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 source lines can a molecular interface carry, and what fixes the spin-polarization amplitude once its physical transfer packet is known? This paper answers both questions without appending an independent polarization coefficient. The paper-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 finite certificate for one chirality-odd generator line in a declared typed graph class. When one odd generator survives and the mirror rows share one access kernel, the complete normalized energy profiles obey \(\widehat p_{+}(E)=-\widehat p_{-}(E)\), removing exactly \((C-1)(N-1)\) profile degrees of freedom across \(C\) curves and \(N\) energy bins. The amplitude theorem For any full-rank two-channel transfer packet \(M\), define its positive Gram operator \(Q=MM^\dagger\) and principal logarithm \[ \boxed{ \log Q=\alpha I+\mathbf r\!\cdot\!\boldsymbol\sigma, \qquad \rho_{\rm out}=\frac{Q}{\operatorname{Tr}Q} =\frac12\left(I+\tanh|\mathbf r|\,\widehat{\mathbf r}\!\cdot\!\boldsymbol\sigma\right). } \] The output polarization is therefore unique: \(\mathbf P=\tanh|\mathbf r|\widehat{\mathbf r}\). A positive common gain changes only \(\alpha\); it cannot change \(\mathbf r\) or \(\mathbf P\). Common-rail serial packets add their signed rapidities, while a reversed local face subtracts by signed count. This closes the amplitude functional. The material-specific construction of \(M\) from chemistry, interface dynamics, contacts, and the detector camera remains the next source theorem. A raw-current coordinate \[ \boxed{ \Xi_{\rm odd} =\frac14\log\frac{I_{+,R}I_{-,S}}{I_{-,R}I_{+,S}}. } \] This coordinate cancels independent positive gains on the two enantiomer rows. The release prints its exact current gradient for covariance propagation, a passive-access bound, the singular-current rule, and the eligibility conditions under which molecular length or signed defects may be read as serial completed packets. What is already observed The 2026 bifacial-ladder values \(+90\pm5\%\) and \(-88\pm5\%\) provide high-separation activation of a chirality-odd sector, while the digitized mirror curves have cosine similarity 0.9976 in the declared discovery window. Those observations motivated the theorem and remain permanently classified as Newton-Protocol discovery constructors. The public Source Data archive does not contain the raw device-current arrays needed for the new covariance-level test, and its displayed 16-mer is a structural visualization rather than a certified transport count. The sealed prospective decision A companion preregistration tests a molecular-length and facial-defect series in two tiers. Tier S tests held-out additive Log-Gram rapidity after one training slope. Tier K freezes the stronger zero-target-fit coefficient \(1/[4\pi\cos(\pi/8)]\). Failure of Tier K rejects that fixed-gap conjecture without erasing the exact Log-Gram theorem; a qualified held-out pass earns the registered zero-target-fit verdict. Public anchors Paper concept DOI Sealed CISS Log-Gram test Bifacial-ladder discovery paper Quantum Traction Theory: Main Book QTT Observatory QTT Derivation Atlas QTT Lexicon