Foundation Primitive Recognition Calculus Prime Axis Coherence Log Char Mul

The additive character

A log-character is a way to assign a real number to every positive whole number by first assigning a number to each prime. The rule is plain: take the prime factorization of a number, multiply each prime's assigned value by how many times that prime appears, and add the results. For example, if the prime 2 is assigned 3 and the prime 3 is assigned 5, then the number 12, which factors as 2 squared times 3, gets the value 2 times 3 plus 5, or 11.

The declaration logChar_mul proves that this construction behaves exactly as the name suggests: the value assigned to a product of two nonzero numbers is the sum of the values assigned to each factor. So the log-character of 6 is the log-character of 2 plus the log-character of 3. This is the defining property of a multiplicative character: it turns multiplication into addition. The theorem holds for any choice of prime weights, with no further conditions imposed.

This result is one of three pieces that the framework's machine-checked library of formal theorems assembles into a single statement called prime axis coherence. The other two pieces say that different prime weights always produce different log-characters, and that a log-character follows a single power law against a reference scale exactly when its prime weights are all proportional to that scale's weights. Together they describe a moment of freedom followed by collapse: the prime axes start independent, and a coherence condition later synchronizes them into one common scale.

What logChar_mul does not claim is any particular choice of prime weights. It is a structural theorem about the construction, not a statement about which numbers the primes should carry. The theorem also does not claim that every function from whole numbers to reals that respects multiplication arises this way; it proves that this specific construction has the multiplicative property. Finally, it does not by itself establish the power-law alignment: that requires the separate theorem powerLaw_iff_aligned, which brings in the reference scale and the alignment condition.

/-- **Freedom.** Every weight assignment extends to a multiplicative character:
the log-character is additive on products of nonzero naturals. The prime axes
are independent; no relation among them is forced before coherence enters. -/
theorem logChar_mul (a : ℕ → ℝ) {m n : ℕ} (hm : m ≠ 0) (hn : n ≠ 0) :
    logChar a (m * n) = logChar a m + logChar a n := by
  unfold logChar
  rw [Nat.factorization_mul hm hn]
  rw [Finsupp.sum_add_index']
  · intro p; simp
  · intro p k1 k2; push_cast; ring

/-- **Freedom.** Every weight assignment extends to a multiplicative character:
the log-character is additive on products of nonzero naturals. The prime axes
are independent; no relation among them is forced before coherence enters. -/
theorem logChar_mul (a : ℕ → ℝ) {m n : ℕ} (hm : m ≠ 0) (hn : n ≠ 0) :
    logChar a (m * n) = logChar a m + logChar a n := by
  unfold logChar
  rw [Nat.factorization_mul hm hn]
  rw [Finsupp.sum_add_index']
  · intro p; simp
  · intro p k1 k2; push_cast; ring

What this page does not claim

No particular assignment of numbers to primes is claimed or derived. The theorem does not claim that every multiplicative function on the naturals arises from this prime-weight construction. The power-law alignment result is not established by logChar_mul alone, but by the separate theorem powerLaw_iff_aligned.

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/Foundation/PrimitiveRecognitionCalculus/PrimeAxisCoherence.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: