Executable shape search · no stored target comparison

Find the hidden shape before naming the number

The checker does not begin with an earlier numeral. It searches for a nonempty space whose transformation changes every position and reaches the whole space. Reversibility must then follow.

The intrinsic shape rules

Find a nonempty space X and a function F : X → X. The first two rules define the shape. The third must follow and is audited:

F(x) ≠ x
y = x or y = F(x)
derived: F(F(x)) = x
Search finite candidate shapes

Candidate labels D = {0,1,2,3,4,5,6,7,8,9}

Current shape certificate

waiting for a candidate
No candidate has been checked. Acceptance depends only on the shape rules above.

The recipe world

Transform the shape detector into frequencies

After the complete search, the acceptance signal has one spike. Its discrete Fourier transform has equal magnitude at every frequency, while phase carries the spike's address.

Frequency coefficients: arrow = phase, every magnitude = 1

Inverse transform: the unique spike returns

Finite-transform boundary: with ten labels this signature locates the spike modulo ten. The declared candidate box selects its displayed representative. The continuous transform on the integers gives the related phase-winding picture developed in the tutorial.

Proof boundary: the program exhausts the declared finite candidate box. The accompanying structural proof is stronger: choose any position x; fixed-point freedom supplies a different partner F(x); transitivity says every position is one of those. Applying transitivity from F(x) back to x proves that the transformation undoes itself. The probability panel is not used by the checker.