Gravity Seven Gaps Horizon Ledger Preflight Horizon Area Mirror Scaling Admissib

A scaling check

The declaration horizonAreaMirror_scaling_admissible is a small, machine-checked theorem about the area of a black hole horizon. In the framework's library, the horizon area of a Schwarzschild black hole is defined as 4πRs², where Rs is the radius. The theorem states that if you multiply the radius by any positive number λ, the new radius λ·Rs is still a valid horizon radius. This is a purely mathematical statement about the formula for area, not a physical claim about what happens to a real black hole.

The proof is a one-line algebraic fact: the product of two positive numbers is positive. It is a ledger, a discrete record of events, only in the sense that the framework's library keeps formal records of such proofs. The theorem does not say that horizons come in discrete sizes, that area is quantized, or that any particular gap exists between allowed areas. It only confirms that the scaling operation preserves the basic condition that the radius be positive.

This theorem matters because it is part of a larger check on a proposed mechanism. The mechanism imagined that horizon area comes in discrete steps, with a specific gap of 4·ln(φ)·ℓ_P², where φ is the golden ratio and ℓ_P is the Planck length. The scaling theorem, together with a companion result showing the area map achieves every positive real number, shows that no such gap is forced by the framework's existing capital. The proposed quantization fails a falsifier at the current formalization level; the mechanism is not derived, only a model.

In Recognition Science, the framework models physical structure from a ledger of recognition events. Here, the framework's library proves a kinematic fact about the area formula, and it proves an asymptotic limit about Fibonacci entropy gaps, but it does not prove the existence of discrete horizon states. Those states remain an open target. The scaling theorem is a boundary marker: it shows what the capital does and does not force, and it keeps the honest fraction of the mechanism at zero.

/-- **P1 (scaling family exists, admissibility).** The capital's ONLY
admissibility condition on a Schwarzschild horizon configuration is `0 < Rs`
(every theorem in `BlackHoleEntropy.lean` quantifies over exactly this), and
it is preserved by every positive scaling. So the family
`Rs ↦ λ·Rs (λ > 0)` stays inside the admissible class. -/
theorem horizonAreaMirror_scaling_admissible (lam Rs : ℝ)
    (hlam : 0 < lam) (hRs : 0 < Rs) : 0 < lam * Rs :=
  mul_pos hlam hRs

/-- **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

/-- 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 black hole horizons are actually quantized. The theorem does not claim that the golden ratio appears in any physical area spectrum. The theorem does not claim that the framework derives the existence of discrete horizon states.

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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