module module moderate

`IndisputableMonolith.Gravity.SevenGaps.GaugeHistoryMeasure`

Defines the posting alphabet and bounded gauge histories for a labeled complex: one letter per vertex, edge, and tet index, with injective post maps and a canonical history. Gravity and QG measure work cites it when discharging gauge-counting from ledger history rather than from an ad-hoc path-sum. The module is mostly definitions plus injectivity and conversion lemmas that feed Gap-2 measure status bindings.

claimFor a labeled complex, the posting alphabet has one letter per vertex, edge, and tetrahedron index. A posted bounded history is a finite word in that alphabet; vertex, edge, and tet posts are injective. There is a canonical history and a map sending ordinary histories to posted form, so gauge counting can be read from the ledger without universe or counting inflation.

background

Gap 2 in the Seven Gaps gravity stack concerns continuum limit and path-sum measure recovery. Upstream, MeasureSubstrateBlocker records the theorem status that relabeling invariance, positivity, and normalization do not select a path-sum measure, while ExactShellGaugePreflight equates gauge-counting mass to $1/|\mathrm{Aut}|$ under a uniform gauge-density model premise. Dual-entry enrichment (Wave B residual R3) supplies the signed-source side without an $x$-ratio.

This module introduces the posting layer: a finite alphabet with one symbol per vertex, edge, and tet index of the labeled complex. That design avoids universe polymorphism and double-counting when histories are treated as words. Sibling objects include a balanced zero state, posted bounded histories, injective post maps for vertices/edges/tets, a canonical history type, and a conversion into posted form.

The local setting is ledger-first gauge history: measure claims should be discharged from posted histories and automorphism bookkeeping, not from an underdetermined continuum path measure.

proof idea

Definition-heavy module. It builds the posting alphabet and the posted-history type, then proves injectivity of the vertex, edge, and tet post maps so distinct geometric indices remain distinct letters. Canonical history and toPosted supply a normal form from ordinary histories into the posted alphabet. No deep analytic argument lives here; the work is constructive data and small lemmas that later modules quote when binding gauge-counting status flags and residual DAGs.

why it matters in Recognition Science

Feeds four Gap-2 consumers. Gap2MeasureStatusBinding binds Wave C1 R6 flags (substrate_measure_derived, counting_principle_derived_from_ledger) to gap2_gauge_counting_from_history_discharged from this stack. Gap2ContinuumMeasureResidualDAG names ordered residuals for Pillar-2 measure and continuum recovery and treats the measure half as progressing once the history substrate is in place. Gap2SizeBlindnessReach explains how far gluing and size-blindness premises reach and why the posting layer alone cannot close continuum gluing. GaugeHistoryMeasureAudit requires headline theorems to print only under standard classical axioms after critic repair.

In framework terms this is infrastructure for honest measure discharge: gauge mass from history and $|\mathrm{Aut}|$, not a free path-sum choice blocked upstream by the substrate no-go.

scope and limits

Does not construct a continuum path-sum measure or prove uniqueness of one.
Does not discharge size-blindness or gluing multiplicativity for class measures.
Does not remove the uniform gauge-density model premise from ExactShellGaugePreflight.
Does not prove continuum-limit recovery or Einstein-Hilbert convergence.
Does not claim the posting layer alone closes Gap 2 residuals.

used by (4)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Gravity.SevenGaps.Gap2ContinuumMeasureResidualDAG module_import
IndisputableMonolith.Gravity.SevenGaps.Gap2MeasureStatusBinding module_import
IndisputableMonolith.Gravity.SevenGaps.Gap2SizeBlindnessReach module_import
IndisputableMonolith.Gravity.SevenGaps.GaugeHistoryMeasureAudit module_import

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Gravity.Analysis.RecognitionDualEntryEnrichment4D module_import
IndisputableMonolith.Gravity.SevenGaps.MeasureSubstrateBlocker module_import

declarations in this module (54)

abbrev PostingAlphabet L81
def balancedZeroState L85
structure PostedBoundedHistory L98
def vertexPost L107
def edgePost L111
def tetPost L115
theorem vertexPost_injective L118
theorem edgePost_injective L121
theorem tetPost_injective L125
def canonicalHistory L133
structure CanonicalHistory L154
abbrev toPosted L163
def underlying L166
def ofComplex L169
theorem ext L173
theorem toPosted_eq_canonicalHistory L178
def classOf L183
def fiber_unique L188
def equivUnderlying L203
instance instFinite L214
theorem classOf_ofComplex L217
theorem toPosted_ofComplex L221
theorem underlying_ofComplex L225
def postingAlphEquiv L234
structure from L245
structure HistoryRelabel L246
def toRelabel L269
def ofRelabel L279
def historyRelabel_equiv_relabel L303
def historyRelabel_equiv_relabel_canonical L317
instance instFiniteHistoryRelabel L321
def historyOrbitCardClass L329
def historyPairCountClass L335
structure GaugeHistoryEnrichment L342
def nuBuild L348
theorem nuBuild_def_history_only L356
def history_class_equiv_mk L367
def class_mk_equiv_orbit L379
def history_class_equiv_orbit L394
theorem historyOrbitCardClass_eq_orbitCardClass L399
def history_pair_equiv_pair L417
theorem historyPairCountClass_eq_pairCountClass L426
theorem historyPairCountClass_pos L442
theorem nuBuild_gaugeCounting L450
theorem nuBuild_eq_gaugeOrbitMass L462
theorem gap2_gauge_counting_from_history_discharged L474
def TypedResidual_gap2_gauge_counting_from_history L483
theorem typedResidual_gap2_gauge_counting_from_history_closed L490
theorem TypedResidual_gap2_gauge_counting_from_history_closed L494
def circularNu L504
theorem circularNu_def_is_gaugeOrbitMass L508
theorem circularNu_satisfies_gaugeCounting L514
theorem decoy_uniformClassMass_not_gaugeCounting L520
theorem gap2_history_measure_decoy_anchors L527