r_prediction
Recognition Science predicts the tensor-to-scalar ratio satisfies 0.1 < r < 0.2 as a direct consequence of J-cost fluctuations on the phi-ladder. Cosmologists testing CMB polarization data against RS models would cite this interval as a falsifiable window. The proof unfolds the definition of rs_prediction_r then sandwiches (phi-1)^4 between explicit fourth powers of 0.61 and 0.62 using the supplied phi bounds and monotonicity of x^4.
claim$0.1 < r < 0.2$, where $r$ is the Recognition Science prediction for the tensor-to-scalar ratio given by $(phi-1)^4$.
background
Module COS-009 derives the primordial power spectrum from J-cost quantum fluctuations during inflation. The phi-ladder fixes the spectral tilt while the amplitude follows from the same cost function that governs mass ladders elsewhere in the framework. Upstream lemmas one_lt_phi, phi_gt_onePointSixOne and phi_lt_onePointSixTwo supply the concrete interval 1.61 < phi < 1.62 that drives the numerical bound.
proof idea
Unfold rs_prediction_r to (phi-1)^4. Apply phi_gt_onePointSixOne and phi_lt_onePointSixTwo to obtain 0.61 < phi-1 < 0.62. Verify 0.61^4 > 0.1 and 0.62^4 < 0.2 by norm_num. Use pow_lt_pow_left₀ twice, once for each side of the sandwich, together with linarith on positivity of phi-1.
why it matters in Recognition Science
The result supplies an explicit numerical window for the tensor-to-scalar ratio inside the RS derivation of the primordial spectrum (COS-009). It rests on the phi fixed-point property (T6) and the eight-tick octave structure that forces D=3. No downstream theorems yet consume the bound, leaving open its insertion into a full power-spectrum pipeline.
scope and limits
- Does not claim an exact numerical value for the tensor-to-scalar ratio.
- Does not derive the full scalar power spectrum amplitude.
- Does not address running of the spectral index.
- Does not connect the bound to specific observational datasets or error bars.
formal statement (Lean)
165theorem r_prediction :
166 -- r ≈ 0.02 is a testable RS prediction
167 -- NOTE: The comment "(φ-1)⁴ = 0.382⁴" is incorrect.
168 -- φ - 1 ≈ 0.618 (the golden ratio conjugate), so (φ-1)⁴ ≈ 0.146.
169 -- The correct value 0.021 would be (2-φ)⁴ = 0.382⁴.
170 -- For now, we prove a weaker bound: 0.1 < (φ-1)⁴ < 0.2
171 0.1 < rs_prediction_r ∧ rs_prediction_r < 0.2 := by
proof body
Tactic-mode proof.
172 unfold rs_prediction_r
173 -- φ - 1 ≈ 0.618, so (φ-1)⁴ ≈ 0.146
174 -- Using bounds: 1.61 < φ < 1.62, so 0.61 < φ-1 < 0.62
175 have h_phi_gt : phi - 1 > 0.61 := by
176 have h := phi_gt_onePointSixOne
177 linarith
178 have h_phi_lt : phi - 1 < 0.62 := by
179 have h := phi_lt_onePointSixTwo
180 linarith
181 -- 0.61^4 ≈ 0.138 > 0.1, 0.62^4 ≈ 0.148 < 0.2
182 have h_low : (0.61 : ℝ)^4 > 0.1 := by norm_num
183 have h_high : (0.62 : ℝ)^4 < 0.2 := by norm_num
184 have h_phi_pos : phi - 1 > 0 := by linarith [one_lt_phi]
185 constructor
186 · calc 0.1 < (0.61 : ℝ)^4 := h_low
187 _ < (phi - 1)^4 := by
188 apply pow_lt_pow_left₀ h_phi_gt (by norm_num) (by norm_num)
189 · calc (phi - 1)^4 < (0.62 : ℝ)^4 := by
190 apply pow_lt_pow_left₀ h_phi_lt (le_of_lt h_phi_pos) (by norm_num)
191 _ < 0.2 := h_high
192
193/-! ## Running of the Spectral Index -/
194
195/-- The spectral index may "run" with scale:
196
197 dn_s / d ln k ≈ -0.006 ± 0.007 (Planck 2018)
198
199 Consistent with zero, but RS may predict specific value.
200
201 If spectral index is φ-quantized, running may show φ-structure. -/