Masses Mass Genesis T10 Load Reachability Audit Single Phase Multiplicity Compos

The scaling lemma

The declaration singlePhaseMultiplicity_compositePosting is a lemma about a specific kind of composite posting, which in this framework means a weighted combination of eight phase postings. The lemma states that if you take a single phase, put a rational number c of postings there, and leave all other phases empty, the result is exactly the same as scaling the unit posting by c. This is a precise algebraic identity: the composite posting at time t equals c times the unit posting at t.

The consequence is a simple formula for the load, which is the squared norm of the neutralized posting, a measure of its size. The load of this composite posting is 2c². This follows because neutralization is linear and the load is a squared norm. The lemma is proved in the machine-checked library of formal theorems, and it is the first step in a larger audit of what loads are reachable.

The audit shows a striking fact. The set of loads reachable by composite postings includes 2c² for every rational c. This set is dense in the nonnegative real numbers: for any positive target load L and any positive tolerance ε, there is a rational c such that 2c² is within ε of L. In other words, composite postings can approach any required load, including the loads the mass law predicts, to any desired precision.

Yet the same audit proves that these loads never equal the required load exactly. The required load is missed exactly and approached to every tolerance. This is not a repair of the wall that says no composite posting equals the target; the wall stands. What changes is the interpretation. Exact equality of real numbers is not what carrying a mass spectrum requires, and no comparison in the library tests it. The wall has zero width.

What the lemma does not claim is therefore as important as what it proves. It does not claim that the ledger can carry the mass spectrum, because the audit only concerns exact equality, not approximation. It does not claim that the wall is broken or that the mass law is wrong. It does not claim that the framework has derived the mass spectrum from first principles. The lemma is a measurement of the wall's strength, not a repair of it.

/-- The composite posting it names is the unit posting scaled by `c`. -/
theorem singlePhaseMultiplicity_compositePosting (c : ℚ) :
    compositePosting (singlePhaseMultiplicity c)
      = fun t => (((c : ℝ)) : ℂ) * unitPosting t := by
  funext t
  simp only [compositePosting, unitPosting]
  have hzero : ∀ p ∈ Finset.univ, p ≠ (0 : Fin 8) →
      (((singlePhaseMultiplicity c p : ℚ) : ℝ) : ℂ) * commitSettlementWindow p t = 0 := by
    intro p _ hp
    simp [singlePhaseMultiplicity, hp]
  rw [Finset.sum_eq_single (0 : Fin 8) hzero (by simp)]
  simp [singlePhaseMultiplicity]

/-- **Its load is `2 c ^ 2`.** Neutralization is linear and the load is a squared
norm, so the rational multiplicity comes out squared against the unit posting's
load of `2`. -/
theorem singlePhaseMultiplicity_load (c : ℚ) :
    normSq8 (neutralize (compositePosting (singlePhaseMultiplicity c)))
      = 2 * (c : ℝ) ^ 2 := by
  rw [singlePhaseMultiplicity_compositePosting, normSq8_neutralize_smul,
    unitPosting_load]
  ring

/-- **For every tolerance, a composite posting lands inside it.** The target may
be any positive real, the required anchor load included. -/
theorem exists_compositePosting_within {L ε : ℝ} (hL : 0 < L) (hε : 0 < ε) :
    ∃ n : Fin 8 → ℚ,
      |normSq8 (neutralize (compositePosting n)) - L| < ε := by
  obtain ⟨c, hc⟩ := exists_rat_sq_within hL hε
  exact ⟨singlePhaseMultiplicity c, by rw [singlePhaseMultiplicity_load]; exact hc⟩

/-- **The wall, measured.** The required anchor load is missed exactly, and
approached to every tolerance. Both halves in one statement, so neither can be
quoted without the other. -/
theorem required_load_missed_exactly_and_approached_to_every_tolerance
    (s : Anchor.Sector) (r Z : ℤ) (hZ : 0 ≤ Z)
    (hp : Int.fib (Anchor.r0 s + r - 13)
            + Z * Int.fib (Anchor.r0 s + r - 14) ≠ 0) :
    (∀ n : Fin 8 → ℚ,
        MassLaw.predict_mass s r Z / 8
          ≠ normSq8 (neutralize (compositePosting n)))
      ∧ (∀ ε : ℝ, 0 < ε → ∃ n : Fin 8 → ℚ,
          |normSq8 (neutralize (compositePosting n))
            - MassLaw.predict_mass s r Z / 8| < ε) := by
  refine ⟨required_load_ne_compositePosting s r Z hZ hp, fun ε hε => ?_⟩
  have hpos : 0 < MassLaw.predict_mass s r Z / 8 := by
    have := MassLaw.predict_mass_pos s r Z
    linarith
  exact exists_compositePosting_within hpos hε

What this page does not claim

The ledger can carry the mass spectrum. The wall that no composite posting equals the target is broken. The framework has derived the mass spectrum from first principles.

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/T10LoadReachabilityAudit.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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