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KO42 — the 1.287 Hz, in the mathematics

KO42 is how the HulyaPulse modulation

R(t) = 1 + α·sin(2π·f·t + φ₀)      f = 1/τ,  τ = 777,000,777 ns,  α = 10⁻³

enters a computation. There are exactly three mathematically honest ways to put the pulse into the math, and KO42 names all three. Each is implemented once, in ko42Modulation.ts, and verified against an independent oracle.

One constant, integer phase

τ = 777,000,777 ns is the primitive; f := 1/τ (display 1.287 Hz) and f·τ = 1 exactly. Phase is computed in integer nanoseconds (φ = (t_ns mod τ_ns)/τ_ns), so it never float-drifts and repeats over exactly one Zeqond at any time — even at epoch scale. The runtime α is the bounded 10⁻³ (so |α·sin| ≤ 0.1% by construction); the published paper value 0.00129 is retained for rendering the symbolic equation only (DISPLAY.ALPHA_K).

KO42.gauge — the pulse as an exact change of time (recommended core)​

Treat R(t) as a reparametrization of time, not a force. Zeq computational time s advances as ds = R(t)·dt, with the closed form

s(t) = t − (α/2πf)·[cos(2πf·t + φ₀) − cos φ₀]

Because R ≥ 1 − α > 0, s(t) is strictly increasing and globally invertible, and because f·τ = 1, s(t + τ) = s(t) + τ exactly — a Zeqond of wall time is a Zeqond of Zeq time. The print boundary is gauge-invariant. Every observable at a matching physical time is unchanged: this is the change-of-variables theorem, not an approximation.

Verified: integrating x″ = −ω₀²x to the same physical T with pulse-modulated steps vs uniform steps gives error ratio 1.00 — zero α-dependent bias. The zeqond advance computed in integer-ns is exactly τ to float epsilon at any t. For Hamiltonian systems the exact form is the Poincaré/Sundman time transformation (Hairer–Lubich–Wanner ch. VIII); the nano-zeqond engines' dt = dtSim·R(t) is its first-order version.

This is the honest reading of "the framework's clock and its math are the same opinion": the pulse is the framework's proper computational time — which is exactly what the state-contract architecture already treats the Zeqond as.

KO42.drive — the pulse as an explicit parametric drive​

If you want the 1.287 Hz to change the equation and do real work, drive a parameter with it — an explicitly labelled model term:

ODE  :  x″ + ω₀²·[1 + ε·sin(2πf·t)]·x = 0        (Mathieu)
PDE  :  □ϕ + μ²(r)·[1 + ε·sin(2πf·t)]·ϕ + λϕ³ = J (driven master eq. core)

This is genuine Floquet mathematics — parametric resonance, the same structure as cosmological preheating. The framework computes checkable things a wrapper cannot fake:

  • Resonance occurs at ω₀ = n·π·f → for f = 1/τ, n × 0.6435 Hz (first tongue π·f ≈ 4.0433 rad/s). Everything else stays bounded.
  • Growth rate μ = ε·ω₀/4 (Landau–Lifshitz §27).

Measured (ε = α = 10⁻³): at resonance μ = 9.75×10⁻⁴ vs theory 1.011×10⁻³ (ratio 0.964); off-resonance the excursion stays ~ε with no growth. In the nano-zeqond engine this is the opt-in driveEps option; ε = 0 leaves the default arithmetic byte-identical.

Claim scope (binding)

Drive results are true statements about Zeq-driven systems — a model family. They say nothing about undriven nature. The zeqond-average removes the drive at first order only; for nonlinear systems the O(ε²) ≈ ε²/2 secular term survives — compute and report it, never assume zero.

KO42.probe — the pulse as a falsifiable estimator​

Turn the assertion into an instrument. For any residual r(t) = data − S(t) (measurement minus the standard prediction), the modulation amplitude at f is the lock-in

ε̂ = (2/T)·|∫₀ᵀ r(t)·e^(−2πif·t) dt|,   noise floor  σ_ε ≈ σ·√(2/n)

Report ε̂ ± σ_ε. A detection would be extraordinary; an upper bound is a legitimate, publishable result. Verified: the estimator recovers an injected ε = 3×10⁻³ buried in σ = 0.05 noise, above the floor.

What KO42 does not claim​

Per the framework's own Constants Charter, the constants are definitional (like the SI second): no universality, no cosmological origin, no proper-time modulation of nature is asserted by the runtime. KO42 is the honest machinery for putting a chosen clock into computation — provably (gauge), explicitly (drive), or measurably (probe). It never silently multiplies a physics answer by a time-dependent factor; that anti-pattern (v4.0 value × R(t)) was removed and does not return.

In code​

import {
  integerPhase, carrier, gaugeTime, gaugeWallStep,
  driveFactor, resonantOmega, floquetGrowthRate,
  lockInAmplitude, lockInNoiseFloor,
} from "@zeq/api-core/lib/ko42Modulation";

// gauge: advance Zeq-time by dsSim, exact wall step
const dtWall = gaugeWallStep(dsSim, tSeconds);

// drive: resonance + growth for a tuned oscillator
const w0 = resonantOmega(1);            // ≈ 4.0433 rad/s (0.6435 Hz)
const mu = floquetGrowthRate(w0, 1e-3); // ε·ω₀/4

// probe: recover a modulation amplitude from residuals
const { epsHat } = lockInAmplitude(residualSeries, dtSeconds);

Derivation and full benchmark: MODULATION-REINTRODUCTION-DESIGN.md and audits/phase12-ko42-modulation/ in the framework repo (7/7 oracle, 8/8 unit).