Encyclopedia Gravity Gravity Seven Gaps Horizon Ledger Preflight Ledger Boundary Cost No Uniform Gap

ARTICLE 5 claims 5 theorems

Gravity Seven Gaps Horizon Ledger Preflight Ledger Boundary Cost No Uniform Gap

A machine-checked theorem shows that, within the framework's own model, a black hole horizon's boundary cost can be rescaled to any nearby value, blocking any claim of a fixed, uniform gap.

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The scaling argument

The declaration ledger_boundary_cost_no_uniform_gap is a theorem in the framework's machine-checked library of formal theorems. It concerns the ledger, a discrete record of recognition events. In this model, the horizon of a black hole is represented as a bipartition of a finite lattice, and its boundary cost is a real number derived from the ledger's structure. The theorem states that for any such ledger, any bipartition with a positive boundary cost, and any positive number g, there exists a scaling factor λ ≥ 1 such that the boundary cost of the scaled ledger is different from the original, yet the absolute difference is less than g.

In plain terms, the boundary cost can be made arbitrarily close to its original value by scaling the ledger, while still being distinct. This directly establishes that the set of achievable boundary costs has no uniform gap. There is no positive number g that separates one achievable cost from all others. The proof is constructive: it uses the fact that scaling the ledger by λ scales the boundary cost linearly, a fact also proven in the library as scaleLedger_boundaryCost.

This result is part of a larger preflight investigation into a proposed mechanism for quantized black hole horizon areas. The framework's capital, or existing formal theorems, treats horizon area as a continuous real number, not a discrete quantity. This theorem is a key part of the formal refutation of the quantization mechanism at the current level of formalization. It shows that the scaling family of ledgers blocks any uniform gap in the boundary cost spectrum, just as the scaling family of horizon areas blocks any uniform gap in the area spectrum.

The theorem does not claim that the boundary cost is continuous, nor does it claim that a discrete spectrum is impossible in principle. It only shows that, within the framework's current formal model, no uniform gap is forced. The theorem also does not establish the existence of any specific physical quantization; it is a purely formal result about the consequences of the ledger's scaling properties.

