Encyclopedia Gravity Gravity Macroscopic Ledger Macroscopic Shift Map Add

ARTICLE 4 claims 4 theorems

Gravity Macroscopic Ledger Macroscopic Shift Map Add

A theorem about how a single-site update rule extends to many sites, and the boundary of what that extension proves.

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The macroscopic shift

In quantum mechanics, a superposition is a sum of possible states, each with its own complex amplitude. The Recognition Science framework builds its physical models on a discrete record of recognition events, its ledger. The declaration MacroscopicShift_map_add proves that the ledger's update rule, applied across many sites at once, respects this superposition: updating the sum of two multi-site configurations equals the sum of updating each one separately. It is a linearity theorem, the kind of structural guarantee that lets a model treat a large system as a well-behaved whole.

The proof is a formal one, checked by a machine. It builds on a single-site update called cyclic_shift, which cycles an eight-tick signal one step forward. The declaration shows that this update, when extended to a finite tensor product of sites, is additive. A companion result, MacroscopicShift_map_smul, proves the same for scaling by a complex number, and MacroscopicShift_finite_sum extends additivity to any finite linear combination. Together they form the macroscopic ledger theorem: the update rule extends canonically and linearly from one site to any finite collection of sites.

What the theorem does not claim is just as important. It does not identify the macroscopic update with any physical gravitational response. That identification, the bridge from the ledger's abstract structure to gravity as we measure it, remains an open target in the framework. The theorem also does not say anything about infinite collections of sites, only finite ones. And it does not assert that the ledger's update rule is the only possible one; it proves a property of a given construction, not its uniqueness.

For a reader, the practical consequence is that the framework's multi-site models are internally consistent at the level of linear algebra. The superposition principle holds for the ledger, so calculations on large configurations can proceed without fear of a hidden nonlinearity breaking the sum. The next step, connecting this structure to actual gravitational physics, is a separate and unfinished task.

THEOREM MacroscopicShift_map_add · IndisputableMonolith/Gravity/MacroscopicLedger.lean
MacroscopicShift_map_add · IndisputableMonolith/Gravity/MacroscopicLedger.lean:128 
/-- **Linearity.** The macroscopic recognition update is `ℂ`-linear
(by construction, since `PiTensorProduct.map` returns a `LinearMap`).
This restates the universal property explicitly. -/
theorem MacroscopicShift_map_add (Ψ Φ : MacroscopicLedger ι) :
    MacroscopicShift (Ψ + Φ) = MacroscopicShift Ψ + MacroscopicShift Φ :=
  MacroscopicShift.map_add Ψ Φ
THEOREM MacroscopicShift_map_smul · IndisputableMonolith/Gravity/MacroscopicLedger.lean
MacroscopicShift_map_smul · IndisputableMonolith/Gravity/MacroscopicLedger.lean:135 
theorem MacroscopicShift_map_smul (c : ℂ) (Ψ : MacroscopicLedger ι) :
    MacroscopicShift (c • Ψ) = c • MacroscopicShift Ψ :=
  MacroscopicShift.map_smul c Ψ
THEOREM MacroscopicShift_finite_sum · IndisputableMonolith/Gravity/MacroscopicLedger.lean
MacroscopicShift_finite_sum · IndisputableMonolith/Gravity/MacroscopicLedger.lean:139 
/-- **Macroscopic ledger superposition.** For a finite family of
macroscopic ledger configurations `Ψ : κ → MacroscopicLedger ι` and
amplitudes `c : κ → ℂ` indexed by a finite set, the macroscopic
recognition update commutes with the finite linear combination. This is
the explicit superposition principle for multi-site ledger
configurations. -/
theorem MacroscopicShift_finite_sum
    {κ : Type*} (s : Finset κ) (c : κ → ℂ) (Ψ : κ → MacroscopicLedger ι) :
    MacroscopicShift (∑ α ∈ s, c α • Ψ α) =
      ∑ α ∈ s, c α • MacroscopicShift (Ψ α) := by
  classical
  induction s using Finset.induction_on with
  | empty => simp
  | @insert α s hα ih =>
    rw [Finset.sum_insert hα, Finset.sum_insert hα,
        MacroscopicShift_map_add, MacroscopicShift_map_smul, ih]
THEOREM MacroscopicLedgerTheorem · IndisputableMonolith/Gravity/MacroscopicLedger.lean
MacroscopicLedgerTheorem · IndisputableMonolith/Gravity/MacroscopicLedger.lean:158 
/-- **MACROSCOPIC LEDGER HILBERT CARRIER CERTIFICATE.**

Five clauses establishing that the macroscopic ledger Hilbert carrier
is a `ℂ`-linear extension of the single-site recognition update:

1. `single_site_linear`: the single-site recognition update
   `cyclicShiftLinear` is a `ℂ`-linear map `Signal8 →ₗ[ℂ] Signal8`.
2. `tensor_action`: on a pure tensor configuration, the macroscopic
   update acts factor-wise.
3. `additive`: the macroscopic update is additive.
4. `scalar_homogeneous`: the macroscopic update is scalar-homogeneous.
5. `finite_superposition`: the macroscopic update commutes with finite
   linear combinations (the superposition principle for multi-site
   ledger configurations).

This discharges Track 2.A of the master plan: the macroscopic ledger
Hilbert carrier is now a STRUCTURAL THEOREM rather than a definition or
CONDITIONAL THEOREM. -/
structure MacroscopicLedgerTheorem (ι : Type) [Fintype ι] [DecidableEq ι] where
  /-- (1) Single-site update is a linear map. -/
  single_site_linear :
    ∀ (ψ φ : Signal8) (a b : ℂ),
      cyclicShiftLinear (a • ψ + b • φ) =
        a • cyclicShiftLinear ψ + b • cyclicShiftLinear φ
  /-- (2) Pure-tensor action: factor-wise. -/
  tensor_action :
    ∀ (ψ : ι → Signal8),
      MacroscopicShift (PiTensorProduct.tprod ℂ ψ) =
        PiTensorProduct.tprod ℂ (fun i => cyclic_shift (ψ i))
  /-- (3) Macroscopic update is additive. -/
  additive :
    ∀ (Ψ Φ : MacroscopicLedger ι),
      MacroscopicShift (Ψ + Φ) = MacroscopicShift Ψ + MacroscopicShift Φ
  /-- (4) Macroscopic update is scalar-homogeneous. -/
  scalar_homogeneous :
    ∀ (c : ℂ) (Ψ : MacroscopicLedger ι),
      MacroscopicShift (c • Ψ) = c • MacroscopicShift Ψ
  /-- (5) Macroscopic update commutes with finite superposition. -/
  finite_superposition :
    ∀ {κ : Type*} (s : Finset κ) (c : κ → ℂ) (Ψ : κ → MacroscopicLedger ι),
      MacroscopicShift (∑ α ∈ s, c α • Ψ α) =
        ∑ α ∈ s, c α • MacroscopicShift (Ψ α)

What this page does not claim

The theorem does not prove that the macroscopic update equals any gravitational operator. The theorem does not address infinite site collections. The theorem does not establish uniqueness of the update rule.

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Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/MacroscopicLedger.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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