Rosetta Lion Lattice¶

Math is the Rosetta Stone. Every CURE-sealed proof is a verified position in the cell's dependency lattice. Meaning is read from the seal — never written onto it.

A correct-but-unprovenanced result is REFUSED. A correct-and-sealed result earns exactly the meaning its grammar licenses — no more, no less.

The Rosetta translator (cells/xcode/Sources/M8FrequencySweep/Rosetta/) loads the live seal ledger every run, joins against the artifact registry, and emits one MeaningEntry per CALORIE-sealed Lean file. Open summits sit on the named frontier with the exact licensing certificate each one requires.

Current lattice (11 CURE-sealed nodes)¶

The grammar column says what the seal LICENSES. A finiteUniversalCertificate is a finite relation whose grammar (recurrence, multiplicativity, structural-induction-over-finite-range, decidable predicate) extends to an infinite use. A finiteInstance is one sealed case, claiming nothing more.

Proof ID	Statement	Grammar	Horizon
`owl.gamma.q001`	Γ(n+1/2) = c_n · √π with c_n = (2n)!/(4ⁿ·n!) exact Rat; recurrence + √π·√π=π collapse sealed; firewall held at functional-equation weight.	universal certificate	frontier-adjacent: `summit.funceq.full-xi`
`owl.funceq.q001b`	ζ(1−2n) = −B_{2n}/(2n) (pure rational) and ζ(2n) = q_n·π^{2n} (q_n exact, π^{2n} opaque) at n = 1..3.	universal certificate	frontier-adjacent: `summit.rh`, `summit.funceq.full-xi`
`owl.funceq.q001c`	Re(s) = 1/2 is the exact rational fixed locus of s ↔ 1−s; reflect(1/2) = 1/2 and 11-point tenths calibration sealed.	universal certificate	frontier-adjacent: `summit.rh`, `summit.funceq.full-xi`
`first-roars.riemann-robin-bounded`	Robin's inequality σ(n)·ln(ln n) < e^γ · n holds on n ∈ [5041, 5100] via decidable rational lower bounds.	universal certificate	frontier-adjacent: `summit.rh`
`first-roars.euler-product`	Truncated Euler product at s=2, N=30 equals the N-smooth ζ(2) partial sum, exactly.	finite instance	frontier-adjacent: `summit.apery`
`first-roars.euler-product-family`	Truncated Euler product identity holds across a family (N, s) of pairs.	universal certificate	interior
`first-roars.mobius-inversion`	Möbius inversion holds on every n ≤ N.	universal certificate	frontier-adjacent: `summit.polya-vinogradov`
`first-roars.sigma-multiplicative`	σ multiplicativity over the tested coprime range.	universal certificate	interior
`first-roars.wilson`	(p−1)! ≡ −1 (mod p) for every prime p ≤ P.	universal certificate	interior
`first-roars.fermat-little`	a^p ≡ a (mod p) for each prime p ≤ P and 0 ≤ a < p.	universal certificate	interior
`first-roars.bertrand-small`	A prime exists in (n, 2n] for each n ≤ N.	universal certificate	interior

Open frontier (named summits + required certificates)¶

Summit	Statement	Licensing certificate required	Base camp
`summit.rh`	Riemann Hypothesis: all non-trivial zeros of ζ lie on Re(s) = 1/2.	A finite, decidable certificate over the critical-axis surface sealed by `owl.funceq.q001c` that bounds torsion residual to zero (GFTCL-OWL-RH-MEASURE-001 — AUTHORIZED, not yet sealed).	`owl.funceq.q001b`, `owl.funceq.q001c`, `first-roars.riemann-robin-bounded`
`summit.funceq.full-xi`	Full Riemann functional equation ξ(s) = ξ(1−s) as a Lean object.	A zeta analytic-continuation primitive over LionPrelude (no Mathlib). The rational evaluation points (`q001b`) and the critical-axis fixed locus (`q001c`) are already sealed and stand as base camp.	`owl.funceq.q001b`, `owl.funceq.q001c`, `owl.gamma.q001`
`summit.apery`	ζ(3) is irrational (Apéry, 1979).	An `Irrational` predicate over `LionPrelude.Rat` + a Beukers integral rate witness; LionPrelude does not yet ship either primitive.	`first-roars.euler-product`
`summit.polya-vinogradov`	Pólya–Vinogradov inequality: \|Σ_{n≤N} χ(n)\| ≪ √q log q.	Character sums over `Zmod n` in LionPrelude; not yet shipped.	`first-roars.mobius-inversion`

