Foundation Pair Kernel Gap2a Remaining Physical Equality Residual

The remaining physical equality

In Recognition Science, the framework builds physical structure from a discrete record of recognition events, called the ledger. The framework's library, a machine-checked collection of formal theorems, has already established the carriers and the bridge that connects them. What remains is a single, sharply defined question: which real number is the physical source scale? The module called PairKernelGap2aRemainingPhysicalEqualityResidual proves that the current premises do not answer this question, and that the smallest missing piece is exactly one equality statement.

The module works with two candidate source scales, called candidate A and candidate B. Both candidates satisfy all the current premises after the common-type bridge is in place. The theorem currentPremisesAfterCommonTypeBridge_admit_distinct_candidates proves that there exist two distinct real numbers that both satisfy the premises. This means the premises alone cannot select a unique scale. The module then shows that the remaining physical equality, the statement that the physical source is identified with a unique cotangent coordinate, is equivalent to each of the other candidate selection principles, and that it selects candidate B while rejecting candidate A.

The central result is that this remaining physical equality is unforced. The theorem forcesRemainingPhysicalEqualityFromAfterBridge_unforced proves that the current premises do not imply the remaining physical equality. The module also proves that any law forced uniformly by the current premises cannot be scale-breaking. This is a precise statement about what the framework can and cannot do at this stage: it can rule out scale-breaking laws, but it cannot yet pick the scale.

The module names this missing piece the smallest missing physical primitive. It is exactly the remaining physical equality, not a strictly weaker parent. The structure MissingRemainingPhysicalEqualityPrimitive bundles the proof that this equality is the atom, that it is equivalent to the other candidate selection principles, that it is unforced by the post-bridge package, that both banked candidates are admitted, and that the exact J-conjugate coordinate and a decoy unit both fail to satisfy it. The theorem remainingPhysicalEquality_gap2a_residual certifies that this bundle is complete.

What this means in plain language is that the framework has reached a precise boundary. It has proved that the current premises are not enough to decide the physical source scale, and it has isolated the exact statement that would decide it. The module does not reopen earlier bridges, does not introduce new axioms, and does not touch constants. It banks the residual as a theorem-backed fact: the gap is real, and it is exactly one equality wide.

theorem currentPremisesAfterCommonTypeBridge_admit_distinct_candidates :
    ∃ sourceScale₁ sourceScale₂ : ℝ,
      sourceScale₁ ≠ sourceScale₂ ∧
        CurrentPremisesAfterCommonTypeBridge sourceScale₁ ∧
        CurrentPremisesAfterCommonTypeBridge sourceScale₂ :=
  currentPremisesWithUniqueRealCotangent_admit_distinct_candidates

theorem forcesRemainingPhysicalEqualityFromAfterBridge_unforced :
    ¬ ForcesRemainingPhysicalEqualityFromAfterBridge :=
  remainingPhysicalEquality_unforced_after_commonTypeBridge

/-- Any law forced uniformly by the post-bridge package cannot select a unique
scale among the banked candidates. -/
theorem afterBridgeForcedLaw_cannot_be_scaleBreaking
    (law : ℝ → Prop)
    (hforced :
      ∀ sourceScale : ℝ,
        CurrentPremisesAfterCommonTypeBridge sourceScale →
          law sourceScale) :
    ¬ IsScaleBreakingAttachmentLaw law := by
  intro hbreaking
  rcases hbreaking with ⟨selected, ⟨_hselPrem, _hselLaw⟩, honly⟩
  have hA := hforced _
    candidateA_satisfies_currentPremisesAfterCommonTypeBridge
  have hB := hforced _
    candidateB_satisfies_currentPremisesAfterCommonTypeBridge
  have hAselected :
      candidateA_sourceMagnitudeExpr.eval = selected :=
    honly _
      ⟨candidateA_satisfies_currentRecognitionSourcePremises, hA⟩
  have hBselected :
      candidateB_sourceMagnitudeExpr.eval = selected :=
    honly _
      ⟨candidateB_satisfies_currentRecognitionSourcePremises, hB⟩
  exact
    candidates_select_distinct_magnitudes
      (hAselected.trans hBselected.symm)

theorem remainingPhysicalEquality_gap2a_residual :
    Gap2aRemainingPhysicalEqualityResidualCert where
  common_type_bridge_available := commonTypeBridge_cert
  distinct_candidates_admitted :=
    currentPremisesAfterCommonTypeBridge_admit_distinct_candidates
  no_unique_scale :=
    currentPremisesAfterCommonTypeBridge_do_not_select_unique_scale
  atom_iff_native_dual := remainingPhysicalEquality_iff_nativeActionDual
  atom_iff_pulled_back_equals_noether :=
    fun sourceScale a b hab =>
      remainingPhysicalEquality_iff_pulledBackEqualsNoether
        sourceScale a b hab
  atom_unforced := forcesRemainingPhysicalEqualityFromAfterBridge_unforced
  forced_laws_not_scale_breaking := afterBridgeForcedLaw_cannot_be_scaleBreaking
  candidateA_fails_atom := remainingPhysicalEquality_rejects_candidateA
  candidateB_satisfies_atom := remainingPhysicalEquality_selects_candidateB
  missing_parent := missingRemainingPhysicalEqualityPrimitive
  does_not_claim_gap2a_closure := trivial

What this page does not claim

This module does not prove that candidate A or candidate B is the actual physical scale. This module does not introduce a new axiom or modify the Constants.lean file. This module does not claim that the remaining physical equality is derivable from the current premises.

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/PairKernelGap2aRemainingPhysicalEqualityResidual.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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