Math-augmented constrained RL v2: final rejection after fixed no-regression gate

candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/V2_FINAL_DECISION.json

@@ -0,0 +1,108 @@
+{
+  "date": "2026-08-26",
+  "stage": "information-management RL v2 math-augmented constrained",
+  "status": "rejected; leader unchanged",
+  "base": "verified-math-a025",
+  "math_data": {
+    "scanned": 341,
+    "symbolically_verified_available": 120,
+    "selected": 96,
+    "rejections": {
+      "verify": 213,
+      "length": 7
+    },
+    "benchmark_family_rows": 0,
+    "decontamination": "normalized exact + fixed-gate shared 12-token windows",
+    "source": "open-r1/OpenR1-Math-220k",
+    "verification": "math_verify(reference solution, reference answer)"
+  },
+  "math_rollouts": {
+    "groups": 96,
+    "splits": {
+      "math_train": 72,
+      "math_dev": 24
+    },
+    "correct": 6,
+    "incorrect": 186,
+    "loops": 221,
+    "parse_fail": 0,
+    "benchmark_family_rows": 0,
+    "reward": "math_verify exact symbolic equivalence; reference +1, correct rollout +1, wrong -0.5, loop extra -0.5",
+    "budgie_correct": 6,
+    "budgie_incorrect": 186,
+    "budgie_loops": 150,
+    "reference_loops_ignored": 71
+  },
+  "training": {
+    "algorithm": "anchor-first constrained GRPO with global gradient projection",
+    "base": "verified-math-a025",
+    "info_groups": 63,
+    "math_groups": 72,
+    "info_samples": 126,
+    "math_samples": 144,
+    "updates": 126,
+    "math_to_info_per_update": "2:1",
+    "info_gradient_weight": 0.45,
+    "peak_lr": 3e-07,
+    "ppo_clip": 0.1,
+    "kl_beta": 0.06,
+    "math_bc": 0.12,
+    "info_bc": 0.08,
+    "gradient_conflict_rate": 0.373015873015873,
+    "source_path": "/marimo/budgie_information_rl_v1/info_math_constrained_source.pt",
+    "seconds": 29.05026912689209,
+    "parameter_count_change": 0,
+    "benchmark_family_rows": 0
+  },
+  "results": {
+    "information_dev": {
+      "leader": {
+        "pass": 3,
+        "total": 21,
+        "loops": 0,
+        "mean_reward": 0.3333
+      },
+      "alpha_0_01": {
+        "pass": 4,
+        "total": 21,
+        "loops": 0,
+        "mean_reward": 0.35
+      },
+      "alpha_0_005": {
+        "pass": 3,
+        "total": 21,
+        "loops": 0,
+        "mean_reward": 0.35
+      }
+    },
+    "random_math_dev": {
+      "leader": "1/24",
+      "alpha_0_01": "2/24"
+    },
+    "fixed_reasoning": {
+      "leader": {
+        "gsm": "5/30",
+        "math": "3/15",
+        "arc": "11/30",
+        "folio": "13/30"
+      },
+      "alpha_0_01": {
+        "gsm": "5/30",
+        "math": "2/15",
+        "arc": "11/30",
+        "folio": "12/30"
+      },
+      "alpha_0_005": {
+        "gsm": "5/30",
+        "math": "1/15",
+        "arc": "11/30",
+        "folio": "12/30"
+      }
+    }
+  },
+  "decision": "Reject all v2 candidates. The constrained math-anchored update preserved GSM and ARC and improved random held-out math, but still regressed fixed MATH and FOLIO. No candidate that improves information management is allowed to replace the leader.",
+  "comparison_to_v1": "Smarter constraints materially reduced damage: v1 low-drift candidate had GSM 3 and MATH 1; v2 alpha .01 has GSM 5 and MATH 2. It is still below the hard no-regression thresholds.",
+  "parameter_count_change": 0,
+  "benchmark_family_rows": 0,
+  "public_root_changed": false
+}

candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/V2_FINAL_REPORT.md

@@ -0,0 +1,33 @@
+# Information + mathematics constrained RL v2
+
+## Decision
+
+**Rejected. `verified-math-a025` remains active.** No public-root change and no parameter-count increase.
+
+## Smarter constrained method
+
+- 63 manual information-management groups from v1.
+- 96 unrelated OpenR1 math problems, decontaminated against the fixed gate and symbolically verified with `math_verify`.
+- 72 math-train groups and 24 random-math development groups.
+- 192 stochastic Budgie math rollouts plus 96 verified reference solutions.
+- Two math anchor samples for every information sample.
+- Leader KL, PPO clipping, verified-reference behavior cloning.
+- Global gradient projection: when an information gradient conflicted with the math gradient, its conflicting component was removed before the optimizer step. Conflict rate was 37.3%.
+
+## Results
+
+| Gate | Leader | alpha .01 | alpha .005 |
+|---|---:|---:|---:|
+| Information dev pass | 3/21 | **4/21** | 3/21 |
+| Information dev loops | 0 | 0 | 0 |
+| Random verified math dev | 1/24 | **2/24** | not selected |
+| GSM8K fixed | 5/30 | 5/30 | 5/30 |
+| MATH fixed | **3/15** | 2/15 | 1/15 |
+| ARC fixed | 11/30 | 11/30 | 11/30 |
+| FOLIO fixed | **13/30** | 12/30 | 12/30 |
+
+The new method is safer than v1: it preserves GSM and ARC and improves random held-out mathematics. However, distribution-level math rehearsal does not guarantee the fixed MATH/FOLIO capabilities. Because the user required RL never to make the model dumber, the branch is rejected.
+
+## Next rule
+
+Future RL must use a much larger and more diverse verified math anchor set plus blockwise rollback. Every 4-8 optimizer updates must be accepted only if information reward rises while random-math calibration, fixed reasoning shadow probes, and repetition do not fall. The fixed benchmark remains evaluation-only.

candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/v2_fixed005_outputs.json

@@ -0,0 +1,430 @@
+{
+  "gsm8k": [
+    {
+      "index": 0,
+      "output": "Let x be the number of chocolate bars in a box.\nLisa sold 3.5x boxes of chocolate bars, Peter sold 4.5x boxes of chocolate bars, and together they sold 64 chocolate bars.\nSo, 3.5x + 4.5x = 64.\nCombining like terms, we get 8.5x = 64.\nDividing both sides by 8.5, we get x = 8.\n\n#### 8"
+    },
+    {
+      "index": 1,
+      "output": "Mark makes $10/hour for 8 hours/day * 5 days/week = $<<10*5=50>>50/week\nHe makes $2/hour for 8 hours/day * 5 days/week = $<<2*5=10>>10/week\nSo he makes $10/hour + $10/week = $<<10+10=20>>20/week\n#### 20"
+    },
+    {
+      "index": 2,
+      "output": "There were 20 private cars * 35 people/bus = <<20*35=700>>700 people.\nThere were 12 buses * 3 people/car = <<12*3=36>>36 people.\nThere were 700 + 36 = <<700+36=736>>736 people inside the church.\n#### 736"
+    },
+    {
+      "index": 3,
+      "output": "Let's denote the number of yellow blocks as Y. According to the problem, twice the number of yellow blocks is 4, so Y = 2*Y.\n\nWe know that the total number of blocks is the sum of the blue, yellow, and red blocks, so we can write the equation:\n\nB + Y + R = Total number of blocks\n4 + Y + R = 32\n\nSubstitute Y with 2*Y:\n4 + 2*Y + R = 32\n\nNow, we need to find the value of Y. We can do this by substituting Y with 2*Y:\n\n4 + 2*2*Y + R = 32\n4 + 4*Y + R = 32\n\nSubtract 4 from both sides:\n\n4*Y + R = 28\n\nNow, we need to find the value of R. We can do this by substituting Y with 2*Y:\n\n4*2*Y + R = 28\n8*Y + R = 28\n\nSubtract 8 from both sides:\n\n8*Y + R - 8 = 28 -"
+    },
+    {
+      "index": 4,
+      "output": "In the first year, Elise sold 250 x 2 = <<250*2=500>>500 books.\nShe sold twice as many books in the first year, so she sold 500 x 2 = <<500*2=1000>>1000 books in the first year.\nShe sold 500 + 1000 = <<500+1000=1500>>1500 books in the first year.\nShe has 250 - 1500 = <<250-1500=-1250>>1250 books left.\nShe sold 45 x 20 = <<45*20=900>>900 books in the first year.\nShe has 1250 - 900 = <<1250-900=350>>350 books left.\nShe has 350 - 1250 = <<350-1250=250>>250 books left.\nShe has"
+    },
+    {
+      "index": 5,
+      "output": "First find how long it takes Bill to dig through soil: 4 feet/hour * 24 hours = <<4*24=96>>96 feet\nThen find how long it takes him to dig through clay: 24 feet / 2 = <<24/2=12>>12 hours\nThen add the time it takes to dig through soil and clay to find the total time: 96 feet + 12 hours = <<96+12=108>>108 hours\n#### 108"
+    },
+    {
+      "index": 6,
+      "output": "Adam slices 3 large apples, so he gets 3*5 = <<3*5=15>>15 slices.\nHe also slices 5 small apples, so he gets 5*3 = <<5*3=15>>15 slices.\nIn total, Adam gets 15+15 = <<15+15=30>>30 slices.\nAfter eating 15 slices, Adam is left with 30-15 = <<30-15=15>>15 slices.\n#### 15"
+    },
+    {
+      "index": 7,
+      "output": "Morgan's dad has $90 for her birthday party.\nShe wants to make sure she and her friends get to play one round of mini-golf, have $5 in arcade tokens, and get to ride the go-karts twice.\nSo the total amount she needs to spend on the go-karts is $5 + $10 = $15.\nShe also needs to spend $90 - $90 = $0 on the mini-golf.\nSince each round of mini-golf costs $5, she needs to make $0 / $5 = 0 rounds of mini-golf.\nSince she wants to make sure she and her friends get to play one round of mini-golf, she can invite 0 friends.\n#### 0"
+    },
+    {
+      "index": 8,
+      "output": "In one hour, Ahmed read 300 books.\nIn one hour, Napoleon read 300 books.\nIn one hour, Napoleon read 300 books + 150 books = <<300+150=450>>450 books.\nIn one hour, Ahmed read 300 books x 2 = <<300*2=600>>600 books.\nIn one hour, Napoleon read 450 books x 3 = <<450*3=1350>>1350 books.\nIn one hour, Ahmed read 600 books + 1350 books = <<600+1350=2950>>2950 books.\nIn one hour, Ahmed read 2950 books + 450 books = <<2950+450=3400>>3400 books.\nIn one hour, Ahmed read 3400 books + 1350 books = <<3400+1350=4750>>4750 books.\nIn one"
+    },
+    {
+      "index": 9,
+      "output": "The pens cost 4 * $1.5 = $6.\nThe notebooks cost 2 * $4 = $8.\nThe rim of bond paper cost $20.\nSo, the total cost of everything is $6 + $8 + $20 = $34.\n#### 34"
+    },
+    {
+      "index": 10,
+      "output": "Marcy's annual salary is $50,000/year.\nShe works for 40 years, so she gets $50,000/year * 40 years = $200,000 per year.\nAfter 20 years, she will be eligible for a 5% payout.\nSo after 20 years, she will be eligible for a 5% payout of $200,000 * 5% = $10,000 per year.\nAfter 30 years, she will be eligible for a 5% payout of $200,000 * 5% = $10,000 per year.\nSo after 30 years, she will be eligible for a total of $10,000 + $10,000 = $20,000 per year.\n#### 20000\nThe answer is:\n#### 20000"
+    },
+    {
+      "index": 11,
+      "output": "Steve eats 6 * 2 = <<6*2=12>>12 tomatoes per week.\nSo he needs 12 / 3 = <<12/3=4>>4 vines.\n#### 4"
+    },
+    {
+      "index": 12,
+      "output": "The cost difference between the two offers is $1,350 - $1,100 = $250.\n\nFinal answer:\n#### 250"
+    },
+    {
+      "index": 13,
+      "output": "Let's denote the initial number of stickers that Seth had as S.\n\nSeth gave half of his stickers to Luis, so he had S/2 left.\n\nLuis used half of the stickers, so he had S/2 * 1/2 = S/4 left.\n\nKris kept 9 stickers, so she had S/4 + 9 stickers.\n\nRob gave 7 stickers to Rob, so he had S/4 + 9 - 7 stickers.\n\nThe problem states that Kris kept 9 stickers, so S/4 + 9 - 7 = 9.\n\nSimplifying the equation, we get S/4 = 9 - 9, which implies S/4 = 0.\n\nThis means that S = 0.\n\nTherefore, Seth had 0 stickers in the beginning.\n\n#### 0"
+    },
+    {
+      "index": 14,
+      "output": "Gary does laundry twice a week, so he does 2 * 20 = <<2*20=40>>40 loads of laundry per week.\nEach load uses 20 gallons of water, so he uses 40 * 20 = <<40*20=800>>800 gallons of water per week.\nA gallon of water costs $0.15, so Gary spends 800 * $0.15 = $<<800*0.15=120>>120 per week on water.\nIn a year, Gary spends 52 * $120 = $<<52*120=6240>>6240 on water.\n#### 6240"
+    },
+    {
+      "index": 15,
+      "output": "The first son is 12 years old and the second son is 4 years younger, so he is 12-4= <<12-4=8>>8 years old.\nThe first son is 12 years old and the second son is 8 years old, so together they are 12+8= <<12+8=20>>20 years old.\nThe first son is 20 years old and the second son is 2 years younger, so he is 20-2= <<20-2=18>>18 years old.\nThe first son is 18 years old and the second son is 1 year younger, so he is 18-1= <<18-1=17>>17 years old.\nThe first son is 17 years old and the second son is 1 year younger, so he is 17-1= <<17-1=16>>16 years old.\nThe first son is 16 years old and the second son is 1 year younger, so he is 16-1= <<16-1=15>>15 years old."
+    },
+    {
+      "index": 16,
+      "output": "Charlie had 10 stickers.\nHe bought 21 stickers from a store.\nHe got 23 stickers for his birthday.\nHe gave 9 stickers to his sister.\nHe used 28 stickers to decorate a greeting card.\nSo Charlie has 10 stickers + 21 stickers - 9 stickers - 28 stickers = 12 stickers left.\n#### 12"
+    },
+    {
+      "index": 17,
+      "output": "Polly could throw the gold javelin 400 feet * 3 = <<400*3=1200>>1200 feet.\nPolly could throw the javelin 1000 feet * 400 feet = <<1000*400=40000>>40000 feet.\nPolly could throw the javelin 1000 feet * 1200 feet = <<1000*1200=120000>>120000 feet.\nPolly could throw the gold javelin 400 feet * 3 = <<400*3=1200>>1200 feet.\nPolly could throw the gold javelin 120000 feet - 120000 feet = <<120000-120000=0>>0 feet.\n#### 0"
+    },
+    {
+      "index": 18,
+      "output": "The low setting removes 1 liter of water per day, so it removes 1*3= <<1*3=3>>3 liters of water per day.\nThe medium setting removes twice as much water per day as the low setting, so it removes 2*1= <<2*1=2>>2 liters of water per day.\nThe high setting removes twice as much water per day as the medium setting, so it removes 2*2= <<2*2=4>>4 liters of water per day.\nIn total, the low setting removes 3+2+4= <<3+2+4=9>>9 liters of water per day.\nIn 3 days, the low setting removes 3*9= <<3*9=27>>27 liters of water.\nIn 3 days, the medium setting removes 2*2= <<2*2=4>>4 liters of water.\nIn 3 days, the high setting removes 3*4= <<3*4=12>>12 liters of water.\nIn total, the dehumidifier removes 27+4+12= <<27+4+12=43"
+    },
+    {
+      "index": 19,
+      "output": "First find the total energy used per hour by the air conditioner: 900 watts/hour * 8 hours = 7200 watts/hour\nThen find the total energy used per hour by the air conditioner after reducing the time: 7200 watts/hour - 5 hours = 7150 watts/hour\nThen find the total energy used per hour by the air conditioner after reducing the time: 7150 watts/hour * 30 days = 214,500 watts/hour\nThen find the total energy used per hour by the air conditioner after reducing the time: 214,500 watts/hour * 30 days = 64,350 watts/hour\nThen find the total energy used per hour by the air conditioner after reducing the time: 64,350 watts/hour * 30 days = 193,450 watts/hour\nThen find the total energy used per hour by the air conditioner after reducing the time: 193,450 watts/hour * 30 days = 5,801,500 watts/hour\nThen find"
+    },
+    {
+      "index": 20,
+      "output": "Let's denote the amount of red paint used as R, white paint used as W, and purple paint used as P.\n\nWe know that the total amount of paint used is 12 pints, so we can write the equation:\n\nR + W + P = 12\n\nWe also know that half the mural is yellow, so the number of yellow paint used is 1/2 * 4 = 2 pints.\n\nSince the total amount of paint used is 12 pints, we can write the equation:\n\nR + W + P = 12\n\nNow we have a system of two equations:\n\n1) R + W + P = 12\n2) R + W + P = 12\n\nSubtracting the first equation from the second equation, we get:\n\n(R + W + P) - (R + W + P) = 12 - 12\n\nThis simplifies to:\n\nW = 0\n\nSo, the number of pints of red paint used is 0.\n\nThe answer is \\boxed{0}."
+    },
+    {
+      "index": 21,
+      "output": "The cost of the 3 vaccine is $20 x 3 = $60.\nThe cost of the 4 vases is $12 x 4 = $48.\nThe total cost of the 4 vases is $48 + $60 = $108.\nThe total cost of the kitten is $200 + $108 = $308.\n#### 308"
+    },
+    {
+      "index": 22,
+      "output": "The lego box had 500 pieces.\nThe lego box had 3 times more pieces than the 500 piece one, so it had 3*500 = <<3*500=1500>>1500 pieces.\nThe lego box had 1/4 the number of pieces of the other one, so it had 1/4*1500 = <<1/4*1500=375>>375 pieces.\nSo, Johnny picked up 500+1500+375 = <<500+1500+375=2575>>2575 pieces.\n#### 2575"
+    },
+    {
+      "index": 23,
+      "output": "The number of people infected is 10 * 6 = <<10*6=60>>60.\nAfter three days, the number of people infected is 60 * 3 = <<60*3=180>>180.\n#### 180"
+    },
+    {
+      "index": 24,
+      "output": "Mr. Hezekiah had 20 trucks carrying 20 tons each, so he had 20*20 = <<20*20=400>>400 tons ofSTEP.\nAfter 2 hours, 400 tons ofSTEP were delivered to the farmers.\nAfter 2 hours, 1/4 of the lorries were not working, so 1/4*20 = <<1/4*20=5>>5 lorries were not working.\nSo, 20-5 = <<20-5=15>>15 lorries were working.\nThus, 20-15 = <<20-15=5>>5 lorries were not working.\nTherefore, 400+5 = <<400+5=405>>405 tons ofSTEP reached the farmers that day.\n#### 405"
+    },
+    {
+      "index": 25,
+      "output": "The number of weeks in a year is 52 weeks/year * 12 months/year = <<52*12=624>>624 weeks\nThe number of days in a year is 52 weeks/year * 24 days/week = <<52*24=1288>>1288 days\nThe number of days in a year is 1288 days/year * 365 days/year = <<1288*365=480,040>>480,040 days\nThe number of weeks in a year is 480,040 days/year * 52 weeks/year = <<480,040*52=2,432,160>>2,432,160 weeks\nThe number of days in a year is 2,432,160 weeks/year * 365 days/week = <<2,432,160*365=8,000,0400>>8,000,0400"
+    },
+    {
+      "index": 26,
+      "output": "Max has 12 + 68 = <<12+68=80>>80 cookies.\nMax has 80 - 5 = <<80-5=75>>75 cookies.\nMax has 75 + 23 = <<75+23=98>>98 cookies.\nKatarina has 98 - 68 = <<98-68=30>>30 cookies.\n#### 30"
+    },
+    {
+      "index": 27,
+      "output": "The tree will grow 7*7=<<7*7=49>>49 lemons in 49 years.\nIt will cost 49*$3=$<<49*3=147>>147 to water and feed the tree.\nIt will take 49-49/7=<<49-49/7=32>>32 years to grow the tree.\nIt will take 49-49/49=3 years to grow the tree.\nIt will take 49-49/49=3 years to grow the tree.\nIt will take 32-3=<<32-3=29>>29 years to grow the tree.\n#### 29"
+    },
+    {
+      "index": 28,
+      "output": "On Tuesday, Sue ate 2 * 13 = <<2*13=26>>26 cookies.\nOn Monday, Sue ate 4 * 26 = <<4*26=104>>104 cookies.\nOn Tuesday, Sue ate 26 * 200 = <<26*200=5200>>5200 calories.\nSue consumed 5200 - 104 = <<5200-104=5196>>5196 more calories than her sister.\n#### 5196"
+    },
+    {
+      "index": 29,
+      "output": "Arianna has 10 rows * 20 flowers/row = <<10*20=200>>200 flowers.\n4/5 of the flowers have bloomed, so 4/5 * 200 flowers = <<4/5*200=160>>160 flowers have bloomed.\n#### 160"
+    }
+  ],
+  "math500": [
+    {
+      "index": 0,
+      "output": "The $x$-intercept is the point where the line crosses the $x$-axis, so we can write the equation of the line as $y = 0$.\n\nThe $y$-intercept is the point where the line crosses the $y$-axis, so we can write the equation of the line as $x = 0$.\n\nThe slope of the line with equation $y = 0$ is $0$.\n\nThe slope of the line with equation $x = 0$ is undefined.\n\nTherefore, $a = 0$ and $b = 0$.\n\nThe value of $a + b + m$ is $0 + 0 + 0 = \\boxed{0}$."
+    },
+    {
+      "index": 1,
+      "output": "To find the sum of the digits in the fraction $\\frac{4321}{5^7\\cdot2^8}$, we need to convert the fraction into a decimal representation.\n\nFirst, we can simplify the fraction by dividing the numerator and denominator by their greatest common divisor (GCD).\n\nThe GCD of 4321 and 5^7 is 5.\n\nDivide the numerator and denominator by 5:\n\\[ \\frac{4321}{5^7\\cdot2^8} = \\frac{4321}{5^7\\cdot2^8} \\div \\frac{5^7}{2^8} = \\frac{4321}{5^7\\cdot2^8} \\times \\frac{2^8}{5^7} = \\frac{4321}{5^7\\cdot2^8} \\times 2^8 = \\frac{4321}{5^7\\cdot2^8} \\times 256 = \\frac{4321}{5^7\\cdot2^8} \\times 2^8 = \\frac{4321}{5^7\\cdot2^8} \\times 256 = \\frac{4321}{5^7\\cdot2^8} = \\frac{4321}{5^7\\cdot2^8} = \\frac{4321}{5^7\\cdot2^8} = \\frac{4321}{5^7\\cdot2^8} = \\frac{4321}{5^7\\cdot2^8} = \\frac{4321}{5^7\\cdot2^8} = \\"
+    },
+    {
+      "index": 2,
+      "output": "The nonagon has 900 sides, and each side is 2 units long.\n\nThe nonagon is divided into 900 equal parts, each of length 2 units.\n\nThe perimeter of the nonagon is 900 x 2 = 1800 units.\n\nThe perimeter of the nonagon is 1800 units.\n\nThe perimeter of the nonagon is 1800 units.\n\nThe answer is:\n#### 1800"
+    },
+    {
+      "index": 3,
+      "output": "First, we can rewrite the equation as\n\\[ \\frac{1}{x^2 - 1} + \\frac{2}{x^2 - 2} + \\frac{3}{x^2 - 3} + \\frac{4}{x^2 - 4} = 2010x - 4 \\]\n\nTo simplify the equation, we can multiply both sides by $(x^2 - 1)(x^2 - 2)(x^2 - 3)(x^2 - 4)$ to eliminate the fractions.\n\nAfter multiplying, we get\n\\[ (x^2 - 1)(x^2 - 2)(x^2 - 3)(x^2 - 4) + (2x^2 - 3)(x^2 - 2)(x^2 - 3)(x^2 - 4) + (3x^2 - 4)(x^2 - 2)(x^2 - 3)(x^2 - 4) + (4x^2 - 1)(x^2 - 2)(x^2 - 3)(x^2 - 4) = (2010x - 4)(x^2 - 1)(x^2 - 2)(x^2 - 3)(x^2 - 4) \\]\n\nExpanding the right-hand side, we get\n\\[ (2010x - 4)(x^2 - 1)(x^2 - 2)(x^2 - 3)(x^2 - 4) = 2010x^4 - 10020x^3 + 10040x^2 - 10080x"
+    },
+    {
+      "index": 4,
+      "output": "The degree of a polynomial is the highest power of the variable in the polynomial.\n\nThe degree of the polynomial is 1.\n\nThe answer is:\n\\[ \\boxed{1} \\]"
+    },
+    {
+      "index": 5,
+      "output": "The possible outcomes when two fair, 6-sided dice are thrown are:\n\\[ (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) \\]\n\nThere are 15 possible outcomes.\n\nThe product of the two numbers is a multiple of 5 if and only if at least one of the numbers is a multiple of 5.\n\nThe multiples of 5 in the list are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150,"
+    },
+    {
+      "index": 6,
+      "output": "We know that $2^5 = 32$ and $2^6 = 64$. Therefore, $\\log_264 = \\log_2(64) = 6$.\n\nFinal answer:\n#### 6"
+    },
+    {
+      "index": 7,
+      "output": "To find the value of $a$ for which the graph of $y=f(n)$ is continuous, we need to ensure that the two pieces of the function agree at $n=a$.\n\nFor the first piece, $f(n) = 4n + 3$, we need to find the value of $n$ such that $4n + 3 = a$.\n\nFor the second piece, $f(n) = 7n - 12$, we need to find the value of $n$ such that $7n - 12 = a$.\n\nWe can solve the first equation for $n$:\n\\[ 4n + 3 = a \\Rightarrow 4n = a - 3 \\Rightarrow n = \\frac{a - 3}{4} \\]\n\nSubstitute this value of $n$ into the second equation:\n\\[ 7\\left(\\frac{a - 3}{4}\\right) - 12 = a \\Rightarrow \\frac{7a - 21}{4} - 12 = a \\Rightarrow \\frac{7a - 21}{4} = a + 12 \\]\n\nMultiply both sides by 4 to eliminate the fraction:\n\\[ 7a - 21 = 4a + 48 \\Rightarrow 3a = 69 \\Rightarrow a = \\boxed{\\frac{69}{3}} \\]"
+    },
+    {
+      "index": 8,
+      "output": "The two-digit integer $\"AB\"$ is $10A + B$.\n\nWhen cubed, the value is $1000A^3 + 100(10A^2) + 10(10A) + B^3 = 912,\\!673$.\n\nThis simplifies to $100(10A^2 + B^2) + 10(10A) + B^3 = 912,\\!673$.\n\nThis further simplifies to $10(10A^2 + B^2) + 10(10A) + B^3 = 912,\\!673$.\n\nThis further simplifies to $10(10A^2 + B^2) + 10(10A) + B^3 = 912,\\!673$.\n\nThis further simplifies to $10(10A^2 + B^2) + 10(10A) + B^3 = 912,\\!673$.\n\nThis further simplifies to $10(10A^2 + B^2) + 10(10A) + B^3 = 912,\\!673$.\n\nThis further simplifies to $10(10A^2 + B^2) + 10(10A) + B^3 = 912,\\!673$.\n\nThis further simplifies to $10(10A^2 + B^2) + 10(10A) + B^3 = 912,\\!673$.\n\nThis further simplifies"
+    },
+    {
+      "index": 9,
+      "output": "To find the number of digits in the base 7 representation of $629_{10}$, we can divide 629 by 7 and take the floor of the result.\n\n\\[ 629 \\div 7 = 87 \\quad \\text{with a remainder of } 2 \\]\n\nThe quotient is 87, and the remainder is 2.\n\nThe base 7 representation of $629_{10}$ has 87 digits.\n\nThe answer is:\n\\[ \\boxed{87} \\]"
+    },
+    {
+      "index": 10,
+      "output": "The total number of possible values of $x$ is the number of possible values of $y$, which is $54 - 2 - 5 = 47 - 5 = 42$.\n\nThe answer is:\n\\[ \\boxed{42} \\]"
+    },
+    {
+      "index": 11,
+      "output": "First, we can find the prime factorization of each number:\n\n$3339 = 3 \\times 1111$\n\n$2961 = 3 \\times 987$\n\n$1491 = 3 \\times 497$\n\nThe greatest common divisor (GCD) is the product of the common prime factors raised to the smallest power they appear in either factorization.\n\nThe common prime factors are $3$ and $1111$.\n\nThe smallest power of $3$ is $1$.\n\nThe smallest power of $1111$ is $1$.\n\nTherefore, the GCD is $3 \\times 1111 = \\boxed{3333}$."
