4 Range cores and low-rank interfaces
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After storage and propagation have been organized, the remaining question is which subspace has actually been resolved. Local ranks govern the width of phase messages. Orthogonal factor updates carry that information into a small core with the same nonzero spectrum when the full range has been captured. An incomplete range requires an additional certificate, developed next.
4.1 Phase-local rank
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Let \(R\) be a newly generated Krylov residual and
Theorem 4.1: Minimal exact phase width
Let
Every phase-supported basis spanning \(\operatorname{range}(R_h)\) has at least \(r_h\) vectors, and a rank-revealing QR or compact SVD attains exactly \(r_h\).
This justifies nonuniform Krylov block widths by phase. Here rank means exact algebraic rank. Discarding a direction by a floating singular-value threshold is an approximation and requires its own error bound.
4.2 Low-rank interface messages
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If an inter-phase coupling has exact factorization
of rank \(r_{hg}\), then
This representation reduces storage and communication. It does not automatically reduce wall-clock time unless the factors are applied through batched or fused kernels. A Python reference implementation can lose to dense BLAS despite lower arithmetic and storage.
4.3 Exact QR-core updates
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Suppose
and append a new block \(U\). Set
Here the columns of \(Q_\perp\) form an orthonormal basis of the nonzero residual range. Zero residual directions are omitted.
Theorem 4.2: Exact QR-core update
Therefore the nonzero eigenvalues of \([A,U][A,U]^\top\) are exactly the nonzero eigenvalues of the small core Gram, with multiplicity. If \(\operatorname{range}(Q)=\operatorname{range}(A)\), the compact residual update also spans exactly \(\operatorname{range}([A,U])\). In that case the core has full row rank and its Gram matrix has no additional zero eigenvalues.
Proof
The first block product is \(QR=A\). The second is \(QC+Q_\perp R_\perp=QC+F=U\). The stated orthogonality conditions make \([Q,Q_\perp]\) an isometry on the compact core space. Left multiplication by this isometry preserves all nonzero singular values of the core, and squaring them yields the nonzero Gram eigenvalues. If \(Q\) spans exactly the old range and \(Q_\perp\) spans exactly the residual range, their combined range is precisely that of the appended factor. The core then has full row rank. Without this exact-range hypothesis, unused basis directions can introduce zero eigenvalues into the core Gram.
This is useful for overlapping genomic windows and small incremental variant updates.
4.4 Range-core spectral compression
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Suppose an orthonormal basis \(Q_r\) captures the factor range:
Then
Theorem 4.3: Exact range core
The nonzero eigenvalues of
are exactly the nonzero eigenvalues of
with multiplicity. If \(\operatorname{range}(Q_r)=\operatorname{range}(A)\), then \(BB^\top\) is positive definite on the core space and its entire spectrum is the nonzero spectrum of \(K\).
Proof
Since \(A=Q_rB\) and \(Q_r^\top Q_r=I\), one has \(AA^\top=Q_r(BB^\top)Q_r^\top\). The map \(x\mapsto Q_rx\) is an isometry from the core space onto \(\operatorname{range}(Q_r)\). On that range the two operators are conjugate, and \(AA^\top\) vanishes on its orthogonal complement. Hence their nonzero eigenvalues agree with multiplicity. If the captured range equals \(\operatorname{range}(A)\), then \(B\) has full row rank, so \(BB^\top\) is positive definite. Without that equality, the core may retain additional zero eigenvalues.
Thus the computational pipeline can terminate its data-streaming stage after the range is captured:
4.5 An exact core calculation
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Consider the orthonormal range basis and core factor
Direct multiplication gives
The large operator has eigenvalues \(4,2,0\). Its two-dimensional core has exactly the nonzero eigenvalues \(4,2\). The Gaussian quadratic form therefore has the law \(4Z_1^2+2Z_2^2\), whether it is computed from the full factor, the full Gram matrix, or the core.
Why the nonzero-spectrum qualification matters
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Take instead
The captured subspace contains the range of \(A\), but is larger than that range. Both Gram matrices have spectrum \(\{1,0\}\). Their nonzero spectrum is \(\{1\}\). Thus the nonzero spectrum of \(AA^\top\) cannot be identified with the entire spectrum of \(BB^\top\). Choosing the compact basis \(Q_r=(1,0)^\top\) instead gives \(B=(1,0)\) and the one-dimensional core Gram \((1)\). The extra zero records an unused basis direction, not an additional random contribution. This is why range containment and exact-range equality have different conclusions in Theorem 4.3.
The first example in this section has active rank two. It separates algebraic exactness from backend coverage. The spectral reduction and analytic packet theory remain valid, but the implemented directed positive-tail backend described in Chapter 9 requires active rank greater than two. Capturing a range exactly does not remove a later numerical method's hypotheses.