Information law verification python-flint 0.6.0, arb precision 256 bits == 1. Random worlds == Each world: 2-5 world states, 2-4 register states, random detailed-balance undriven channels (some parallel), random act and register channels. -- general: 300 worlds, 3183 system states, seed 20261011 stationary 300/300 PASS (1) sigma_world = floor - I_act + S_A 300/300 PASS (2) sigma_reg = I_learn + S_R 300/300 PASS (3) I_learn = I_act + I_noise 300/300 PASS (3') S_U = floor + I_noise 300/300 PASS (4) term signs exact 300/300 PASS (4) I_noise >= 0 certified 300/300 PASS world fuel certified below its floor: 1/300 largest saving floor - sigma_world = 0.00078866348 with I_act = 0.011831090 -- blind memory (register rates do not read x): 100 worlds, 1058 system states, seed 20261012 stationary 100/100 PASS (1) sigma_world = floor - I_act + S_A 100/100 PASS (2) sigma_reg = I_learn + S_R 100/100 PASS (3) I_learn = I_act + I_noise 100/100 PASS (3') S_U = floor + I_noise 100/100 PASS (4) term signs exact 100/100 PASS (4) I_noise >= 0 certified 100/100 PASS (V4) I_learn <= 0 certified 100/100 PASS world fuel certified below its floor: 0/100 corollary: never below the floor PASS -- unused memory (act rates do not read m): 100 worlds, 1060 system states, seed 20261013 stationary 100/100 PASS (1) sigma_world = floor - I_act + S_A 100/100 PASS (2) sigma_reg = I_learn + S_R 100/100 PASS (3) I_learn = I_act + I_noise 100/100 PASS (3') S_U = floor + I_noise 100/100 PASS (4) term signs exact 100/100 PASS (4) I_noise >= 0 certified 100/100 PASS (V5) sigma_world = floor + Phi_A 100/100 PASS world fuel certified below its floor: 0/100 corollary: never below the floor PASS == 2. The demon world: closed form and the exchange rate == World {0,1}, undriven flips at rate 1 (pi_c uniform); register copies x with accuracy a = 9/10; held law p = (3/4, 1/4); one act channel, in register state 1, moves x from 1 to 0 and carries the whole holding current 1/2. kappa sigma_world floor - sigma_world (floor - sigma_world)/I_act S_A <= sum j_A^2/kappa 1 -0.3465735903 0.8958797346 0.8154648768 PASS 10 -0.5249110622 1.074217207 0.9777946393 PASS 100 -0.5468123736 1.096118518 0.9977300720 PASS 1000 -0.5490562068 1.098362351 0.9997724971 PASS 10000 -0.5492811450 1.098587289 0.9999772446 PASS 1000000 -0.5493058943 1.098612039 0.9999997724 PASS closed forms, decided exactly (kappa-independent): floor = (p0 - p1) ln(p0/p1) 0.5493061443 PASS I_act = (p0 - p1) ln(a/(1-a)) 1.098612289 PASS I_noise = (2a - 1) ln(a/(1-a)) 1.757779662 PASS I_learn = (p0 - p1 + 2a - 1) ln(a/(1-a)) 2.856391951 PASS limits as kappa grows: sigma_world -> floor - I_act = -0.5493061443 (the act channel returns fuel) sigma_reg -> I_learn = 2.856391951 total -> floor + I_noise = 2.307085806 == 3. Archipelago, three islands: the exchange rate reached in a Xypher world == Weather builds every bridge at rate 1 and wears it at rate b_c = 3 (cold store). Held world law p(G) ~ 2^h(G) b_c^-|G|, h = home bridges standing (islands 0-1, 0-2). Register: a sensor reading h(G) with accuracy 4/5. Act channels toggle home bridges. -- blind agent (no register), act currents matched to cancel the weather (Horowitz-Zhou-England): at kappa = 10^6: sigma_world = 0.8317768507, floor = 0.8317766167, I_act = 0 with no register there is no information (I_act = 0): the floor is its limit. -- informed agent: each home build is carried in the register state that reads the pre-build count, so every build consumes ln(2a/(1-a)) = ln 8 of information. kappa sigma_world floor - sigma_world (floor - sigma_world)/I_act S_A <= sum j_A^2/kappa 1 -1.452394655 2.284171272 0.9153784906 PASS 10 -1.640413618 2.472190235 0.9907268310 PASS 100 -1.661215875 2.492992492 0.9990633069 PASS 1000 -1.663319260 2.495095876 0.9999062354 PASS 10000 -1.663529834 2.495306450 0.9999906226 PASS 1000000 -1.663552999 2.495329616 0.9999999062 PASS held p is the same in both agents; floor(p) = 0.8317766167 I_act = 2.495329850, I_noise = 5.240192685, I_learn = 7.735522535 as kappa grows: sigma_world -> floor - I_act = -1.663553233; total -> floor + I_noise = 6.071969302 == 4. Archipelago, three islands: a sensing agent with fixed-force fuel == Same weather (b_c = 3). Fuel from a hot store (b_h = 2): build a home bridge at rate 1, return it at rate b_h. Register: sensor of h(G), steps toward h at rate 4, away at rate 1. Identity check only: these agents are not compared; that is the next experiment. -- LOCAL fuel, no register theorem identities (1)-(4): PASS fuel flow J = 2/7 (exact); act-channel entropy flow = -J ln b_h: PASS sigma_world = J ln(b_c/b_h): PASS sigma_world 0.1158471737 floor 0.05209187337 I_act 0 I_noise 0 I_learn 0 S_A 0.06375530038 S_R 0 -- informed fuel, only while the sensor reads a home bridge down theorem identities (1)-(4): PASS fuel flow J = 30822/96473 (exact); act-channel entropy flow = -J ln b_h: PASS sigma_world = J ln(b_c/b_h): PASS sigma_world 0.1295413801 floor 0.06606685360 I_act 0.1814705608 I_noise 0.2765558674 I_learn 0.4580264282 S_A 0.2449450873 S_R 0.9822384649 == Summary == checks failed: 0 OVERALL PASS