ledger_forces_separation
The closed flux condition on any ledger forces existence of a balanced parity encoding such that no decoder recovers the hidden bit from a proper subset of positions. Researchers examining ledger-induced computation-recognition gaps would cite the result when exploring information barriers in dual-complexity models. The tactic proof supplies the standard encoder from BalancedParityHidden and reduces the assumed decoder to a contradiction with the omega_n_queries lower bound.
claimLet $L$ be any ledger obeying closed flux conservation ($L$.closed_flux($γ$)=0 for all $γ$). Then there exists an encoding $e: {0,1} → ({1,…,n} → {0,1})$ such that, for every bit $b$ and every index set $M$ with $|M| < n/2$, no decoder $d$ satisfies $d$(restrict($e(b)$, $M$)) = $b$ for all masks $R$.
background
The declaration lives inside a scaffold module whose module documentation states it is exploratory and outside the verified certificate chain. Core supporting definitions are the hidden-mask encoder (which returns the mask $R$ when the bit is false and its complement when true), the restrict projection onto a queried index set $M$, and the omega_n_queries theorem asserting that any universally correct decoder must examine all $n$ positions under an adversarial mask. The ledger hypothesis is the closed-flux form of the balanced predicate from LedgerForcing. Upstream MagnitudeOfMismatch.forces supplies the symmetry-forcing context that the proof invokes indirectly through the ledger structure.
proof idea
The tactic proof opens with intro L hflux, instantiates BalancedParityHidden.enc as the witness, then enters the universal quantifiers over $b$, $M$ and $hM$. After a classical assumption of a decoder it obtains $M$.card < $n$ by transitivity, applies omega_n_queries to produce the negation of a universal decoder, and finally shows the assumed decoder contradicts that negation by simplification on the restricted encoding.
why it matters in Recognition Science
The result supplies the information-hiding step that the module documentation identifies as the key insight linking double-entry ledger structure to complexity separations. It is cited in the module's hypothetical SAT separation and recognition-scale P ≠ NP claims, though no downstream theorems yet depend on it. The declaration therefore touches the open question of whether ledger assumptions can be lifted to unconditional complexity statements while remaining outside the core T0–T8 forcing chain and Recognition Composition Law.
scope and limits
- Does not establish an unconditional P versus NP separation.
- Does not apply outside the specific ledger and balanced-parity model.
- Does not yield explicit time or query complexity bounds.
- Does not close the scaffolding status of the surrounding module.
formal statement (Lean)
279theorem ledger_forces_separation :
280 -- The ledger's double-entry structure creates the separation
281 ∀ (L : IndisputableMonolith.LedgerUnits.Ledger),
282 -- Closed flux conservation (T3)
283 (∀ γ, L.closed_flux γ = 0) →
284 -- Forces balanced encoding
285 (∃ encoding : Bool → Fin n → Bool,
286 ∀ b M (hM : M.card < n / 2),
287 -- Cannot distinguish without enough measurements
288 ¬(∃ decoder, ∀ R,
289 decoder (BalancedParityHidden.restrict (encoding b) M) = b)) := by
proof body
Tactic-mode proof.
290 intro L hflux
291 -- The ledger structure forces information hiding
292 use BalancedParityHidden.enc
293 intro b M hM
294 -- Apply the adversarial bound
295 classical
296 intro h
297 rcases h with ⟨decoder, hdec⟩
298 have hMn : M.card < n := lt_of_lt_of_le hM (Nat.div_le_self _ _)
299 have : ¬ (∀ (b : Bool) (R : Fin n → Bool),
300 decoder (BalancedParityHidden.restrict (BalancedParityHidden.enc (n:=n) b R) M) = b) := by
301 simpa using (BalancedParityHidden.omega_n_queries (n:=n) M decoder hMn)
302 exact this (by intro b' R'; simpa using hdec R')
303
304/-- Empirical validation scaffold -/