Masses Mass Genesis T10 Unit Free Anchor Wall Exists Unit Rescaling Below Charge

The unit-dependent ceiling

In the Recognition Science account, a predicted mass is a number of energy units, like saying an object weighs 5 kilograms. The number itself depends on the scale you choose. The declaration exists_unit_rescaling_below_charged_threshold is a formal theorem stating that for any pattern in the theory, you can pick a positive rescaling factor for the mass unit, effectively choosing a new scale, so that the rescaled predicted mass is less than 256. The proof is simple: since every predicted mass is positive, dividing the unit by that mass makes the rescaled value exactly 1, which is certainly below 256.

This theorem is a statement about the fragility of a numerical ceiling. The framework's library also proves a companion result for the threshold 64, and the 256 version follows directly from it. The significance is that a wall described by numbers like 64 or 256 is not an absolute obstruction. It is an artifact of the unit in which masses happen to be expressed. If you read the electron's mass in a different unit, its predicted value can be made to sit below any fixed ceiling you like. The theorem shows this for every pattern, not just a few.

What survives a change of unit is not a number but a ratio. A separate theorem in the same library, settledAnchor_mass_amplitude_ratio, establishes that for two patterns, the product of one predicted mass and the square of the other's posting amplitude equals the reverse product. When you divide through, every quantity becomes a ratio, and ratios do not change when you rescale the unit. This ratio law leads to a stronger result: two species with different predicted masses cannot both post at the same amplitude, and no choice of mass unit can make two different masses agree. The electron and the muon, which sit on different rungs of the same ladder, are the concrete example the framework runs the argument on.

In plain terms, the ceiling theorem says a numerical wall can be moved by changing your ruler. The ratio theorem says a structural wall, one about proportions, cannot. The framework's library makes both points with machine-checked proofs, and the contrast between them is the lesson: when a claim depends on a number, check whether the number depends on your units.

/-- The same for the sharper `256` threshold carried by the charged spectrum. -/
theorem exists_unit_rescaling_below_charged_threshold (ψ : LightPattern (Fin 8)) :
    ∃ k : ℝ, 0 < k ∧ k * predictedMass ψ < 256 := by
  obtain ⟨k, hk, hlt⟩ := exists_unit_rescaling_below_ceiling ψ
  exact ⟨k, hk, by linarith⟩

/-- **The ceiling wall does not survive a change of mass unit.** For every pattern
there is a positive rescaling of the mass unit placing its rescaled predicted mass
below the Boolean ceiling. Nothing about the pattern is used except that its
predicted mass is positive, so this applies to the whole charged spectrum. -/
theorem exists_unit_rescaling_below_ceiling (ψ : LightPattern (Fin 8)) :
    ∃ k : ℝ, 0 < k ∧ k * predictedMass ψ < 64 := by
  have hpos : 0 < predictedMass ψ := predictedMass_pos ψ
  refine ⟨1 / predictedMass ψ, div_pos one_pos hpos, ?_⟩
  have hcancel : 1 / predictedMass ψ * predictedMass ψ = 1 := by
    field_simp
  rw [hcancel]
  norm_num

/-- The same for the sharper `256` threshold carried by the charged spectrum. -/
theorem exists_unit_rescaling_below_charged_threshold (ψ : LightPattern (Fin 8)) :
    ∃ k : ℝ, 0 < k ∧ k * predictedMass ψ < 256 := by
  obtain ⟨k, hk, hlt⟩ := exists_unit_rescaling_below_ceiling ψ
  exact ⟨k, hk, by linarith⟩

/-- **The ratio law.** Two load-normalized patterns with settled anchors relate
their predicted masses and their posting amplitudes by `m * b² = m' * a²`. Every
quantity here is a ratio once the equation is divided through, so the statement is
untouched by a change of mass unit. -/
theorem settledAnchor_mass_amplitude_ratio
    {a b : ℝ} {ψ χ : LightPattern (Fin 8)}
    (hψ : LoadNormalizedToTopology ψ) (hwψ : SettledAnchorWindow a ψ)
    (hχ : LoadNormalizedToTopology χ) (hwχ : SettledAnchorWindow b χ) :
    predictedMass ψ * b ^ 2 = predictedMass χ * a ^ 2 := by
  rw [loadNormalized_settledAnchor_predictedMass_eq hψ hwψ,
    loadNormalized_settledAnchor_predictedMass_eq hχ hwχ]
  ring

/-- **The ratio wall does survive.** A positive rescaling of the mass unit cannot
make two different predicted masses agree, so the obstruction of section 1 is
untouched by any unit identification. -/
theorem distinct_masses_stable_under_unit_rescaling
    {k : ℝ} (hk : 0 < k) {ψ χ : LightPattern (Fin 8)}
    (hne : predictedMass ψ ≠ predictedMass χ) :
    k * predictedMass ψ ≠ k * predictedMass χ := fun h =>
  hne (mul_left_cancel₀ (ne_of_gt hk) h)

What this page does not claim

This theorem does not claim that any mass value is physically impossible, only that a particular numerical ceiling is not unit-independent. It does not claim that the ratio law itself is a statement about measured masses, only about the framework's predicted masses. It does not claim that the electron and muon are the only species where the ratio obstruction applies.

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/T10UnitFreeAnchorWall.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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