ledger_fraction_numerator_offset_forced
plain-language theorem explainer
The theorem shows that any rational offset n in the numerator family ((W + E_total) + n)/(4 E_passive) must be zero to match the canonical ledger fraction. Researchers closing mass perturbations in the Recognition Science O4 channel cite this to fix the two-weight numerator. The proof substitutes the topology counts W = 17, E_total = 12, E_passive = 11, rewrites via arithmetic identities, and applies linear arithmetic to force n = 0.
Claim. Let $n$ be rational. If $((W + E_{total} + n)/(4 E_{passive})) = 29/44$, then $n = 0$.
background
The module bridges J-cost perturbations to the mass layer by certifying the two-channel form and the explicit radiative decomposition used in refined_shift. Ledger_fraction is the canonical ratio 29/44 fixed by mass topology counts; W denotes the active weight 17, E_total the total energy count 12, and E_passive the passive edge count 11. Upstream arithmetic lemmas from ArithmeticFromLogic supply associativity and commutativity of addition, while Anchor defines E_passive as passive_field_edges D and mass_topology_counts supplies the explicit integers.
proof idea
The proof extracts W = 17, E_total = 12, E_passive = 11 from mass_topology_counts. It rewrites the hypothesis using add_assoc, add_comm, add_left_comm together with multiplication commutativity and associativity lemmas to obtain ((29 + n)/44) = 29/44. Linear arithmetic then concludes n = 0.
why it matters
This result fixes the numerator offset inside the ledger fraction family and is invoked directly by base_shift_numerator_offset_forced to close the base shift under the canonical 2W term. It completes one step of the O4 perturbative closure certified by the module, linking the J-cost(1 + α) channel to the eight-tick octave and D = 3 structure from the forcing chain.
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