Cosmology Dark Energy Dilution From Jcost Aggregate Hadamard Mul

The multiplicative shadow

In the Recognition Science framework, the cost of recognition is not a free choice. A proved theorem forces any cost function satisfying five plain conditions to take the form J(x) = (x + 1/x)/2 - 1. From that cost function, the framework builds a ledger: a discrete record of recognition events. When the framework models a system as passing through several independent channels, it aggregates the per-channel costs into a single occupancy, a number describing how much of the original signal survives. The aggregate is defined as exp of the sum of per-channel log-costs, a form the framework calls the log-additive ledger.

The declaration aggregate_hadamardMul, a theorem in the framework's machine-checked library of formal theorems, states a clean algebraic fact about this aggregate. If you have two vectors of positive per-channel costs, x and y, and you multiply them componentwise, then the aggregate of the product equals the product of the aggregates. In symbols: aggregate α (hadamardMul x y) = aggregate α x * aggregate α y. The proof is two lines: log-costs add, and exp turns sums into products. This is the multiplicative shadow of J-cost log-additivity, a phrase the framework's own documentation used as a prose gesture before this theorem made it precise.

The theorem matters because a downstream derivation about dark energy dilution had been carrying a premise called composes, which asserted that occupancy across independent dimensions multiplies. That premise was an axiom, a structural assumption. This theorem discharges it: for any occupancy that is the uniform-weight log-aggregate of a single per-channel survival ratio, the composes law is now a proved theorem, not an assumption. The construction builds a full dilution structure whose composes field is supplied by proof. The premise count does not drop to zero; it is a substitution, not a reduction.

What the theorem does not claim is just as important as what it proves. It does not force the physical dark-energy occupancy to have the channel-aggregate form. That identification, occupancy equals exp of an additive log-ledger over independent channels, is a model-level structural choice, not a derivation. The per-channel self-similar attenuation, the condition that one channel attenuates by the reciprocal self-similarity fixed point, remains the surviving physical premise. The theorem replaces an abstract axiom with a logically equivalent concrete form, grounding it in the J-cost API. It is not premise-free, and it does not derive the dilution from nothing.

/-- **The J-cost aggregate is multiplicative over independent channels.**
Log-costs add (`Cost.Ndim.dot_log_hadamardMul`), so aggregates multiply. -/
theorem aggregate_hadamardMul {n : ℕ} (α x y : Cost.Ndim.Vec n)
    (hx : ∀ i, 0 < x i) (hy : ∀ i, 0 < y i) :
    Cost.Ndim.aggregate α (Cost.Ndim.hadamardMul x y)
      = Cost.Ndim.aggregate α x * Cost.Ndim.aggregate α y := by
  unfold Cost.Ndim.aggregate
  rw [Cost.Ndim.dot_log_hadamardMul α x y hx hy, Real.exp_add]

/-- **THE `composes` PREMISE, DISCHARGED.** Occupancy across `m + n` independent
channels is the product of the block occupancies. Proof: the log-ledger is
additive over disjoint channel blocks, and `exp` turns sums into products
(`Real.exp_add`) — the multiplicative shadow of J-cost log-additivity, now a
theorem instead of an axiom. -/
theorem occOf_composes (m n : ℕ) :
    C.occOf (m + n) = C.occOf m * C.occOf n := by
  rw [occOf_eq_exp, occOf_eq_exp, occOf_eq_exp, ← Real.exp_add]
  congr 1
  push_cast
  ring

/-- The occupancy through `n` independent recognition channels: the uniform-weight
J-cost log-aggregate of the per-channel survival ratio. MODEL-level
identification: this form (exp of an additive log-ledger over independent
channels) is what replaces the abstract `composes` axiom. -/
def occOf (n : ℕ) : ℝ :=
  Cost.Ndim.aggregate (fun _ : Fin n => 1) (fun _ => C.s)

What this page does not claim

The theorem does not prove that dark energy actually has the channel-aggregate form; that identification remains a model choice. The theorem does not reduce the premise count of the dilution derivation; it substitutes the composes axiom with a logically equivalent concrete form. The theorem does not derive the per-channel self-similar attenuation premise; that condition remains untouched as a physical input.

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/Cosmology/DarkEnergyDilutionFromJCost.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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