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theorem proved term proof

exponential_form_uniqueness_ode_principle

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formal statement (Lean)

 257theorem exponential_form_uniqueness_ode_principle :
 258    True := trivial

proof body

Term-mode proof.

 259
 260/-! ## Summary of What This Module Proves
 261
 2621. **Structural properties** of the exponential form:
 263   * `alphaInv_def`: α⁻¹ is the exponential expression (unfold)
 264   * `alphaInv_positive`: α⁻¹ > 0
 265   * `exp_factor_bounded`: the exponential factor is in (0, 1] when f_gap ≥ 0
 266
 2672. **Log-coordinate structure**:
 268   * `log_alphaInv_seed_ratio`: ln(α⁻¹/α_seed) = -f_gap/α_seed (linear in f_gap)
 269   * `log_alphaInv_eq`: ln(α⁻¹) = ln(α_seed) - f_gap/α_seed
 270
 2713. **Differential structure**:
 272   * `deriv_alphaInv_of_gap`: dα⁻¹/df_gap = -α⁻¹/α_seed (the defining ODE)
 273   * `logarithmic_derivative_constant`: d ln(α⁻¹)/df_gap = -1/α_seed
 274     (constant logarithmic rate)
 275
 2764. **Leading-order consistency**:
 277   * `alphaInv_linear_term`: at f_gap=0, α⁻¹ = α_seed
 278   * `alphaInv_linear_rate`: at f_gap=0, dα⁻¹/df_gap = -1
 279
 280## What's NOT proved
 281
 2821. **Uniqueness from structural principles**: the formula is defined, not
 283   derived. `exponential_form_from_constant_log_derivative` states a
 284   candidate uniqueness but is not proved.
 2852. **Forcing of the constant log-derivative**: why RS requires
 286   (d/df_gap) ln(α⁻¹) to be constant (and specifically = -1/α_seed)
 287   remains a BRIDGE claim between the formalism and physics.
 288
 289## Residual Openness
 290
 291The exponential form of α⁻¹ is best understood as a STRUCTURAL CHOICE
 292inherited from the J-cost's log-coordinate behavior, rather than as a
 293derived consequence. It is a natural choice given:
 294- The J-cost's exponential/cosh structure in log coordinates.
 295- The requirement of positivity for all f_gap values.
 296- The linear leading-order behavior α⁻¹ ≈ α_seed - f_gap.
 297- The scale-free running (constant logarithmic derivative).
 298
 299But NONE of these individually force the exponential form uniquely without
 300additional assumptions. The integer 44 IS forced (proved in
 301`alpha_44_forcing.md`). The exponential form is PLAUSIBLE but not uniquely
 302forced in the current Lean.
 303
 304This is documented in `epistemic_layers.md` as a BRIDGE claim.
 305
 306-/
 307
 308end
 309
 310end AlphaExponentialForm
 311end Constants
 312end IndisputableMonolith

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