[
Verse 1
]:

Two‑to‑one mapping, binary domain,  
Hidden period, xor‑chain.  
Initial ket zero, tensor ket zero,  
Hilbert space span, uniform hero.  

Hadamard on first --- superposition wide,  
Oracle U‑f, second register tied.  
Entangled outputs, f of x equals f of x xor s,  
Clusters emerge, quantum finesse.  

Second Hadamard on the primary array,  
Measure auxiliary --- dot product null, mod two display.

[
Chorus
]:

Unmask the period, linear trials,
Orthogonal vectors, classics defiled.
Xor concealment, function’s veil,
Simon’s protocol --- exponential prevail.

[
Verse 2
]:

Function f from n‑bits to n‑bits concealed,  
One‑to‑one on cosets, subgroup revealed.  
Superposed inputs query oracle en masse,  
Post‑oracle state, sum over x, ket x, ket f of x amassed.  

Quantum Fourier on first collapses to y,  
Orthogonal to s, modular sky.  
Repeat order n, independent samples call,  
Gaussian elimination resolves s from the scrawl.

[
Bridge --- nu‑metal plunge 
]:

State evolution, ket zero tensor ket zero ascends,  
Hadamard tensor identity, coherence extends.  
Oracle entangles, conditional f appends,  
Hadamard again, measurement bends.  

Essential for hidden subgroups’ quest,  
Cryptographic lattices put to the test.  
Direct delineation, orthogonal span,  
Precise extraction, revelation grand.

[
Breakdown --- tempo drop, distorted samples
]:

Forsake  exhaustive pairwise yoke,
Adopt ensemble’s coherent invoke. 
No classical drudgery, no combinatorial toil,  
Quantum disclosure, periodic coil.  

No duplication, no protraction, oracle’s decree,  
Periodic truth in entanglement’s sea.

[
Verse 3
]:

From  superconducting qubits to photonic frames,  
Simon’s schema endures in experimental domains.  
Quantum cryptanalysis, precision campaigns,  
Unveiling periods with oracle refrains.  

Intrinsic to Fourier sampling’s expanse,  
Harmonic deconstruction, spectral dance.  
Theoretical lodestar, superiority’s chance,  
Reconfiguring hidden periodic trance.

[
Final Chorus
]:

Unmask the period, linear trials,
Orthogonal vectors, classics defiled.
Xor concealment, function’s veil,
Simon’s protocol --- exponential prevail.

[
Outro / cadence
]:

Extract --- Orthogonalize --- Resolve --- Transmute.
Extract --- Orthogonalize --- Resolve --- Transmute.

Periodic constructs, forever demystified.

SIMON'S PERIODIC ENIGMA [R1]

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