What changes when the phase turns?
Turn the phase wheel. In this worked state, two outcomes each stay at 50%. Then advance the five and six position clocks to see when they return together.
Begin with one allowed lattice example.
The word 11000011 is a valid codeword in the stated construction. Lift its entries to an integer vector r, then divide by √2. Its squared length is two, so it is one E₈ root.
- Carry forward
- The specified code and Construction A scaling
- Added at this step
- The stated E₈ code construction and explicit chart
- Operation
- Lift 11000011 to r/√2
- Result
- One E₈ root with squared norm 2
In words: Lift 11000011 to r/√2
Why this step works and what it assumes
The valid word, E₈ lattice, compatible complex structure and explicit chart are supplied together. This example does not turn an arbitrary binary word into a physical quantum state.
Read the original calculationTry the individual controls
Try this. Press Quarter turn and watch the two probabilities. Then advance the five and six position clocks until both return after thirty steps.
Read the two outcomes
This worked state has two equally likely outcomes. Each has probability fifty percent.
Turn the common phase
A quarter turn changes the state’s common phase. The two probabilities stay at fifty percent through all four turns.
Compare two returning clocks
Advance the five position and six position clocks together. After thirty steps, both are back at their starting positions.
Explore the full interactive calculation
From a binary message to a quantum state
In the specified mapping, eight zeros and ones supply eight real coordinates. Pairing them gives four complex amplitudes. Their normalized squared magnitudes give the probabilities of four possible measurement outcomes.
Apply a quarter turn: it changes one common phase while those probabilities stay fixed. The other tabs let you compare the distinct operations of advancing clock phases and changing their labels.
Follow one codeword into a Bell state.
The valid word 11000011 lifts to the integer vector below. In the Construction A scaling r/√2 it is an E₈ root of squared norm two.
The specified chart pairs adjacent real coordinates: zₖ = r₂ₖ₋₁ + i r₂ₖ for k = 1, …, 4. It produces four complex amplitudes, ordered as |00⟩, |01⟩, |10⟩, |11⟩.
After normalization, the two nonzero amplitudes have the same phase. Ignoring that one common phase gives the Bell ray. Measuring in this basis yields 00 or 11 with equal probabilities; the outcomes are perfectly correlated.
0 quarter turns. Every amplitude is multiplied by i at each turn. The four probabilities and the phase independent ray stay the same.
Check the quantum meaning with an operator.
Eigenvalue +1; exact vector check passes. X ⊗ X swaps 00 and 11, leaving their equal superposition unchanged.
What does this prove about the larger dictionary?
The common +1 stabilizers are XX, ZZ and −YY. This is one worked example in the exact sixty ray correspondence. J relates four phase representatives of each Gaussian line. The specified chart and its equivariance are essential; equal counts alone would not produce a quantum dictionary.
This is a finite representation correspondence. A physical preparation or measurement instrument has the premises of the source and reader construction.
Codeword and allowed supportComplex structure JThe exact chart and Pauli checksThe finite dictionary contains sixty rays in fifteen orthogonal four ray contexts. This reader works through one explicit example, including its chart, quarter turn and Pauli operators. The full correspondence and a physical measurement instrument retain their original source premises.
These finite actions make the specified clock arithmetic visible. They do not assign seconds, prove a physical recurrence or identify every operator in the quantum source. The G₃₁ compiler clock and coherent evolution retain their own representations.
Inspect the original actions and source linksHow this fits into the whole theory
The worked state uses the specified mapping from E₈ coordinates. The clock comparison separately shows how finite return counts work.
What this experiment establishes
The finite correspondence requires the stated lattice, compatible complex structure and explicit chart together. It does not by itself implement a physical quantum instrument, universal quantum computation or a clock period in seconds.