The Born rule as the seating law of helical patterns

Quantum mechanics postulates the Born rule; FDT derives it. A pattern meeting a probe is one helix, and it seats whole in the first port whose closure drive overtakes the resident disturbance of the non-empty omnium. Three routes return the same law, Pr(k) = A_k², for every port count, every carrier and every coupling strength.

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The record

What every quantum experiment shows

Send single patterns one at a time into a probe with ports ê₁ … ê_N, a beam splitter, an interferometer or a Stern–Gerlach magnet, and count where each one lands. Every such experiment returns frequencies that follow one law. A pattern prepared in the state ψ lands in port k with frequency

Pr(k) = |⟨ê_k, ψ⟩|²

for light, electrons, neutrons, atoms and molecules alike. Quantum mechanics takes this weight as a postulate, the Born rule of 1926, and attaches no mechanism to it. The record carries more than the port frequencies, and the derivation has to return all of it.

The record the derivation answers to, and where the seating law reads each part.
What the record shows	How the seating law reads it
Single patterns sort into ports with frequency \|⟨ê_k, ψ⟩\|², whatever the carrier.	The pattern’s energy share in port `k` is `A`_k², and it seats there with that frequency (I-⁠4, I-⁠5).
Arrival times of single quanta follow the delivered intensity, through one detector or many.	The seating instant is uniform along the delivered fraction of the stroke and independent of the port.
A pattern found in one port is found there again by a second probe with the same ports.	Phase fixation: the seated pattern keeps its projection onto the seated port’s mode (I-⁠4).
An unexcited mode, read by homodyne detection, shows Gaussian quadratures, independent of phase, identical from mode to mode and unchanged by every passive linear network.	The resident disturbance of the non-empty omnium in that mode (0.1).
A shared state seats in port pairs with frequency \|Ψ_kl\|², and local frequencies never show the distant setting.	One pattern with a shared bivector grade (I-⁠4).

The FDT reading

0.1 One substance, I-⁠4 the quantum face, I-⁠5 closed patterns, P-⁠2 nodal locking

A pattern meeting a probe is one helix. Its projection onto the plane of port k is a point of that plane, ψ_k = A_ke^iφ_k, with A_k the projected radius and φ_k the helical phase (I-⁠4). The response coefficients ε₀ and μ₀ of the omnium make the stored energy a quadratic form, so the energy the pattern brings to port k is the fraction ϖ_k = A_k² of one quantum, and the shares add to one (0.1, 0.1a). A closed pattern is integral: each seating places the whole quantum in exactly one port (I-⁠5). The Born rule answers one question, how often each port seats the pattern, and three routes with different machinery answer it.

The frame function. The helical phase is unresolved at observation, so a port’s seating frequency depends on its share alone. Every pattern seats somewhere, so the frequencies add to one for every share vector, which forces them to be additive in the share. Cauchy’s equation leaves one answer: the frequency is the share.

The energy ledger. A bright stroke of n identical patterns delivers the energy ϖ_knE_q to port k, and that energy arrives as whole quanta. The count in port k is the fraction ϖ_k of the stroke; any other seating law would deliver to the port an energy different from the energy the stroke carries there.

The seating mechanism. The third route supplies the single event. No mode of the omnium is empty: each port mode carries a resident disturbance, the superposition of the medium’s strokes in that mode with their helical phases unresolved (0.1, I-⁠4). Independence across ports and invariance under every change of port frame fix its law. The resident energy Y_k of each port, in units of its mean, is exponentially distributed, independently from port to port. As the stroke is delivered, the closure drive of each port rises in proportion to its share, and the pattern seats, whole, in the first port whose drive reaches its resident energy:

ϒ = −ln(1 − η), ϒ_k = ϖ_kϒ η, the delivered fractionk_* = arg min_k Y_k/ϖ_k, Pr(k_* = k) = ϖ_k = A_k²

This is the nodal locking of P-⁠2 made quantitative: the quantum locks into the port whose resident disturbance is quietest relative to the pattern’s own drive. The drive grows without bound as the undelivered remainder vanishes, so every pattern seats within its own stroke. The same rule returns the time-resolved law, with the seating instant uniform along the delivered fraction, and the update: after seating, the pattern keeps its projection onto the seated port’s mode, and its projections onto the other ports return into the seated closure.

The rule has a picture. The resident shares ϑ_k = Y_k/ΣY_j are spread uniformly over the simplex of shares, and the pattern’s share point ϖ cuts that simplex into cells. For three ports, the cell of port k is the triangle with its apex at ϖ over the edge opposite the vertex ϑ_k = 1, and its area is the fraction ϖ_k of the whole. The Born weights are the barycentric coordinates of the pattern’s share point. For two ports the Hopf map carries the resident disturbance to a point uniform in height on the two-sphere, Archimedes’ hat-box theorem, and the pattern seats in port 1 when it leans further toward port 1 than the resident disturbance does:

Pr(1) = ½(1 + cos Θ) = cos²(Θ/2)

Only the exponential lawAmong resident laws drawn independently from one common law, the exponential is the only one under which each port’s seating frequency depends on its own share alone, at three ports or more: Yellott’s theorem. A Gamma law of shape two at ϖ = (0.2, 0.3, 0.5) seats at (0.140, 0.271, 0.590). The figure lets you switch the law.

InterferenceThe shares in the probe’s own frame are blind to every helical phase. A probe whose ports are rotated reads shares that depend on the relative phases φ_k − φ_j: interference is this frame dependence of the shares of one helix.