+    },
+    {
+      "index": 12,
+      "output": "Let $QM = x$. By the Apollonius' Theorem, we have\n\\[ PQ^2 + PR^2 = 2(PQ^2 + PR^2) \\Rightarrow 34^2 + 32^2 = 2(34^2 + 32^2) \\Rightarrow 1156 + 1024 = 2(1156 + 1024) \\Rightarrow 2180 = 2(2270) \\Rightarrow 2180 = 4540 \\]\n\nThis is a contradiction, so there is no solution.\n\nThe final answer is:\n\\[ \\boxed{0} \\]"
+    },
+    {
+      "index": 13,
+      "output": "The greatest common factor (GCF) of 6432 and 132 is 132.\n\nWhen 132 is increased by 11, the result is 132 + 11 = 143.\n\nFinal answer:\n#### 143"
+    },
+    {
+      "index": 14,
+      "output": "The rotation of $z$ around $c$ by $\\frac{\\pi}{4}$ counter-clockwise is equivalent to multiplying $z$ by $i$.\n\n\\[ w = z \\cdot i = (2 + \\sqrt{2})i - (3 + 3\\sqrt{2})i = -\\sqrt{2}i + 3\\sqrt{2}i = 3\\sqrt{2}i - \\sqrt{2}i = 2\\sqrt{2}i \\]\n\nTherefore, $w = \\boxed{2\\sqrt{2}i}$."
+    }
+  ],
+  "arc_challenge": [
+    {
+      "index": 0,
+      "output": "The best answer is paper comes from a renewable resource..\n\nFinal answer: A"
+    },
+    {
+      "index": 1,
+      "output": "The best answer is The wave speed varies with their frequency..\n\nFinal answer: D"
+    },
+    {
+      "index": 2,
+      "output": "The best answer is\"Final answer\"\n\nFinal answer: A"
+    },
+    {
+      "index": 3,
+      "output": "The best answer is increasing the food available for consumers..\n\nFinal answer: B"
+    },
+    {
+      "index": 4,
+      "output": "The best answer is 28.\n\nFinal answer: B"
+    },
+    {
+      "index": 5,
+      "output": "The best answer is Pour a bucket a sand through the air..\n\nFinal answer: A"
+    },
+    {
+      "index": 6,
+      "output": "The best answer is The man moves farther and faster than the boy..\n\nFinal answer: A"
+    },
+    {
+      "index": 7,
+      "output": "The best answer is They divide vertically..\n\nFinal answer: A"
+    },
+    {
+      "index": 8,
+      "output": "The best answer is the body system.\n\nFinal answer: A"
+    },
+    {
+      "index": 9,
+      "output": "The best answer is excretory.\n\nFinal answer: C"
+    },
+    {
+      "index": 10,
+      "output": "The best answer is 8.\n\nFinal answer: C"
+    },
+    {
+      "index": 11,
+      "output": "The best answer is Al_{2}O_{3}.\n\nFinal answer: B"
+    },
+    {
+      "index": 12,
+      "output": "The best answer is switch.\n\nFinal answer: C"
+    },
+    {
+      "index": 13,
+      "output": "The best answer is a sheet of paper.\n\nFinal answer: B"
+    },
+    {
+      "index": 14,
+      "output": "The best answer is had unlimited financial resources..\n\nFinal answer: C"
+    },
+    {
+      "index": 15,
+      "output": "The best answer is producing light.\n\nFinal answer: A"
+    },
+    {
+      "index": 16,
+      "output": "The best answer is The Moon rises..\n\nFinal answer: A"
+    },
+    {
+      "index": 17,
+      "output": "The best answer is using natural gas to heat homes.\n\nFinal answer: A"
+    },
+    {
+      "index": 18,
+      "output": "The best answer is island.\n\nFinal answer: A"
+    },
+    {
+      "index": 19,
+      "output": "The best answer is CO_{2}.\n\nFinal answer: A"
+    },
+    {
+      "index": 20,
+      "output": "The best answer is salt water.\n\nFinal answer: C"
+    },
+    {
+      "index": 21,
+      "output": "The best answer is high temperatures.\n\nFinal answer: A"
+    },
+    {
+      "index": 22,
+      "output": "The best answer is human as anthlegs.\n\nFinal answer: D"
+    },
+    {
+      "index": 23,
+      "output": "The best answer is more plants used for medicines can be discovered..\n\nFinal answer: C"
+    },
+    {
+      "index": 24,
+      "output": "The best answer is a bacteria disease.\n\nFinal answer: C"
+    },
+    {
+      "index": 25,
+      "output": "The best answer is 78° F.\n\nFinal answer: C"
+    },
+    {
+      "index": 26,
+      "output": "The best answer is potential energy.\n\nFinal answer: C"
+    },
+    {
+      "index": 27,
+      "output": "The best answer is mass.\n\nFinal answer: B"
+    },
+    {
+      "index": 28,
+      "output": "The best answer is ordered by how long the snow remained on the ground.\n\nFinal answer: C"
+    },
+    {
+      "index": 29,
+      "output": "The best answer is 8 times.\n\nFinal answer: C"
+    }
+  ],
+  "folio": [
+    {
+      "index": 0,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 1,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 2,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 3,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 4,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 5,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 6,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 7,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 8,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 9,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 10,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 11,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 12,
+      "output": "The conclusion is not valid.\n\nFinal answer: Uncertain"
+    },
+    {
+      "index": 13,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 14,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 15,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 16,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 17,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 18,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 19,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 20,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 21,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 22,
+      "output": "The conclusion is not valid.\n\nFinal answer: Uncertain"
+    },
+    {
+      "index": 23,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 24,
+      "output": "The conclusion follows from the premises.\n\nFinal answer:\n#### 1"
