LedgerSuperpositionTheorem
plain-language theorem explainer
Packages T1 of Gravity from Recognition IV: the one-tick recognition update on the eight-tick complex signal carrier is ℂ-linear (on pairs and finite sums) and preserves the canonical Hermitian inner product and its induced norm. Gravity and quantum-channel authors cite it as the formal witness that coherent ledger superpositions are physical and evolve unitarily under recognition. The structure is a pure interface; its canonical inhabitant wires four already-proved lemmas.
Claim. A witness that the cyclic one-tick recognition update $U$ on the eight-tick carrier $\mathrm{Signal}_8 := (\mathrm{Fin}\,8 \to \mathbb{C})$ satisfies: (i) $U(a\psi + b\varphi) = a\,U\psi + b\,U\varphi$ for all $\psi,\varphi$ and $a,b\in\mathbb{C}$; (ii) $U(\sum_{\alpha\in s} c_\alpha L_\alpha) = \sum_{\alpha\in s} c_\alpha\,U(L_\alpha)$ for finite sums; (iii) $\langle Uf, Ug\rangle_8 = \langle f,g\rangle_8$ and hence $\|Uf\|_8 = \|f\|_8$, where $\langle f,g\rangle_8 = \sum_{k=0}^{7}\overline{f(k)}\,g(k)$.
background
Module Gravity IV anchors two load-bearing theorems of Gravity from Recognition IV: The Quantum Channel. T1 (Ledger Superposition) asserts that the recognition state space is a complex Hilbert carrier and that the one-tick update preserves coherent superpositions.
Signal8 is the local abbreviation for the eight-tick analytic carrier Fin 8 → ℂ, identified with the canonical object from ComplexStructureForcing. The Hermitian product inner8 is the standard sum $\sum_k \overline{f(k)} g(k)$. The update itself is the cyclic shift on that eight-slot register (the eight-tick octave of the forcing chain).
Upstream foundation modules (SchrodingerDerivation, RecognitionOperator, ComplexStructureForcing) already supply linearity and unitarity of the shift; this structure only names the bundled T1 interface. No new RS-internal axioms are introduced.
proof idea
This declaration is a structure (interface), not a proved theorem. It has no proof body: four fields record the mathematical obligations.
The canonical inhabitant ledgerSuperpositionTheorem fills them by direct assignment: pair linearity from ledger_superposition_preserved, finite-sum linearity from ledger_superposition_finite_sum, inner-product preservation from recognition_update_inner_preserved, and norm preservation from recognition_update_norm_preserved. Non-emptiness is then the one-line wrapper ledgerSuperpositionTheorem_inhabited.
why it matters
This is the formal content of paper T1: "ledger superpositions are physical and are preserved by recognition evolution." Downstream, ledgerSuperpositionTheorem inhabits the structure and ledgerSuperpositionTheorem_inhabited records Nonempty, so later gravity and quantum-channel developments can cite a single named witness rather than four scattered lemmas.
In the RS forcing chain the carrier is the eight-tick octave (T7); complex structure and the inner product come from ComplexStructureForcing. T1 is the bridge from that discrete Hilbert carrier to the quantum-channel reading of gravity: once the update is linear and isometric, superpositions of definite ledger configurations remain physical under recognition evolution. T2 (cost-gradient functoriality) in the same module builds on the free linear extension story that this Hilbert-carrier package makes legitimate.
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