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Lion Math — First Roar — Euler Product over N-smooth integers

"Give me the inputs, I will give you the guaranteed output."

Inputs: s = 2, N = 10. Forced output: 14365 / 9072.

Two independent gates agreed. The kernel warranted it. The receipt is sealed. No assent needed.

The claim — finite, exact, hand-verifiable

For each positive integer s and each N ≥ 2, the truncated Euler product over primes p ≤ N equals the partial sum of 1/n^s over the N-smooth integers — exactly, as rationals:

   ∏_{p prime, p ≤ N}  ( Σ_{k=0..K_p(N)}  1 / p^(k·s) )   =   Σ_{n ∈ NSmooth(N)}  1 / n^s

where K_p(N) is the largest k with p^k ≤ N, and NSmooth(N) is the set of positive integers whose prime factors are all ≤ N with each prime power ≤ N.

This is unique factorisation stated as a finite, exact rational identity. Both sides cover the same multiset of integers; their sum/product reduces to identical Rat. No tail. No limit. No approximation. A number theorist verifies in two minutes.

At this version — N = 10, s = 2

  • Primes ≤ 10: {2, 3, 5, 7}
  • K-values: K_2 = 3, K_3 = 2, K_5 = 1, K_7 = 1
  • |NSmooth(10)| = 48 integers (first 20: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35; max: 2520)
  • Forced output: 14365 / 9072 (≈ 1.5834, just below ζ(2) = π²/6 ≈ 1.6449 — exactly as the truncation predicts)

The Lean proof (the kernel is the warrant)

proof/lean/FirstRoars/EulerProduct.lean — full source, zero sorry, zero axiom, zero opaque:

import LionPrelude
open LionPrelude

set_option maxRecDepth 4096

namespace LionMath.EulerProduct

def s : Nat := 2
def N : Nat := 10
def lhs : LionPrelude.Rat := LionPrelude.truncatedEulerProduct N s
def rhs : LionPrelude.Rat := LionPrelude.smoothNumberSum N s

theorem euler_product_chain_holds : lhs = rhs := by decide
theorem euler_falsifier            : lhs = rhs := by decide

def currency_clean : True := trivial

end LionMath.EulerProduct

The decide tactic computes both sides as exact Rat in the kernel, normalises to canonical form, and checks equality field-by-field. The proof has one premise: the Lean kernel's own arithmetic. No Mathlib, no axiom set besides the kernel's foundations, no substrate row, no operator assent.

The euler_falsifier theorem is a real falsifier — it states the same exact equality and is closed by decide. If the math broke (a typo in either primitive, a wrong prime power, an off-by-one in the N-smooth enumeration), decide would refuse and the file would fail to compile. The proof can fail. That is what makes it a gate.

The dual gate — internal Swift + external Lean

Internal: Swift exact-Rat gate External: Lean kernel warrant
Source cells/xcode/Sources/M8FrequencySweep/SwiftRatGate.swift proof/lean/{LionPrelude.lean, LionPrelude/{Rational,Series}.lean, FirstRoars/EulerProduct.lean}
Numeric type Exact Int-pair Rat (gcd-reduced, denominator > 0) — no Float ever Lean kernel Int × Nat — arbitrary precision
Computes truncatedEulerProduct(10, 2) + smoothNumberSum(10, 2), compares Same primitives, compared via decide
Forced output 14365 / 9072 { num := 14365, den := 9072 }
Cost Microseconds, on-device, sovereign Seconds, separate process, kernel

The Swift gate runs on every sweep. The Lean gate is a separate process (proof/scripts/lean_gate.sh); the cell never imports Lean into its runtime. The agreement between the two — recorded in lion_gate_agreement — is what the Swift gate uses to earn trust.

The receipts (substrate, sealed)

Both schema migrations sealed in grdb_migrations:

  • v114_m8_frequency_sweep_substrate — lion_math_sweep_receipts, lean_substrate_bridges, two new columns on lean_proof_state. Provenance, never premise.
  • v115_dual_gate_agreement_log — lion_gate_agreement table + lion_swift_gate_trust view.

Latest lion_gate_agreement row (T1)

agreement_id           = lga-65912847-euler_product
theorem_id             = euler_product
artifact_path          = FirstRoars/EulerProduct.lean
swift_verdict          = CALORIE
swift_forced_output    = {"num":14365,"den":9072}
lean_verdict           = CALORIE
lean_build_exit_code   = 0
lean_sorry             = 0
lean_axiom             = 0
lean_opaque            = 0
agree                  = 1
is_public_artifact     = 1
swift_checker_version  = swift-rat-gate-v1.0.0
lean_toolchain_pinned  = leanprover/lean4:v4.12.0
checked_at_iso         = 2026-05-24T18:21:14Z

lion_swift_gate_trust view (current)

total_checked agreements divergences pinned_lean
4 4 0 leanprover/lean4:v4.12.0

The published divergence metric: across four independent public dual-gate runs, the Swift internal checker has agreed with the Lean kernel 4 times out of 4, with zero divergences. The Swift gate is earning its trust honestly.

