Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A singular history covariance can have nonunique coefficients

Example

Let Ω={−1,1} have equal probability, let Z(ω)=ω, and set Xt=Z for every t∈Z. For n=2 and h=1, the history covariance matrix is singular and the normal equations have multiple coefficient vectors, but they all give the unique predictor class [Z].

Facts & Assumptions

Given: The two-point uniform probability space, Z(ω)=ω, Xt=Z for all integer t, and the prediction target X3 with history X1,X2.

[F1]

A real random variable on a finite probability space is a map from outcomes to R; event probabilities and expectations are the corresponding finite weighted sums. Finite probability spaces, outcome weights, events, and event probabilities Real random variables on finite probability spaces and their finite distributions Expectation of a real random variable on a finite probability space

[F2]

Variance and covariance are expectations of centered products. Variance, standard deviation, and covariance on a finite probability space

[F3]

A process is weakly stationary when its second moments are finite, its mean is constant, and its covariance depends only on the lag. Strict and weak stationarity for integer-indexed time series

[F4]

Finite affine L2 projection gives a unique minimizing class, and its covariance normal equations may have multiple coefficient solutions that represent that same class. Normal equations for best affine L2 prediction

Verification

1.1givenF1F3

Each outcome has weight 1/2, so [F1] gives E[Z]=0 and E[Z2]=1. Every Xt=Z is square-integrable with mean zero, and every finite-dimensional law is shift-invariant because each coordinate is the same Z; hence X is weakly stationary.

2.1givenF2F4step 1.1algebra

The reversed history is (Y1,Y2)=(X2,X1)=(Z,Z) and the target is X3=Z. Direct covariance calculation gives Γ=(1111) and γh=(1,1)T; by [F4] the normal equations reduce to a1+a2=1, so the matrix is singular and has solutions (1,0)T and (0,1)T.

3.1F2F4step 2.1∎

The intercept is zero because every variable has mean zero; either displayed solution gives forecast a1Z+a2Z=Z and zero error, so they represent the same unique minimizing class by [F4].

Depends on

Used by

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Dependency tree · two levels

26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources