22. Learning several charts without inventing several worlds
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The exact atlas identities constrain the meaning of a fitted model, but independently trained predictors need not satisfy them. Here is a simple coherent reconciliation with a precise objective.
Fix a finite-dimensional represented space \(X\). It may be the observed signal space or a predictive quotient, but it must be the same space for all atlases in this section. Let \(A_a:X\to C_a\) and \(S_a:C_a\to X\) satisfy \(S_aA_a=I\). Let \(G_a:C_a\to C_a\) be linear fitted maps. Pull them back to object space:
For positive weights summing to one, define
Proposition 21: Coherent reconciliation and coordinate-stable transport
The family \(\widetilde G_a\) is consistent and satisfies every atlas intertwining identity. The operator \(\overline L\) is the unique minimizer of
over linear maps on \(X\), where \(\|\cdot\|_F\) is the Frobenius norm. For any target operator \(L_*\) and any chosen operator norm,
Further, the norm \(\|c\|_a=\|S_ac\|\) on \(\mathcal C_a\) makes every exact transport \(T_{ba}=A_bS_a\) an isometry. A represented transition \(A_bLS_a\) has exactly the same induced Lipschitz constant in these norms as \(L\) has in object space.
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Proof
Coherence follows from Proposition 2 because every chart realizes the same \(\overline L\). Expanding the squares gives
The weighted cross term vanishes, proving the unique minimizer. The target bound follows from the triangle inequality. The pullback expression is a norm because \(S_a\) is inverse to \(A_a\) on \(\mathcal C_a\). Finally, \(\|T_{ba}c\|_b=\|S_bA_bS_ac\|=\|S_ac\|\). The same identity for a transition, and the bijection between \(X\) and \(\mathcal C_a\), establish equality of Lipschitz constants. \(\square\)
This reconciliation is ordinary weighted least squares in a common semantic space. It does not prove that consensus is correct, reduce statistical error under arbitrary dependence, or turn deterministic recodings into independent evidence. The output must also be reanalyzed into a realizable coefficient vector. An independently fitted redundant output can have an additional inconsistency residual \((I-A_aS_a)G_aA_ax\) before reconciliation.
For Euclidean coefficient errors, conditioning remains important. If \(A_a\) has smallest singular value at least \(m_a>0\) and largest singular value at most \(M_a\), its canonical synthesis \(A_a^\dagger\) has norm at most \(1/m_a\), and \(\|T_{ba}\|\leq M_b/m_a\). An arbitrary left inverse can behave much worse away from the consistent coefficient space, so the canonical-synthesis qualification matters.
A learned atlas can therefore be parameterized within prescribed singular-value bounds, with synthesis tied to its inverse or pseudoinverse. This gives a construction contract. It is not a claim that every unconstrained training procedure respects those bounds.
When atlases change over time but represent the same state, using the pullback norms avoids multiplying arbitrary chart condition numbers at every step. Exact transport telescopes to the direct transport. Numerical synthesis errors and genuinely inconsistent predictions remain defects and must still be bounded.