Gravity Seven Gaps Dynamic Structure Function Blocker Exists Fixed Background If

The fixed-background limit

In the Recognition Science framework's study of gravity on a discrete lattice, a central object is the Dirac structure function, a slot in the bracket that determines how two observables interact. The framework's library has an exact construction that places a fixed, site-dependent weight into this slot, and it can carry that weight's smearing shape to the continuum limit. The question is whether such a fixed background weight can represent a genuinely dynamic inverse metric, one that changes as the phase-space point moves.

The theorem exists_fixed_background_iff_phaseSpaceConstant settles this with a crisp equivalence: a fixed background weight can represent a phase-space-dependent inverse metric at every point if and only if that metric is actually phase-space constant. The forward direction is the substantive one. If a fixed weight matches a dynamic metric at every phase-space point, then the metric takes the same value at any two points, so it is constant. The reverse direction is immediate: if the metric is constant, choose that constant as the background weight.

The library then exhibits a concrete two-site example, concreteDynamicInverseMetric, defined as 1 plus the square of the configuration coordinate. This metric is positive everywhere, and it takes different values at two explicit phase-space points (1 at one, 2 at another). Since it is not phase-space constant, the theorem implies no fixed background weight can represent it. This is not a failure of the fixed-background construction; it is a precise boundary on what that construction can do.

In Recognition Science, this result certifies a distinction in the program's seven-gap roadmap for gravity. The fixed-background bracket, despite its exact lattice identity and continuum reach, cannot by itself be the full dynamic Dirac structure function that full ADM gravity requires, where the inverse spatial metric in that slot must vary with the canonical data. The theorem does not construct the missing dynamic Hamiltonian; it names that construction as a separate remaining obligation, along with an HKT rigidity statement, in the target Gap5DynamicDiracAndHKTRigidityTarget.

What the result changes is the shape of the remaining work. It converts a vague worry about whether the fixed background might suffice into a sharp, checkable criterion: any candidate dynamic inverse metric that the framework's bracket must handle either is constant, in which case the fixed background already covers it, or it varies, in which case a genuinely phase-space-dependent Hamiltonian construction is required. The path forward is now a defined target rather than an open question.

/-- THEOREM (exact characterization). A candidate inverse metric admits one
fixed background representation at all phase points exactly when it is
phase-space constant. -/
theorem exists_fixed_background_iff_phaseSpaceConstant
    (g : PhaseSpace n → ZMod n → ℝ) :
    (∃ w : ZMod n → ℝ, FixedBackgroundRepresents w g) ↔
      PhaseSpaceConstant g := by
  constructor
  · rintro ⟨w, hw⟩
    exact fixed_background_represents_only_constant w g hw
  · intro hg
    let x₀ : PhaseSpace n := (fun _ => 0, fun _ => 0)
    refine ⟨g x₀, ?_⟩
    intro x j
    exact hg x₀ x j

/-- THEOREM. The concrete inverse-metric candidate is everywhere positive. -/
theorem concreteDynamicInverseMetric_pos
    (x : PhaseSpace 2) (j : ZMod 2) :
    0 < concreteDynamicInverseMetric x j := by
  unfold concreteDynamicInverseMetric
  positivity

/-- THEOREM. The concrete metric takes different values at two explicit
phase-space points on the two-site lattice. -/
theorem concreteDynamicInverseMetric_witness :
    concreteDynamicInverseMetric zeroPhasePoint (0 : ZMod 2) = 1 ∧
      concreteDynamicInverseMetric unitConfigurationPoint (0 : ZMod 2) = 2 := by
  norm_num [concreteDynamicInverseMetric, zeroPhasePoint, unitConfigurationPoint]

/-- 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)

/-- OPEN. Gap 5 requires both the phase-space-dependent Dirac construction
and an HKT rigidity theorem.  The current `HamW` theorem and its continuum
smearing result supply neither conjunct. -/
def Gap5DynamicDiracAndHKTRigidityTarget (n : ℕ) [NeZero n] : Prop :=
  PhaseSpaceDependentDiracPremise n ∧ HKTRigidityStatement n

What this page does not claim

This does not claim that the fixed-background bracket is wrong or useless; it remains exact and has continuum reach. This does not claim that a dynamic Hamiltonian construction exists; it only names it as a target. This does not claim that the HKT rigidity statement is proved; it is recorded as a separate obligation.

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

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