IndisputableMonolith.Gravity.SevenGaps.DynamicStructureFunctionBlocker
On the periodic lattice, a fixed background weight can represent a Hamiltonian structure function only when that function is phase-space constant. The module builds a concrete dynamic inverse metric that varies with canonical data, proves it is nowhere constant, and concludes no fixed background represents it. Gravity and QG constraint-algebra workers cite it as the blocker separating frozen weights from true dynamics. The argument is an existence witness plus an iff between fixed-background representability and phase-space constancy.
claimA lattice inverse metric $G$ is phase-space constant if its value at every site is independent of the canonical data. A fixed background weight $w$ represents $G$ only if $G$ is phase-space constant. There exists a concrete positive dynamic inverse metric $G_{\mathrm{dyn}}$ that is not phase-space constant; hence no fixed $w$ represents $G_{\mathrm{dyn}}$. Equivalently, a fixed background represents some inverse metric if and only if that metric is phase-space constant.
background
This module sits in the QG Seven-Gaps campaign, Pillar 1 (constraint algebra), downstream of the background-weighted hypersurface bracket. That upstream development generalizes the frozen hypersurface-deformation bracket to a Hamiltonian density whose stiffness slot carries a fixed lattice weight $w:\mathbb{Z}/n\mathbb{Z}\to\mathbb{R}$, and derives the exact lattice bracket in which $w$ appears as a structure function.
Phase-space constancy means: varying the canonical configuration and momentum data never changes the inverse-metric value at any site. Fixed-background representation means the structure function of the weighted Hamiltonian is exactly some prescribed $w$ independent of phase-space point. The module introduces a concrete dynamic inverse metric on a minimal phase space (zero phase point and unit configuration point as witnesses) that is strictly positive yet non-constant under those variations.
The local goal is negative: show that the weighted-bracket framework cannot absorb genuinely dynamic structure functions by freezing them into $w$.
proof idea
Define phase-space constancy and the predicate that a fixed background represents a given inverse metric. Prove the one-direction implication: any inverse metric represented by a fixed $w$ must be phase-space constant (by independence of $w$ from canonical data). Establish the converse existence direction as an iff: a fixed background represents some inverse metric precisely when that metric is phase-space constant.
Construct an explicit positive dynamic inverse metric on the lattice phase space, verify positivity at the witness points, and show it takes distinct values when the canonical data change, hence is not phase-space constant. Conclude that no fixed background represents this concrete metric. A final declaration records that the weighted Hamiltonian carries a background structure function in the sense of the upstream bracket module, tying the blocker to that algebra.
why it matters in Recognition Science
The module is the dynamic-structure-function blocker for Seven-Gaps Pillar 1. Upstream, WeightedHypersurfaceBracket locks the C10 live bet: the weight $w$ emerges cleanly in the lattice Ham-Ham bracket. This file shows the complementary limitation: that $w$ is frozen; anything that actually depends on phase-space data lies outside fixed-background representability.
Downstream it is imported by FullTheoryLedger, the Phase 0c machine-checked status record of the full-theory campaign (one boolean flag per pillar benchmark, flipped only when the target is kernel-checked and critic-passed). The blocker supplies the precise obstruction that the ledger and later dynamic-extension work must either accept (keep $w$ fixed) or discharge (promote the structure function to a dynamical field). In the broader Recognition gravity stack it separates kinematic weighting from true back-reaction on the lattice constraint algebra.
scope and limits
- Does not construct a fully dynamical metric as a field equation or evolution law.
- Does not alter the weighted Ham-Ham bracket identities proved upstream.
- Does not claim continuum GR or Ashtekar structure functions on manifolds.
- Does not rule out state-dependent weights outside the fixed-background predicate.
- Does not evaluate numerical gravity observables or observational bounds.
used by (1)
depends on (1)
declarations in this module (20)
-
def
PhaseSpaceConstant -
def
FixedBackgroundRepresents -
theorem
fixed_background_represents_only_constant -
theorem
exists_fixed_background_iff_phaseSpaceConstant -
def
concreteDynamicInverseMetric -
def
zeroPhasePoint -
def
unitConfigurationPoint -
theorem
concreteDynamicInverseMetric_pos -
theorem
concreteDynamicInverseMetric_witness -
theorem
concreteDynamicInverseMetric_not_constant -
theorem
no_fixed_background_represents_concrete -
def
HamWHasBackgroundStructureFunction -
theorem
HamW_has_background_structure_function -
def
BackgroundWeightedContinuumReach -
theorem
background_weighted_continuum_reach -
structure
PhaseSpaceDependentHamiltonianConstruction -
def
backgroundHamiltonianConstruction -
def
PhaseSpaceDependentDiracPremise -
def
Gap5DynamicDiracAndHKTRigidityTarget -
theorem
gap5_background_weight_blocker