Finite Source-Graph Uniqueness and Normalized Energy-Shape Theorems for Chiral Spin Filtering

A finite-graph test of mirror-antisymmetric spin filtering How many independent chirality-odd sources can a molecular interface carry, and when must complete mirror energy profiles collapse to one shape? This paper answers both questions with a finite source theorem rather than a fitted spin-amplitude model. The native source-graph theorem is \[ \boxed{ \dim W_{ab} = \#\left\{[\Gamma]\in\mathcal C/\mathcal K: \chi_{ab}\big|_{\operatorname{Stab}(\Gamma)}\equiv1 \right\}. } \] It is a necessary and sufficient certificate for one chirality-odd generator line in a declared finite typed graph class. The legal class, graph equivalence, reversal action, discrete choices, discarded variables, and refinement rule are all printed and machine checked. When one odd generator survives and the mirror rows share one access kernel, \[ \mathcal A_{c,E}\big|_{W_{ab}} =\lambda_c\mathcal A_E\big|_{W_{ab}}, \qquad \lambda_c>0, \] the complete normalized profiles obey \[ \boxed{ \widehat p_{+}(E)=-\widehat p_{-}(E). } \] For \(C\) profiles sampled in \(N\) bins, this removes exactly \[ \boxed{ CN-(N+C-1)=(C-1)(N-1) } \] target-wise degrees of freedom. Constant detector gain, contact area, and independently measured enantiomeric dilution cancel. No numerical CISS amplitude or fitted angular coefficient enters the relation. High-separation empirical activation The published bifacial-ladder values \(+90\pm5\%\) and \(-88\pm5\%\) reject the chirality-even/no-reversal branch by \(25.17\) quadrature-combined quoted-uncertainty units under the printed-error convention, while their exact-antisymmetry residual is only \(0.283\). Independent heptahelicene data reject the same branch by \(14.38\) units. The digitized ladder energy profiles have mirror-shape cosine \(0.997560\), stable over the complete declared extraction-sensitivity grid. \[ \boxed{ p_{\rm odd}(E)=\frac{p_+(E)-p_-(E)}{2}\ne0 } \] is therefore directly activated across distinct molecular platforms. The finite graph theorem then asks the sharper question: is the multiplicity of that observed odd sector exactly one? The crystallographic vector test For 3-MCHO/Cu(643), the terrace line is fixed before the spin curves are read: \[ \widehat{\mathbf t} =\frac{(6,4,3)\times(1,1,1)}{ \left\|(6,4,3)\times(1,1,1)\right\|} =\frac{(1,-3,2)}{\sqrt{14}}. \] The corresponding parameter-free edge reflection is \[ \boxed{ D_t=I-2\Pi_{\rm edge}, \qquad D_x=\operatorname{diag}(-1,1,1). } \] The public Cu figure places \(80.98\%\) of the digitized squared odd contrast along this independently constructed terrace direction. The paper specifies the full-covariance two-statistic test that separately asks whether chirality is present and whether the exact \(D_x\) transformation survives. The decisive four-cell experiment \[ \boxed{ \mathbf P_{10}=\frac{\mathbf d_m^{(+)}+\mathbf d_m^{(-)}}{2}, \qquad \mathbf P_{11}=\frac{\mathbf d_m^{(+)}-\mathbf d_m^{(-)}}{2}. } \] Using both molecular enantiomers on both Cu(643) surface enantiomorphs resolves the molecule-only and molecule-surface mixed sectors that no fixed-surface measurement can separate. The protocol includes finite-purity inversion, common-frame axial transport, blinding, randomization, replication, graph refinement, and full covariance. The ladder and Cu rows are permanently declared Newton-Protocol discovery constructors. The held-out tests are graph-refinement stability, covariance-complete \(D_x\), normalized-shape transfer, and the four-cell sector inversion. Reconstruction packet The release includes the PDF and LaTeX source, machine-readable digitized traces, primary-source hashes, extraction sensitivity grids, formal theorem and mutation checks, three adversarial scientific audits, and a complete SHA-256 manifest. Public anchors Paper concept DOI Quantum Traction Theory: Main Book v10.01, especially pp. 48, 80--82, 263--266, and 912--916 QTT Observatory QTT Derivation Atlas QTT Lexicon Source/readout ontology