ph006_probability_certificate
plain-language theorem explainer
PH-006 certifies that probability equals the fiber structure of lossy projections from a deterministic reality possessing a unique positive J-minimizer. Foundations researchers and philosophers of physics cite it to unify frequentist, Bayesian, propensity, and logical interpretations as partial views of the same projection shadow. The term proof directly assembles the unique defect-zero witness, the projection_lossy theorem, and the fiber partition lemmas.
Claim. There exists a unique positive real number $x$ such that the defect function vanishes at $x$. For every finite-resolution observer, the projection map from underlying states to observed outcomes is many-to-one. The fibers of this projection partition the real line, and every fiber is nonempty.
background
The module resolves the meaning of probability in Recognition Science as J-cost projection weight. Reality remains deterministic with a unique J-minimizer at each ledger step; observers possess finite positive resolution and therefore see only coarse-grained projections of the underlying state. Probability is epistemic: the weight of an outcome equals the measure of its preimage fiber under the projection map.
proof idea
The term constructs the four conjuncts explicitly. The first conjunct supplies the unique positive defect-zero point by exhibiting 1 via defect_at_one and proving uniqueness via defect_zero_iff_one. The second conjunct applies the upstream projection_lossy theorem. The third and fourth conjuncts invoke the sibling lemmas fibers_cover and each_fiber_nonempty to establish the partition property and fiber nonemptiness.
why it matters
This certificate completes the PH-006 step by showing probability is the projection shadow of deterministic ledger dynamics rather than an ontic primitive. It unifies the four classical interpretations exactly as stated in the module doc-comment: frequentist frequencies track fiber measures, Bayesian credence tracks resolution, propensity tracks the J-cost landscape, and logical structure tracks the partition. The result anchors downstream claims that the Born rule emerges from the same projection geometry.
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