Holography Edge Sector Bridge Sector Of Mem Admissible Sectors

The sector map

The declaration sectorOf_mem_admissibleSectors is a formal theorem in the Recognition Science framework's machine-checked library of formal theorems. It proves a specific combinatorial fact about a map called sectorOf. This map takes a configuration of four raw boundary bits, called edges, and sends it to a canonical representative of its symmetry orbit, which is the sector label. The theorem states that whenever such a configuration satisfies a closure constraint (a parity condition, meaning the XOR of the four bits is fixed), the resulting sector is always one of the four admissible sectors. In plain language, it proves that the sector label is a deterministic function of the edge bits alone: given the edges, the sector is fully determined, with no independent information of its own.

The proof is by exhaustive finite check, a decide tactic in the Lean kernel, meaning it is axiom-clean and verified by the computer. The theorem is one of two adjudicating facts in a larger argument about whether a proposed area law, H + log 4, double-counts information. The argument is that since the sector is entirely recoverable from the edges, adding a separate log 4 term for the sector on top of an entropy per event H counts the same information twice. A companion theorem, sector_is_lossy_quotient_of_closed, shows the map is exactly two-to-one: there are eight closed edge configurations but only four sectors, so the sector carries strictly less information (log 4) than the edges (log 8).

What this theorem does not claim is equally important. It does not decide whether the physically correct pixel-area formula is 4H (using raw pre-closure edge capacity) or 3H (using post-closure realized information). That remains a physical question about which substrate the area attaches to. The theorem also does not prove that one edge corresponds to one independent recognition event; that identification remains an unformalized physical assertion. The theorem only closes the combinatorial question: the sector is a projection of the edges, never a free-standing degree of freedom.

/-- **THEOREM (well-definedness, by `decide`).** `sectorOf` sends every ledger-closed
edge configuration into `admissibleSectors`: the canonical orbit representative of a
closed configuration is itself closed and is a sector representative. Closure is
`D₄`-invariant (permuting which vertex holds which bit does not change the XOR of the
4 bits), verified here by exhaustive finite check rather than assumed. -/
theorem sectorOf_mem_admissibleSectors :
    ∀ c : FaceCfg, closed c = true → sectorOf c ∈ admissibleSectors := by decide

/-- **The double-count kill (by `decide`).** The closed-edge substrate carries `log 8`
of information; the sector quotient carries only `log 4`, exactly half. Since sector
identity is entirely recoverable from (a strict compression of) the edge bits, adding
an independent `log 4` term on top of a per-edge entropy `H` double-counts information
the edges already contain. This is the precise combinatorial content behind killing
`H + log 4` as a candidate area law: `admissibleSectors` is a 2-to-1 lossy quotient of
`closedConfigs`, not a free-standing degree of freedom. -/
theorem sector_is_lossy_quotient_of_closed :
    admissibleSectors.card * 2 = closedConfigs.card := by decide

/-- **THEOREM (well-definedness, by `decide`).** `sectorOf` sends every ledger-closed
edge configuration into `admissibleSectors`: the canonical orbit representative of a
closed configuration is itself closed and is a sector representative. Closure is
`D₄`-invariant (permuting which vertex holds which bit does not change the XOR of the
4 bits), verified here by exhaustive finite check rather than assumed. -/
theorem sectorOf_mem_admissibleSectors :
    ∀ c : FaceCfg, closed c = true → sectorOf c ∈ admissibleSectors := by decide

What this page does not claim

This theorem does not decide whether the physically correct pixel-area formula is 4H or 3H. This theorem does not prove that one edge is one independent recognition event. This theorem does not resolve which lattice model, unshared-edge or shared-vertex, is physically correct for a horizon.

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/Holography/EdgeSectorBridge.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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