Deriving Born rule from Quantum Traction Theory Axiom 1-7

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Main Equation

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Plain English: every time the system and the detector land on the same tick – same “Reality Dimension (or world-cell) address” , the device records an outcome. Count how often outcome r happens across many such co-locations; that frequency equals the textbook number \langle\Psi|E_r|\Psi\rangle. The “probability” is just the limit of counting real events.

Why this is different from textbooks

  • No probability postulate. Textbooks assume the Born rule. QTT derives it from a tally of co-location events in a Planck-scale ledger.
  • Measurement isn’t magic – It’s defined. “Collapse” becomes a mundane update: we co-located on tick T, so we wrote a record. No spooky action, no global jump—just local bookkeeping.
  • Wavefunction = projection. The lab wave is a visible projection of deeper tick-level dynamics; randomness is epistemic (from mixing/averaging), not fundamental.

Special, testable predictions of the QTT rewrite

These are concrete ways the QTT picture can be probed or distinguished from “Born-as-postulate.”

  1. Gated-detector bias (address window). If your detector only accepts a subset of ticks (a gate window G), the observed frequencies shift in a precise way: \displaystyle p_r(G)=\frac{\langle\Psi|\,G^{1/2}E_r\,G^{1/2}|\Psi\rangle}{\langle\Psi|\,G\,|\Psi\rangle}\,. Prediction: tightening or delaying the gate changes rates via G, not by “altering the state.” Widen the gate and you recover the usual Born weight.
  2. Inter-trial spacing effect. If trials are spaced closer than the apparatus’s reset time \tau_{\rm reset}, residual tick-overlap causes small, quantitative deviations from the asymptotic frequency that vanish like e^{-\Delta\tau/\tau_{\rm reset}}. (Space trials widely and the deviations disappear.)
  3. Universal half-angle factor in symmetric splittings. In fully symmetric two-branch experiments, QTT predicts a geometric amplitude projection factor \displaystyle I_{\rm clk}=\cos\!\big(\tfrac{\pi}{8}\big)=0.9239\ldots that fixes subtle intensity ratios when the instrument is perfectly balanced. (This enters as a relative amplitude factor; flux normalization is preserved.)
  4. Coincidence-timing law. Two-detector coincidence rates depend on the overlap of their tick-gates: \displaystyle p_{r,s}(\Delta)=\frac{\langle\Psi|\,(G_A\!\star_\Delta G_B)^{1/2}\,(E_r\!\otimes F_s)\,(G_A\!\star_\Delta G_B)^{1/2}|\Psi\rangle}{\langle\Psi|\,(G_A\!\star_\Delta G_B)|\Psi\rangle}\,, where \star_\Delta shifts and overlaps the two gates by a delay \Delta. Prediction: move the coincidence window and the joint frequencies follow this overlap law.
  5. Convergence-rate signature. Because outcomes are sums of i.i.d. co-location indicators, frequencies converge as O(1/\sqrt{N}) with an extra, measurable prefactor set by the gate-overlap variance. Tune gates → tune the prefactor; the Born limit itself remains the same.
  6. Context stability. If two measurement contexts use the same effective gate G, QTT predicts the same asymptotic frequencies even if the hardware differs. (It’s the ledger-gate that matters, not the brand of detector.)
  7. Delayed-choice clarity. Delayed or advanced settings don’t retro-cause outcomes; they only change which co-locations are admitted by G. QTT reproduces standard delayed-choice results but predicts the minute rate shifts when timing windows change.
  8. Robust “no-signalling.” Because co-locations are local and counted, marginal frequencies on one wing are invariant under remote gate tweaks (after averaging). QTT matches the quantum “no-signalling” theorem for all practical gates.

Testings:

  • Gate-sweep test: vary the detector’s acceptance window width and delay; verify that p_r(G) follows the overlap law above and saturates to the standard Born weight as G\!\to\!I.
  • Spacing test: decrease the inter-trial interval below \tau_{\rm reset} and watch deviations shrink as e^{-\Delta\tau/\tau_{\rm reset}} when spacing is lengthened.
  • Symmetry test: in a perfectly balanced splitter, look for the predicted \cos(\pi/8) amplitude ratio in carefully normalized branch intensities (with total flux conserved).

FAQ (one screen)

Does this “change quantum mechanics”?
No change to the lab predictions—QuantumTraction explains them. The Born numbers come from counting co-locations in a deeper ledger rather than assuming a probability axiom.

Isn’t this just hidden variables?
No. The “tick” is a local address gate in Reality Dimension, not a global hidden parameter. QTT keeps microcausality and “no-signalling,” and reproduces standard interference.

Where’s the mystery?
Gone. Instead of a magical collapse, we have a precise admission rule (the gate) and a frequency theorem (the box above).

Coming in the next book release

We’ll publish the step-by-step proofs, the gate overlap calculus, and new experimental proposals in the next version of the book.

Get updates — quantumtraction.org/the-book

Shareable snippet

Born rule, no postulate: “Count the co-locations.”
\boxed{\lim_{N\to\infty}\tfrac{1}{N}\sum \mathbf{1}\{\text{co\text{-}loc}_r\}=\langle\Psi|E_r|\Psi\rangle}
One ledger, one law, zero knobs.

Published by Quantum Traction Theory

Ali Attar

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