Masses Mass Genesis T10 Directed Posting Modular Seed Sqrt Eigenvalue Eq Sqrt Mo

The square-root eigenvalue

The declaration sqrtEigenvalue_eq_sqrt_modularEigenvalue is a formal theorem in the framework's machine-checked library of formal theorems. It states that the square root of the relative modular operator's eigenvalue equals the eigenvalue of the square-root operator. In plain terms, for a specific four-sector algebra, the number you get by taking the square root of a ratio of weights is the same as the ratio of the square roots of those weights. This is a standard property of square roots applied to a particular operator, but the declaration proves it for the exact construction used in the mass-genesis seed.

The construction is a directed-posting algebra. The sector space has four states, representing two phases (commit and clear) across two rungs (0 and 1). The one posting event is a matrix unit that maps the commit sector to the clear sector at the base rung. It is nilpotent and has no right inverse, so the posting cannot be undone. The event set is the cone of nonnegative real multiples of that unit, making the ledger append-only. The weights before and after the posting are built from a single event functional, the per-phase load quantum a², with the pre-posting ladder carrying a factor of φ (the golden ratio) per rung and the post-posting weight adding one more a² quantum at the base rung.

The theorem's content is that the relative modular operator, defined by the standard formula Δ(X) = ρ' X ρ⁻¹, has eigenvalues {1, 2, φ, 2/φ} with multiplicities 4, 8, 2, and 2 over the 16 matrix units. Its square root has eigenvalues that are the square roots of these, in particular √2 on the posted rung's diagonal units. The declaration sqrtEigenvalue_eq_sqrt_modularEigenvalue confirms that the eigenvalue of the square-root operator is indeed the square root of the eigenvalue of the original operator. This is not a new physical law but a precise algebraic identity within this specific model.

What the declaration does not claim is broader. It does not claim that the square root of two is forced by the posting alone; the control theorem no_posting_diagonal_eigenvalue_one records that without the deposit every diagonal eigenvalue is 1. The √2 content is exactly the posting's signature. It also does not claim that √2 can be generated from φ alone. Two separate theorems prove this: two_not_zpow_phi shows that 2 is no integer power of φ, and sqrt_two_not_inQPhi shows that √2 does not lie in the field Q(φ). The non-rational modular spectrum arises from the pair, the ledger's φ profile against the deposit, not from any deposit forcing a square root.

In Recognition Science, this seed is a building block for mass genesis. The framework models masses on a φ-power ladder, and this theorem shows that a single directed posting introduces a √2 scale that cannot be reduced to the golden ratio. The declaration is a theorem about a model, not a derivation of a physical constant. It is a precise, machine-checked statement about a specific algebraic construction, and its significance lies in what it rules out: that the posting's effect could be expressed using only powers of φ.

/-- The square-root modular eigenvalue on `E_{ij}`. -/
def sqrtEigenvalue (a : ℝ) (i j : Sector) : ℝ :=
  Real.sqrt (omegaPost a i) / Real.sqrt (omegaPre a j)

/-- **PROPOSITION 2: the relative modular operator is not scalar.** The
posted rung's diagonal eigenvalue is `2`, the unposted rung's is `1`, and no
single scalar can be both. -/
theorem relativeModular_not_scalar {a : ℝ} (ha : a ≠ 0) :
    ¬ ∃ c : ℂ, ∀ X : M4, relativeModular a X = c • X := by
  rintro ⟨c, hc⟩
  have h00 := hc (Matrix.single 0 0 (1 : ℂ))
  rw [relativeModular_eigen, eigen_00 ha] at h00
  have h22 := hc (Matrix.single 2 2 (1 : ℂ))
  rw [relativeModular_eigen, eigen_22 ha] at h22
  have key : ∀ (d : ℝ) (i : Sector),
      ((d : ℝ) : ℂ) • Matrix.single i i (1 : ℂ) = c • Matrix.single i i (1 : ℂ) →
      ((d : ℝ) : ℂ) = c := by
    intro d i h
    have e := congr_fun (congr_fun h i) i
    simp [Matrix.single] at e
    exact e
  have e1 := key 2 0 h00
  have e2 := key 1 2 h22
  have h21 : ((2 : ℝ) : ℂ) = ((1 : ℝ) : ℂ) := by rw [e1, e2]
  norm_num at h21

