Foundation Public Spine Linking Assembly Target Of Arc Acyclic

The arc-acyclic target

In topology, an arc is a continuous path between two points, and a space is acyclic in a given dimension when it has no holes of that size: every loop can be shrunk to a point. The declaration target_of_arcAcyclic is a theorem in the framework's machine-checked library of formal theorems. It states that if, in every dimension other than three, the complement of an arc in a high-dimensional sphere has no holes in the relevant sense, then three is the only dimension where a certain kind of nontrivial linking can occur.

The theorem is conditional: it does not assert that those arc-complements are acyclic. That is a separate frontier claim, not yet proved. What the theorem establishes is the bridge: assuming that frontier, the framework's public spine can conclude that nontrivial linking forces dimension three. The proof proceeds by reducing the high-dimensional cases to a known vanishing result, with dimensions zero and one already unconditional.

This matters because it is part of a campaign to replace an earlier axiom. The framework previously relied on a postulate, linking_requires_D3, that simply asserted the conclusion. The new target aims to derive the same conclusion from a hypothesis about arc-complements, which is a stronger and more explanatory route. The theorem shows the derivation works if the hypothesis holds; it does not show the hypothesis itself.

What the declaration does not claim is just as important. It does not prove that arc-complements are acyclic in any dimension. It does not establish the physical recognition-to-linking bridge that would connect this topological result to physical space. And it does not, by itself, force three spatial dimensions in the physical world; it forces a mathematical conclusion conditional on an unproved frontier.

/-- The campaign target, conditional on the arc-complement frontier: a
fully inhabited `AlexanderLinkingBridge`, bypassing the
`DimensionForcing.linking_requires_D3` axiom on the public spine. -/
theorem target_of_arcAcyclic
    (harc : ∀ D, 2 ≤ D → D ≠ 3 →
      LinkingVanishingHighDim.ArcComplementsAcyclic D) :
    PublicSpine.target_D3_from_nonencoding_linking :=
  PublicSpine.bridge_of_forces_D3 (forces_D3_of_arcAcyclic harc)

/-- The binder's uniqueness half, conditional on the arc-complement
frontier.  Dimensions `0`, `1` are unconditional (`LinkingVanishingLowDim`);
dimensions `2` and `≥ 4` are the Mayer-Vietoris reduction of
`LinkingVanishingHighDim`. -/
theorem forces_D3_of_arcAcyclic
    (harc : ∀ D, 2 ≤ D → D ≠ 3 →
      LinkingVanishingHighDim.ArcComplementsAcyclic D) :
    ∀ D, PublicSpine.DetectsNontrivialLinking D → D = 3 :=
  fun D hdet => LinkingVanishingHighDim.forces_D3_of_arcAcyclic harc D hdet

/-- The campaign target, conditional on the arc-complement frontier: a
fully inhabited `AlexanderLinkingBridge`, bypassing the
`DimensionForcing.linking_requires_D3` axiom on the public spine. -/
theorem target_of_arcAcyclic
    (harc : ∀ D, 2 ≤ D → D ≠ 3 →
      LinkingVanishingHighDim.ArcComplementsAcyclic D) :
    PublicSpine.target_D3_from_nonencoding_linking :=
  PublicSpine.bridge_of_forces_D3 (forces_D3_of_arcAcyclic harc)

What this page does not claim

The theorem does not prove that arc-complements are acyclic in any dimension. It does not establish the physical bridge from recognition to linking. It does not force three spatial dimensions in the physical world without the frontier hypothesis.

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/Foundation/PublicSpineLinkingAssembly.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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