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IndisputableMonolith.Gravity.LedgerSuperposition

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Gravity-side module that equips the eight-tick complex signal carrier with a linear ledger superposition structure and proves that recognition updates preserve the inner product and norm. Physicists working the quantum-channel or macroscopic-ledger tracks cite it for the cost-gradient channel and C-linear extension lemmas. The argument is structural: finite sums of Signal8 states, inner-product identities, and a packaged theorem record.

claimOn the eight-tick carrier $\mathrm{Signal}_8 \simeq (\mathbb{F}_8 \to \mathbb{C})$, finite linear superpositions of ledger states are well-defined; the recognition update acts $\mathbb{C}$-linearly and preserves the standard Hermitian inner product and norm. The module packages this as a ledger-superposition theorem used by macroscopic gravity and BMV phase calculations.

background

Recognition Science forces an eight-tick period ($T7$, $2^D$ with $D=3$) on the ledger. The cyclic shift on that period is not diagonalizable over $\mathbb{R}$, so Complex Structure Forcing promotes the carrier to $\mathrm{Signal}_8 : \mathrm{Fin},8 \to \mathbb{C}$. Schrödinger Derivation then builds the continuum wave equation on that same carrier.

This gravity module sits one layer up: it treats finite superpositions of single-site $\mathrm{Signal}_8$ states as the microscopic Hilbert data for a ledger, introduces the natural inner product $\langle\cdot,\cdot\rangle_8$, and studies how the recognition operator acts on those superpositions. Sibling names (inner product, finite-sum closure, update-norm preservation) mark the elementary linear-algebra layer before macroscopic or multi-mass constructions.

Local abbreviations identify the analytic eight-tick signal with the canonical foundation type, so gravity proofs reuse the forced complex structure rather than re-deriving it.

proof idea

Definition-and-lemma layer, not a single deep proof. Signal8 is aliased to the foundation carrier; an eight-component Hermitian inner product is defined componentwise. Finite sums of ledger amplitudes are closed under the carrier operations. Recognition-update lemmas show the update is linear and preserves both the inner product and the induced norm (unitarity on the discrete carrier). Those facts are collected into a theorem record (LedgerSuperpositionTheorem) with an inhabited instance so downstream modules can assume the package rather than reassemble the pieces.

why it matters in Recognition Science

Track-2 gravity needs a Hilbert carrier on which recognition acts linearly before one can talk about macroscopic ledgers or gravitational phases. MacroscopicLedger imports this module to discharge Track 2.A: the recognition update extends canonically and $\mathbb{C}$-linearly from the single-site Signal8 carrier to the macroscopic ledger Hilbert space.

BMVPositive (Gravity IV, Theorem 3) uses the linear cost-gradient channel defined here: in the two-mass two-branch protocol the channel produces a branch-dependent gravitational phase whose entangling combination $\Delta\phi$ is generically nonzero, hence a non-product joint state. Without superposition preservation and norm control on Signal8, that phase calculation has no carrier.

Landmarks in play: T7 eight-tick octave, forced complex structure, and the Schrödinger derivation that already fixed Signal8 as the analytic state space.

scope and limits

used by (2)

From the project-wide theorem graph. These declarations reference this one in their body.

depends on (3)

Lean names referenced from this declaration's body.

declarations in this module (20)