-- Independent exact-rational resource certificate in Lua 5.3+.
-- Run with `lua languages/resources.lua` or `texlua languages/resources.lua`.
-- This checks the cubic waveform by polynomial integration rather than by
-- calling the Python closed-form resource formula. No floating-point proof.
assert(math.type and math.type(1)=="integer", "Lua integer arithmetic required")
-- Reserve the most negative integer so abs/negation are always representable.
local MAX=math.maxinteger
local function checked(a)
  assert(math.type(a)=="integer" and a>=-MAX and a<=MAX,"integer range exceeded")
  return a
end
local function imul(a,b)
  checked(a);checked(b)
  assert(b==0 or math.abs(a)<=MAX//math.abs(b),"integer multiplication overflow")
  return a*b
end
local function iadd(a,b)
  checked(a);checked(b)
  if b>0 then assert(a<=MAX-b,"integer addition overflow")
  elseif b<0 then assert(a>=-MAX-b,"integer addition overflow") end
  return a+b
end
local function gcd(a,b) a=math.abs(a);b=math.abs(b);while b~=0 do a,b=b,a%b end;return a end
local function rat(a,b)
  b=b or 1;checked(a);checked(b);assert(b~=0);if b<0 then a,b=-a,-b end
  local g=gcd(a,b);return {a//g,b//g}
end
local function add(a,b)
  local g=gcd(a[2],b[2]);return rat(iadd(imul(a[1],b[2]//g),imul(b[1],a[2]//g)),imul(a[2]//g,b[2]))
end
local function neg(a)return {-a[1],a[2]}end
local function sub(a,b)return add(a,neg(b))end
local function mul(a,b)
  local g=gcd(a[1],b[2]);local h=gcd(b[1],a[2])
  return rat(imul(a[1]//g,b[1]//h),imul(a[2]//h,b[2]//g))
end
local function div(a,b)return mul(a,rat(b[2],b[1]))end
local function eq(a,b)return a[1]==b[1]and a[2]==b[2]end
local function fmt(a)return tostring(a[1]).."/"..tostring(a[2])end
local Z=rat(0)
local function pder(a,dt)
  local r={};for j=2,#a do r[j-1]=div(mul(rat(j-1),a[j]),dt)end
  if #r==0 then r={Z}end;return r
end
local function padd(a,b)
  local r={};for j=1,math.max(#a,#b)do r[j]=add(a[j]or Z,b[j]or Z)end;return r
end
local function pscale(a,s)local r={};for j,v in ipairs(a)do r[j]=mul(v,s)end;return r end
local function integral(a,dt)
  local r=Z;for j,v in ipairs(a)do r=add(r,div(v,rat(j)))end;return mul(r,dt)
end
local function square(a)
  local r={};for j=1,2*#a-1 do r[j]=Z end
  for i,x in ipairs(a)do for j,y in ipairs(a)do r[i+j-1]=add(r[i+j-1],mul(x,y))end end
  return r
end
local low,high=rat(2,5),rat(9)
local mean,T,ca=rat(403,83),rat(249,50),rat(1,5)
local segments={{rat(7,10),low,low},{rat(1),low,high},{rat(79,50),high,high},
                {rat(1),high,low},{rat(7,10),low,low}}
local force,heat,meanint=Z,Z,Z
for _,seg in ipairs(segments)do
  local dt,l,h=seg[1],seg[2],seg[3];local diff=sub(h,l)
  local q={l,Z,mul(rat(3),diff),mul(rat(-2),diff)}
  local dq=pder(q,dt);local ddq=pder(dq,dt)
  local x={sub(q[1],mean),q[2],q[3],q[4]}
  local v=padd(padd(ddq,pscale(dq,ca)),x)
  force=add(force,integral(square(v),dt))
  heat=add(heat,mul(ca,integral(square(dq),dt)))
  meanint=add(meanint,integral(q,dt))
end
force,heat,meanint=div(force,T),div(heat,T),div(meanint,T)
assert(eq(force,rat(1815714302,6027875)),"force certificate failed")
assert(eq(heat,rat(14792,2075)),"heat certificate failed")
assert(eq(meanint,mean),"mean certificate failed")
print("PASS: exact average force squared = "..fmt(force))
print("PASS: exact average acoustic heat = "..fmt(heat))
print("PASS: exact stiffness mean = "..fmt(meanint))
print("All operations used normalized integer rational arithmetic with overflow guards.")