phi_ladder_energy_strictly_increasing
The energy stored on successive rungs of the φ-ladder increases strictly with rung index. Engineers and physicists deriving recognition-based bounds on chemical versus nuclear storage scales would cite this monotonicity to separate the hierarchies. The short term proof reduces the inequality to the positivity of the coherence energy together with the strict growth of positive powers of φ under the given integer ordering.
claimLet $E(n) = E_0 · φ^n$ denote the energy stored on rung $n$ of the φ-ladder, where $E_0 > 0$ is the coherence energy per recognition event and $φ > 1$ is the golden ratio. Then $n < m$ implies $E(n) < E(m)$.
background
In the EN-004 module energy storage is constructed from the φ-ladder: each integer rung $n$ stores energy $E_0 · φ^n$, where $E_0$ is the positive coherence quantum $E_{coh storage}$. The positivity of this base energy follows from the theorem that unfolds the definition to a positive power of φ. The golden ratio satisfies $1 < φ$ by the lemma one_lt_phi imported from Constants and PhiSupport. The local theoretical setting derives practical storage limits from the J-cost function, with chemical and nuclear scales separated by integer powers of φ.
proof idea
The proof is a one-line wrapper. It unfolds the definition of rung energy to expose the product of the positive coherence energy with φ raised to the rung index, applies the left-multiplication inequality for positive factors, and closes with the zpow monotonicity lemma using $1 < φ$ and the hypothesis $n < m$.
why it matters in Recognition Science
This monotonicity is invoked inside rs_energy_storage_hierarchy to obtain the three-scale energy ordering (chemical < nuclear < mass-energy) and is listed as a verified component in the en004_certificate. It supplies the missing step that higher rungs on the φ-ladder always yield strictly larger energies, aligning with the Recognition Science claim that storage scales are quantized on the ladder and separated by factors of φ. No open scaffolding questions are directly addressed.
scope and limits
- Does not quantify the absolute numerical value of the coherence energy.
- Does not extend the ordering to non-integer rung indices.
- Does not incorporate variations in J-cost beyond the pure ladder structure.
- Does not address relativistic or quantum-field corrections to the stored energy.
Lean usage
have h : phi_rung_energy n < phi_rung_energy m := phi_ladder_energy_strictly_increasing n m hnmformal statement (Lean)
191theorem phi_ladder_energy_strictly_increasing (n m : ℤ) (h : n < m) :
192 phi_rung_energy n < phi_rung_energy m := by
proof body
Term-mode proof.
193 unfold phi_rung_energy
194 apply mul_lt_mul_of_pos_left _ E_coh_storage_pos
195 exact zpow_lt_zpow_right₀ one_lt_phi h
196
197/-! ## §VI. Fundamental Bound Summary -/
198
199/-- The RS energy storage hierarchy theorem.
200 Derives three distinct energy scales from the φ-ladder structure. -/