Gravity Seven Gaps Horizon Ledger Preflight Ledger Capacity Mirror Scaling

A scaling identity

The declaration ledgerCapacityMirror_scaling proves a basic algebraic identity about a capacity bound. In the framework's vocabulary, a ledger is a discrete record of recognition events, and a capacity bound is a real number that limits how much can be stored. The theorem states that for any real numbers λ, A, and ℓ₀, the expression (λ²·A)/ℓ₀² equals λ²·(A/ℓ₀²). This is a consequence of the commutative, associative, and distributive laws of arithmetic; the proof in the machine-checked library of formal theorems is a single ring simplification. It is a theorem about real numbers, not about any physical state.

The identity matters because it is one link in a chain that refutes a specific proposal. The proposal was that black hole horizon area comes in discrete chunks, with a gap of 4·ln(φ)·ℓ_P², where φ is the golden ratio and ℓ_P the Planck length. This would produce a comb of absorption frequencies. The scaling identity, together with a companion theorem showing the area map achieves every positive real value, implies that no positive gap separates achievable horizon areas. If the area can be scaled continuously to any positive value, a fixed gap cannot exist. The theorem scaling_family_blocks_ledger_gap formalizes this: for any positive gap g and any positive area A, there exists a radius whose area differs from A by less than g. The scaling family therefore blocks the gap.

The same logic applies to the ledger form. If you scale a recognition ledger by a factor λ ≥ 1, the boundary cost scales linearly, as shown in the theorem scaleLedger_boundaryCost. The theorem ledger_boundary_cost_no_uniform_gap then proves that the achievable boundary-cost spectrum has no uniform gap either. For any proposed gap, a scaled ledger produces a boundary cost within that gap of the original. This is a positive result: the capital forces the absence of the quantization, at the current formalization level.

What the declaration does not claim is broader. It does not claim that horizon area is quantized, nor that the Fibonacci-based comb mechanism is correct. The mechanism's load-bearing hypotheses, such as the existence of a discrete horizon state class and the identification of entropy with the logarithm of a Fibonacci count, are not derived from the framework's capital. They are recorded as named targets, and the consistency witness that satisfies them is a witness by fiat, not a derivation. The status record horizonCombPreflightStatus lists the flags plainly: the scaling family exists, the area gap is not forced, the discrete state class is absent, and the mechanism is not forced. The declaration is a small algebraic step in a larger, honest accounting of what is proved and what remains open.

/-- The capital's capacity bound (`LedgerCapacityLimit A ell0 = A/ell0²`,
BlackHoleEntropy.lean:32) is itself scale-covariant: capacity of a λ²-scaled
area is λ² times the capacity. A real-valued bound cannot quantize the
spectrum. -/
theorem ledgerCapacityMirror_scaling (lam A ell0 : ℝ) :
    (lam ^ 2 * A) / ell0 ^ 2 = lam ^ 2 * (A / ell0 ^ 2) := by
  ring

/-- **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

/-- **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

What this page does not claim

The declaration does not claim that horizon area is quantized; it proves the opposite for the current formalization. It does not claim the Fibonacci-based comb mechanism is correct; the mechanism's load-bearing hypotheses remain targets. It does not claim the golden ratio φ is derived from the framework in this context; the constant appears in the proposed gap, not in the theorem's conclusion.

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

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