Gravity Seven Gaps Path Sum Probes Freudenthal Bounded Complex Tet Verts

The torus attachment

The recognition framework, a system that models physical structure from a discrete record of events, builds many of its objects from a periodic grid of tetrahedra, the simplest 3D shapes. The declaration freudenthalBoundedComplex_tetVerts is a small but precise result about one such grid: it proves that the corner-to-tetrahedron incidence map of this grid is exactly the same as the one in a standard construction called the canonical periodic Freudenthal torus. In plainer terms, it verifies that when the framework attaches this torus to its path-sum state space, the basic combinatorial wiring of which tetrahedron touches which corner is preserved exactly, with no accidental reordering or duplication.

This attachment is part of a larger probe, labeled C3, whose purpose is to connect the torus to a space where path sums, weighted sums over paths, are studied. The probe proves the counts of vertices, edges, and tetrahedra (N³, 7N³, and 6N³ for a side length N), and it proves that the edge-endpoint and tetrahedron-corner maps match the canonical ones. It also proves that the translation group of the grid, the shifts that move the whole pattern by a fixed amount, embeds into the relabeling automorphisms of the attached object. This means that for a grid of side N, there are at least N³ distinct ways to relabel the object while preserving its structure, a fact that forces any unnormalized path-sum contribution from this torus to have modulus at most 1/N³.

What the declaration does not claim is as important as what it proves. It does not prove that the attached object is simplicial, meaning that its tetrahedra meet only along shared faces, edges, or vertices. The probe explicitly records that this is not established. It also does not preserve the assignment of which edge sits inside which tetrahedron, nor the metric geometry of each tetrahedron; the state space class treats tetrahedra as equilateral at a fixed scale by convention, not by proof. And it makes no claim about measures, limits, or the value of any path sum. The probe is explicitly non-flag-bearing: it records provenance and checks for a landmine, it does not assert a physical or analytic result.

The landmine check is the payoff. Because the translations embed, any future claim that an unnormalized torus contribution is nonvanishing or dominant must account for the 1/N³ suppression or be rejected as potentially 0 = 0. This is a concrete, machine-checked warning that prevents a whole class of overclaims. The declaration thus does real work: it pins down exactly what is known about this attachment, and exactly what remains open, so that later steps in the framework cannot quietly assume more than is proved.

/-- Tetrahedron-corner incidence is inherited verbatim from the canonical
encoder. -/
theorem freudenthalBoundedComplex_tetVerts (N : ℕ) [NeZero N] :
    (freudenthalBoundedComplex N).tetVerts = canonicalTetVerts N N N := rfl

/-- **LANDMINE, count form.**  `|Aut(T_N)| ≥ N ^ 3`. -/
theorem autCard_ge_translations (N : ℕ) [NeZero N] :
    N ^ 3 ≤ Nat.card (Aut (freudenthalBoundedComplex N)) := by
  have h := Nat.card_le_card_of_injective (translationAut N)
    (translationAut_injective N)
  rwa [Nat.card_eq_fintype_card, card_vertex] at h

/-- Outcome record for probes C3 and C6.  No `True` shells; every flag is
forced by `rfl` below. -/
structure ProbeStatus where
  torus_attached_to_state_space : Bool
  counts_and_incidence_preserved : Bool
  edge_in_tet_slots_preserved : Bool
  per_tet_metric_preserved : Bool
  simpliciality_of_image_proved : Bool
  translations_embed : Bool
  unnormalized_mu_torus_claims_admissible : Bool

What this page does not claim

The declaration does not prove that the attached torus is simplicial. It does not preserve the edge-in-tetrahedron assignment or the per-tetrahedron metric geometry. It makes no claim about measures, limits, or the value of any path sum.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: