Encyclopedia Gravity Gravity Seven Gaps Horizon Ledger Preflight Horizon Comb Preflight Status Flags

ARTICLE 5 claims 5 theorems

Gravity Seven Gaps Horizon Ledger Preflight Horizon Comb Preflight Status Flags

A machine-checked status report on a proposed black hole frequency pattern: what the framework's capital forces, and what it does not.

Verify this page

The Preflight Verdict

The declaration horizonCombPreflightStatus_flags is a machine-checked status report. It records, in one compact structure, the verdict on a proposed physical mechanism: the idea that a black hole's horizon area comes in discrete steps, with a specific gap tied to the golden ratio, producing a predictable comb of absorption frequencies. The report does not assert the mechanism is true. It asserts exactly which parts of the mechanism the framework's existing capital forces, and which parts it does not.

The core finding is a negative one. The framework's capital, the machine-checked library of formal theorems, treats horizon area as a continuous real number. A theorem proves that for any positive area and any tiny gap you name, there is a valid horizon with an area that differs from the first by less than that gap. This means no minimum positive gap between achievable areas is forced. The same conclusion holds for the ledger's boundary cost, the framework's discrete record of recognition events at the horizon: scaling the ledger can produce a boundary cost arbitrarily close to any other, so no uniform gap exists there either. The proposed quantization is not derived; at the current formalization level, it is positively refuted.

The report does record one genuine theorem: if a discrete horizon state class exists, and its microstate counts follow the Fibonacci sequence, and entropy is the logarithm of that count, then the gap in entropy tends to the logarithm of the golden ratio, and the area gap tends to 4 times the Planck area times that logarithm. This is a conditional result, an if-then chain. The conditions themselves are named targets, not established facts. A witness shows the conditions are satisfiable, but that is a consistency check, not a derivation.

The status report also explicitly records that the mechanism is not forced, and that an older, dead idea involving echoes is not being revived. The declaration's value is its precision: it separates what is proved from what is hoped, and it names the missing ingredients as open targets. It does not predict an observable effect; it measures how much of a model the framework's theorems can support.

THEOREM horizonArea_achieves_every_positive · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean
horizonArea_achieves_every_positive · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean:180 
/-- Every positive real is an achieved horizon area of an admissible
configuration: take `Rs = √(A/(4π))`. The achievable area spectrum is the
full ray `(0, ∞)`. -/
theorem horizonArea_achieves_every_positive (A : ℝ) (hA : 0 < A) :
    ∃ Rs : ℝ, 0 < Rs ∧ schwarzschildHorizonAreaMirror Rs = A := by
  have h4pi : (0 : ℝ) < 4 * Real.pi := by positivity
  refine ⟨Real.sqrt (A / (4 * Real.pi)),
    Real.sqrt_pos.mpr (div_pos hA h4pi), ?_⟩
  simp only [schwarzschildHorizonAreaMirror]
  rw [Real.sq_sqrt (le_of_lt (div_pos hA h4pi))]
  have hpi : Real.pi ≠ 0 := Real.pi_ne_zero
  field_simp
THEOREM scaling_family_blocks_ledger_gap · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean
scaling_family_blocks_ledger_gap · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean:193 
/-- **P1 VERDICT (continuum form): the scaling family blocks the ledger area
gap.** For every claimed gap `g > 0` and every achieved area `A > 0` there is
an ADMISSIBLE configuration whose area differs from `A` but by less than `g`.
Hence the existing capital forces NO quantized area spectrum — in particular
not `ΔA = 4·ln(φ)·ℓ_P²` — and the comb mechanism FAILS gate P1 at the current
formalization level. This is the kernel-checked no-go the preflight was gated
on. -/
theorem scaling_family_blocks_ledger_gap (g A : ℝ) (hg : 0 < g) (hA : 0 < A) :
    ∃ Rs : ℝ, 0 < Rs ∧
      schwarzschildHorizonAreaMirror Rs ≠ A ∧
      |schwarzschildHorizonAreaMirror Rs - A| < g := by
  obtain ⟨Rs, hRs, hArea⟩ :=
    horizonArea_achieves_every_positive (A + g / 2) (by linarith)
  refine ⟨Rs, hRs, ?_, ?_⟩
  · rw [hArea]
    intro h
    linarith
  · rw [hArea, show A + g / 2 - A = 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 area_gap_tendsto · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean
area_gap_tendsto · IndisputableMonolith/Gravity/SevenGaps/HorizonLedgerPreflight.lean:414 
/-- **THEN (kernel-checked).** Under the model hypotheses the per-level AREA
gap converges to `4·ln(φ)·ℓ_P²` — the comb mechanism's target gap, reached
here ONLY as the asymptotic consequence of the inserted hypotheses. -/
theorem area_gap_tendsto (M : HorizonCombModel) :
    Filter.Tendsto (fun n => M.area (n + 1) - M.area n)
      Filter.atTop (nhds (4 * M.lP2 * Real.log Constants.phi)) := by
  have h := (M.entropy_gap_tendsto).const_mul (4 * M.lP2)
  refine h.congr fun n => ?_
  simp only [area]
  ring
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 declaration does not claim the proposed absorption comb is a real physical effect. It does not claim the golden ratio gap is derived from the framework's axioms. It does not claim the dead echo-train idea has been revived.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:

MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND