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theorem

embed_strictMono_of_one_lt

proved
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Why this theorem is linked from Quantum Error Correction Assisted Axion Search in CMOS Spin Qubit Arrays unclear

Pith linked this Lean declaration because the review connected a specific passage in the paper to this theorem. The relation tag says how strong that connection is; it is not a generic placeholder.

optimal logical GHZ size k_L^* = 1/(2(γ_eff + γ_cor)T_seg)

Relation between the paper passage and the cited Recognition theorem.

module
IndisputableMonolith.Foundation.ArithmeticFromLogic
domain
Foundation
line
684 · github
papers citing
404 papers (below)

plain-language theorem explainer

The declaration shows that the orbit embedding of the recovered natural numbers into positive reals is strictly monotone when the generator value exceeds one. Workers reconstructing arithmetic structures from the Law of Logic would reference this result to guarantee order preservation under the embedding. The argument reduces immediately to the right-to-left direction of the companion iff statement for the same embedding.

Claim. Let $γ$ be a generator with value greater than 1. The embedding map from the Peano structure on the orbit to the positive reals is strictly monotone.

background

Generator denotes a positive real number distinct from 1, extracted from the non-triviality of any comparison operator satisfying the Law of Logic. The embed function maps the identity element of the recovered naturals to 1 and each successor step to multiplication by the generator value, producing the orbit under repeated multiplication. The recovered naturals themselves arise as the abstract Peano structure on the orbit of any such generator, as constructed in the preceding lemmas of this module.

proof idea

The proof is a one-line wrapper that invokes the right-to-left implication of the companion iff lemma on the given strict inequality between embedded values.

why it matters

This result secures the order-preserving property required for the recovery of natural-number arithmetic from the functional equation. It directly supports the module summary that the Peano axioms become theorems once the orbit structure is in hand. The declaration leaves open the generator-free characterization of the orbit and the direct definition of order on the recovered naturals, both flagged for later modules.

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papers checked against this theorem (showing 30 of 404)

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