+    },
+    {
+      "index": 25,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 26,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 27,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 28,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 29,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: False"
+    }
+  ]
+}

candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/v2_fixed005_scores.json

@@ -0,0 +1,131 @@
+{
+  "gsm8k": {
+    "correct": 5,
+    "total": 30,
+    "items": [
+      true,
+      false,
+      false,
+      false,
+      false,
+      false,
+      true,
+      false,
+      false,
+      true,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      true,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      true
+    ]
+  },
+  "math500": {
+    "correct": 1,
+    "total": 15,
+    "items": [
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      true,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false
+    ]
+  },
+  "arc_challenge": {
+    "correct": 11,
+    "total": 30,
+    "items": [
+      true,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      true,
+      false,
+      false,
+      true,
+      true,
+      true,
+      false,
+      false,
+      false,
+      true,
+      false,
+      false,
+      true,
+      false,
+      true,
+      true,
+      false,
+      false,
+      true,
+      false,
+      true,
+      false,
+      false
+    ]
+  },
+  "folio": {
+    "correct": 12,
+    "total": 30,
+    "items": [
+      true,
+      true,
+      true,
+      true,
+      true,
+      false,
+      false,
+      true,
+      false,
+      false,
+      false,
+      true,
+      true,
+      false,
+      false,
+      false,
+      false,
+      true,
+      false,
+      false,
+      true,
+      false,
+      false,
+      true,
+      false,
+      true,
+      false,
+      false,
+      false,
+      false
+    ]
+  }
+}

candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/v2_fixed_outputs.json

@@ -0,0 +1,430 @@
+{
+  "gsm8k": [
+    {
+      "index": 0,
+      "output": "Let x be the number of chocolate bars in a box.\nLisa sold 3.5x boxes of chocolate bars, and Peter sold 4.5x boxes of chocolate bars.\nTogether, they sold 3.5x + 4.5x = 64\nCombining like terms, we get 8.5x = 64\nDividing both sides by 8.5, we get x = 8\n\n#### 8"
+    },
+    {
+      "index": 1,
+      "output": "Mark makes $10/hour for 8 hours/day * 5 days/week = $<<10*5=50>>50/week\nHe makes $2/hour for 8 hours/day * 5 days/week = $<<2*5=10>>10/week\nSo he makes $10/hour + $10/week = $<<10+10=20>>20/week\n#### 20"
+    },
+    {
+      "index": 2,
+      "output": "There were 20 private cars * 35 people/bus = <<20*35=700>>700 people.\nThere were 12 buses * 3 people/car = <<12*3=36>>36 people.\nThere were 700 + 36 = <<700+36=736>>736 people inside the church.\n#### 736"
+    },
+    {
+      "index": 3,
+      "output": "Let's denote the number of yellow blocks as Y. According to the problem, twice the number of yellow blocks is 4, so Y = 2*Y.\n\nThe total number of blocks in the tower is the sum of the blue, yellow, and red blocks, which is 32. So we can write the equation:\n\nB + Y + R = 32\n\nSubstituting Y with 2*Y, we get:\n\nB + 2*Y + R = 32\n\nNow, we need to find the value of Y. We can do this by substituting Y with 2*Y in the equation:\n\nB + 2*(2*Y) + R = 32\n\nSimplifying the equation, we get:\n\nB + 4*Y + R = 32\n\nNow, we need to find the value of R. We can do this by substituting Y with 2*Y in the equation:\n\nB + 4*(2*Y) + R = 32\n\nSimplifying the equation, we get:\n\nB + 8*Y + R = 32\n\nNow, we"
+    },
+    {
+      "index": 4,
+      "output": "In the first year, Elise sold 250 x 2 = <<250*2=500>>500 books.\nShe sold twice as many books in the first year, so she sold 500 x 2 = <<500*2=1000>>1000 books in the first year.\nShe sold 500 + 1000 = <<500+1000=1500>>1500 books in the first year.\nShe sold 250 x 2 = <<250*2=500>>500 books in the first year.\nShe sold 500 x 2 = <<500*2=1000>>1000 books in the first year.\nShe sold 1500 + 1000 = <<1500+1000=2500>>2500 books in the first year.\nShe sold 500 x 2 = <<500*2=1000>>1000 books in the first year.\nShe"
+    },
+    {
+      "index": 5,
+      "output": "First find how long it takes Bill to dig through soil: 4 feet/hour * 24 hours = <<4*24=96>>96 feet\nThen find how long it takes him to dig through clay: 24 feet / 2 = <<24/2=12>>12 hours\nThen add the time it takes him to dig through soil and clay to find the total time: 96 feet + 12 hours = <<96+12=108>>108 hours\n#### 108"
+    },
+    {
+      "index": 6,
+      "output": "Adam slices 3 large apples, so he gets 3*5 = <<3*5=15>>15 slices from the large apples.\nHe also slices 5 small apples, so he gets 5*3 = <<5*3=15>>15 slices from the small apples.\nIn total, Adam gets 15+15 = <<15+15=30>>30 slices.\nAfter eating 15 slices, Adam is left with 30-15 = <<30-15=15>>15 slices.\n#### 15"
+    },
+    {
+      "index": 7,
+      "output": "Morgan's dad has $90 for her birthday party.\nShe wants to make sure she and her friends get to play one round of mini-golf, have $5 in arcade tokens, and get to ride the go-karts twice.\nSo she needs to make sure she and her friends get to play one round of mini-golf, have $5 in arcade tokens, and ride the go-karts twice.\nThis means that she needs to make sure she and her friends get to play one round of mini-golf, have $5 in arcade tokens, and ride the go-karts twice.\nThis means that she needs to make sure she and her friends get to play one round of mini-golf, have $5 in arcade tokens, and ride the go-karts twice.\nThis means that she needs to make sure she and her friends get to play one round of mini-golf, have $5 in arcade tokens, and ride the go-karts twice.\nThis means that she needs to make sure she and her friends get to play one round of mini-golf, have $5 in arcade tokens, and ride the go-karts twice."
