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KO42 — The Metric Tensioner

KO42 is the framework's precision contract. It guarantees every composition of operators stays within ≤0.1% error of the ground-truth analytical result. Without it, Zeq would be just another numerical library.

The equation​

KO42.1 (automatic, enforced by the kernel):
  ds^2 = g_{mu nu} dx^mu dx^nu + alpha × sin(2 pi × 1.287 t) × dt^2

KO42.2 (manual, user-controlled):
  ds^2 = g_{mu nu} dx^mu dx^nu + beta  × sin(2 pi × 1.287 t) × dt^2

The extra term is a small, Zeqond-periodic perturbation of the time-time component of the metric. It rings at exactly the HulyaPulse.

What it does​

Given any composition C_1 o C_2 o ... o C_n of operators, KO42 computes the induced numerical error band. If the band exceeds 0.1%, the kernel:

  • KO42.1 mode: increases alpha until the error band tightens under 0.1%, or splits the computation across more Zeqonds.
  • KO42.2 mode: returns the error band to the caller as cko.error_band and marks the CKO as over_band: true. The caller decides whether to accept it or re-run with a different operator composition.

Either way, the ≤0.1% bound is either honored or visible. There is no silent degradation.

Why 0.1%​

0.1% is the bound beyond which most physical models start to diverge from reality faster than measurement error. It's also the threshold where double-precision floating-point arithmetic starts to accumulate non-trivial roundoff across long compositions. Below it, the framework is indistinguishable from analytical ground truth for all reasonable inputs; above it, we stop calling it a framework and start calling it an approximation.

Interaction with the 7-step protocol​

Step 1 of the wizard protocol is "KO42 is mandatory." This means:

  • You cannot opt out of KO42 for any protocol call.
  • You cannot compose more than 3 operators plus KO42 in a single call (step 2 — operator limit ≤ 4).
  • You cannot submit parameters that would force the error band above 0.1% without explicit acknowledgment (step 4 — precision imperative).

The kernel enforces all three. Attempts to bypass are returned as KO42ViolationError.

Inspecting KO42 in CKOs​

Every CKO includes KO42 provenance:

{
  "protocol": "zeq-propagation",
  "zeqond_at": 72380014.8,
  "operators": ["QM9", "NM23", "GR31"],
  "ko42": {
    "mode": "KO42.1",
    "alpha": 0.00129,
    "error_band": 0.00043,
    "within_bound": true
  },
  "E": { ... },
  "signature": "..."
}

error_band: 0.00043 means 0.043% — well inside the 0.1% bound. This is the typical case.

Theoretical background​

KO42 is derived in Section 4 of the Zeq paper (DOI 10.5281/zenodo.18158152). The key result is that a sinusoidal perturbation of the metric at exactly the HulyaPulse frequency creates a resonant averaging over Zeqond windows that damps numerical error without introducing bias — the average of sin(2 pi × 1.287 t) over one Zeqond is exactly zero, so the perturbation vanishes in the ground-truth result but persists in the error correction.

This is why the framework's clock and its math are the same opinion: you cannot decouple KO42 from the HulyaPulse without losing the ≤0.1% guarantee.