Gravity Continuum Manifold Emergence Jcost Is Euclidean Metric

The metric from the cost

In the Recognition Science framework, reality is modeled as a discrete record of events, a ledger of recognitions where each entry carries a forced cost. The framework's central cost function, J(x) = (x + 1/x)/2 - 1, measures the price of a single recognition event. The theorem jcost_is_euclidean_metric establishes what happens to this cost when the recognition steps become very small: it becomes the standard Euclidean metric, the familiar measure of distance in flat space.

Formally, the theorem states that for any small number ε (with |ε| < 1), the logarithm of the cost function, J_log(ε), is approximated by ε²/2, with an error no larger than |ε|⁴/20. This is a precise, machine-checked statement about the local shape of the cost function. It means that for tiny displacements, the cost of recognition grows quadratically with the displacement, exactly as the squared distance does in Euclidean geometry. The second derivative of the cost at the point of zero displacement is exactly 1, which sets the scale of this quadratic behavior.

This local quadratic behavior is the seed of a larger structure. The framework's library shows that when this cost is applied to a lattice of points, the cost of a neighboring point's deviation becomes a discrete version of the Laplacian, the operator that measures how a function curves. In the limit as the lattice spacing goes to zero, this discrete Laplacian converges to the continuous Laplacian. From this, the framework derives the Minkowski interval of special relativity, with its characteristic negative sign for time, and then, through a weak-field approximation, the curved spacetime metric of general relativity. The theorem jcost_is_euclidean_metric is the first, foundational step in this chain: it provides the spatial part of the metric.

The theorem does not claim that the cost function itself is a metric. It establishes that the cost function's local, quadratic approximation behaves like a Euclidean metric. It does not claim that the full, nonlinear cost function is Euclidean; the approximation holds only for small displacements. It also does not claim that the derivation of the Einstein field equations is complete within the framework; that step relies on established external mathematics, specifically the Cheeger-Müller-Schrader result on Regge calculus, as the framework's own documentation notes. The theorem is a precise statement about a local approximation, and it is the foundation upon which the framework builds its account of spacetime.

/-- **THEOREM (J-Cost = Metric)**: J-cost is quadratic at leading order.
    The quadratic form ε²/2 IS the Euclidean distance-squared in log-ratio space. -/
theorem jcost_is_euclidean_metric (ε : ℝ) (hε : |ε| < 1) :
    |J_log ε - ε ^ 2 / 2| ≤ |ε| ^ 4 / 20 :=
  jcost_quadratic_leading ε hε

/-- **THEOREM (Metric Normalization)**: J''(1) = 1 sets the canonical scale.
    The spatial metric tensor is g_ij = δ_ij at each site, up to O(ε²). -/
theorem metric_normalization : deriv (deriv Jcost) 1 = (1 : ℝ) :=
  deriv2_Jcost_one

/-- The J-cost on neighbor pairs gives the lattice Laplacian (from ContinuumLimit). -/
theorem jcost_neighbor_is_laplacian (f : LatticeField 3) (x : Fin 3 → ℤ)
    (h_small : ∀ k : Fin 3,
      |f (shift_plus k x) - f x| < 1 ∧ |f (shift_minus k x) - f x| < 1) :
    |neighbor_cost f x -
      ∑ k : Fin 3, ((f (shift_plus k x) - f x) ^ 2 / 2 +
                     (f (shift_minus k x) - f x) ^ 2 / 2)| ≤
    ∑ k : Fin 3, (|f (shift_plus k x) - f x| ^ 4 / 20 +
                   |f (shift_minus k x) - f x| ^ 4 / 20) :=
  jcost_gives_laplacian_structure f x h_small

What this page does not claim

The full nonlinear cost function is a Euclidean metric; the theorem only establishes a local quadratic approximation. The derivation of the Einstein field equations is complete within the framework; it relies on external mathematics. The theorem provides a physical mechanism for the emergence of spacetime; it establishes a mathematical approximation.

Verify this page

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

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