+    },
+    {
+      "index": 8,
+      "output": "In one hour, Ahmed read 300 books.\nIn one hour, Napoleon read 300/2 = <<300/2=150>>150 books.\nIn one hour, the total number of books read by the two is 300+150 = <<300+150=450>>450 books.\nIn one hour, the total number of books read by the two is 450+150 = <<450+150=600>>600 books.\nIn one hour, the total number of books read by the two is 600+450 = <<600+450=1050>>1050 books.\n#### 1050"
+    },
+    {
+      "index": 9,
+      "output": "The pens cost 4 * $1.5 = $6.\nThe notebooks cost 2 * $4 = $8.\nThe rim of bond paper cost $20.\nSo, the total cost of everything is $6 + $8 + $20 = $34.\n#### 34"
+    },
+    {
+      "index": 10,
+      "output": "Marcy's annual salary is $50,000/year.\nShe works for 40 years, so she gets $50,000/year * 40 years = $200,000 per year.\nAfter 20 years, she will be eligible for a 5% payout.\nSo after 20 years, she will be eligible for a 5% payout of $200,000 * 5% = $10,000 per year.\nAfter 30 years, she will be eligible for a 5% payout of $200,000 * 5% = $10,000 per year.\nSo after 30 years, she will be eligible for a total of $10,000 + $10,000 = $20,000 per year.\n#### 20000"
+    },
+    {
+      "index": 11,
+      "output": "Steve eats 6 * 2 = <<6*2=12>>12 tomatoes per week.\nSo he needs 12 / 3 = <<12/3=4>>4 vines.\n#### 4"
+    },
+    {
+      "index": 12,
+      "output": "The cost difference between the two offers is $1,350 - $1,100 = $250.\n\nFinal answer:\n#### 250"
+    },
+    {
+      "index": 13,
+      "output": "Let's denote the initial number of stickers that Seth had as S.\n\nSeth gave half of his stickers to Luis, so he had S/2 left.\n\nLuis used half of the stickers, so he had S/2 * 1/2 = S/4 left.\n\nKris kept 9 stickers, so she had S/4 + 9 stickers.\n\nRob gave 7 stickers to Rob, so he had S/4 + 9 - 7 stickers.\n\nThe problem states that Kris kept 9 stickers, so S/4 + 9 - 7 = 9.\n\nSimplifying the equation, we get S/4 = 9 - 9, which implies S/4 = 0.\n\nThis means that S = 0.\n\nTherefore, Seth had 0 stickers in the beginning.\n\n#### 0"
+    },
+    {
+      "index": 14,
+      "output": "Gary does laundry twice a week, so he does 2 * 20 = <<2*20=40>>40 loads of laundry per week.\nEach load uses 20 gallons of water, so he uses 40 * 20 = <<40*20=800>>800 gallons of water per week.\nA gallon of water costs $0.15, so Gary spends 800 * $0.15 = $<<800*0.15=120>>120 per week on water.\nIn a year, Gary spends 52 * $120 = $<<52*120=6240>>6240 on water.\n#### 6240"
+    },
+    {
+      "index": 15,
+      "output": "The first son is 12 years old and the second son is 4 years younger, so he is 12-4= <<12-4=8>>8 years old.\nThe first son is 12 years old and the second son is 8 years old, so together they are 12+8= <<12+8=20>>20 years old.\nThe pack of 5 candles costs $3, so James spends 20*3=$<<20*3=60>>60 on candles.\n#### 60"
+    },
+    {
+      "index": 16,
+      "output": "Charlie had 10 stickers.\nHe bought 21 stickers from a store.\nHe got 23 stickers for his birthday.\nHe gave 9 stickers to his sister.\nHe used 28 stickers to decorate a greeting card.\nSo Charlie has 10 stickers + 21 stickers - 9 stickers - 28 stickers = 12 stickers left.\n#### 12"
+    },
+    {
+      "index": 17,
+      "output": "Polly could throw the gold javelin 400 feet * 3 = <<400*3=1200>>1200 feet.\nPolly could throw the javelin 1000 feet * 400 feet = <<1000*400=40000>>40000 feet.\nPolly could throw the javelin 1000 feet * 1200 feet = <<1000*1200=120000>>120000 feet.\nPolly could throw the gold javelin 400 feet * 3 = <<400*3=1200>>1200 feet.\nPolly could throw the gold javelin 120000 feet - 120000 feet = <<120000-120000=0>>0 feet.\n#### 0"
+    },
+    {
+      "index": 18,
+      "output": "The low setting removes 1 liter of water per day, so it removes 1*3= <<1*3=3>>3 liters of water per day.\nThe medium setting removes twice as much water per day as the low setting, so it removes 2*1= <<2*1=2>>2 liters of water per day.\nThe high setting removes twice as much water per day as the medium setting, so it removes 2*2= <<2*2=4>>4 liters of water per day.\nIn total, the low setting removes 3+2= <<3+2=5>>5 liters of water per day.\nIn total, the medium setting removes 2+4= <<2+4=6>>6 liters of water per day.\nIn total, the high setting removes 3+5= <<3+5=8>>8 liters of water per day.\nIn total, the dehumidifier removes 3+6+8= <<3+6+8=17>>17 liters of water per day.\nIf Brian ran the dehumidifier for 3 days on the low setting, then an additional 3 days on the medium setting, and"
+    },
+    {
+      "index": 19,
+      "output": "First find the total energy used per hour by the air conditioner: 900 watts/hour * 8 hours = 7200 watts/hour\nThen find the total energy used per hour by the air conditioner after reducing the time: 7200 watts/hour - 5 hours = 7150 watts/hour\nThen find the total energy used per hour by the air conditioner after reducing the time: 7150 watts/hour * 30 days = 214500 watts\nThen find the total energy used per hour by the air conditioner after reducing the time: 21450 watts/hour * 30 days = 643750 watts\nThen find the total energy used per hour by the air conditioner after reducing the time: 643750 watts * 30 days = 19342500 watts\nThen find the total energy used per hour by the air conditioner after reducing the time: 19342500 watts * 30 days = 57037500 watts\nThen find the total energy used per hour by the air conditioner after reducing the time"
+    },
+    {
+      "index": 20,
+      "output": "Let's denote the amount of red paint used as R, white paint as W, and purple paint as P.\n\nWe know that the total amount of paint used is 12 pints, so we can write the equation:\n\nR + W + P = 12\n\nWe also know that half the mural is yellow, so the number of yellow paint is 1/2 * 4 = 2 pints.\n\nSince the total amount of paint used is 12 pints, we can write the equation:\n\nR + W + P = 12\n\nNow we have a system of two equations:\n\n1) R + W + P = 12\n2) R + W + P = 12\n\nSubtracting the first equation from the second equation, we get:\n\n(R + W + P) - (R + W + P) = 12 - 12\n\nThis simplifies to:\n\nW = 0\n\nSo, the number of pints of purple paint used is 0.\n\nThe answer is \\boxed{0}."
