Masses Mass Genesis T10 Pattern Anchor Equality T10 Raw Mode Source Data Nonempt

The settled anchor

A light pattern is a discrete record of eight phase values around a cycle, and an anchor is its first window: the two values at the start of the cycle. A settled anchor is a specific identification: the pattern's first window equals what a settled commitment posts, scaled by a real amplitude a. The framework's machine-checked library of formal theorems shows that if a pattern has a settled anchor and its posting amplitude equals the pattern's primitive positive-stationary factor amplitude, then the full raw-mode source interface is inhabited. In plain terms, all four premises that the raw canonical mode needs are present: a legal first-edge post, the ledger-to-photon support bridge, anchor neutrality, and the amplitude equation. The declaration t10RawModeSourceData_nonempty_of_settledAnchor is the theorem that this interface is nonempty under those two conditions.

The theorem's content is sharpest in what it leaves as a residual. For a settled anchor, the raw canonical mode holds if and only if the posting amplitude equals the primitive positive-stationary factor amplitude. That equivalence is proved exactly: the residual between settlement and the rigidity anchor premise is precisely the amplitude equation, nothing else. The support bridge, which was previously named as missing, is supplied by the settled shape, and anchor neutrality is supplied by settlement neutrality. The theorem does not claim that the primitive factor amplitude equals the unit posting amplitude; those are two distinct quantities, and the equality between them is not asserted.

The consequence is a rigidity result. Two Q3-closed patterns with the same topology whose anchors are settled windows satisfying the amplitude equation are equal. This means matter-pattern rigidity now fires from settlement data plus the amplitude equation alone, without any further unproved bridge. The settled anchor is the identification that makes the raw mode source data available, and the theorem certifies that the interface is inhabited whenever the two named inputs hold.

/-- Given a settled anchor and the amplitude equation, the full
`T10RawModeSourceData` interface (legal first-edge post, support bridge,
neutrality, amplitude) is inhabited. -/
theorem t10RawModeSourceData_nonempty_of_settledAnchor
    {a : ℝ} {ψ : LightPattern (Fin 8)}
    (hwin : SettledAnchorWindow a ψ)
    (hamp : a = primitivePositiveStationaryFactorAmplitude ψ) :
    Nonempty (T10RawModeSourceData ψ) := by
  obtain ⟨L, L', hpost⟩ := phase01_postingRealizable
  refine ⟨⟨L, L', hpost, ?_, settledAnchor_neutral hwin, ?_⟩⟩
  · intro _ i h0 h1
    exact settledAnchor_tail_zero hwin i h0 h1
  · rw [settledAnchor_phase0 hwin, hamp]

/-- **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

/-- **Rigidity from settlement.** Two Q3-closed patterns with the same
topology whose anchors are settled windows satisfying the amplitude
equation are EQUAL. Matter-pattern rigidity now fires from settlement data
plus the amplitude equation alone. -/
theorem q3Pattern_unique_of_sameTopology_settledAnchor
    {ψ χ : LightPattern (Fin 8)}
    (Eψ : Q3ClosedPatternEvidence ψ)
    (Eχ : Q3ClosedPatternEvidence χ)
    (htop : ψ.topology = χ.topology)
    {a b : ℝ}
    (hψwin : SettledAnchorWindow a ψ)
    (hχwin : SettledAnchorWindow b χ)
    (hψamp : a = primitivePositiveStationaryFactorAmplitude ψ)
    (hχamp : b = primitivePositiveStationaryFactorAmplitude χ) :
    ψ = χ :=
  q3Pattern_unique_of_sameTopology_rawCanonicalPositiveStationary
    Eψ Eχ htop
    (settledAnchor_rawCanonicalMode hψwin hψamp)
    (settledAnchor_rawCanonicalMode hχwin hχamp)

What this page does not claim

The primitive factor amplitude equals the unit posting amplitude. The settled anchor identification itself is derived rather than an input. The raw canonical mode holds without the amplitude equation being satisfied.

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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