impossible_infinite_cost
No real number x ≤ 0 equals the value of any configuration. Modal geometry researchers cite this result to mark the lower boundary of the possibility space. The proof is a one-line term-mode contradiction that invokes the built-in positivity field on every configuration and closes with linear arithmetic.
claimFor every real number $x$ with $x ≤ 0$, there is no configuration $c$ such that the value field of $c$ equals $x$.
background
In ModalGeometry, configurations are taken from the ILG structure whose value field is required to be strictly positive. The J-cost, introduced in PhiForcingDerived, diverges to infinity as the argument approaches zero from above, so the region x ≤ 0 is excluded from the possibility space. Upstream lemmas supply the ledger factorization that calibrates J and the spectral emergence that fixes the discrete tier structure underlying all admissible values.
proof idea
The term proof introduces the witness pair ⟨c, hc⟩, extracts the positivity fact 0 < c.value from the Config structure, and obtains an immediate contradiction with the hypothesis x ≤ 0 by linarith.
why it matters in Recognition Science
The theorem supplies the lower edge of the possibility ball used throughout ModalGeometry. It realizes the boundary statement that ∂P is a limit point where J(x) → ∞ rather than an attainable configuration, thereby closing the forcing chain step that separates admissible costs from the infinite-cost region. No downstream uses are recorded yet.
scope and limits
- Does not construct any explicit configuration with positive value.
- Does not compute the numerical value of J at the boundary.
- Does not extend the result to complex or vector-valued costs.
- Does not address the rate at which J diverges.
formal statement (Lean)
252theorem impossible_infinite_cost (x : ℝ) (hx : x ≤ 0) :
253 ¬∃ c : Config, c.value = x := by
proof body
Term-mode proof.
254 intro ⟨c, hc⟩
255 have : 0 < c.value := c.pos
256 linarith
257
258/-- **BOUNDARY OF POSSIBILITY**: The limit of the possible.
259
260 ∂P = {x : x → 0⁺} where J(x) → ∞
261
262 This is NOT a configuration, but a limit of configurations. -/