IndisputableMonolith.Meta.ConstructiveNote
The module defines computable real numbers inside the Recognition Science meta layer. It supplies a predicate together with instances showing that pi, phi, and the basic arithmetic operations on them admit algorithmic rational approximations to any precision. Researchers building constructive derivations of the forcing chain cite these definitions to keep all constants algorithmically accessible. The module consists of a collection of definitions and supporting computability statements with no proof bodies.
claimA real number $x$ is computable if there exists an algorithm that, given any natural number $n$, outputs a rational $q$ satisfying $|x - q| < 2^{-n}$.
background
Recognition Science fixes all constants in native units where the fundamental time quantum is the tick τ₀ = 1 from the imported Constants module. The present module adds a constructive layer by introducing the predicate of computability for real numbers. It then records that the golden-ratio fixed point phi, the number pi, and the field operations preserve computability, ensuring that quantities arising on the phi-ladder remain algorithmically tractable.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the constructive foundation required by the Recognition Science forcing chain T0–T8 and the Recognition Composition Law. It guarantees that the self-similar fixed point phi and the eight-tick octave remain inside the computable reals, thereby supporting downstream statements about spatial dimension D = 3 and the alpha band. No open questions are closed here.
scope and limits
- Does not prove general theorems about computable reals beyond the listed instances.
- Does not treat non-computable reals or their measure-theoretic properties.
- Does not connect computability statements to explicit physical predictions or mass formulas.
depends on (1)
declarations in this module (15)
-
class
Computable -
theorem
pi_computable -
theorem
phi_computable -
lemma
two_zpow_pos -
instance
rational_computable -
theorem
computable_neg -
theorem
computable_add -
theorem
computable_sub -
theorem
computable_mul -
theorem
computable_div -
theorem
computable_pow -
theorem
computable_log -
theorem
alpha_seed_computable -
theorem
log_phi_computable -
theorem
curvature_computable