Exercise PLAN-509

The unique parity split combines into the fixed all-zero balanced block f(A)/(2c); the other control port is the difference and can coincide with the sum on degenerate spectra. The factor2 is a fixed-scheme accounting identity, not exclusive success or universal optimality.

1. The parity averages are even/odd polynomials, sum to f, and are uniquely recovered by adding/subtracting f(x) and f(-x).
2. Definite-parity transforms equal the desired functions of Hermitian A, with input normalization/rescaling and compatible common-c branch encodings made explicit.
3. A new balanced selector controls the two completed parity encodings on the same signal workspace.
4. After recombination, the clean b0 port is (E+O)/(2c), the clean b1 port is (E-O)/(2c), and remaining terms are orthogonal to the clean signal subspace.
5. The fixed chosen block is f(A)/(2c) with input-dependent success probability; zero target output cannot be normalized conditionally.
6. The1/2 follows from fixed Hadamard matrix elements, not exclusive-success-port or global optimality; allowed degenerate spectra and alternative weighted schemes delimit the claim.

The polynomial split is f_e(x)=(f(x)+f(-x))/2 and f_o(x)=(f(x)-f(-x))/2. Direct substitution shows f_e is even, f_o is odd, and f_e+f_o=f. If f=e+o with e even and o odd, then f(-x)=e(x)-o(x), so adding/subtracting recovers these same formulas; uniqueness is as polynomials, not a claim that their evaluations on one spectrum uniquely determine them.

For the finite-dimensional Hermitian operator in the quantum circuit setting, write A=V diag(lambda_i)V†. An even polynomial's right singular-value transform equals its eigenvalue function because f_e(|lambda|)=f_e(lambda); an odd polynomial's balanced transform supplies the sign of lambda, giving f_o(lambda). Choose any unitary sign completion at zero eigenvalues; f_o(0)=0 makes that choice immaterial. Hence the parity-respecting construction implements the desired two functions, not an arbitrary mixed-parity singular-value function of a non-Hermitian matrix.

Input access and scale are real obligations. If A is not already given as an exact normalized block, choose a known rho>0 with ||A||<=rho and let B=A/rho. Obtain an exact block encoding of B and use the definite-parity polynomials g_e(x)=f_e(rho*x) and g_o(x)=f_o(rho*x). They retain their parities and g_e(B)=f_e(A), g_o(B)=f_o(A). A finite Hermitian matrix has such a rho. Concretely, for S=sqrt(I-B^2), the block matrix U_B=[[B,S],[S,-B]] is unitary: spectral functional calculus gives BS=SB and B^2+S^2=I, so its square is I and its clean block is B. Expanding each g in its finite definite-parity Chebyshev basis, implementing the corresponding qubitized blocks, and using phase-aware coefficient LCU gives each parity branch at a positive coefficient-norm factor (with a separate zero-block construction when necessary). This is existence/construction mathematics, not an efficient oracle-synthesis guarantee from the word Hermitian alone. Chebyshev coefficients, coefficient preparation, SELECT and input-oracle work must actually be supplied and charged in any implementation; approximation would require an additional error contract. Nothing here assumes ||A||<=1 without this normalization step.

Put E=f_e(A) and O=f_o(A). Implement the two parity branches at compatible clean signal registers s and a common positive scale c, meaning J†U_e J=E/c and J†U_o J=O/c with J|psi>=|0^r>_s|psi>. The scale need not be minimal. Workspace padding preserves a clean block. If existing nonzero-branch factors are c_e,c_o, take c>=max(c_e,c_o) and explicitly attenuate their accepted blocks by c_e/c or c_o/c, for example by a clean-qubit rotation whose00 matrix element is that ratio. Its clean register is included in s before combining; equalization/padding is not free and is not part of the assertion that the COMBINER adds just one qubit. A zero branch can be represented by a unitary with zero accepted block, for instance X on a signal qubit, at any chosen positive c; never divide by its zero coefficient norm.

Introduce exactly one new selector b and set SELECT=|0><0|_b tensor U_e+|1><1|_b tensor U_o, with V=(H_b tensor I)SELECT(H_b tensor I). For every normalized input |psi>, the complete projected-port transcript is V|0>_b|0^r>_s|psi> = |0>_b|0^r>_s (E+O)|psi>/(2c) + |1>_b|0^r>_s (E-O)|psi>/(2c) + |G_psi>, where (I_b tensor <0^r|_s tensor I)|G_psi>=0. Thus projection onto the FIXED b=0,s=0 port gives f(A)/(2c), a block encoding with declared factor2c. It is a projected block, not an assertion that the full unitary output has no garbage. Its success probability is ||f(A)psi||^2/(4c^2); if f(A)psi=0 it is0 and there is no normalized conditional output. Otherwise conditioning on this port gives f(A)psi/||f(A)psi||, with the exact coherent phase retained.

The coefficient1/2 is forced by this particular balanced H-SELECT-H and this chosen projector: each branch contributes the product of two Hadamard entries1/sqrt(2). It cannot be changed while leaving those operations and that accepted port fixed. This is the precise sense of unavoidable in the naive combination. It is not a universal lower bound on block factors or a state-independent claim about probability loss. The alternative control port contains (E-O)/(2c), not generically the sum, but it need not be useless or distinct on every allowed spectrum. For f(x)=1+x and A=0, a genuinely parity-mixed polynomial has E=I,O=0, and BOTH signal-clean control ports carry I/(2c). For E=0 they differ only by a minus sign. Conversely E=O gives a zero difference port. Polynomial parity alone does not preclude any of these operator degeneracies.

Other schemes can use different accepted projectors, exploit known zero/equal branches, or use coefficient-weighted LCU: factors c_e,c_o give sum factor c_e+c_o using amplitudes sqrt(c_e/(c_e+c_o)) and sqrt(c_o/(c_e+c_o)), rather than2*max(c_e,c_o). That comparison does not remove the literal1/2 from the fixed common-c balanced selected block. A claimed minimal normalization or a complete gate, input-preparation, postselection/amplification or readout complexity theorem would require separate analysis.

1. Derives and proves uniqueness of the polynomial even/odd split, distinguishing polynomial uniqueness from degeneracies after evaluating on spec(A). (1 weight)
2. Explains why definite-parity SVT gives the corresponding eigenvalue function for Hermitian A, including null eigenvalues; makes any A/rho input normalization, polynomial rescaling and oracle/preparation/equalization work explicit rather than silently assuming a unit-normalized available input. (1 weight)
3. Uses compatible clean signal registers and common c>0 to calculate H-SELECT-H: chosen b0,s0 block f(A)/(2c), other clean control block (f_e(A)-f_o(A))/(2c), flag-orthogonal garbage and the actual input-dependent success probability/zero-output boundary. (1 weight)
4. Explains the forced1/2 only for this fixed balanced common-c selected-port scheme. Refutes exclusive-success-port/global-optimality overclaims with an allowed degenerate-spectrum example and separates weighted LCU, workspace equalization, implementation and amplification costs. (1 weight)

No aligned Lean view has been admitted for this native core in this textbook checkpoint. This is not a claim that the underlying mathematics lacks a formal proof. Contributions must preserve this exact target contract and split.

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