Holography Coefficient Bridge Coefficient Of Multiplicity

The coefficient's scope

In the Bekenstein-Hawking formula for black hole entropy, the factor 1/4 is the ratio between a screen's area and its entropy. Recognition Science approaches this ratio from a discrete model: a holographic screen is a collection of small patches, each patch can be in one of sixteen configurations, and a closure condition selects which configurations are physically allowed. The framework's machine-checked library of formal theorems proves that the ratio of patches to entropy must be either 1/4 or 3/4, depending on a single integer multiplicity, the number of recognition events, or discrete ledger entries, that one closed patch realizes.

The theorem coefficient_of_multiplicity states this cleanly: for any multiplicity m, the ratio is m divided by 4. The proof is a direct calculation from the model's definitions: the number of allowed sectors is 4, so the ratio is m over 4. This is not a numerical coincidence; it follows from the structure of the model. The theorem pins the coefficient to exactly two rational values because the model's closure map has rank 1 and nullity 3, meaning one constraint is active and three degrees of freedom remain free. Both branches are proven: if the multiplicity is 1, the ratio is 1/4, the Bekenstein value; if the multiplicity is 3, the ratio is 3/4, giving a coefficient of 4/3.

The framework does not claim to know which branch is correct. The choice between multiplicity 1 and multiplicity 3 is an open physical question, isolated as a single selector. The theorem bekenstein_of_selector shows that if the multiplicity equals the closure rank (1), the Bekenstein ratio follows; the theorem kappa_four_thirds_branch shows the alternative. Everything downstream of this choice is proved, but the choice itself is not. The identification of one closed patch with one recognition event is an unformalized physical assertion, not a theorem in the library.

What this means for the reader is that the framework has reduced a large question about holographic entropy to a single yes/no decision about what a closed patch represents. The mathematics is settled; the physics is not. This is an honest boundary: the framework proves the structure of the answer, but the answer itself remains a target for further work.

/-- Coefficient as an explicit function of the (open) event multiplicity: for any
multiplicity `m`, the pixel-to-sector ratio is `m / 4`. The whole coefficient question is
thus reduced to the single integer `m`. -/
theorem coefficient_of_multiplicity (m : ℕ) :
    (m : ℚ) / (admissibleSectors.card : ℚ) = (m : ℚ) / 4 := by
  rw [recognition_sector_count]; norm_num

/-- **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

The framework does not claim that the Bekenstein value 1/4 is the correct one. The framework does not claim that a closed patch's multiplicity is known. The framework does not claim to derive the fine-structure constant or any other coupling constant.

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)

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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