base_shift_denominator_forced
plain-language theorem explainer
Equating the base shift to 2W plus the term (W + E_total) over (k times E_passive) forces the positive integer k to equal 4. Mass spectra calculations in the Recognition Science framework cite this when closing the O4 perturbative coefficient in the J-cost channel. The proof reduces the equality to a prior ledger fraction lemma by simplification and linear arithmetic casting.
Claim. Let $k$ be a positive integer. If the base shift equals $2W + (W + E_{total})/(k E_{passive})$, then $k=4$.
background
The Mass-Layer J-Cost Perturbation Bridge module defines base_shift as twice the W term plus the ledger fraction, where the fraction is the ratio (W + E_total) over (k times E_passive). E_passive is the passive edge count abbrev equal to 11 from the D-cube edge structure. The canonical arithmetic object from ArithmeticOf supplies the realization-independent Peano initiality used in the reduction.
proof idea
The tactic proof first simplifies base_shift to isolate the ledger fraction. Linear arithmetic equates the supplied fractional expression to the ledger fraction over reals, then casts the equality back to rationals. It concludes by exact application of the ledger_fraction_denominator_forced lemma on the positivity hypothesis.
why it matters
This theorem closes the denominator forcing step for the base shift in the J-cost perturbation, feeding the ratio_29_over_11k_forces_k_eq_four and ledger_fraction_denominator_scale_forced siblings. It certifies the O4 coefficient forcing package in the module and aligns with the Recognition Science eight-tick octave and spectral emergence constraints on mass topology.
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