Local Evidence Order Determines the Critical Exponent

Abstract

Linear event mass and quadratic evidence have distinct prime thresholds.

本节不声称 α=1/2 与 Riemann 临界线具有解析等价、零点等价或物理因果关系。这里只存在「二次证据导致指数折半」的结构类比。

Definition 1.1 (First-event mass).

$$m(s)(p)=p^{-s}$$

Formalization. D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.firstEventMass (✓ std3).

Source. Repository-derived.

Commentary.

The named mass is the local probability formula p to the power minus s. A separate theorem checks its event-probability semantics.

Definition 1.2 (Quadratic statistical energy).

$$E(delta)(i)=delt{a}_i\cdot delt{a}_i$$

Formalization. D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.quadraticStatisticalEnergy (✓ std3).

Source. Repository-derived.

Commentary.

For any index type, the energy at one coordinate is the square of its local deviation. No prime structure enters this definition.

Theorem 1.3 (First-event mass is the zeta activation probability).

$$P_s(V_p>0)=m(s)(p)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.firstEventMass_eq_activation_probability (✓ std3). ∎

Source. Repository-derived.

Commentary.

Above exponent one, the normalized zeta law exists. Its event that the p-adic exponent is positive has exactly the named first-event mass.

Theorem 1.4 (First-event mass has threshold one).

$$\forall s,Summable\left(m(s)\right)\iff 1<s$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.firstEventMass_summable_iff_one_lt (✓ std3). ∎

Source. Repository-derived.

Commentary.

The prime-indexed activation masses are summable exactly when s is strictly greater than one. Prime distribution is load-bearing.

Theorem 1.5 (Quadratic energy doubles the exponent).

$$E(m(alpha))=m(2\cdot alpha)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.quadratic_prime_energy_eq_firstEventMass (✓ std3). ∎

Source. Repository-derived.

Commentary.

For the deviation p to the power minus alpha, squaring changes the same prime family to exponent two alpha.

Theorem 1.6 (Quadratic energy has threshold one half).

$$\forall alpha,Summable\left(E(m(alpha))\right)\iff \frac{1}{2}<alpha$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.quadratic_prime_energy_summable_iff_half_lt (✓ std3). ∎

Source. Repository-derived.

Commentary.

The doubled exponent is above one exactly when alpha is above one half. The exact iff still uses the prime-series theorem.

Theorem 1.7 (Accumulated order determines the critical exponent).

$$\forall s,Summable\left(m(s)\right)\iff 1<s\wedge \left(\forall alpha,Summable\left(E(m(alpha))\right)\iff \frac{1}{2}<alpha\wedge 1\ne \frac{1}{2}\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.local_evidence_order_critical_thresholds (✓ std3). ∎

Source. Repository-derived.

Commentary.

The same prime spectrum gives threshold one for linear activation mass and one half for quadratic evidence. The thresholds are unequal.

Theorem 1.8 (First-event mass diverges at and below one).

$$\forall s,s\le 1\Rightarrow \neg Summable\left(m(s)\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.firstEventMass_at_most_one_not_summable (✓ std3). ∎

Source. Repository-derived.

Commentary.

Every exponent at most one lies on the nonsummable side, including the boundary itself and all nonpositive exponents.

Theorem 1.9 (Zero exponent is constant and divergent).

$$\forall p,m(0)(p)=1\wedge \neg Summable\left(m(0)\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.firstEventMass_zero (✓ std3). ∎

Source. Repository-derived.

Commentary.

At exponent zero every prime contributes one, so the infinite prime family is not summable.

Theorem 1.10 (Zero deviation has summable energy).

$$Summable\left(E(0)\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.quadraticStatisticalEnergy_zero_summable (✓ std3). ∎

Source. Repository-derived.

Commentary.

On every index type, the zero deviation has identically zero quadratic energy and is summable.

Theorem 1.11 (Finite prime truncations are summable).

$$\forall S,s,Summable\left(m(s)ch{i}_S\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.finite_prime_truncation_summable (✓ std3). ∎

Source. Repository-derived.

Commentary.

Restricting to finitely many primes is summable at every exponent. This degeneration uses finite support, not prime distribution.

Theorem 1.12 (Empty and singleton energy families are summable).

$$\forall delta:\varnothing \to \mathbb{R},Summable\left(E(delta)\right)\wedge \forall delta:Unit\to \mathbb{R},Summable\left(E(delta)\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.quadratic_energy_empty_and_unit_summable (✓ std3). ∎

Source. Repository-derived.

Commentary.

Every family on the empty type or the one-element unit type has finite support, so its quadratic energy is summable.

Theorem 1.13 (The one-half boundary diverges).

$$\neg Summable\left(E(m(\frac{1}{2}))\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.quadratic_prime_energy_one_half_not_summable (✓ std3). ∎

Source. Repository-derived.

Commentary.

At alpha equal to one half, quadratic energy becomes reciprocal-prime mass, so the boundary itself is not summable.

References

• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.finite_prime_truncation_summable
• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.firstEventMass
• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.firstEventMass_at_most_one_not_summable
• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.firstEventMass_eq_activation_probability
• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.firstEventMass_summable_iff_one_lt
• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.firstEventMass_zero
• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.local_evidence_order_critical_thresholds
• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.quadraticStatisticalEnergy
• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.quadraticStatisticalEnergy_zero_summable
• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.quadratic_energy_empty_and_unit_summable
• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.quadratic_prime_energy_eq_firstEventMass
• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.quadratic_prime_energy_one_half_not_summable
• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/LocalEvidenceOrderThreshold.quadratic_prime_energy_summable_iff_half_lt
• Dependency: D5/S3/Analytic/ZetaEntropyPlane/PrimeEvidenceSharpThreshold