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theorem

reality_from_one_distinction

proved
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Why this theorem is linked from Placing the Near-Earth Object Impact Probability in Context unclear

Pith linked this Lean declaration because the review connected a specific passage in the paper to this theorem. The relation tag says how strong that connection is; it is not a generic placeholder.

We derive an impact frequency of NEOs 140m and larger, using the NEOMOD2 NEO population model and JPL Horizons.

Relation between the paper passage and the cited Recognition theorem.

From ∃ x y, x ≠ y the structure of reality is forced.

module
IndisputableMonolith.Foundation.RealityFromDistinction
domain
Foundation
line
88 · github
papers citing
12652 papers (below)

plain-language theorem explainer

Any inhabited carrier K admitting a non-trivial distinction forces the complete Recognition Science chain: absolute floor on K and on Bool, spacetime emergence with Lorentzian signature, time as orbit, and the algebraic derivations of c=1, ℏ and G from φ. Researchers deriving physics from bare distinguishability cite this as the master bundling result. The term-mode proof assembles the RealityCertificate by direct application of the upstream lemmas for each field.

Claim. Let $K$ be an inhabited type admitting a distinction $h : ∃ x,y : K, x ≠ y$. Then a RealityCertificate $K$ exists whose fields are the distinction $h$, the absolute-floor witness supplied by bare distinguishability, the native logic-realization instance, the absolute floor on Bool, the spacetime-emergence certificate, the light-cone condition, the time-as-orbit certificate, and the constant derivations $c=1$, $ℏ$ algebraic in $φ$, $G$ algebraic in $φ$.

background

RealityCertificate is the structure packaging the deliverables of the forcing chain for a carrier $K$. Its fields record the distinction hypothesis, the absolute-floor witness (Nonempty (AbsoluteFloorWitness $K$)), the logic-realization instance, the unconditional Bool absolute floor, spacetime emergence with Lorentzian signature, light-cone condition, time-as-orbit certificate, and the algebraic expressions for the constants. The module supplies the master theorem that threads absolute-floor closure, universal instantiation from distinction, time-as-orbit, spacetime emergence, and constant derivations into one deliverable. Upstream, absolute_floor_of_bare_distinguishability states that bare distinguishability supplies the nontrivial specification part of the absolute-floor witness, while bool_absolute_floor realizes it on the minimal carrier Bool.

proof idea

The proof is a term that constructs the RealityCertificate by supplying each field from an upstream result. The absolute_floor field is filled by absolute_floor_of_bare_distinguishability applied to $h$; native_realization by UniversalInstantiationFromDistinction.exists_logicRealization_of_distinction $K$ $h$; bool_floor by bool_absolute_floor; spacetime by spacetime_emergence_cert_nonempty; light_cone by lightlike_iff_speed_c; time_orbit by timeAsOrbitCert_inhabited; and the constants by c_rs_eq_one, ℏ_algebraic_in_φ, G_algebraic_in_φ respectively.

why it matters

This declaration bundles the forcing chain from one distinction into a single named theorem. It is invoked by reality_from_bool to produce the certificate on Bool and by terminalArrow_exists to obtain the terminal arrow for any distinguished carrier. Within the framework it completes the master forcing chain from the bare hypothesis of distinguishability through absolute-floor closure, spacetime emergence, and φ-derived constants, as described in the module documentation.

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papers checked against this theorem (showing 30 of 12652)

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