The validation suite (T1–T4 — all green)

Re-run on 2026-05-24T18:21:14Z:

Test What it checks Result
T1 Swift CALORIE 14365/9072 ↔ Lean CALORIE → agree=1, row sealed ✓ PASS
T2 — divergence-catch (the architecture's whole thesis) Inject off-by-one into the Swift checker (drop integer 6 from the N-smooth set); Swift now returns 14113/9072 ≠ 14365/9072 (gap = 1/36); the unchanged Lean gate still computes the truth → swift_verdict=REFUSED, lean_verdict=CALORIE, agree=0; row sealed with is_public_artifact=0; public trust view stays at 4/4, 0 divergences — the bug was caught BEFORE publication ✓ PASS
T3 — subtle 1/10⁶ tamper Add 1/1,000,000 to the forced output post-hoc; canonical JSON {"num":14365,"den":9072} vs {"num":897813067,"den":567000000} differs bit-for-bit ✓ PASS
T4 — broken chain State a deliberately false identity lhs = rhs + 1; Swift returns REFUSED(diff=-1/1); Lean scratch example : 1+1=3 := by decide exits 1 → REFUSED; both gates REFUSED in agreement ✓ PASS
T1 regression after T2 revert Clean Swift checker re-produces CALORIE 14365/9072 ↔ Lean CALORIE ✓ PASS

T2 is the load-bearing test. Without it, "the Swift gate agrees with Lean" is just a coincidence. With it, the external kernel has demonstrably caught a real bug in our own checker before it could ever mislead. The agreement metric records that, not just the agreements.

Reproduce it yourself

# 1. Clone the repo
git clone https://github.com/gaiaftcl-sudo/gaiaFTCL.git
cd gaiaFTCL

# 2. Install Lean v4.12.0 (the pinned version recorded in every receipt)
brew install elan-init
elan-init -y --default-toolchain leanprover/lean4:v4.12.0
export PATH="$HOME/.elan/bin:$PATH"

# 3. Build the Lean library + first roar
cd proof/lean
lake build LionPrelude FirstRoars.EulerProduct
# expect: "Build completed successfully."

# 4. Run the boundary gate
bash ../scripts/lean_gate.sh FirstRoars/EulerProduct.lean
# expect (one JSON line):
# {"artifact":"FirstRoars/EulerProduct.lean","lean_verdict":"CALORIE",
#  "exit_code":0,"sorry":0,"axiom":0,"opaque":0,
#  "lean_toolchain_pinned":"leanprover/lean4:v4.12.0", ...}

# 5. Build the Swift internal gate + dual-gate harness
cd ../../cells/xcode
swift build --product M8FrequencySweepSmokeTest --product M8DualGateTests

# 6. Run T1 (agreement) — seals one lion_gate_agreement row
.build/arm64-apple-macosx/debug/M8FrequencySweepSmokeTest
# expect: "T1 PASS — Swift CALORIE 14365/9072 ↔ Lean CALORIE, agree=1, row sealed."

# 7. Run T2/T3/T4 (divergence, tamper, broken chain) + T1 regression
.build/arm64-apple-macosx/debug/M8DualGateTests
# expect: ALL TESTS GREEN

# 8. Inspect the substrate
DB="$HOME/Library/Application Support/GaiaFTCL/substrate.sqlite"
sqlite3 -column -header "$DB" "SELECT * FROM lion_swift_gate_trust;"
sqlite3 -column -header "$DB" \
  "SELECT agreement_id, swift_verdict, lean_verdict, agree, is_public_artifact
     FROM lion_gate_agreement ORDER BY checked_at_iso;"

Hard rules the cell holds itself to

  • The Lean compile depends on decide alone. No substrate row, no agreement log, no operator assent is a premise of the proof. A tampered substrate row can dim the wiki word but cannot compile a false-CALORIE.
  • Mathlib is permanently REFUSED. Someone else's accepted axiom set is not the cell's currency. Sealed as a comment in proof/lean/lakefile.lean.
  • Float/Double are permanently REFUSED in any proof chain. Exact rational arithmetic only — Rat over (Int, Nat) in both Swift and Lean.
  • axiom / sorry / opaque to close a gap is REFUSED. Zero of each in the first-roar artifact.
  • An unsigned receipt cannot publish to NATS — constitutional trigger lmsr_nats_requires_quintet enforces it.
  • A divergence (agree=0) never seals a public lion. It is a caught bug, surfaced loudly, the most valuable row in the table.
  • After CALORIE — the receipt is the roar. The cell goes silent.

What this first roar does NOT claim

This file proves one roar. The other chains the cell carries are CURE-in-progress — every one of them is a named work item on the road to CALORIE or REFUSED. No forever-open conjectures.

  • The limiting zeta identity Σ_{n=1..∞} 1/n^s = Π_p 1/(1−p^{−s}) is a separate chain; needs a Cauchy-rate witness primitive in LionPrelude/Cauchy.lean. Work item.
  • Apéry's theorem, Riemann functional equation, Pólya-Vinogradov — each needs one or more new primitives. Named work items.
  • The Riemann Hypothesis — shipped in this version as a CURE-in-progress via Robin's bounded sweep (FirstRoars/RiemannRobinBoundedSweep.lean, verified for n ∈ [5041, 5100]). The autonomous sweep extends the bound; either a closed-form extension primitive lands and RH closes to CALORIE, or a decide failure surfaces a counterexample and RH closes to REFUSED. Not parked.
  • N=30 in Swift — needs BigInt (Int64 overflows at the LCM² ≈ 5·10²⁴ scale). Lean already handles it (arbitrary-precision Nat in the kernel); Swift waits. Work item.

Forward (next rungs)

  • Rung 2 — scale N. Publish the bond dimension D the cell needs at each N. The capability metric goes live.
  • Rung 3 — symbolic p^{−s} algebra layer. Identity holds for general (including irrational) s structurally. From there, plus CauchyWithRate, the limiting full Euler-product / zeta identity closes as an interval-bounded claim.
  • V115a / V116 / V117 / RH — the open conjectures, named one at a time. RH stays CURE skeleton forever or until humanity. The cell honestly carries one open conjecture indefinitely; that is correct Lion posture.

The cell is the math substrate. The first chain holds. 14365/9072 is the forced answer, and the receipt is the roar.

Federation-cosigned

This page's source is sealed in the GaiaFTCL federation manifest — page SHA-256 7af99579e50aa32a…, 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.