THEOREM ledger_boundary_cost_no_uniform_gap · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean
ledger_boundary_cost_no_uniform_gap · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean:255 
/-- **P1 VERDICT (ledger form): no uniform gap in the horizon boundary-cost
spectrum.** For every claimed gap `g > 0`, every recognition ledger with
positive horizon boundary cost admits an axiom-preserving scaling whose
boundary cost is distinct but within `g`. The discrete-lattice capital does
not quantize horizon cost. -/
theorem ledger_boundary_cost_no_uniform_gap
    {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
    (L : RecognitionLedger.RecognitionLedger Λ)
    (P : RecognitionLedger.SubstrateBipartition Λ)
    (hB : 0 < RecognitionLedger.boundaryCost L P)
    (g : ℝ) (hg : 0 < g) :
    ∃ (lam : ℝ) (hlam : 1 ≤ lam),
      RecognitionLedger.boundaryCost (scaleLedger L lam hlam) P
        ≠ RecognitionLedger.boundaryCost L P ∧
      |RecognitionLedger.boundaryCost (scaleLedger L lam hlam) P
        - RecognitionLedger.boundaryCost L P| < g := by
  set B := RecognitionLedger.boundaryCost L P with hBdef
  have hBne : B ≠ 0 := ne_of_gt hB
  have hlam : 1 ≤ 1 + g / (2 * B) := by
    have hpos : 0 < g / (2 * B) := div_pos hg (by linarith)
    linarith
  have hexp : (1 + g / (2 * B)) * B = B + g / 2 := by
    field_simp
  refine ⟨1 + g / (2 * B), hlam, ?_, ?_⟩
  · rw [scaleLedger_boundaryCost, ← hBdef, hexp]
    intro h
    linarith
  · rw [scaleLedger_boundaryCost, ← hBdef, hexp,
      show B + g / 2 - B = g / 2 from by ring, abs_of_pos (half_pos hg)]
    linarith
THEOREM ledger_boundary_cost_no_uniform_gap · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean
ledger_boundary_cost_no_uniform_gap · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean:255 
/-- **P1 VERDICT (ledger form): no uniform gap in the horizon boundary-cost
spectrum.** For every claimed gap `g > 0`, every recognition ledger with
positive horizon boundary cost admits an axiom-preserving scaling whose
boundary cost is distinct but within `g`. The discrete-lattice capital does
not quantize horizon cost. -/
theorem ledger_boundary_cost_no_uniform_gap
    {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
    (L : RecognitionLedger.RecognitionLedger Λ)
    (P : RecognitionLedger.SubstrateBipartition Λ)
    (hB : 0 < RecognitionLedger.boundaryCost L P)
    (g : ℝ) (hg : 0 < g) :
    ∃ (lam : ℝ) (hlam : 1 ≤ lam),
      RecognitionLedger.boundaryCost (scaleLedger L lam hlam) P
        ≠ RecognitionLedger.boundaryCost L P ∧
      |RecognitionLedger.boundaryCost (scaleLedger L lam hlam) P
        - RecognitionLedger.boundaryCost L P| < g := by
  set B := RecognitionLedger.boundaryCost L P with hBdef
  have hBne : B ≠ 0 := ne_of_gt hB
  have hlam : 1 ≤ 1 + g / (2 * B) := by
    have hpos : 0 < g / (2 * B) := div_pos hg (by linarith)
    linarith
  have hexp : (1 + g / (2 * B)) * B = B + g / 2 := by
    field_simp
  refine ⟨1 + g / (2 * B), hlam, ?_, ?_⟩
  · rw [scaleLedger_boundaryCost, ← hBdef, hexp]
    intro h
    linarith
  · rw [scaleLedger_boundaryCost, ← hBdef, hexp,
      show B + g / 2 - B = g / 2 from by ring, abs_of_pos (half_pos hg)]
    linarith
THEOREM scaleLedger_boundaryCost · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean
scaleLedger_boundaryCost · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean:245 
/-- Boundary (horizon) cost of the scaled ledger is exactly `lam` times the
original: the scaling family acts CONTINUOUSLY on the horizon ledger content
while preserving every ledger axiom. -/
theorem scaleLedger_boundaryCost {Λ : Type*} [Fintype Λ] [DecidableEq Λ]
    (L : RecognitionLedger.RecognitionLedger Λ) (lam : ℝ) (hlam : 1 ≤ lam)
    (P : RecognitionLedger.SubstrateBipartition Λ) :
    RecognitionLedger.boundaryCost (scaleLedger L lam hlam) P
      = lam * RecognitionLedger.boundaryCost L P := by
  simp only [RecognitionLedger.boundaryCost, scaleLedger, Finset.mul_sum]
THEOREM schwarzschildHorizonAreaMirror · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean
schwarzschildHorizonAreaMirror · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean:142 
/-- Definitional MIRROR of the sealed capital's horizon area
(`Relativity/Compact/BlackHoleEntropy.lean:19`): `A(Rs) = 4·π·Rs²`.
Same formula, restated here because the Relativity subtree is sealed.

CAVEAT (critic 2026-07-15): the identity between this mirror and the
sealed definition is INSPECTION-VERIFIED only, not kernel-checked (the
sealed-import guard forbids stating the equality in Lean). Any edit to
the sealed `HorizonArea` formula silently invalidates the P1 no-go
below; keep the two textually in sync. -/
def schwarzschildHorizonAreaMirror (Rs : ℝ) : ℝ := 4 * Real.pi * Rs ^ 2
THEOREM horizonCombPreflightStatus_flags · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean
horizonCombPreflightStatus_flags · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean:580 
/-- Status record (rfl-forced; documentation, not new mathematics). -/
theorem horizonCombPreflightStatus_flags :
    HorizonCombPreflightStatus.p1_scaling_family_exists_at_current_formalization
        horizonCombPreflightStatus = true ∧
    horizonCombPreflightStatus.p1_area_gap_forced = false ∧
    HorizonCombPreflightStatus.p2_discrete_horizon_state_class_in_capital
        horizonCombPreflightStatus = false ∧
    HorizonCombPreflightStatus.p3_asymptotic_entropy_gap_theorem_landed
        horizonCombPreflightStatus = true ∧
    horizonCombPreflightStatus.p3_exact_area_gap_derived = false ∧
    horizonCombPreflightStatus.p4_transition_capital_exists = false ∧
    horizonCombPreflightStatus.mechanism_forced = false ∧
    horizonCombPreflightStatus.echo_discriminator_revived = false :=
  ⟨rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl⟩

What this page does not claim

The theorem does not claim that the boundary cost spectrum is continuous. The theorem does not claim that a discrete spectrum is impossible in principle, only that it is not forced by the current model. The theorem does not establish the existence of any specific physical quantization of horizon area.

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/HorizonLedgerPreflight.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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