Information Dna Storage Density Rs

The density ledger

DNA storage density is a measure of how much digital information can be packed into a physical mass of DNA. The landmark 2012 experiment by Church, Gao, and Kosuri stored about 5.5 petabits per gram, which is roughly 215 × 10¹⁵ bytes per gram. That number is a measurement: a laboratory result, not a theoretical prediction. The practical appeal is obvious: a gram of DNA can hold far more data than any conventional hard drive, and the medium is stable for centuries under the right conditions.

The Recognition Science framework takes a different angle. It models the density as a cost, a number that measures how far a candidate storage density deviates from an ideal reference density. The framework's central object is the cost function, which assigns a penalty to any ratio of two densities. The framework proves three general facts about this cost: it is zero when the two densities are equal, it is never negative for positive densities, and a certain threshold value involving the golden ratio is positive. These are the only claims the machine-checked library establishes for this page.

The module does not prove that DNA storage density equals any specific value. The research note that inspired the module suggests a structural relationship: the framework's cost might apply to the ratio of a measured density to an ideal one. But the module defines the cost without specifying what the two densities are. It is a template, not a result about DNA. The note itself acknowledges this: the paragraph recording the intended direction is a research note, not a theorem.

What the module does establish, in plain language, is that the cost function behaves sensibly. If you compare a density to itself, the cost is zero. If you compare two positive densities, the cost is never negative. And the golden ratio threshold is a positive number. These facts are proved for the general cost function, not for DNA specifically. The module is a placeholder, a scaffold that becomes meaningful only when someone defines what the two densities are in the context of DNA storage.

In Recognition Science, this is how the framework works: it provides the cost structure, and the subject matter supplies the definitions. The module is honest about its limits. It does not claim to have derived the density of DNA storage. It claims only that the cost function has certain properties, which would hold for any subject to which the cost is applied. The reader should take the module as a proof of the cost function's basic hygiene, not as a proof about DNA.

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]

What this page does not claim

The module does not prove any specific value for DNA storage density. The module does not establish that DNA storage density follows a golden ratio scaling. The module does not define what the reference density e is.

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/Information/DNA_Storage_Density_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.

Derived articles

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