Chemistry Photosynthesis2 From Jcost Psii Qycert

A certificate with a narrow scope

Photosynthesis converts sunlight into chemical energy, and its first step, photosystem II, has a measured maximum quantum yield of about 0.88 in a dark-adapted state. That number means roughly 88 percent of absorbed light quanta drive the initial charge separation. The declaration PSII_QYCert, in the framework's machine-checked library of formal theorems, was intended to connect this biological yield to a mathematical cost function, but the certificate itself proves something far more general and far less specific.

The certificate is a structure, a bundle of three proved statements about a function called domainCost, which is defined as Jcost (m / e), where m and e are real numbers. The three facts are: the cost is zero when m equals e (and e is nonzero); the cost is never negative when both m and e are positive; and the quantity phi minus 3/2 is positive, where phi is the golden ratio. These are general properties of the cost function, not biological results. The certificate proves them for any real inputs, with no reference to photons, chlorophyll, or any physical quantity.

The docstring attached to the module records the intended research direction: that the maximum quantum yield of photosystem II equals 1 minus J(phi), which evaluates to about 0.882, matching the measured 0.88. But the docstring itself states plainly that this is a research note, not a result. The Lean code proves nothing specific to photosynthesis, because domainCost is defined without any connection to m and e as biological variables. What would make this a theorem about photosystem II is a separate definition of m and e in terms of the physics of the system, and that definition does not appear.

The certificate is shared verbatim across 2383 sibling modules, a template stated once and universally quantified in a central module. This means the photosynthesis-specific content is entirely in the narrative, not in the formal proof. The certificate establishes that the cost function has certain algebraic properties, and that a threshold involving the golden ratio is positive. It does not establish that photosystem II's yield is 0.88, nor that the cost function models any aspect of photosynthesis.

What a reader can take away is a clear separation between a proved algebraic fact and a hoped-for application. The framework's library proves the three general statements rigorously. The connection to photosynthesis remains a research aspiration, recorded in prose, awaiting a formal definition that ties the abstract cost to the biological system. Until that definition exists, the certificate's name overstates its content.

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The certificate does not prove that the maximum quantum yield of photosystem II is 0.88 or 0.882. The certificate does not establish that the cost function Jcost models any aspect of photosynthesis. The research note in the docstring is not a proved result.

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/Chemistry/Photosynthesis2FromJCost.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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