Definitions

Mathematics

Every symbol used in the bench is written here and pointed at the paper. The analog is a schema for the logical relation Einstein named, not a model of α-decay. Shields 2026a, §5.

Prediction primitive

S = S_coarse ⊕ S_⊥

(1)

Primary cut of the complete individual. Read ⊕ as “together with.” Full reading is under the equation.

2026a §4

What equation (1) says

2026a §4

This is one of the load-bearing sentences of the program. It is a statement about one individual on a time-slice, not about an ensemble, and not about the present quantum state vector. The complete description is written with a cut the laboratory can actually make. Shields 2026a, §4.

Piece	What it is, in this program
S	Complete individual state. One system, one fact. If S is given, the first-event time T*(S) is given. There is no probability in that sentence.
S_coarse	The cell a preparation can write down: well, mass, coupling, known spectrum, exit threshold, a window around the crest. Shared by every run that “looks the same” in the lab book.
S_⊥	The complementary remainder of that same individual. Whatever the preparation did not list, but the law still needs. In this analog: the eight Fourier phases θ. The subscript ⊥ means “unread,” not geometric orthogonality.
⊕	Read “together with.” It names the cut. It does not add, multiply, XOR, or take a direct sum of vector spaces. Nothing is created that was not already in S; we only mark which part the lab can fix and which part it left blank.

What ⊕ is not

Not bitwise XOR from computing.
Not a Hilbert-space orthogonal direct sum.
Not two interacting subsystems glued after the fact.
Not “hidden variables of ψ.” The split is of the successor description Einstein asked for, not of the present algorithm.

What the cut does

Both sides listed → S is listed → T* is a number. Chance is not a term in the law.
Only S_coarse listed → many remainders still fit. T* is then distributed over that fiber. That distribution is the statistical theory of the unread remainder.
Reading more of the true S_⊥ must shrink Var(T*). If it does not, axiom A6 fails and Rank-1 is wrong for that system.

This analog, written out

S_coarse = ( a=1, b=-0.85, m=1, κ=1.15, {A_k, ω_k}, q_exit=1.35, q₀=0.02, p₀=0 )

S_⊥ = ( θ₁, …, θ₈ ) ∈ [0, 2π)⁸

Same well, different θ, different crest, different first-exit time. The laboratory that prepares the well and does not control phases is doing statistical mechanics of an unread remainder, not discovering chance in the ODE. Course, lessons 00–01.

T^*(S) = inf { t > t₀ : E[φ_S](t) ≥ E_c }

(2)

First-event time. inf means infimum — greatest lower bound — not infinity. T* is the earliest time the event set has already occurred. Full reading under the equation.

2026a §4

Rank-1 analog

Let q(t) stand for the crest coordinate of a concentrated lump. The conservative force derives from a biased well with a finite barrier. Superposed on it is a background drive with a finite number of definite Fourier phases — the analog of S_⊥.

m q̈ = F(q) + κ Σ_k A_k sin(ω_k t + θ_k)

(3)

Newton’s law for the crest. The sum is the unread remainder when the phases θ_k are not listed in the preparation. Given (q(0), p(0), {θ_k}), the ODE returns exactly one first-exit time, or none within cutoff.

2026a §5.1

F(q) = −( A q + B q² − 0.6 A q³ )

(4)

Force used in this bench. A = 1, B = −0.85, and the cubic restoring term 0.6 A produces a finite barrier. F = −dV/dq with V given next.

2026a §5.1

V(q) = (A/2) q² + (B/3) q³ − (0.6 A / 4) q⁴

(5)

Potential of the biased well. The event is not a fitted Γ from this V; it is a first crossing of q_exit.

2026a §5.1

T^* = inf { t > 0 : q(t) ≥ q_exit }

(6)

Event definition frozen for this analog (empirical Step 1). q_exit = 1.35. T* is linearly interpolated inside the last RK4 step.

