Reflected Growth Pair and the Second-Order Negative Spectrum

Abstract

Identify the reflected-pair signed determinant with the negative second-derivative eigenvalue.

Definition 1.1 (The positive-rate branch).

Lean statement: D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.positiveRateBranch

Formalization. D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.positiveRateBranch (✓ std3).

Source. Repository-derived.

Commentary.

The first branch is the rate-delta coordinate of the frozen reflected growth pair. The name refers to the signed generator rate and does not assume that delta itself is positive.

Definition 1.2 (The reflected negative-rate branch).

Lean statement: D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.negativeRateBranch

Formalization. D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.negativeRateBranch (✓ std3).

Source. Repository-derived.

Commentary.

The second branch is generated by minus delta. Together the two branches retain the orientation that a symmetric sum later forgets.

Definition 1.3 (The negative second-derivative observer).

Lean statement: D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.negativeSecondDerivative

Formalization. D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.negativeSecondDerivative (✓ std3).

Source. Repository-derived.

Commentary.

The observer applies minus the second iterated derivative at the selected parameter value. It provides the operator chart in which minus delta squared becomes an eigenvalue.

Theorem 1.4 (The signed determinant is the negative second-order eigenvalue).

$$\forall delta:\mathbb{R},t:\mathbb{R},\mathrm{negativeSecondDerivative}(\mathrm{positiveRateBranch}(delta),t)=\mathrm{reflectionPairSignedDeterminant}(delta)\cdot \mathrm{positiveRateBranch}(delta,t)\wedge \mathrm{negativeSecondDerivative}(\mathrm{negativeRateBranch}(delta),t)=\mathrm{reflectionPairSignedDeterminant}(delta)\cdot \mathrm{negativeRateBranch}(delta,t)\wedge \mathrm{negativeSecondDerivative}(\mathrm{reflectedGrowthSum}(delta),t)=\mathrm{reflectionPairSignedDeterminant}(delta)\cdot \mathrm{reflectedGrowthSum}(delta,t).$$

Proof. Machine-checked in Lean as D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.reflected_growth_pair_second_order_spectrum (✓ std3). ∎

Source. Repository-derived.

Commentary.

Both reflected exponential branches satisfy the same second-order equation. Applying the negative second derivative therefore multiplies each branch by the signed determinant minus delta squared.

The same identity holds for the branch-forgetting symmetric sum. This is a general scalar spectral statement and introduces no zeta or Riemann-hypothesis premise.

Theorem 1.5 (The symmetric observer is first-order blind at the center).

$$\forall delta:\mathbb{R},\mathrm{iteratedDeriv}(1,\mathrm{reflectedGrowthSum}(delta),0)=0.$$

Proof. Machine-checked in Lean as D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.reflected_growth_sum_first_derivative_at_zero (✓ std3). ∎

Source. Repository-derived.

Commentary.

At the reflection center, the two opposite first derivatives cancel. The symmetric observer therefore loses the sign of the branch orientation.

Theorem 1.6 (The second-order reading is twice the squared split).

$$\forall delta:\mathbb{R},\mathrm{iteratedDeriv}(2,\mathrm{reflectedGrowthSum}(delta),0)=2\cdot delt{a}^2.$$

Proof. Machine-checked in Lean as D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.reflected_growth_sum_second_derivative_at_zero (✓ std3). ∎

Source. Repository-derived.

Commentary.

The second derivative at the reflection center equals two delta squared. Thus the first branch label is forgotten while the magnitude of the split remains observable at even order.

Theorem 1.7 (Every nonzero split has a strictly positive second-order reading).

$$\forall delta:\mathbb{R},delta\ne 0\Rightarrow 0<\mathrm{iteratedDeriv}(2,\mathrm{reflectedGrowthSum}(delta),0).$$

Proof. Machine-checked in Lean as D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.reflected_growth_sum_second_order_visible_of_ne_zero (✓ std3). ∎

Source. Repository-derived.

Commentary.

For nonzero delta, the centered second-order symmetric signal is strictly positive. This is a local detectability result, not a theorem that an arbitrary finite zeta observer isolates such a split.

References

• Truth anchor: D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.negativeRateBranch
• Truth anchor: D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.negativeSecondDerivative
• Truth anchor: D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.positiveRateBranch
• Truth anchor: D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.reflected_growth_pair_second_order_spectrum
• Truth anchor: D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.reflected_growth_sum_first_derivative_at_zero
• Truth anchor: D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.reflected_growth_sum_second_derivative_at_zero
• Truth anchor: D5/S3/Analytic/Adelic/ReflectedGrowthPairSecondOrderSpectrum.reflected_growth_sum_second_order_visible_of_ne_zero
• Dependency: D5/S3/Analytic/Adelic/ReflectedGrowthPairNegativeSquare