module module moderate

`IndisputableMonolith.Gravity.SevenGaps.HorizonLedgerPreflight`

Preflight layer for the SevenGaps gravity program: a local mirror of the sealed Schwarzschild horizon area $A(R_s)=4\pi R_s^2$, plus scaling lemmas that relate horizon area to recognition-ledger boundary cost. Anyone citing the P1 no-go on a uniform ledger gap at horizons will land here. The argument is definitional mirroring plus elementary positive scaling, not a deep existence proof.

claimLocal mirror of the sealed horizon-area formula $A(R_s)=4\pi R_s^2$, together with scaling maps on area and ledger capacity, showing that every positive area is achieved and that no uniform positive gap can separate ledger boundary cost from the horizon-area family.

background

Recognition gravity books gravitational structure on the recognition ledger: a cost-carrying bookkeeping object whose continuum limit supplies the gravitational action. Horizon thermodynamics in the sealed Relativity subtree uses the classical Schwarzschild area $A(R_s)=4\pi R_s^2$. That subtree is import-sealed, so this module restates the same formula as an inspection-synced mirror rather than importing the sealed definition.

The module sits in Gravity.SevenGaps and pulls Constants (RS time quantum and related units), RecognitionLedger (ledger cost and capacity), and interval logarithm numerics for rigorous bounds when logs appear in scaling or capacity estimates. Sibling names indicate a scaling family on both the area mirror and a ledger-capacity mirror, plus a boundary-cost comparison under rescaling.

Caveat from the module doc: equality between this mirror and the sealed HorizonArea is inspection-verified only; the sealed-import guard forbids a kernel-checked identity. Edits to the sealed formula silently break the downstream no-go unless the texts stay in sync.

proof idea

Definitional core: restate $A(R_s)=4\pi R_s^2$ as a local mirror. Scaling lemmas show the area mirror is homogeneous of degree two in the radius and hits every positive real. Parallel scaling is defined on ledger capacity and on a scaled ledger's boundary cost. The blocking lemma assembles these: any candidate uniform positive gap between ledger boundary cost and the horizon-area family is contradicted by the scaling orbit, which can drive the area (and the paired ledger quantities) through every positive magnitude. Interval-log support is available where logarithmic comparisons need certified bounds. No deep analytic existence machinery; the force comes from surjectivity of the positive scaling family.

why it matters in Recognition Science

This module is the preflight gate for SevenGaps arguments that compare black-hole horizon area to recognition-ledger boundary cost. It supplies the area mirror and the scaling no-go (ledger_boundary_cost_no_uniform_gap and scaling_family_blocks_ledger_gap) that prevent a fixed positive ledger gap from surviving at every horizon scale. Downstream consumers are not yet wired in the graph (used_by is empty), so the immediate role is structural: keep the sealed Relativity area formula usable inside Gravity without breaking the sealed-import boundary, and clear a uniform-gap obstruction before finer horizon-patch or Fibonacci-witness constructions (HorizonPatchClassTarget, fibPatchWitness). In the broader RS gravity story, the ledger is the action bookkeeping; ruling out a scale-invariant cost gap at horizons protects consistency between classical area law and recognition cost.

scope and limits

Does not kernel-check equality with the sealed Relativity HorizonArea definition; sync is inspection-only.
Does not derive $A=4\pi R_s^2$ from RS first principles; it mirrors the classical formula.
Does not prove a full black-hole entropy law or fix Boltzmann counting on the ledger.
Does not by itself close SevenGaps; it only blocks a uniform boundary-cost gap via scaling.
Does not currently show downstream used_by edges; parent theorems are not yet linked in-graph.

depends on (3)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Gravity.RecognitionLedger module_import
IndisputableMonolith.Numerics.Interval.Log module_import

declarations in this module (33)

theorem below L121
def schwarzschildHorizonAreaMirror L151
theorem horizonAreaMirror_scaling L157
theorem horizonAreaMirror_scaling_admissible L168
theorem ledgerCapacityMirror_scaling L176
theorem horizonArea_achieves_every_positive L183
theorem scaling_family_blocks_ledger_gap L200
def scaleLedger L228
theorem scaleLedger_boundaryCost L248
theorem ledger_boundary_cost_no_uniform_gap L260
structure HorizonPatchClassTarget L303
def fibPatchWitness L317
theorem fib_ratio_tendsto_phi L336
theorem log_fib_gap_tendsto_log_phi L345
structure HorizonCombModel L378
def entropy L392
def area L398
theorem entropy_gap_tendsto L403
theorem area_gap_tendsto L417
def horizonCombModelWitness L429
def AreaGapTarget L440
def combFrequencyGM L454
theorem combFrequencyGM_pos L456
theorem combFrequencyGM_bounds L467
def kerrCombOffset L488
theorem kerrCombOffset_eq L493
def modelTransitionFrequency L505
theorem model_area_gap_gives_kerr_comb L511
theorem schwarzschild_comb_frequency L523
def AdjacentSectorTransitionNonzero L540
structure HorizonCombPreflightStatus L550
def horizonCombPreflightStatus L570
theorem horizonCombPreflightStatus_flags L581