Materials Fuse Filament3 From Jcost

FDM Print Strength from J-Cost

Fused deposition modeling (FDM) is the common 3D printing process where a thermoplastic filament is melted and laid down in layers. A known weakness is that the bond between layers is weaker than the material itself; interlayer strength typically reaches only 50 to 80 percent of the bulk material's strength. The module materials fuse filament3 from jcost in the Recognition Science framework attempts to model this bond fraction using a cost function.

The framework's central object is the cost, a measure of the price of recognition, defined as J(x) = (x + 1/x)/2 - 1. The module defines a domain cost as J(m/e), where m and e are real numbers. The machine-checked library of formal theorems proves three general facts about this cost: it vanishes when m equals e, it is nonnegative for positive inputs, and the threshold phi - 3/2 is positive, where phi is the golden ratio. These are properties of the cost function itself, not of any printing process.

In plain language, the module proves that the cost is zero when the two quantities are equal, never negative when both are positive, and that a certain threshold involving the golden ratio is positive. The research note attached to the module goes further, suggesting that the ideal layer bond fraction equals 1 - J(phi) = 88.2 percent, and that a poor print with void fraction J(phi)^2 = 1.4 percent leads to 80 to 85 percent strength. But the formal proofs do not establish these claims, because the module never defines what m and e mean in terms of filament, layer, or bond.

In Recognition Science, the framework models physical structure from the forced cost of recognition. Here the framework's library shows that the cost function has the three basic properties, and the module packages them into a certificate structure. What the module does not do is connect the cost to any measurable property of a printed part. The step from J(m/e) to interlayer bond strength requires a definition of m and e in the subject's own terms, which the module does not provide.

The consequence is a clear boundary: the formal part is sound but empty of physics, while the physics lives in an unformalized research note. A reader should not take the 88.2 percent bond fraction as a proved result. It is a hypothesis about how the cost might apply to FDM, awaiting a definition that ties the abstract quantities to the real process.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The 88.2 percent bond fraction is not a proved result; it is an unformalized research note. The module does not establish any property specific to FDM printing or interlayer bonding. The certificate structure does not add physical content beyond the three general cost properties.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Materials/FuseFilament3FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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