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Status: exact $E_{47}$ projector combined with verified HBr constants and an exploratory numerical routing construction.

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First-principles formalism

The molecular side supplies measured or tabulated HBr constants. Independently, the canonical finite-dimensional operator is

$$ K=(C-6I)(C-30I) $$

on

$$ V_2^{\otimes3}. $$

Let the columns of $N\in\mathbb C^{125\times47}$ form an orthonormal basis of $\ker K$. Then

$$ P_E=NN^\dagger $$

satisfies

$$ P_E^2=P_E, $$

$$ P_E^\dagger=P_E, $$

$$ KP_E=0, $$

and

$$ \operatorname{Tr}(P_E)=47. $$

Logical deductions

• The spectral projector is exact and independent of HBr.
• HBr constants can be used as inputs to an explicitly defined numerical experiment.
• Multiplication by $125$, reduction modulo $125$, or routing a residue to $47$ is an algorithmic encoding chosen by the modeler.
• A small projector residual verifies the linear algebra, not a molecular law.