Masses Coherence Exponent

Coherence exponent

The masses coherence exponent is the integer 5, written into the energy scale of particle masses as E_coh = phi^(-5), where phi is the golden ratio. In Recognition Science the exponent is not chosen. It is forced by a constraint on the dimension D: both D and 2^D must be Fibonacci numbers. The module proves that the only positive integer satisfying this is D = 3, since 3 = F4 and 8 = F6 are both Fibonacci. The Fibonacci recurrence then gives 8 - 3 = 5 = F5, and this difference is defined as the coherence exponent.

What the module establishes is that the value 5 is structurally determined, not a free parameter. The definition of the coherence exponent is the difference between the octave 2^D and the dimension D. With D = 3, the octave is 8, and the difference is 5. The theorem coherence_exponent_unique packages the whole result: D = fib 4, octave = fib 6, coherence_exponent = fib 5, and coherence_exponent = 5. The proof is a direct computation in Lean, using the Fibonacci values 3, 5, and 8.

The module also checks the constraint for other dimensions. D = 1 works, since 1 and 2 are both Fibonacci. But D = 2 fails because 4 is not Fibonacci, D = 5 fails because 32 is not Fibonacci, and D = 8 fails because 256 is not Fibonacci. The uniqueness claim is therefore not an assumption; it is a established fact about the finite list of Fibonacci numbers up to 1597.

The consequence is that the energy scale phi^(-5) sits at a fixed rung of the phi-power ladder. This is the same constant that appears as hbar in RS-native units, where hbar = phi^-5. The coherence exponent is the link between the Fibonacci constraint on dimension and the numerical value of the reduced Planck constant in the framework.

THEOREM E_coh_eq · IndisputableMonolith/Masses/CoherenceExponent.lean

THEOREM D_3_fibonacci_constraint · D_2_fails · D_5_fails · D_8_fails · IndisputableMonolith/Masses/CoherenceExponent.lean

THEOREM fibonacci_deficit · coherence_exponent_from_fibonacci · IndisputableMonolith/Masses/CoherenceExponent.lean

THEOREM coherence_exponent_unique · IndisputableMonolith/Masses/CoherenceExponent.lean

What this page does not claim

This answer does not claim that the Fibonacci constraint itself is derived from more basic recognition principles. This answer does not claim that the coherence exponent determines any measured particle mass directly. This answer does not claim that D = 3 is the only dimension satisfying the constraint in any extended sense beyond the finite Fibonacci list.

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/Masses/CoherenceExponent.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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