recognition_update_inner_preserved
plain-language theorem explainer
The one-tick recognition update (cyclic shift on eight-slot complex signals) preserves the canonical Hermitian inner product on Signal8. Anyone citing T1 of Gravity from Recognition IV needs this as the unitarity half of ledger superposition. The proof unfolds the sum definition, expands both Fin-8 sums, and cancels by ring after the shift merely permutes summands.
Claim. For all eight-tick analytic signals $f,g:\mathrm{Fin}\,8\to\mathbb{C}$, if $U$ denotes the one-tick cyclic shift (recognition update), then $\langle Uf, Ug\rangle_8 = \langle f,g\rangle_8$, where $\langle f,g\rangle_8=\sum_{k}\overline{f(k)}\,g(k)$.
background
The module anchors Theorem 1 (Ledger Superposition) from Gravity from Recognition IV: The Quantum Channel. The carrier is Signal8, the eight-tick analytic signal space $\mathrm{Fin},8\to\mathbb{C}$, forced by the eight-tick octave (T7) and the complex structure on the recognition ledger. The one-tick recognition update is the cyclic shift on those eight slots.
The canonical Hermitian product is the standard $\ell^2$ sum $\langle f,g\rangle_8=\sum_{k:\mathrm{Fin},8}\overline{f(k)},g(k)$ from ComplexStructureForcing. T1 asserts that this carrier is a complex Hilbert space on which the update is $\mathbb{C}$-linear and inner-product preserving, so coherent superpositions of definite ledger configurations remain physical under time evolution.
Upstream, inner8 is exactly that sum; cyclic_shift is the spectral one-tick operator reused from MacroscopicLedger/DFT8. No new RS axioms are introduced.
proof idea
Direct definitional comparison. Rewrite both sides as the Foundation inner8 sum, unfold inner8 and cyclic_shift, expand each side with Fin.sum_univ_eight, then simp and ring. The cyclic shift only reindexes the eight summands, so the totals agree. No external lemmas beyond the Fin-8 sum expansion are required.
why it matters
This is clause T1.iii (inner-product preservation) of Ledger Superposition. It is wired into the master witness ledgerSuperpositionTheorem as the inner_preserved field, and it immediately yields the norm-preservation corollary by specializing both arguments. Together with linearity (T1.i–ii), it shows the recognition update is a unitary $\mathbb{C}$-linear operator on the eight-tick Hilbert carrier, so superpositions are preserved rather than decohered by the ledger tick.
In the broader forcing chain this sits on T7 (eight-tick octave) and the complex-structure forcing that supplies Signal8 and inner8. It is one of the four load-bearing formal anchors of the Gravity IV quantum-channel paper; without it the claim that coherent ledger superpositions are physical states under recognition update would be incomplete.
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