Math · h-of-x

The first equation.

Sundog's public math starts with a narrow inverse: if an eligible photo gives the 22° halo as a scale ruler and both parhelia are visible, the side-glint offset recovers hidden sun altitude. This page keeps the handle, the measurement, and the failure boundary in the same room.

Interactive handle

Move the hidden altitude.

The sun altitude is hidden in the photograph. What moves in the image is the parhelion offset, measured in units of the 22° halo radius.

Sun altitude h 25.0°

The slider moves the hidden altitude. The sketch updates the measurement you would see in the photograph.

normalized offset x = 1.103

pixel example 132.4 px

recovered h(x) 25.0°

Eligibility

The boundary is part of the math.

h-of-x is valuable because it says where it can be used. Without that gate, a pretty inverse becomes a storytelling machine.

Allowed

Bilateral parhelia, independent 22° halo anchor, and a photo route that does not infer the ruler from the answer.

Not enough

One flanking spot, a named arc with no calibrated route, or a rendered layer that only supplies vocabulary.

Structural stop

CZA (circumzenithal arc) cutoff near 32.2°, tangent-family merge near 29°, and coverage-gated supralateral or tangent routes stay outside this handle unless their own receipts promote them.

Current readout

How h-of-x sits inside the larger apparatus.

Geometry

Promoted handle

The parhelion-offset inverse is the current image-recoverable handle. The seven-photo calibration pass keeps it narrow and inspectable.

Falsifier

Failure first

The structural boundary map asks whether traceable systems fail where the closed-form inverse loses identifiability.

Alignment

One surface, not the whole field

Bayes, mesa, mirror alignment, and game workbenches test the same posture under other substrates; they do not enlarge this optical claim by rhetoric.

Generality

Same discipline, harder substrates

The generality umbrella carries the same boundary-first habit into Riemann, Yang-Mills, Navier-Stokes C1, ARC, P-vs-NP, and Three-Body transfer tests.

Faraday

The same move, made exact

In classical electromagnetism the hidden-state-from-a-visible-trace move comes out exact. Shadow Faraday closes the induction loop from a local reading alone; its Phase 7 boundary audit then finds the sharp edge — an Aharonov–Bohm setup where the local field shows nothing, yet a loop around it reads the hidden flux exactly. A visible trace carries a hidden state — until the trace has to become a loop.

Origin: the H(x) story Open atlas math Define parhelion Open boundary map Open alignment primer Open generality umbrella Open Faraday receipt Read geometry roadmap