Alien Math · Quantum Wizard · QW-001 to QW-008
Quantum mechanics never cancels the bill
Eight effects that sound like cheating: probability flowing backwards, a spin-1/2 that reads as spin 100, a bomb found without touching it, a signal that outruns its own bandwidth. Each one is real. Each one moves the cost somewhere you weren't looking.
How these were run. All simulated on CPU with numpy, scipy and mpmath. Each run's record says how it could have failed and what it can't catch. One gap: the predictions were written in working notes before each run but filed to disk afterwards, in the same session, so they are not pre-registered in the strict sense the analog lab uses. One prediction failed and stays failed (ring revivals, below).
The board
Bends and bills
| run | what bends | our number | where the bill is paid |
|---|---|---|---|
| QW-001 | Waves moving only rightward push probability leftward (backflow) | cBM = 0.0384506 ± 10−6, independent method | At most 3.85% of the probability, only inside a time window; the net flow still goes forward |
| QW-002a | The same thing on a ring | 0.0329 at N = 256, rising toward cBM, never above it | Revivals don't help; a full revival kills backflow outright |
| QW-002b | Clockwise-only light turns counterclockwise in patches | Reversed pockets: 3.6% of the area, 0.4% of the light, all wrapped around +1 vortices | The reversal lives where there is almost nothing |
| QW-003 | A spin-1/2 reads out as spin 100 (weak values) | Pointer shift −99.99g at Aw = −100 | Postselection keeps 1 run in 10,000, and total information is exactly the standard limit |
| QW-003 | …but the survivors carry the information | 0.01% of runs keep 99.99% of the Fisher information | The pointer can never move more than its own width |
| QW-004 | Watching freezes evolution (Zeno) | 99.75% frozen with 1,000 checks | Each check must be fast and clean |
| QW-004 | Watching at the wrong rate speeds decay up (anti-Zeno) | 7.0× faster than Fermi's golden rule | Only with a structured, detuned environment |
| QW-004 | Find a bomb without touching it | 99.9% lossless; 73% at 1% loss per cycle | Loss per cycle sets the ceiling |
| QW-005 | A signal wiggles faster than its fastest frequency (superoscillation) | 2, 4 and 8× the band limit | Amplitude dims by exactly aN, about 0.5 to 1.2 decades per extra wiggle |
| QW-006 | Tunnelling time stops growing with barrier width (Hartman) | Delay saturates at 2.000; implied speed 8× at width 16 | Transmission 5×10−14; no signal front outruns light |
| QW-007 | Three pigeons, two holes, no two share (quantum pigeonhole) | P(same box) = 0 for every pair, n = 3 to 6 | Postselection succeeds 2−n of the time |
| QW-008 | A qubit refutes macrorealism (Leggett–Garg) | K3 = 1.5 against a bound of 1; survives any dephasing below Ω | The margin collapses fast and needs non-invasive measurement |
QW-001 · the most publishable thing here
The backflow constant moved, and we can say which way
A free particle with only rightward momentum can still have probability flowing left for a while. The largest fraction that can flow back is the Bracken–Melloy constant. Penz and colleagues (2006) gave 0.0384517. A 2025 paper (arXiv 2505.13184) revised it to between 0.038450556 and 0.038450568, with bounds it calls conjectural.
We used a different method: Gauss–Legendre truncation of the backflow operator at 31 cutoffs up to U = 114, then extrapolation in 1/U. It lands on 0.0384506 ± 10−6, siding with the 2025 value. That is a lean, not a decision: higher-order fits scatter by 2×10−6.
Why we are not announcing a closed form. Feeding the estimate to mpmath.identify returns a tidy formula. It also returned a tidy formula for 60 out of 60 random 7-digit control numbers. At this precision a closed form means nothing; a real hunt needs about 20 digits. The next step is to subtract the eigenfunction's known tail analytically, which should reach 9 or 10 digits.
The ring version (QW-002a) rises toward the line constant and never crosses it. The prediction was that revivals would push it higher. They don't, and that miss stays recorded.
QW-003 · weak values
Throw away 99.99% of the data, keep 99.99% of the information
Weakly couple a spin to a pointer, then keep only the rare runs where the spin ends up nearly opposite to how it started. The pointer in those runs moves 100 times further than a spin-1/2 should allow. That is the weak value, and it is real. It also never beats the quantum limit: the total information per trial is exactly 1.0000 in every case. What it does do is concentrate almost all of that information into the 0.01% of runs you keep, which matters when the detector is the bottleneck. This is the idea behind picoradian beam-deflection sensing (Dixon and colleagues, 2009), so it is known physics, shown cleanly.
QW-004 · Zeno, both ways
Watching freezes things, and watching wrong speeds them up
Check a turning qubit often enough and it stops turning: 99.75% frozen with 1,000 checks. Check it at the wrong rate, against a detuned environment, and it decays 7× faster than the golden rule predicts. Same knob, turned the other way. The Zeno bomb tester finds an absorber without a photon touching it 99.9% of the time with no loss, 73% at 1% loss per cycle, against 25% for the textbook single pass.
The Observer Notebook has live versions of both: the Zeno effect and the bomb you never touch.
A run of the Zeno and weak-value circuits on real quantum hardware is done and will be added after review.
QW-005 to QW-008
Faster than the band, faster than light, and three pigeons
- Superoscillation. A band-limited signal can wiggle 2, 4 or 8 times faster than its highest frequency over a short stretch. The amplitude there drops by exactly aN.
- Hartman effect. The tunnelling delay stops growing with barrier width and saturates at 2.000, implying a speed 8 times light at width 16. Transmission is 5×10−14 and the signal front never outruns light. The first attempt was wrong (energy filtering looked like a negative delay) and the fix is recorded.
- Quantum pigeonhole. With the right pre- and post-selection, no two of n pigeons share a box, for every pair, n = 3 to 6. Postselection succeeds 2−n of the time.
- Leggett–Garg. A single qubit scores K3 = 1.5 where any classical "it was always somewhere" story allows 1, and the violation survives any dephasing slower than the drive.
The angular backflow pockets in QW-002b always wrap oppositely wound vortices. Topology nearly forces that, so it is a consistency check, not a discovery. Optical backflow has already been observed in light.
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