Open the claim ↗

Fill cells 000, 011, 101, 110. Look at all three faces. Now empty it and fill 001, 010, 100, 111. Look at all three faces again.

If every face agrees, must the far corner agree?
Read all answer branches

What that answer leaves out

Yes, the faces determine everything

Pairwise projections do not determine joint corner occupancy.

Excluded world. The odd-parity world has the same faces and an occupied far corner; it is left out.

Write the limit in your own words

No, inspect the corner as well

Keep the triple readout alongside the matching pairs.

Excluded world. An arrangement with unequal pair shadows is excluded from this construction.

Carry the limit forward

Only if the beads look alike

Matching appearance does not add a joint constraint.

Excluded world. The alternate parity world made from identical beads is left out.

Write the limit in your own words

The example measures a typical gap

This is a constructed possibility proof.

Excluded world. An observed system with a different joint distribution is left out.

Write the limit in your own words

Move the device

four occupied distinct binary corners; uniform pair shadows; atomic parity switch only

Model scope: SELF-AUDITED. Read the exact constraints and limits.

The rendered comparison

Even arrangement

Even xy orthographic projection: each pair bin is occupied.
XY
Even xz orthographic projection: each pair bin is occupied.
XZ
Even yz orthographic projection: each pair bin is occupied.
YZ
Even far-corner inset: empty.
Far corner

Odd arrangement

Odd xy orthographic projection: each pair bin is occupied.
XY
Odd xz orthographic projection: each pair bin is occupied.
XZ
Odd yz orthographic projection: each pair bin is occupied.
YZ
Odd far-corner inset: occupied.
Far corner

The face images are rendered orthographic bead silhouettes. The corner view clips away the opposite layer; the lattice is hidden for these measurements. This constructed example permits matching pairs and different triple occupancy.

Projection receipt and non-claims

What does this fail to establish?

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  • how large the gap is in practice. Both arrangements are constructed. This is a possibility proof and must be said out loud as one.
  • These are self-audited digital models, not independently verified physical devices.
  • No learner efficacy, transfer, accessibility equivalence, or vendor performance has been measured.
  • Unbounded and temporal devices remain text exercises; no faithful state machine is asserted for them.
  • The receipt exercise has no supplied totals or ceiling, so it supports no computed denominator here.
  • Atomic parity switching only. Loose-bead arrangements and emptying intermediates are not certified.
  • Text reports the projected and corner occupancies, but equivalence for non-sighted learners has not been user-tested.
Return to the claim

Without the device

Look away from the object, then answer the question again. This is a prompt for reflection, not a measured learning outcome.

If every face agrees, must the far corner agree?
  • Yes, the faces determine everything
  • No, inspect the corner as well
  • Only if the beads look alike
  • The example measures a typical gap