I-⁠6The port share is not a density coordinate. ϖ_k reaches both ends of its range, 0 and 1, while every α lives on the open interval. None of ϖ, ϑ, Y, η or ϒ stands in a force, optical or density slot.

The seating race. Drag the pattern’s share point ϖ inside the triangle, then seat patterns. In each seating every port draws a fresh resident energy Y_k, the closure drives rise in proportion to the shares, and the first drive to reach its threshold seats the pattern whole. Each dot is a seating’s resident share point, coloured by the port that seated it; it lands in that port’s cell. The bars count the seatings against the shares, which are the cell areas. Switch the resident law to watch every law but the exponential miss the shares, and open the two-port view for the Hopf picture. The draws are simulated live in the browser.

I-⁠4, I-⁠5, P-⁠2

The tests

T-⁠1 to T-⁠4 Convergence tests against the record

The derivation recovers a law the record already contains, so its four tests are convergence tests, and each names the result that would end it.

T-⁠1
The port law. Seating frequency A_k² at every complete port set, for every carrier and every coupling. The record agrees. A reproducible departure from A_k² at a complete port set ends the derivation.
T-⁠2
The seating instant. Uniform along the delivered fraction, independent of the port count and of the port. The record agrees: arrival-time histograms follow the delivered intensity through one port or many. An arrival-time profile that depends on the port count ends the seating rule.
T-⁠3
The joint law. A pattern shared by two probes seats in the port pair (k, l) with frequency |Ψ_kl|², and local frequencies stay blind to the distant frame. The record agrees. A local frequency that follows the distant frame ends the joint reading.
T-⁠4
Coupling blindness. Frequencies are independent of the coupling strength and of the common resident scale, because the coupling sets only how fast η advances. The record agrees: Born weights do not move with detector gain or bias once losses are counted as a port. A gain-dependent split at a complete port set ends the universality reading.

Five of the ten numerical checks that accompany the paper.
Check	What it tests	Result
V1	The port law at 2, 3, 5 and 8 ports, 2 × 10⁶ seatings each	Largest departure from ϖ_k: 2.7, 4.6, 6.2 and 5.5 × 10⁻⁴, against a one-sigma scale near 3 × 10⁻⁴
V4	Two ports, cos²(Θ/2), at Θ = 0.3, 1.0, 2.0 and 2.8 rad	0.9777, 0.7710, 0.2927, 0.0291 against 0.9777, 0.7702, 0.2919, 0.0289
V5	The seating instant at three ports	Mean delivered fraction at seating 0.5007, 0.5008 and 0.5006 by port
V6	Uniform resident energies at ϖ = (0.2, 0.8)	Port 1 seats at 0.1248; the port law requires 0.2000
V9	The resident law at ϖ = (0.2, 0.3, 0.5)	Exponential (0.1999, 0.3001, 0.5000); Gamma of shape two (0.1398, 0.2707, 0.5895); inverse-exponential (0.1696, 0.2974, 0.5330)

Readings declined

Use 6 Data yes, interpretation not necessarily

Standard reading

|ψ_k|² is postulated as the probability of outcome k, with no mechanism behind it.

FDT reading

The same weight is the seating frequency of a deterministic helical pattern: the squared projected radius is the pattern’s energy share in the port, and the seating law returns it (I-⁠4, I-⁠5).

Measurement reduces the state discontinuously.

Measurement is phase fixation: a closed pattern seats whole in one port and keeps its projection onto that port (I-⁠4, I-⁠5).

A superposed system has no definite properties until it is measured.

One helix has definite projections onto every port; what is unresolved at the seating instant is the helical state of the resident disturbance (I-⁠4).

The outcome of a single measurement is irreducibly random.

A single seating is fixed by the pattern and the resident disturbance together, and the universe is deterministic (I-⁠4).

Correlated outcomes of a shared state are an influence between two systems.

There is one pattern with a shared bivector grade: phase fixation at one probe fixes the projection the other receives, while local frequencies stay blind to the distant frame (0.1b, I-⁠4).

The shot noise of an unexcited mode is the fluctuation of an empty vacuum.

It is the resident disturbance of the non-empty omnium in that mode; its measured statistics are accepted in full and are the resident law (0.1).

Convergence

Use 2 Same source, same data

The limiting map is the identity. The FDT seating frequency of port k is A_k² = |⟨ê_k, ψ⟩|², the consensus Born weight, with no approximation and no regime limit, for every port count, every carrier reading and every coupling strength. The seating instant is uniform along the delivered fraction, the consensus arrival-time law of a single-quantum stroke; the update is the projection onto the seated port; mixtures read the statistical operator; and shared patterns return the joint weights |Ψ_kl|² with local frequencies blind to the distant frame. The derivation recovers every measured outcome frequency of the record to the precision at which that record agrees with the consensus law.

FDT adds the resident disturbance and the seating rule, so a single seating is fixed by ψ and Y together. An observable departure would need a handle on Y below its law, and there is none: reading Y means seating a pattern in that mode, which meets a fresh resident disturbance drawn by the same law, and preparing Y means injecting a known disturbance, which is a pattern and seats on its own. At this interface FDT supplies no additional input that changes an observable, and no divergence is claimed.

Two pieces sit outside this derivation and leave every frequency in it unchanged: the rate kernel by which a given probe’s coupling advances the delivered fraction, which is to be derived (O-⁠12), and the port modes of a given probe, which are modes of the confinement operator, to be derived (O-⁠9). Only their orthogonality enters here.