Holography Seam Cycle Carrier Jcost Exp Even

The evenness of cost

The declaration Jcost_exp_even is a proved theorem in the framework's machine-checked library of formal theorems. It states that for any real number a, the cost function J, evaluated at the exponential of a, equals J evaluated at the exponential of minus a: J(e^a) = J(e^(−a)). In plainer terms, if you measure a separation on a logarithmic scale, the cost of going forward by a certain amount is exactly the same as the cost of going backward by that same amount. The theorem is not an assumption; it is derived from the symmetry property J(x) = J(1/x), which the framework proves for its unique cost function.

The result matters because it connects a purely algebraic property of the cost function to a physical picture. In the framework, the cost of a separation is tied to a recognition cycle, a discrete record of events that repeats after a fixed period. The theorem Jcost_exp_even shows that this cost is even under reversal of the separation, which in turn supports the claim that a certain correlator, a measure of how two points in the cycle relate, is also even. This evenness is not a new symmetry; it is the same reciprocity that forces the cost function itself.

The theorem does not claim that the cost function is even in its raw argument x. It only claims evenness after the exponential map, on the logarithmic ladder. It also does not claim that the physical identification of the seam correlator with the cycle flow is proved; that identification remains a modeling choice, not a theorem. The theorem is a clean, narrow result about the cost function's behavior under reversal, and it is that narrowness that lets it serve as a reliable building block in the framework's larger derivations.

/-- On the exponential ladder the unique recognition cost is EVEN in the log
separation: `J(e^a) = J(e^{−a})`. This is exactly T5 reciprocity `J(x) = J(1/x)`
(`Cost.Jcost_symm`) read at `x = e^a`. Evenness of the seam correlator is the
carrier-level form of this reciprocity, not an independent symmetry. -/
theorem Jcost_exp_even (a : ℝ) :
    Cost.Jcost (Real.exp a) = Cost.Jcost (Real.exp (-a)) := by
  rw [Real.exp_neg]
  exact Cost.Jcost_symm (Real.exp_pos a)

/-- On the exponential ladder the unique recognition cost is EVEN in the log
separation: `J(e^a) = J(e^{−a})`. This is exactly T5 reciprocity `J(x) = J(1/x)`
(`Cost.Jcost_symm`) read at `x = e^a`. Evenness of the seam correlator is the
carrier-level form of this reciprocity, not an independent symmetry. -/
theorem Jcost_exp_even (a : ℝ) :
    Cost.Jcost (Real.exp a) = Cost.Jcost (Real.exp (-a)) := by
  rw [Real.exp_neg]
  exact Cost.Jcost_symm (Real.exp_pos a)

/-- On the exponential ladder the unique recognition cost is EVEN in the log
separation: `J(e^a) = J(e^{−a})`. This is exactly T5 reciprocity `J(x) = J(1/x)`
(`Cost.Jcost_symm`) read at `x = e^a`. Evenness of the seam correlator is the
carrier-level form of this reciprocity, not an independent symmetry. -/
theorem Jcost_exp_even (a : ℝ) :
    Cost.Jcost (Real.exp a) = Cost.Jcost (Real.exp (-a)) := by
  rw [Real.exp_neg]
  exact Cost.Jcost_symm (Real.exp_pos a)

What this page does not claim

The cost function is even in its raw argument x; evenness holds only after the exponential map. The physical identification of the seam correlator with the cycle flow is proved; it remains a modeling choice. The theorem alone establishes the full periodicity of the correlator; that requires the cycle closure property.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Holography/SeamCycleCarrier.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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