/-- **Escape 1: the diagonal ratio `2` is no integer power of `phi`.**
`phi^0 = 1`, `phi^1 < 2`, `phi^k >= phi^2 > 2.5` for `k >= 2`, and
`0 < phi^k < 1` for `k < 0`. So the modular ratio group is not generated by
`phi` alone. -/
theorem two_not_zpow_phi : ¬ ∃ k : ℤ, phi ^ k = 2 := by
  rintro ⟨k, hk⟩
  rcases lt_trichotomy k 0 with hneg | rfl | hpos
  · have h2 : phi ^ k < 1 := by
      have hmk : (1 : ℝ) < phi ^ (-k) := one_lt_zpow₀ one_lt_phi (by omega)
      have hinv : phi ^ k = (phi ^ (-k))⁻¹ := by
        rw [← zpow_neg, neg_neg]
      rw [hinv]
      exact inv_lt_one_of_one_lt₀ hmk
    linarith [hk, h2]
  · rw [zpow_zero] at hk
    norm_num at hk
  · rcases eq_or_lt_of_le (show (1 : ℤ) ≤ k from hpos) with h1 | h2
    · subst h1
      rw [zpow_one] at hk
      linarith [phi_lt_two, hk]
    · have hk2 : phi ^ k = phi ^ (2 : ℤ) * phi ^ (k - 2) := by
        rw [← zpow_add₀ (ne_of_gt phi_pos)]
        congr 1
        omega
      have hge1 : (1 : ℝ) ≤ phi ^ (k - 2) := one_le_zpow₀ one_lt_phi.le (by omega)
      have h25 : (2.5 : ℝ) < phi ^ (2 : ℤ) := by
        rw [zpow_ofNat]
        exact phi_squared_bounds.1
      have hmul : phi ^ (2 : ℤ) * (1 : ℝ) ≤ phi ^ (2 : ℤ) * phi ^ (k - 2) :=
        mul_le_mul_of_nonneg_left hge1 (le_of_lt (zpow_pos phi_pos 2))
      have hgt : phi ^ k > 2 := by
        rw [hk2]
        linarith [hmul, h25]
      linarith [hk, hgt]

/-- **Escape 2: `sqrt(2)` is not in the rational-phi field.** If
`sqrt(2) = p + q phi` with rational `p q`, squaring and using `phi^2 =
phi + 1` with rational independence gives `p^2 + q^2 = 2` and
`2 p q + q^2 = 0`; the first case makes `2` a rational square, the second
makes `10` one. Both are `2 mod 4`, contradiction. -/
theorem sqrt_two_not_inQPhi : ¬ InQPhi (Real.sqrt 2) := by
  rintro ⟨p, q, hpq⟩
  have hsq : ((p : ℝ) + (q : ℝ) * phi) ^ 2 = 2 := by
    rw [← hpq, Real.sq_sqrt (by norm_num)]
  have hexp : ((p ^ 2 + q ^ 2 : ℚ) : ℝ) + ((2 * p * q + q ^ 2 : ℚ) : ℝ) * phi
      = ((2 : ℚ) : ℝ) + ((0 : ℚ) : ℝ) * phi := by
    push_cast
    linear_combination hsq - (q : ℝ) ^ 2 * phi_sq_eq
  obtain ⟨h1, h2⟩ := rat_linear_independent hexp
  have hq0 : q * (2 * p + q) = 0 := by linear_combination h2
  rcases mul_eq_zero.mp hq0 with hqz | hqz
  · subst hqz
    simp at h1
    exact not_rat_sq_of_two_mod_four (by norm_num : 2 % 4 = 2) (by exact_mod_cast h1)
  · have hqeq : q = -2 * p := by linear_combination hqz
    rw [hqeq] at h1
    have h5 : 5 * p ^ 2 = 2 := by linear_combination h1
    have h10 : (5 * p) ^ 2 = 10 := by linear_combination 5 * h5
    exact not_rat_sq_of_two_mod_four (by norm_num : 10 % 4 = 2) (by exact_mod_cast h10)

/-- The post weight is exactly one posting step from the pre weight. -/
theorem omegaPost_is_post_step (a : ℝ) : PostsStep a (omegaPre a) (omegaPost a) := rfl

What this page does not claim

The declaration does not prove that any physical mass equals √2 times another mass. It does not claim that the posting alone forces the square root of two; the control theorem shows the deposit is necessary. It does not establish any property of infinite-dimensional factors or type III₁ classification.

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/T10DirectedPostingModularSeed.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.

Derived articles

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