Gravity Seven Gaps Dynamic Structure Function Blocker Ham W Has Background Struc

The background bracket

In the Recognition Science framework's machine-checked library of formal theorems, a Hamiltonian is a function that generates the time evolution of a physical system. The declaration HamW_has_background_structure_function is a theorem about a specific family of Hamiltonians, each labeled by a fixed weight function w. It proves that for any such weight, the Poisson bracket of two Hamiltonians in this family takes a particular algebraic form: a sum over lattice sites of terms combining the weight, the two functions being bracketed, and the canonical position and momentum variables. This is an exact identity, not an approximation.

The theorem's content is that the bracket has a "background structure function": the weight w sits in the slot where a full theory of gravity would place the inverse spatial metric. The framework's library also proves that this background-weighted bracket reaches the continuum limit: as the lattice spacing shrinks, the discrete sum converges to an integral of the weight times the smearing functions. Both results hold with the weight held fixed as the phase-space point varies.

The limitation is precise. Full ADM gravity requires the inverse spatial metric in that slot to vary with the canonical metric data, which is phase-space dependent. The theorem fixed_background_represents_only_constant proves that a fixed background weight can represent a phase-space-dependent inverse metric only if that metric is actually constant across phase space. The library then exhibits a concrete counterexample: a two-site inverse metric that takes value 1 at one phase point and 2 at another, so it is not constant. The theorem no_fixed_background_represents_concrete proves that no fixed background weight can represent this example. Therefore the background-weighted bracket, despite its exact lattice identity and continuum reach, cannot by itself be the full dynamic Dirac structure function of gravity.

The declaration does not claim that a phase-space-dependent Hamiltonian construction exists. It names that construction as a separate remaining obligation, recorded in the target Gap5DynamicDiracAndHKTRigidityTarget, alongside the existing HKT rigidity statement. The blocker is a certified boundary: it shows what the background-weighted bracket can do, and precisely where its reach ends.

/-- THEOREM. The existing exact bracket theorem supplies the fixed-background
structure-function proposition. -/
theorem HamW_has_background_structure_function (w : ZMod n → ℝ) :
    HamWHasBackgroundStructureFunction w :=
  bracket_HamW_HamW w

/-- 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 declaration does not prove that a phase-space-dependent Hamiltonian construction exists. The declaration does not claim the background-weighted bracket is the full dynamic Dirac structure function of gravity. The declaration does not assert that the concrete two-site inverse metric is physically realistic, only that it is a valid counterexample.

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.

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