Masses Mass Genesis T10 Posting Cascade

The posting cascade

The T10 posting cascade is a framework-internal model of how deposits of mass accumulate. In Recognition Science, a ledger is a discrete record of events, and a deposit is one such event that adds a fixed quantum of weight to a chosen rung of a ladder. The cascade is the sequence of deposits, written as an ordered list of rungs, where each entry adds its quantum in turn. This is a definitional choice, a model of how mass could build up, not a claim about what happens in the physical world.

The framework's machine-checked library of formal theorems proves several structural facts about this cascade. The central result is that the eigenvalue of the cascade, a number that describes how the modular operator scales a matrix entry, equals (φ^n + count(n)) / φ^m, where count(n) is how many times rung n appears in the deposit list. This formula is proved, not assumed. It shows that the eigenvalue depends only on the deposit counts, not on the amplitude of the quantum, and that the diagonal response exceeds 1 exactly at rungs that have received deposits. A single deposit at rung k recovers the spectrum of the earlier single-posting module, so the cascade generalizes that case without losing it.

Four candidate principles for selecting where deposits land all fail, and each failure is a proved theorem. First, conditioning: the step spectrum of a second deposit depends on the prior sequence, but matching a cascade to a single-posting weight profile forces the deposit counts to be a singleton, and the only spectrally distinguished singleton is rung 1, the killed phi-alone shape. So conditioning collapses to the already-killed case. Second, field escape: √φ is not in the field Q(φ), extending the earlier result that √2 is not, but the general case for other k remains open. Third, extremal principles: total excess load equals sequence length regardless of locations, and a single deposit's response contribution is strictly decreasing in the rung number, so the killed shape maximizes response. Fourth, shift equivariance: cyclic-shift equivariance of the empty cascade is false for N ≥ 2, and singleton cascades inherit the noncommutation of the earlier module, so noncommutation does not select among singletons.

In plain language, the module proves a negative result: no principle among these four forces a viable nonempty deposit location from the accumulated modular data and the append-only structure. The cascade verdict is a theorem, not a conjecture. The framework's own structure, as formalized, does not determine where mass deposits land. This is a boundary result: it says what the framework does not claim, and it leaves the question of actual deposit location open for further principles or extensions.

theorem cascadeEigenvalue_eq {N : ℕ} {a : ℝ} (ha : a ≠ 0) (ks : List (Fin N))
    (n m : Fin N) (p q : Fin 2) :
    cascadeEigenvalue a ks (n, p) (m, q) =
      (phi ^ n.val + (depositCount ks n : ℝ)) / phi ^ m.val := by
  have ha2 : a ^ 2 ≠ 0 := pow_ne_zero 2 ha
  have hnum : omegaCascade a ks (n, p) =
      a ^ 2 * (phi ^ n.val + (depositCount ks n : ℝ)) := by
    show postingEventLoad a * phi ^ n.val +
        (depositCount ks n : ℝ) * postingEventLoad a =
      a ^ 2 * (phi ^ n.val + (depositCount ks n : ℝ))
    unfold postingEventLoad
    ring
  have hden : omegaPreN a (m, q) = a ^ 2 * phi ^ m.val := rfl
  unfold cascadeEigenvalue
  rw [hnum, hden, mul_div_mul_left _ _ ha2]

theorem cascade_diagonal_gt_one_iff {N : ℕ} {a : ℝ} (ha : a ≠ 0)
    (ks : List (Fin N)) (n : Fin N) (p : Fin 2) :
    1 < cascadeEigenvalue a ks (n, p) (n, p) ↔ 0 < depositCount ks n := by
  rw [cascadeEigenvalue_diag ha]
  constructor
  · intro h
    have h' : (1 : ℝ) * phi ^ n.val < phi ^ n.val + (depositCount ks n : ℝ) :=
      (lt_div_iff₀ (pow_pos phi_pos _)).mp h
    have : (0 : ℝ) < (depositCount ks n : ℝ) := by linarith
    exact Nat.cast_pos.mp this
  · intro h
    rw [lt_div_iff₀ (pow_pos phi_pos _)]
    have : (0 : ℝ) < (depositCount ks n : ℝ) := Nat.cast_pos.mpr h
    linarith [pow_pos phi_pos n.val]

theorem singleton_cascade_spectrum {N : ℕ} (k : Fin N) :
    cascadeSpectrum ([k] : List (Fin N)) = modularSpectrum k := by
  ext x
  constructor
  · rintro ⟨n, m, rfl⟩
    rw [depositCount_singleton]
    by_cases h : n = k
    · refine Or.inr ⟨m, ?_⟩
      rw [if_pos h, h]
      push_cast
      rw [add_comm (phi ^ k.val) (1 : ℝ)]
    · refine Or.inl ⟨n, m, h, ?_⟩
      rw [if_neg h]
      push_cast
      rw [add_zero]
  · rintro (⟨n, m, hnk, rfl⟩ | ⟨m, rfl⟩)
    · refine ⟨n, m, ?_⟩
      rw [depositCount_singleton, if_neg hnk]
      push_cast
      rw [add_zero]
    · refine ⟨k, m, ?_⟩
      rw [depositCount_singleton, if_pos rfl]
      push_cast
      rw [add_comm (phi ^ k.val) (1 : ℝ)]

theorem total_excess_length_only {N : ℕ} (ks : List (Fin N)) :
    (∑ n : Fin N, depositCount ks n) = ks.length := by
  classical
  induction ks with
  | nil => simp [depositCount]
  | cons k ks ih =>
    have hsplit :
        (∑ n : Fin N, depositCount (k :: ks) n) =
          (∑ n : Fin N, depositCount ks n) + 1 := by
      simp only [depositCount, List.count_cons]
      rw [Finset.sum_add_distrib]
      simp [Finset.sum_ite_eq]
    rw [hsplit, ih, List.length_cons]

What this page does not claim

The cascade verdict does not claim that deposits cannot land anywhere, only that the four tested principles do not force a location. The model does not claim that the deposit sequence is a physical process, only a formal construction. The theorem does not claim that √2 or √φ are the only field-escape cases; the general case remains open.

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/T10PostingCascade.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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