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Explain the Lean theorem `rsSpectrum_bounded` in module `IndisputableMonolith.CrossDomain.CardinalitySpectrum`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

Big AI job. Grok 4.3 reads the canon and writes a Lean-grounded derivation; usually 20 seconds to 2 minutes. Your answer will appear below.
confidence: high in recognition cached

Explanation of rsSpectrum_bounded

(1) What the declaration says in plain English

The theorem rsSpectrum_bounded states that every integer appearing in the list rsSpectrum is at most 3125. In other words, the entire collection of canonical RS cardinalities is bounded above by this value.

(2) Why it matters in Recognition Science

It confirms that the structured numerical spectrum generated by RS primitives (powers and products of the generators 2, 3, 5 and gap45) remains finite and capped at D_config^5. This is part of the broader claim that RS produces an ordered, non-random set of cardinalities rather than an unbounded or arbitrary collection.

(3) How to read the formal statement

The Lean statement is:

∀ n ∈ rsSpectrum, n ≤ 3125
  • ∀ n ∈ rsSpectrum means "for every natural number n that is an element of the list rsSpectrum".
  • n ≤ 3125 asserts the upper bound. The proof is by decide, which evaluates the concrete list membership and comparisons at compile time.

(4) Visible dependencies or certificates in the supplied source

The theorem is referenced inside the CardinalitySpectrumCert structure (field spectrum_bounded). It depends on the definition of rsSpectrum and is accompanied by related facts such as rsSpectrum_length and rsSpectrum_pairwise_lt. The bound 3125 is witnessed by threeOne25_is_D_fifth which equates it to Dconfig^5.

(5) What this declaration does not prove

It does not establish the physical or forcing-chain origin of the listed cardinalities, nor does it prove that 3125 is the tightest possible bound or connect the spectrum to deeper RS structures such as the J-cost functional equation or spacetime emergence. It is a verification of list membership and ordering only.

cited recognition theorems

outside recognition

Aspects Recognition does not yet address:

  • Any connection of the cardinality spectrum to the forcing chain, J-cost uniqueness, or physical constants beyond the numerical verification.
  • Proof that the listed numbers are the only possible RS cardinalities or that the bound is tight in a deeper mathematical sense.

recognition modules consulted

The Recognition library is at github.com/jonwashburn/shape-of-logic. The model is restricted to the supplied Lean source and instructed not to invent theorem names. Treat output as a starting point, not a verified proof.