spacings (> 1.6x the smallest gap; true gaps 10-15 and doubles 24-27 are cleanly separated) back by halving before the median. The executed screen now shows lambda in [13.285, 13.949] (c 0.511-0.537 vs 0.50); the remaining bias decomposes as +0.32 cells integer-crest quantization (cycle mean 13.539 vs module-grid clean fit 13.219, inside the derived <1-cell bound) + 0.32 cells small-sponge reflection (13.219 vs oracle 12.901), all disclosed and gated.

Module certificate

Rung

validated + honest-module

Gates

K sha256-pin + 25-pair mechanical TS strip + executed statics === reference; L 12000 engine ticks of fl(1/120): executed shipped module bit-exact vs statics-built Float32 replica at EVERY wave tick (field u/uPrev, lambda, 132000 wave steps, exactly 1 wave-tick per 2 engine ticks, acc === 0, zero step deficit); M executed render(): HUD string and crest/trough thin-instance buffers === replica at mid-run and final tick, sha-pinned; N screen lambda reconciled through DERIVED ceilings only - strobe period 13 = 26/gcd(660 mod 26, 26) observed minimal and locked from measurement 22/200, executed EMA sits on the closed-form periodic orbit to 0.0 (ceiling 1e-12), orbit convex-bounded by its cycle medians, cycle mean within the derived 1-cell quantization bound of the module-grid fit and within 0.64 cells of the oracle recovery

Tampers

known_value x1.05: exit 1, gates B/C/C'/D/G fail, recovered unchanged (0.499920); cert sha flip: only K fails; cert SUBSTEPS 22->21: K statics MISMATCH and L/M/N fail (replica forked)

Executed final lambda

13.664609

Orbit deviation

0

What it reduces to

Young's two-source interference (Phil. Trans. R. Soc. 94, 1804): nodal hyperbolae at half-integer path difference, λ/2 axis node spacing — plus the d'Alembert/Euler wave-equation dispersion ω = ck (every frequency at one speed c) and, on the lattice, von Neumann's discrete dispersion sin(ωΔt/2) = C·sin(kΔx/2) (Charney–Fjørtoft–von Neumann 1950; Trefethen SIAM Rev. 24, 1982). Non-circular in the estimator sense: the generator contains ONLY the leapfrog stencil and a sinusoidal drive — no λ, no c = λf, no node law, no dispersion formula; the wavelength the medium selects is measured from the raw field and the known c is loaded only to score (tamper test: wrong c fails 5 gates with the recovery unchanged). Distinct from ?world=young (Huygens PHASOR sum, kinematic optics, no medium): here the MEDIUM ITSELF is simulated as a PDE and the interference is of physically propagating waves, with the lattice correction to c = λf predicted and matched. Companion to emwave (1-D Maxwell FDTD → c from μ₀ε₀): waves validates the 2-D scalar-wave layer every ripple-tank analogy in the optics arc stands on.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

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

Oracle

scripts/oracles/waves.reference.json

Sources

T. Young, Phil. Trans. R. Soc. 94, 1 (1804); Halliday, Resnick & Walker, Fundamentals of Physics ch. 17 & 35; L. N. Trefethen, SIAM Review 24, 113 (1982); Charney, Fjørtoft & von Neumann, Tellus 2, 237 (1950).