Holography Record Cost Asymmetry Target Record Cost Asymmetry Holds

The record-cost asymmetry

Bekenstein-Hawking entropy, the thermodynamic entropy of a black hole horizon, is proportional to its area. In the framework's discrete model, the horizon is a finite map between configurations, and the question is which side of that map carries the entropy: the image (the distinct records the map produces) or the kernel (the states it merges together). The declaration target_record_cost_asymmetry_holds establishes, as a proved theorem, that the cost of a ledger counts only the performed distinctions, the image side, and not the collapsed microstates on the kernel side.

The theorem's content is concrete. For a finite map, the record cost, the number of independent binary distinctions the map performs, is the base-2 logarithm of the image's cardinality. A map that distinguishes nothing has record cost zero; a map with two distinct outputs has record cost one. The theorem proves that the eight "free" microstates in the closed fiber all map to the same posted record, so they contribute one distinction, not three. It also proves that the recognition multiplicity, the floor-side count of posted generators, equals this record cost at the relevant scales, and that it differs from the microstate cost, the logarithm of the kernel's size. This is the asymmetry: the ledger charges for what is distinguished, not for what is merged.

This asymmetry resolves a previously open selector: whether horizon entropy attaches to the image or the kernel of the closure map. The theorem proves the image side is forced, given only that ledger cost counts performed distinctions. The rank-nullity identity, which states that the image size times the kernel size equals the domain size, is symmetric and cannot prefer one side; the directionality of the cost function is the sole symmetry-breaker.

In Recognition Science, this result upgrades the earlier conditional derivation to a genuine selection theorem at the floor level. The sole remaining physical input is the identification that horizon thermodynamic entropy equals the record cost of the boundary closure map. Given that premise, the 1/4 coefficient is a theorem; under the counterfactual microstate reading, the same machinery yields 3/4. The theorem thus sharpens the fork between two coherent readings, rather than removing the need for the physical identification.

What the theorem does not claim is equally important. It does not prove that the physical identification itself is true; that remains a named premise, HorizonEntropyIsRecordCost. It does not assert that the microstate reading is thermodynamically incoherent; the module explicitly retracts an earlier claim to that effect. It does not derive the fine-structure constant or any other coupling. The theorem's scope is the discrete, finite model; its power is in showing that, within that model, the record-cost premise uniquely selects the 1/4 coefficient.

theorem target_record_cost_asymmetry_holds : target_record_cost_asymmetry := by
  refine ⟨?_, records_performed, fiber_posts_one_record,
          recordCost_eq_multiplicity_one, recordCost_eq_multiplicity_two,
          bekenstein_selector_from_asymmetry, ?_⟩
  · intro α β _ _ f h
    exact record_zero_general f h
  · intro α β _ _ _ f hconst a₀ hcard
    exact microstate_cost_nonzero_on_constant f hconst a₀ hcard

/-- **Record-zero (general).** A finite map whose image is a subsingleton (`|image| ≤ 1`)
performs no distinction and hence holds zero record cost. A boundary that distinguishes
nothing holds no records. Holds for an ARBITRARY finite map, so it is a theorem, not an
axiom. (Formerly `landauer_zero_general`; see the naming correction in the module header —
this is Bennett memory bookkeeping, not Landauer erasure.) -/
theorem record_zero_general {α β : Type*} [Fintype α] [DecidableEq β] (f : α → β)
    (h : (Finset.univ.image f).card ≤ 1) : recordCost f = 0 :=
  log2_eq_zero_of_le_one h

/-- The one-face closure map's record cost is `1` — the rank, reusing
`CoefficientBridge.closureRank`. -/
theorem recordCost_closed : recordCost (fun c : FaceCfg => PixelLocal.closed c) = 1 := by
  decide

/-- **The symmetry-break (the physical content).** The `2³ = 8` ledger-closed microstates —
the "free bits" whose `log₂ 8 = 3` gives the `κ = 4/3` reading — all map to the SAME posted
record (`closed = true`). So the entire closed fiber performs exactly **one** distinction, not
three. The `κ = 4/3` branch counts *unposted* bulk degeneracy; the ledger floor charges
*performed* distinctions, and this fiber performs one. This is the term static counting lacks:
the image is directional (records produced), the kernel is invisible to the record ledger. -/
theorem fiber_posts_one_record :
    (EdgeSectorBridge.closedConfigs.image (fun c : FaceCfg => PixelLocal.closed c)).card = 1 := by
  decide

/-- **Descent (THEOREM, axiom-clean).** GIVEN the single record-cost identification, the
pixel-to-sector ratio is forced to the Bekenstein `1/4`, with no further premise. -/
theorem bekenstein_coefficient_of_record_cost (m : ℕ)
    (h : HorizonEntropyIsRecordCost m) :
    (m : ℚ) / (admissibleSectors.card : ℚ) = 1 / 4 := by
  unfold HorizonEntropyIsRecordCost at h
  rw [h, recordCost_closed, recognition_sector_count]; norm_num

/-- **The premise does real work: the kernel reading forces `3/4` (`κ = 4/3`).** The closed
fiber has `2³ = 8` microstates, so `microstateCost closed true = log₂ 8 = 3`, and the ratio
is `3/4`. This is precisely the branch the record-cost premise excludes; it is exhibited here
so that the single identification is visibly the fork, not a hidden re-labeling of the
answer. -/
theorem kappa_four_thirds_of_microstate_cost (m : ℕ)
    (h : HorizonEntropyIsMicrostateCost m) :
    (m : ℚ) / (admissibleSectors.card : ℚ) = 3 / 4 := by
  unfold HorizonEntropyIsMicrostateCost at h
  have hmc : microstateCost (fun c : FaceCfg => PixelLocal.closed c) true = 3 := by decide
  rw [h, hmc, recognition_sector_count]; norm_num

What this page does not claim

The physical identification that horizon entropy equals record cost is not proved; it is a named premise. The theorem does not claim the microstate reading is thermodynamically incoherent; that claim was explicitly retracted. The result does not derive any coupling constant such as the fine-structure constant.

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/RecordCostAsymmetry.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.

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