`sensorEulerLedger_net_flow_zero`

proved

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module: IndisputableMonolith.NumberTheory.ConcreteEulerLedger
domain: NumberTheory
line: 194 · github
papers citing: none yet

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IndisputableMonolith.NumberTheory.ConcreteEulerLedger on GitHub at line 194.

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 191  exact finiteEulerLedger_balanced sensor.realPart (sensor_realPart_pos sensor) support.toList
 192
 193/-- The sensor-indexed concrete Euler ledger has zero net flow. -/
 194theorem sensorEulerLedger_net_flow_zero (sensor : DefectSensor)
 195    (support : Finset Nat.Primes) (agent : ℕ) :
 196    LedgerForcing.net_flow (sensorEulerLedger sensor support) agent = 0 := by
 197  unfold sensorEulerLedger
 198  exact finiteEulerLedger_net_flow_zero sensor.realPart (sensor_realPart_pos sensor) support.toList agent
 199
 200/-- Every supported prime contributes its Euler event to the sensor-indexed ledger. -/
 201theorem primeEulerEvent_mem_sensorEulerLedger
 202    (sensor : DefectSensor) {support : Finset Nat.Primes} {p : Nat.Primes}
 203    (hp : p ∈ support) :
 204    primeEulerEvent sensor.realPart (sensor_realPart_pos sensor) p ∈
 205      (sensorEulerLedger sensor support).events := by
 206  unfold sensorEulerLedger
 207  exact primeEulerEvent_mem_finiteEulerLedger
 208    (support := support.toList) (p := p) (by simpa using hp)
 209
 210/-- Every supported prime also contributes the reciprocal Euler event. -/
 211theorem reciprocal_primeEulerEvent_mem_sensorEulerLedger
 212    (sensor : DefectSensor) {support : Finset Nat.Primes} {p : Nat.Primes}
 213    (hp : p ∈ support) :
 214    LedgerForcing.reciprocal (primeEulerEvent sensor.realPart (sensor_realPart_pos sensor) p) ∈
 215      (sensorEulerLedger sensor support).events := by
 216  unfold sensorEulerLedger
 217  exact reciprocal_primeEulerEvent_mem_finiteEulerLedger
 218    (support := support.toList) (p := p) (by simpa using hp)
 219
 220/-- Explicit total J-cost formula for the sensor-indexed concrete Euler ledger. -/
 221theorem sensorEulerLedger_cost_formula (sensor : DefectSensor)
 222    (support : Finset Nat.Primes) :
 223    LedgerForcing.ledger_cost (sensorEulerLedger sensor support) =
 224      2 * (support.toList.map

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