IndisputableMonolith.NumberTheory.ConcreteEulerLedger
ConcreteEulerLedger supplies explicit constructions of prime Euler events and finite balanced ledgers inside the Recognition Science cost framework. It establishes positivity of the real part for defect sensors together with zero net flow on finite prime supports. Number theorists working on Euler-product realizations of RS carrier/sensor structures cite these objects. The module assembles upstream forcing results into concrete definitions and elementary lemmas.
claimFor a defect sensor $s$, the real part satisfies $s > 0$. A finite Euler ledger $L$ over a finite set of primes satisfies net flow zero: the sum of directed flows equals zero.
background
The module lies in the NumberTheory domain and imports LedgerForcing (J-symmetry forces double-entry ledger structure), RecognitionForcing (recognition is forced by the cost foundation), and EulerInstantiation (the Euler product of ζ(s) instantiates the abstract RS carrier/sensor framework from AnnularCost and CostCoveringBridge). It introduces sibling definitions such as primeEulerEvent (an event at a prime carrying J-cost), finiteEulerLedger (a ledger over finite prime support), and sensor_realPart_pos. The supplied doc-comment states that the strip real part of a defect sensor is positive.
proof idea
This is a definition module, no proofs. It consists of a collection of definitions (primeEulerEvent, finiteEulerLedger, etc.) together with short lemmas establishing positivity, ratio inequalities, cost equality to J, and balance properties.
why it matters in Recognition Science
The module feeds DirectedEulerLedger, which packages these finite concrete Euler ledgers into a directed system over finite prime supports and connects the system to ontology-side interfaces from UnifiedRH. It supplies the concrete number-theoretic layer that realizes the Euler-product instantiation of the RS cost structure.
scope and limits
- Does not treat infinite prime supports or continuous limits.
- Does not compute explicit numerical values of J-costs.
- Does not derive physical constants or the alpha band.
- Does not address uniqueness beyond the listed properties.
used by (1)
depends on (3)
declarations in this module (25)
-
theorem
sensor_realPart_pos -
def
primeEulerEvent -
theorem
primeEulerEvent_ratio -
theorem
primeEulerEvent_target -
theorem
primeEulerEvent_ratio_lt_one -
theorem
primeEulerEvent_ratio_ne_one -
theorem
reciprocal_primeEulerEvent_ratio -
theorem
primeEulerEvent_cost_eq_J -
theorem
primeEulerEvent_cost_pos -
def
finiteEulerLedger -
theorem
finiteEulerLedger_balanced -
theorem
finiteEulerLedger_net_flow_zero -
theorem
primeEulerEvent_mem_finiteEulerLedger -
theorem
reciprocal_primeEulerEvent_mem_finiteEulerLedger -
theorem
ledger_cost_foldl_with_offset -
theorem
add_event_cost_formula -
theorem
finiteEulerLedger_cost_formula -
theorem
finiteEulerLedger_cost_formula_J -
def
sensorEulerLedger -
theorem
sensorEulerLedger_balanced -
theorem
sensorEulerLedger_net_flow_zero -
theorem
primeEulerEvent_mem_sensorEulerLedger -
theorem
reciprocal_primeEulerEvent_mem_sensorEulerLedger -
theorem
sensorEulerLedger_cost_formula -
theorem
sensorEulerLedger_identification