Primes Priced By J
A prime is the smallest indivisible unit in arithmetic, and the unique recognition cost assigns its price from one simple measure: its logarithmic size.
The price of a prime
Imagine a ledger, a book that records how much each arithmetic state costs to recognize. A composite number can be opened into smaller factors, so its entries are built from more basic pieces. A prime number is different: it is an indivisible non-vacuum state, the arithmetic counterpart of a single particle. That makes the prime the natural place to ask what one unit of arithmetic should cost.
The answer begins with uniqueness. Suppose a proposed cost function treats a state and its reciprocal alike, assigns zero cost to unity, obeys the required composition rule, meets the calibration condition, and is continuous for positive inputs. The Lean theorem shows that every such function agrees with J at every positive input. J is therefore not a convenient price scale chosen after the fact. The five conditions leave only one scale.
This changes the role of prime numbers in the arithmetic picture. They are no longer only factors in a decomposition theorem. They are the elementary priced units from which the cost of every composite state can be understood. The price is forced before any particular prime is examined, so arithmetic supplies the examples and the uniqueness theorem supplies the rule.
THEOREM law_of_logic_forces_jcost · IndisputableMonolith/Cost/FunctionalEquation.lean
What this page does not claim
That the Riemann hypothesis has been proved. That a self-adjoint Hilbert-space operator realizing the zeta zeros has been constructed.
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/Cost/FunctionalEquation.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:
- How does the prime-priced arithmetic gas produce the Riemann zeta function?
- How does the sum of prime-mode energies relate to the von Mangoldt function?
- What would a real excitation spectrum of the arithmetic gas say about the Riemann hypothesis?
MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND
- THEOREMThe Lean theorem shows that every such function agrees with J at every positive input. law_of_logic_forces_jcost · IndisputableMonolith/Cost/FunctionalEquation.lean