Holography Edge Sector Bridge Sector Of Surjective On Closed

The sector as a projection

A sector, in the Recognition Science framework, is a label attached to a boundary configuration. The question is whether that label carries its own information or merely summarizes information already present in the raw edge bits. The declaration sectorOf_surjective_on_closed settles the second option: it proves that every one of the four admissible sectors is the image of some ledger-closed edge configuration under the deterministic map sectorOf. In plain terms, the sector is a projection of the edges, not an extra degree of freedom added on top of them.

The context matters. Each face configuration has four raw edge bits, but a parity constraint (the ledger-closure condition) removes one degree of freedom, leaving exactly 8 = 2³ closed configurations. The sector map then groups these 8 configurations into 4 orbits under a symmetry, picking the numerically least representative of each orbit. The theorem sector_is_lossy_quotient_of_closed states that the number of sectors times two equals the number of closed configurations: a clean 2-to-1 compression. The sector carries log 4 = 2 log 2 bits of information, while the closed edge substrate carries log 8 = 3 log 2 bits, strictly more.

This is the combinatorial core that kills a proposed area law. If one tried to write the entropy per event as H + log 4, treating the sector as an independent label, the theorem shows this double-counts: the sector's information is already contained in the edge bits. The proof is a finite exhaustive check, done by the decide tactic, so it is axiom-clean and machine-checked. The declaration itself is the surjectivity half: it confirms the map does not miss any sector, which combined with well-definedness makes the image exactly the set of admissible sectors.

What this does not claim is equally important. The theorem does not decide whether the physically correct pixel-area formula is 4H (using pre-closure edge capacity) or 3H (using post-closure realized information). That remains a physical question about which substrate the area attaches to. It also does not prove that one edge equals one independent recognition event; that identification is an unformalized physical assertion. The certificate bundles the combinatorial facts, but the physical interpretation is left open.

/-- **THEOREM (surjectivity, by `decide`). Sectors carry no information beyond the
edge bits.** Every one of the 4 admissible sectors is hit by `sectorOf` from some
ledger-closed edge configuration. Combined with `sectorOf_mem_admissibleSectors`, this
proves `admissibleSectors` is exactly the image of `closedConfigs` under a
deterministic quotient map — a sector is a *projection* of the edges, never an
independent quantity added on top of them. -/
theorem sectorOf_surjective_on_closed :
    admissibleSectors ⊆ closedConfigs.image sectorOf := 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

/-- **Restated as "3 free bits."** Ledger closure on 4 raw edge-bits leaves exactly 3
independent bits, not 4. This is the fact the `4H` vs `3H` live bet hinges on: if pixel
area tracks *realized* (post-closure) information, the count is 3; if it tracks *raw*
(pre-closure) capacity, the count is 4. -/
theorem closed_free_bits : closedConfigs.card = 2 ^ 3 := closed_configs_card

What this page does not claim

The theorem does not decide between the 4H and 3H pixel-area formulas. The theorem does not prove that one edge corresponds to one independent recognition event. The theorem does not apply to a shared-vertex lattice, where the count of free bits may differ.

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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