IndisputableMonolith.NumberTheory.ConcreteEulerLedger
The ConcreteEulerLedger module supplies concrete prime-based realizations of Euler ledgers and defect sensors that instantiate the abstract RS cost and recognition structures. It draws on J-symmetry from LedgerForcing and the Euler product carrier/sensor framework from EulerInstantiation to produce finite balanced ledgers. Researchers constructing number-theoretic models within Recognition Science would cite these objects when building directed systems. The module consists of targeted definitions and lemmas that establish positivity, ratio, and
claimDefines prime Euler events $E_p$ for each prime $p$ together with the associated cost $J(E_p)$ and finite Euler ledgers over finite prime supports $S$ that satisfy double-entry balance and net flow zero.
background
The module sits inside the Recognition Science development that begins with J-symmetry forcing double-entry ledger structure (LedgerForcing) and recognition being forced from the cost foundation (RecognitionForcing). EulerInstantiation shows that the Euler product of the zeta function supplies a concrete carrier/sensor realization of the abstract RS framework. The present module converts those abstract objects into explicit finite constructions indexed by primes, using defect sensors whose real-part positivity is asserted and finite collections whose net flow vanishes.
proof idea
This is a definition module whose lemmas are proved by direct appeal to the imported forcing theorems and to elementary properties of the J-function and prime ratios; the main steps are the verification that sensor real parts are positive, that event ratios lie in (0,1), and that the assembled finite ledger is balanced with zero net flow.
why it matters in Recognition Science
The module supplies the concrete finite Euler ledgers that DirectedEulerLedger packages into a directed system over finite prime supports and then connects to the ontology-side interfaces in UnifiedRH. It therefore completes the concrete-instantiation step of the Euler-product route to the RS cost structure before the directed-system and RH-connection arguments are applied.
scope and limits
- Does not treat infinite prime supports or the full zeta function.
- Does not derive the Riemann hypothesis or any analytic continuation.
- Does not compute explicit numerical values inside the alpha band.
used by (1)
depends on (3)
declarations in this module (25)
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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