Foundation Primitive Recognition Calculus Prcnative Cost Minimality Certificate

The minimality certificate

The cost function in Recognition Science is the price the universe pays for a recognition event. The framework's central theorem proves that any cost function obeying five plain conditions must take a specific form. The proof's sharpness is established by a machine-checked certificate: it shows that each condition is necessary, because dropping any one of them lets a different, non-canonical cost function slip through.

Think of the proof as a stress test. The canonical cost function is the unique solution when all five conditions hold. The certificate asks: what if we remove one condition? For each of the four removals, a counterexample is constructed, a cost function that satisfies the remaining conditions but is not the canonical one. These counterexamples are not abstract: they are concrete functions defined on the rational numbers, and each one is proved to be a genuine alternative.

The first counterexample removes the calibration condition. A function called the Liouville twist multiplies a rational number by a sign determined by the parity of its total prime factor count. This twist satisfies the other conditions but sends 2 to -2, whereas the canonical cost sends 2 to 2. The second counterexample removes the sign condition, yielding a function that takes the absolute value of the canonical output. The third removes the pair condition, producing a function that fails to preserve the product of two prime directions. The fourth removes the reciprocal condition, and a function is built that spikes to zero at 5 and 1/5, violating the canonical behavior there.

Each counterexample is paired with a theorem proving that it is not the canonical cost, and that it does satisfy the weakened hypotheses. Together, these theorems form the minimality certificate: the five conditions are not just sufficient for the canonical cost, they are individually necessary. The final theorem, slim_ledger_minimality_certificate_tagged, packages this into a single tagged statement, marking the whole result as a delta-only strength claim.

This certificate matters because it separates the framework's core from its decoration. If the cost function were a free choice, the framework would be a definition, not a discovery. The certificate shows that the canonical cost is the only one that survives the full set of constraints, and that each constraint is doing real work. A reader can now see exactly which axiom, if weakened, would open the door to a different world.

/-- **Two-point-anchor necessity.** Slim minus two-calibration admits the
Liouville twist, so the anchor cannot be dropped. -/
theorem PRCSlimSansTwoCalibrationUniquenessTarget_refuted :
    ¬ PRCSlimSansTwoCalibrationUniquenessTarget := by
  intro huniq
  exact liouvilleTwistNativeCost_two_not_canonical
    (huniq liouvilleTwistNativeCost
      liouvilleTwistNativeCost_sans_two_hypotheses two)

/-- **Signed-unit necessity.** Base + pairs + zero admit the absolute-value
cost, which the parent module proves fails at the signed unit; the sign field
cannot be dropped. -/
theorem PRCSlimSansSignUniquenessTarget_refuted :
    ¬ PRCSlimSansSignUniquenessTarget := by
  intro huniq
  exact absValueGeneratedNativeCost_negative_one_not_canonical
    (huniq absValueGeneratedNativeCost
      absValueGeneratedNativeCost_sans_sign_hypotheses negativeOneRatio)

/-- **Pair-field necessity.** Base + sign + zero admit the two-adic twist,
which the parent module proves fails prime-pair products at the mixed (2,3)
orbit; the pair field cannot be dropped. -/
theorem PRCSlimSansPairUniquenessTarget_refuted :
    ¬ PRCSlimSansPairUniquenessTarget := by
  intro huniq
  apply twoAdicGeneratedNativeCost_not_prime_pair_product_calibrated
  intro p hp r hr
  exact huniq twoAdicGeneratedNativeCost
    twoAdicGeneratedNativeCost_sans_pair_hypotheses
    (RatioOrbit.mul (primeDirection p hp) (primeDirection r hr))

/-- **RCL-core necessity (bonus).** Slim minus the nonzero RCL admits the
`5`-spike, so the RCL field cannot be dropped either. -/
theorem PRCSlimSansRclUniquenessTarget_refuted :
    ¬ PRCSlimSansRclUniquenessTarget := by
  intro huniq
  exact rclSpikeNativeCost_five_not_canonical
    (huniq rclSpikeNativeCost rclSpikeNativeCost_sans_rcl_hypotheses
      (ratioOrbitOfRat 5))

theorem liouvilleTwistDisplay_two : liouvilleTwistDisplay 2 = -2 := by
  have h2 : liouvilleSign (2 : ℚ) = -1 := by
    have := liouvilleSign_natCast_prime (p := 2) Nat.prime_two
    simpa using this
  rw [liouvilleTwistDisplay, h2]
  norm_num

/-- The certificate deposit. `deltaOnly`: all cost domains and codomains are
`RatioOrbit` and all display arithmetic is discrete; the witnesses are
arithmetically explicit (some parent wrappers use eliminable classical
equality tests), and no completed carrier, continuum object, or continuity
premise appears in any statement or witness (see the header for the
deliberation against the wall convention). -/
theorem slim_ledger_minimality_certificate_tagged :
    PublicSpine.Tagged StrengthTag.deltaOnly SlimLedgerMinimalityCertificate
    where
  holds := slimLedgerMinimalityCertificate_holds

What this page does not claim

This module does not prove that the five conditions are the only possible ones, only that each is necessary for the canonical cost. The counterexamples are not presented as physically realized costs; they are mathematical witnesses to the necessity of each axiom. The certificate does not establish that the canonical cost function is the one realized in nature; that is a separate empirical question.

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/Foundation/PrimitiveRecognitionCalculus/PRCNativeCostMinimalityCertificate.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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