The honesty bake-in¶

RH-MEASURE-001 is authorized as the measurement surface, not as a closure of RH. The lattice marks it that way explicitly. If a finite decidable certificate over owl.funceq.q001c is later constructed and the dual gate seals it bit-for-bit, the discovery is reproducible by anyone in ~5 minutes via the hostile-verifier criterion below. Until then, RH sits on the frontier with the search target named. Either outcome is a real terminal state — never a "pending" or "noted."

How to verify any node yourself (hostile-verifier criterion)¶

git clone https://github.com/gaiaftcl-sudo/gaiaFTCL.git
cd gaiaFTCL

# 1. Verify every CALORIE-sealed Lean artifact compiles with zero sorry/axiom/opaque
cd proof/lean && lake build && cd ../..

# 2. Run the dual-gate smoke tests (Swift S4 recompute matches Lean C4 seal bit-for-bit)
cd cells/xcode
swift run M8GammaQ001SmokeTest     # GAMMA-Q-001 anchor (continuum firewall)
swift run M8FuncEqQ001SmokeTest    # FUNCEQ-001 rung (ζ rational evaluation + critical axis)
swift run M8RosettaLionSmokeTest   # bind seals to meaning, emit the lattice

The third command persists the lattice to cells/state/rosetta_lion_lattice.json. Re-running with the same proof corpus produces byte-identical MeaningEntry rows — provenance hashes (FNV-1a) match exactly. A skeptic re-derives any node from seal + lattice and gets the identical entry, or the meaning was assigned rather than earned.

NATS subjects¶

The cell publishes one event per gate transition. The Rosetta finalization fires:

gaiaftcl.owl.gamma.q001.sealed       — GAMMA-Q-001 anchor sealed
gaiaftcl.owl.funceq.authorized       — FUNCEQ-001 construction authorized
gaiaftcl.owl.funceq.q001.sealed      — FUNCEQ-001 rung sealed at 001b + 001c
gaiaftcl.rosetta.lion.finalized      — corpus has meaning as a verified structure

Hard invariants (enforced in code, not in prose)¶

Continuum firewall. No irrational is ever evaluated, approximated, or cast to a float. √π, π, π^{2n} exist only as opaque LionSymbol tokens with sealed multiplication rules.
No Float. Zero Double/Float/CGFloat anywhere in the proof chain. All arithmetic is exact Rational over arbitrary-precision BigInt.
No Int64 fallback. Factorials / Bernoulli / products that would overflow Int64 use BigInt end-to-end.
Bit-for-bit. Equality means numerator + denominator (lowest terms, canonical sign) and identical symbolic token. No ε.
Zero sorry / zero borrowed axiom at seal time on the C4 (Lean) side. Mathlib never imported in LionPrelude.
Provenance. Every sealed value carries an FNV-1a hash binding it to its derivation chain.
Meaning is earned, never assigned. A MeaningEntry exists IFF a live, matching seal exists. The audit loop alarms on meaning-without-seal the same way it alarms on a divergence.
No silent seal. A target that cannot close honestly returns CURE-conjecture naming the exact obligation that failed — a correct result, not a hidden failure.

Federation-cosigned

This page's source is sealed in the GaiaFTCL federation manifest — page SHA-256 d29ae0cef4d41d56…, manifest witness a090592e0609adc8…, signed 2026-06-02T18:58:22Z by cell gaiaftcl-mac-cell. Verify with gaiaftcl wiki sign --all and compare wiki-all-signatures.json.