Acoustics Room Acoustics Rt60 Rs Rt60 Cert

The RT60 certificate

Reverberation time, RT60, is the number of seconds it takes a sound to decay by 60 decibels after the source stops. In a recording studio the usual range is 0.2 to 0.5 seconds; in a concert hall it runs closer to 1.1 seconds. The Recognition Science framework's ledger, a discrete record of recognition events, offers a way to model such values, and the declaration RT60Cert is the formal object that records what that model actually proves.

The certificate is a structure, a bundle of three proved facts. Each fact is general, not acoustic. The first says that when the two inputs to the domain cost function are equal, the cost is zero. The second says the cost is never negative for positive inputs. The third says the number phi minus 3/2 is positive, where phi is the golden ratio, about 1.618. The certificate packages these three theorems into a single object, and a separate definition shows the object exists.

What the certificate does not do is prove anything about studios or concert halls. The framework's own docstring says so plainly: the Lean code proves nothing specific to room acoustics, because the cost function is defined as J(m/e) without reference to reverberation time. The values in the research note, such as phi cubed times 0.1 seconds giving 0.424 seconds, are a record of where the idea was meant to go, not a result. The certificate is a template, shared verbatim across more than two thousand sibling modules, waiting for a definition of m and e in acoustic terms.

In Recognition Science, the cost function J(x) = (x + 1/x)/2 - 1 is proved unique by a theorem in the framework's machine-checked library of formal theorems. The certificate reuses that general result. What would turn it into a theorem about its subject is a definition of the two quantities m and e in the subject's own terms. Until then, RT60Cert is a proof about cost, not about sound.

structure RT60Cert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty RT60Cert := ⟨cert⟩

What this page does not claim

The certificate does not prove that any studio or hall has a specific reverberation time. The certificate does not derive the value 0.424 seconds from the framework; that value is a research note. The certificate does not establish that the golden ratio governs room acoustics in any measured way.

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/Acoustics/Room_Acoustics_RT60_RS.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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