theta_finetuning
plain-language theorem explainer
The declaration asserts that the probability of the QCD theta parameter satisfying |θ| < 10^{-10} by random selection is approximately 10^{-11}, framing the strong CP fine-tuning puzzle. A particle physicist or cosmologist working on axion models or discrete symmetry resolutions would cite it when quantifying why the observed bound appears unnatural. The proof is a one-line term-mode application of trivial that treats the probability estimate as an accepted fact inside the Recognition Science setting.
Claim. $P( |θ| < 10^{-10} ) ≈ 10^{-11}$ where θ ∈ [0, 2π) is the coefficient of the topological term (g²/32π²) G_μν G̃^μν in the QCD Lagrangian.
background
The module treats the strong CP problem as a direct consequence of eight-tick symmetry. The eight-tick structure (T7) forces discrete phases kπ/4 for k = 0 … 7 via the phase definition, and J-cost minimization on the phi-ladder selects the zero-phase state. Upstream results supply the necessary structures: EightTick.phase gives the discrete phase set, PhiForcingDerived.of encodes the J-cost functional, and LedgerFactorization.of calibrates the underlying multiplicative group. The local setting is that θ must be a multiple of π/4 under 8-tick periodicity, making generic values in [0, 2π) incompatible with the symmetry.
proof idea
The proof is a one-line term-mode wrapper that applies trivial. No lemmas are unfolded; the probability estimate is accepted directly from the module-level statement of the fine-tuning problem.
why it matters
This theorem quantifies the fine-tuning that the Recognition Science mechanism (8-tick symmetry plus J-cost selection) is intended to resolve, as described in the module doc-comment. It sits at the start of the StrongCP development and feeds the later claims that θ = 0 is selected by the discrete phase constraints. The declaration touches the open question of whether the probability estimate itself can be derived from the forcing chain rather than asserted.
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