Holography Coefficient Bridge Target Coefficient Bridge Holds

The coefficient bridge

The Bekenstein-Hawking entropy formula for a black hole, S = A/4 in Planck units, contains a deceptively simple factor of one quarter. In the Recognition Science framework, this factor is not a free parameter. It emerges from a counting problem: how many independent degrees of freedom live on a small patch of a holographic screen. The declaration target_coefficient_bridge_holds is a machine-checked theorem that reduces this counting problem to a single, precisely-stated physical choice.

The framework models a holographic screen as a collection of plaquettes, each with a configuration space of 16 possible states. A closure constraint picks out which configurations are physically allowed. The theorem computes three numbers from these actual sets: the rank of the closure map (1), the nullity, or number of free bits (3), and the total number of bits (4). These satisfy the rank-nullity theorem: 4 = 1 + 3. The coefficient question then becomes: does entropy attach to the single closure event (rank 1), giving the Bekenstein 1/4, or to the three free-bit microstates (nullity 3), giving 3/4?

Both branches are proven exactly. The theorem establishes that the coefficient is either 1/4 or 3/4, nothing else. It does not decide which one is physically realized. That decision rests on an open question, the selector: whether one closed plaquette realizes exactly one recognition event, or three. The framework's library proves everything downstream of that choice, but the choice itself remains a physical input, not a mathematical consequence.

What this means in practice is that the holographic entropy coefficient has been narrowed from a continuum of possibilities to a binary. The next step is not more mathematics but a physical argument about what a recognition event is. Until that argument lands, the framework holds both branches as proven alternatives, with the Bekenstein value as one branch and a 4/3 coefficient as the other.

theorem target_coefficient_bridge_holds : target_coefficient_bridge := by
  refine ⟨closureRank_eq_one, freeBits_eq_three, rank_nullity_add,
          closure_image_times_kernel, ?_, ?_⟩
  · rw [closureRank_eq_one, recognition_sector_count]; norm_num
  · rw [freeBits_eq_three, recognition_sector_count]; norm_num

theorem closureRank_eq_one : closureRank = 1 := by decide

theorem freeBits_eq_three : freeBits = 3 := by decide

theorem rawBits_eq_four : rawBits = 4 := by decide

/-- **Rank-nullity (additive form).** `rawBits = closureRank + freeBits` (`4 = 1 + 3`),
with each side computed independently from the actual sets. -/
theorem rank_nullity_add : rawBits = closureRank + freeBits := by decide

/-- **Bekenstein branch.** Entropy attaches to the closure rank (`m = 1`) ⇒ ratio `1/4`. -/
theorem bekenstein_branch :
    (closureRank : ℚ) / (admissibleSectors.card : ℚ) = 1 / 4 := by
  rw [closureRank_eq_one, recognition_sector_count]; norm_num

/-- **`κ = 4/3` branch.** Entropy attaches to the free-bit nullity (`m = 3`) ⇒ ratio `3/4`
(the coefficient is then `4/3` of Bekenstein). -/
theorem kappa_four_thirds_branch :
    (freeBits : ℚ) / (admissibleSectors.card : ℚ) = 3 / 4 := by
  rw [freeBits_eq_three, recognition_sector_count]; norm_num

What this page does not claim

This answer does not claim that the Bekenstein value 1/4 is the physically realized one. This answer does not claim that the selector question has been answered or derived. This answer does not claim that the framework derives the full Bekenstein-Hawking entropy formula, only the coefficient reduction.

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/Holography/CoefficientBridge.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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