Gravity Seven Gaps Dynamic Structure Function Blocker Concrete Dynamic Inverse M

The dynamic metric gap

In the mathematical description of gravity, the metric describes the geometry of spacetime. In the Hamiltonian formulation of general relativity, the spatial part of this metric is a dynamic variable: it changes from point to point in phase space, the space of all possible configurations and momenta. The framework's machine-checked library of formal theorems studies a discrete, lattice-based version of this structure. A key question is whether a simple, fixed background weight can stand in for this dynamic metric.

The declaration concreteDynamicInverseMetric_not_constant proves that a specific, simple example of such a dynamic inverse metric is not constant. This example, called concreteDynamicInverseMetric (a concrete function that assigns a positive number to each lattice site based on the configuration), takes the value 1 at one phase-space point and the value 2 at another. The theorem states formally that this function is not the same at all points in phase space.

This fact matters because of a separate, already-proved result: a fixed background weight can represent a phase-space-dependent inverse metric only if that metric is constant across phase space. The library's theorem fixed_background_represents_only_constant establishes this. Combining these two results, the framework proves that no fixed background weight can represent the concrete dynamic metric example. This is the blocker: the existing background-weighted bracket, despite its exact lattice identity and continuum reach, cannot by itself be the full dynamic Dirac structure function required for gravity.

The declaration does not claim that the concrete example is physically realistic, nor that it represents actual spacetime. It is a deliberately simple counterexample, a witness to a structural gap. The theorem does not construct the missing dynamic Hamiltonian; it only certifies that the background-weighted approach is insufficient. The library records this as an open target: the construction of a phase-space-dependent Hamiltonian, and the separate question of HKT rigidity, remain to be established.

/-- THEOREM. The positive two-site metric candidate is genuinely
phase-space-dependent. -/
theorem concreteDynamicInverseMetric_not_constant :
    ¬ PhaseSpaceConstant concreteDynamicInverseMetric := by
  intro h
  have hEq := h zeroPhasePoint unitConfigurationPoint (0 : ZMod 2)
  have hw := concreteDynamicInverseMetric_witness
  rw [hw.1, hw.2] at hEq
  norm_num at hEq

/-- THEOREM. If one fixed background weight represents `g` at every
phase-space point, then `g` is phase-space constant. -/
theorem fixed_background_represents_only_constant
    (w : ZMod n → ℝ) (g : PhaseSpace n → ZMod n → ℝ)
    (h : FixedBackgroundRepresents w g) :
    PhaseSpaceConstant g := by
  intro x y j
  rw [← h x j, ← h y j]

/-- THEOREM (concrete no-go). No fixed two-site background weight represents
the concrete dynamic inverse metric at every phase-space point. -/
theorem no_fixed_background_represents_concrete
    (w : ZMod 2 → ℝ) :
    ¬ FixedBackgroundRepresents w concreteDynamicInverseMetric := by
  intro h
  exact concreteDynamicInverseMetric_not_constant
    (fixed_background_represents_only_constant w concreteDynamicInverseMetric h)

What this page does not claim

The concrete example is not a model of physical spacetime; it is a simple mathematical witness. The theorem does not construct the missing phase-space-dependent Hamiltonian, only certifies the gap. The result does not say the background-weighted bracket is wrong, only that it is insufficient for full dynamics.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: