track1D_tt_gram_self_adjoint_endpoint_holds
plain-language theorem explainer
The finite TT Gram operator on the N=5 periodic coefficient space is self-adjoint for the coefficient-space inner product. Track 7 gravity integration cites this as the Track 1.D endpoint that closes the concrete TT projector split for the fixed conformal-plus-longitudinal generator family. The proof is a one-line term applying the TensorShearSector self-adjointness lemma at N=5.
Claim. For all real coefficient maps $a,b$ on the five-slot periodic TT normal-equation index set, if $G$ denotes the finite TT Gram operator on that coefficient space and $\langle\cdot,\cdot\rangle$ its coefficient inner product, then $\langle G a,\, b\rangle = \langle a,\, G b\rangle$.
background
This module is the Track 7 integration-lane receipt for parallel gravity fork handoffs. It records proved endpoints without upgrading the discovery claim, and leaves remaining Track 1 displacement-class leaves as open dependencies.
Track 1.D concerns the transverse-traceless (TT) sector of the discrete gravity analysis. On the fixed N=5 periodic normal-equation index set, one builds a Gram operator from the combined conformal-plus-longitudinal generator family and pairs it with a coefficient-space inner product. Self-adjointness of that Gram map is the algebraic condition that lets the concrete TT projector split close at N=5.
The endpoint proposition is exactly the statement that this Gram operator is symmetric for every pair of coefficient vectors. Spatial dimension D=3 (forced upstream by the T8/T9 chain) sits in the ambient geometry, but the claim itself is finite-dimensional linear algebra on the N=5 coefficient space.
proof idea
One-line term proof. The declaration is definitionally the self-adjointness proposition for the N=5 periodic TT normal-equation Gram operator, and the proof applies periodicTTNormalEquationGram_selfAdjoint5 from TensorShearSector directly. No extra rewriting or case splits appear at this layer; the handoff theorem is a named receipt of that sector lemma.
why it matters
Track 7 packages fork handoffs into a single integration certificate. This theorem supplies the Track 1.D Gram self-adjointness bit consumed by that certificate (forkHandoffIntegrationCert), alongside Schläfli reduction, many-body amplitude-linear lift, Page-capacity, dark-energy w(z), and falsifier-sensitivity endpoints.
In the gravity stack, closing the TT projector split at N=5 is the finite algebraic gate before residual and Bianchi interfaces can treat the TT sector as a well-posed self-adjoint block. The module doc is explicit that this does not upgrade the discovery claim; it only records what the new endpoint proves. Remaining Track 1 displacement-class leaves stay as the next dependency after this handoff.
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