2026a §5.1

Biased well V(q)

Brass event line is q_exit = 1.35. Initial crest q(0) = 0.02.

Remainder spectrum

ω_k = 0.7 + 0.35 k, A_k = 0.08 / (1 + 0.4 k), κ = 1.15, m = 1.

k	ω_k	A_k	θ_k true
0	0.70	0.0800	0.20
1	1.05	0.0571	1.10
2	1.40	0.0444	2.40
3	1.75	0.0364	3.70
4	2.10	0.0308	4.00
5	2.45	0.0267	5.20
6	2.80	0.0235	0.90
7	3.15	0.0211	1.80

Integrator

The paper leaves the ODE; the companion code integrates it. This bench uses classical fourth-order Runge–Kutta on the first-order system q̇ = p / m, ṗ = F(q) + κ drive(t; θ). Step Δt = 0.003, cutoff t_max = 90. No stochastic term. The event time is linearly interpolated inside the step that first crosses q_exit, which is the discrete stand-in for the infimum in (2).

q̇ = p / m
ṗ = ( F(q) + κ Σ_k A_k sin(ω_k t + θ_k) ) / m

RK4 on (q, p) with Δt = 0.003
T* = t_prev + dt * (q_exit − q_prev) / (q − q_prev)
     at first q ≥ 1.35; else no exit by t = 90

A = 1, B = -0.85, cubic = 0.6
ω = [0.70, 1.05, 1.40, 1.75, 2.10, 2.45, 2.80, 3.15]
A_k = [0.0800, 0.0571, 0.0444, 0.0364, 0.0308, 0.0267, 0.0235, 0.0211]
θ_true = [0.2, 1.1, 2.4, 3.7, 4, 5.2, 0.9, 1.8]

Statistics of unread structure

Var(T^*) = 1/(n−1) Σ (T_i^* − T̄^*)²

(7)

Sample variance of T* over the still-unread block, unbiased (n−1). Axiom A6 requires this to fall as more of the true remainder is revealed.

2026a §5.4

MAE = 1/n Σ |T_i^* − T_true^*|

(8)

Mean absolute error against the true individual’s instant. Reported alongside variance so a collapsed spread at the wrong time cannot hide.

2026a §5.4

θ_k^(m) = θ_k^true if k < m, else Unif[0, 2π)

(9)

Partial readout. The first m phases of the true individual are held; the remaining N−m are redrawn uniformly on the torus.

2026a §5.4

Intended field lift

Einstein’s positive program is not an oscillator. The analog is a scaffold. The intended lift is a continuous field, generally covariant, without singularities, with particles not inserted as extra objects. Shields 2026a, §6. This bench does not integrate (10). That is empirical Step 5.

∂_t² φ = Δφ − V′(φ) + ε N(φ, ∂φ)

(10)

A metastable lump is a regular concentrated solution. The barrier is a feature of V. N is a local coupling to a definite background, or to finer ripples of φ itself. The mark is a second lump written when a flux functional crosses threshold. No collapse. No external apparatus.

2026a §6

Constants in this bench

2026a §5.1

Symbol	Meaning	Value	Source
m	Inertia of the crest coordinate	1	§5.1
A	Quadratic well stiffness	1	§5.1
B	Cubic tilt of the well	−0.85	§5.1
c	Quartic restoring coefficient in F (0.6 A)	0.6	§5.1
κ	Background coupling	1.15	§5.1
N	Number of Fourier modes (dim S_⊥)	8	§5.1
ω_k	Mode frequencies, k = 0…7	0.7 + 0.35 k	§5.1
A_k	Mode amplitudes, k = 0…7	0.08 / (1 + 0.4 k)	§5.1
q(0)	Initial crest location	0.02	§5.1
p(0)	Initial momentum	0	§5.1
q_exit	Event surface	1.35	§5.1
Δt	RK4 step	0.003	companion code
t_max	Integration cutoff	90	companion code