Masses Mass Genesis T10 Pattern Anchor Equality Q3 Pattern Unique Of Same Topolo

Pattern uniqueness from settled anchors

In the Recognition Science framework, a ledger (a discrete record of events) gives rise to patterns of light, each with a shape called its topology. The theorem q3Pattern_unique_of_sameTopology_settledOctaveAnchor states a rigidity result: if two such patterns share the same topology, and each has an anchor window that matches the settlement window of a settled legal octave at some posting amplitude, then the patterns are equal, provided each posting amplitude equals the pattern's primitive positive-stationary factor amplitude.

The condition on the anchor window is precise. A light pattern has a settled anchor at posting amplitude a when its first window equals a times the settlement window of the phase-zero commitment. The theorem also has an octave-origin form, where the anchor window is read from the run-origin commitment of a settled legal octave; the two forms agree because every settled octave's origin reads phase zero. The key equivalence is that, for a settled anchor, the raw canonical positive-stationary mode holds if and only if the posting amplitude equals the primitive positive-stationary factor amplitude. The residual between settlement and the rigidity anchor premise is exactly that amplitude equation, nothing else.

This theorem is proved in the framework's machine-checked library of formal theorems. It supplies the previously missing ledger-to-photon support bridge, and it transports settlement neutrality to the pattern anchor. From settlement data plus the amplitude equation alone, matter-pattern rigidity fires: two Q3-closed patterns with the same topology and settled anchors satisfying the amplitude equation are equal.

The theorem does not claim that the primitive factor amplitude equals the unit posting amplitude. The identification that a pattern's anchor window is what one settled commitment posts is carried in shape, but the identification itself and the amplitude equation remain the two named inputs. The theorem also does not claim that any particular pattern exists; it only says that if two patterns meet the conditions, they are the same pattern.

/-- Octave form of rigidity: the anchors may be read from the origins of two
possibly different settled legal octaves. -/
theorem q3Pattern_unique_of_sameTopology_settledOctaveAnchor
    {ψ χ : LightPattern (Fin 8)}
    (Eψ : Q3ClosedPatternEvidence ψ)
    (Eχ : Q3ClosedPatternEvidence χ)
    (htop : ψ.topology = χ.topology)
    {octaveψ octaveχ : Q3SettledLedgerOctave}
    {a b : ℝ}
    (hψwin : SettledOctaveAnchorWindow octaveψ a ψ)
    (hχwin : SettledOctaveAnchorWindow octaveχ b χ)
    (hψamp : a = primitivePositiveStationaryFactorAmplitude ψ)
    (hχamp : b = primitivePositiveStationaryFactorAmplitude χ) :
    ψ = χ :=
  q3Pattern_unique_of_sameTopology_settledAnchor Eψ Eχ htop
    ((settledOctaveAnchorWindow_iff octaveψ a ψ).mp hψwin)
    ((settledOctaveAnchorWindow_iff octaveχ b χ).mp hχwin)
    hψamp hχamp

/-- **The residual is exactly the amplitude equation.** For a settled
anchor, the raw canonical mode holds if and only if the posting amplitude
equals the primitive positive-stationary factor amplitude. Nothing else is
missing between settlement and the rigidity anchor premise. -/
theorem settledAnchor_rawCanonicalMode_iff_amplitude
    {a : ℝ} {ψ : LightPattern (Fin 8)}
    (hwin : SettledAnchorWindow a ψ) :
    AnchorPhaseRawCanonicalPositiveStationaryPrimitiveFactorMode ψ ↔
      a = primitivePositiveStationaryFactorAmplitude ψ := by
  constructor
  · intro hmode
    have hcomp := components_of_rawCanonicalPositiveStationaryMode ψ hmode
    have h0 := hcomp.phase0_amplitude
    rw [settledAnchor_phase0 hwin] at h0
    exact_mod_cast h0
  · exact settledAnchor_rawCanonicalMode hwin

/-- The two forms agree, because every settled octave's origin reads phase
zero. -/
theorem settledOctaveAnchorWindow_iff
    (octave : Q3SettledLedgerOctave) (a : ℝ)
    (ψ : LightPattern (Fin 8)) :
    SettledOctaveAnchorWindow octave a ψ ↔ SettledAnchorWindow a ψ := by
  unfold SettledOctaveAnchorWindow SettledAnchorWindow
  rw [settledOctave_initial_phase_zero octave]

What this page does not claim

The theorem does not claim that the primitive factor amplitude equals the unit posting amplitude. The theorem does not claim that any particular pattern exists; it only asserts equality under the stated conditions. The theorem does not claim that the settled anchor identification itself is derived; it remains a named input.

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/Masses/MassGenesis/T10PatternAnchorEquality.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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