Milestone task: open ExistenceWellPosedness.lean in AXLE and locate the compression contraction proof. The toy compression C_toy (x : ℝ) : ℝ := r_star * x is proved to strictly contract distances by exactly \(r^*\): theorem compression_contracts and theorem compression_strict, both proved (0 sorry) for \(r^* \in [0,1)\). The composite \(\Phi = R \circ K \circ F \circ C_{{\text{{toy}}}}\) is proved to contract by the same factor \(r^*\), because non-expansive maps cannot increase a contraction’s rate — a clean, general argument, not specific to this toy model.
r_star as a generic contraction-ratio variable (variable (r_star : ℝ)), proved abstractly for any \(r^* \in [0,1)\) — it is not, in this file, pinned to the specific certified value \(0.77594059\) from Week 3. Dm3Arithmetic.lean is the file that pins the concrete number. Both files legitimately use the same name for related but distinct objects: an abstract parameter proved-in-general in one file, a concrete certified constant in another. Conflating them would overstate what’s proved — the abstract contraction theorem doesn’t by itself certify that \(0.77594059\) is the right number for this toy system; that’s what certify_rstar.py does, separately.
Submission: state, in your own words, the difference between "the composite map contracts by whatever \(r^*\) turns out to be" (proved in general, ExistenceWellPosedness.lean) and "\(r^*\) for the actual dm³ toy ODE is \(0.77594059\)" (established numerically, certify_rstar.py, cross-checked against the Lean constant in Dm3Arithmetic.lean). This is the same discipline as 101’s Week 8 (\(T^*=2\pi\): structure data vs. geometric derivation) applied to a new pair of objects. Phase I is closed — Week 5 opens Phase II with \(\eta\).
-- dm³ 102 · Week 04 · Milestone III — μ closed out -- ExistenceWellPosedness.lean (abstract, general r*): -- def C_toy (x) := r_star * x -- theorem compression_contracts -- proved, any r* ≥ 0 -- theorem compression_strict -- proved, any r* < 1 -- Φ = R∘K∘F∘C_toy contracts by r* -- proved (non-expansive composition) -- Dm3Arithmetic.lean (concrete, certified value): -- noncomputable def r_star : ℝ := 0.77594059 -- r_star_pos, r_star_lt_one -- proved -- Exercise: these are related but distinct claims. Write one -- paragraph distinguishing "contracts by SOME r*" (general, proved) -- from "r* IS 0.77594059 for THIS system" (numerical, certified). example : True := trivial