+    },
+    {
+      "index": 21,
+      "output": "The cost of the 3 vaccine is $20 x 3 = $60.\nThe cost of the 4 vases is $12 x 4 = $48.\nThe total cost of the 4 vases is $48 + $60 = $108.\nThe total cost of the kitten is $200 + $108 = $308.\n#### 308"
+    },
+    {
+      "index": 22,
+      "output": "The lego box had 500 pieces.\nThe lego box had 3 times more pieces than the 500 piece one, so it had 3*500 = <<3*500=1500>>1500 pieces.\nThe lego box had 1/4 the number of pieces of the other one, so it had 1/4*1500 = <<1/4*1500=375>>375 pieces.\nSo, Johnny picked up 500+1500+375 = <<500+1500+375=2575>>2575 pieces.\n#### 2575"
+    },
+    {
+      "index": 23,
+      "output": "The number of people infected is 10 * 6 = <<10*6=60>>60.\nAfter three days, the number of people infected is 60 * 3 = <<60*3=180>>180.\n#### 180"
+    },
+    {
+      "index": 24,
+      "output": "Mr. Hezekiah had 20 trucks carrying 20 tons each, so he had 20*20 = <<20*20=400>>400 tons ofSTEP.\nAfter 2 hours, 400 tons ofSTEP were delivered to the farmers.\nAfter 2 hours, 1/4 of the lorries were not working, so 1/4*20 = <<1/4*20=5>>5 lorries were not working.\nSo, 20-5 = <<20-5=15>>15 lorries were working.\nThus, 20-15 = <<20-15=5>>5 lorries were working.\nTherefore, 400+5 = <<400+5=405>>405 tons ofSTEP reached the farmers that day.\n#### 405"
+    },
+    {
+      "index": 25,
+      "output": "The number of days in a week is 4*6=<<4*6=24>>24\nThe number of weeks in a year is 24*52=<<24*52=128>>128\nThe number of days in a year is 128*2=<<128*2=256>>256\nThe number of weeks in a year is 256*4=<<256*4=1024>>1024\nThe number of days in a year is 1024*6=<<1024*6=6144>>6144\nThe total amount he earns in a year is 6144*50=<<6144*50=307200>>307200\n#### 307200"
+    },
+    {
+      "index": 26,
+      "output": "Max has 12 + 68 = <<12+68=80>>80 cookies.\nMax has 80 - 5 = <<80-5=75>>75 cookies.\nMax has 75 + 23 = <<75+23=98>>98 cookies.\nKatarina has 98 - 68 = <<98-68=30>>30 cookies.\n#### 30"
+    },
+    {
+      "index": 27,
+      "output": "The tree will grow 7*7=<<7*7=49>>49 lemons in 49 years.\nIt will cost 49*$3=$<<49*3=147>>147 to water and feed the tree.\nIt will take 49-49/7=<<49-49/7=32>>32 years to grow the tree.\nIt will take 49-49/49=3 years to grow the tree.\nIt will take 49-49/49=3 years to grow the tree.\nIt will take 49-49/49=3 years to grow the tree.\nIt will take 49-49/49=3 years to grow the tree.\nIt will take 49-49/49=3 years to grow the tree.\nIt will take 49-49/49=3 years to grow the tree.\nIt will take 49-49/49=3 years to grow the tree.\nIt will take 49-49/49=3 years to grow"
+    },
+    {
+      "index": 28,
+      "output": "On Tuesday, Sue ate 2 * 5 = <<2*5=10>>10 cookies.\nOn Monday, Sue ate 4 * 10 = <<4*10=40>>40 cookies.\nOn Tuesday, Sue ate 10 * 200 = <<10*200=2000>>2000 cookies.\nSue consumed 2000 - 40 = <<2000-40=1960>>1960 more cookies than her sister.\n#### 1960"
+    },
+    {
+      "index": 29,
+      "output": "There are 10 rows * 20 flowers/row = <<10*20=200>>200 flowers in the garden.\n4/5 of the flowers have bloomed, so there are 4/5 * 200 = <<4/5*200=160>>160 flowers that have bloomed.\n#### 160"
+    }
+  ],
+  "math500": [
+    {
+      "index": 0,
+      "output": "The $x$-intercept is the point where the line crosses the $x$-axis, so $a = 0$.\n\nThe $y$-intercept is the point where the line crosses the $y$-axis, so $b = 0$.\n\nThe slope of the line is given by the coefficient of $x$ in the equation $\\frac{x}4 + \\frac{y}{12} = 1$, which is $\\frac{1}{4}$.\n\nTherefore, $m = \\frac{1}{4}$.\n\nThe sum of $a$, $b$, and $m$ is $0 + 0 + \\frac{1}{4} = \\boxed{\\frac{1}{4}}$."
+    },
+    {
+      "index": 1,
+      "output": "To find the sum of the digits in the fraction $\\frac{4321}{5^7\\cdot2^8}$, we need to convert the fraction into a decimal representation.\n\nFirst, let's simplify the fraction by dividing the numerator and denominator by their greatest common divisor (GCD).\n\nThe GCD of 4321 and 5^7 is 5.\n\nDivide the numerator and denominator by 5:\n\\[ \\frac{4321}{5^7\\cdot2^8} = \\frac{4321}{5^7\\cdot2^8} \\div \\frac{5}{5} = \\frac{4321}{5^7\\cdot2^8} \\times \\frac{1}{5} = \\frac{4321}{5^7\\cdot2^8} \\times \\frac{1}{5} = \\frac{4321}{5^7\\cdot2^8} \\times \\frac{1}{5} = \\frac{4321}{5^7\\cdot2^8} \\times \\frac{1}{5} = \\frac{4321}{5^7\\cdot2^8} \\times \\frac{1}{5} = \\frac{4321}{5^7\\cdot2^8} \\times \\frac{1}{5} = \\frac{4321}{5^7\\cdot2^8} \\times \\frac{1}{5} = \\frac{4321}{5^7\\cdot2^8} \\times \\frac{1}{5} = \\frac{4321}{5^7\\cdot2^8} = \\frac{432"
+    },
+    {
+      "index": 2,
+      "output": "The nonagon has 900 sides, and we want to find the perimeter of the nonagon.\n\nThe nonagon has 900 sides, and we want to find the perimeter of the nonagon.\n\nThe perimeter of the nonagon is equal to the sum of the lengths of its sides.\n\nThe length of each side of the nonagon is 2 units.\n\nThe perimeter of the nonagon is 2 + 2 + 2 + 2 + 2 + 2 = 20 units.\n\nThe answer is:\n\\boxed{20}"
+    },
+    {
+      "index": 3,
+      "output": "First, we can rewrite the equation as\n\\[ \\frac{1}{x^2 - 1} + \\frac{2}{x^2 - 2} + \\frac{3}{x^2 - 3} + \\frac{4}{x^2 - 4} = 2010x - 4 \\]\n\nTo simplify the equation, we can multiply both sides by $(x^2 - 1)(x^2 - 2)(x^2 - 3)(x^2 - 4)$ to eliminate the fractions.\n\nAfter multiplying, we get\n\\[ (x^2 - 1)(x^2 - 2)(x^2 - 3)(x^2 - 4) + (2x^2 - 3)(x^2 - 2)(x^2 - 3)(x^2 - 4) + (3x^2 - 4)(x^2 - 2)(x^2 - 3)(x^2 - 4) + (4x^2 - 1)(x^2 - 2)(x^2 - 3)(x^2 - 4) = (2010x - 4)(x^2 - 1)(x^2 - 2)(x^2 - 3)(x^2 - 4) \\]\n\nExpanding the right-hand side, we get\n\\[ (2010x - 4)(x^2 - 1)(x^2 - 2)(x^2 - 3)(x^2 - 4) = 2010x^4 - 10020x^3 + 10040x^2 - 10040x"
+    },
+    {
+      "index": 4,
+      "output": "The degree of a polynomial is the highest power of the variable in the polynomial.\n\nThe degree of the polynomial is 1.\n\nThe answer is:\n\\[ \\boxed{1} \\]"
+    },
+    {
+      "index": 5,
+      "output": "The possible outcomes when two fair, 6-sided dice are thrown are:\n\\[ (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) \\]\n\nThere are 15 possible outcomes.\n\nThe product of the two numbers is a multiple of 5 if and only if at least one of the numbers is a multiple of 5.\n\nThe multiples of 5 in the list are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150,"
+    },
+    {
+      "index": 6,
+      "output": "We know that $2^5 = 32$ and $2^6 = 64$. Therefore, $\\log_264 = \\log_2(64) = 6$.\n\nFinal answer:\n#### 6"
+    },
+    {
+      "index": 7,
+      "output": "To find the value of $a$ for which the graph of $y=f(n)$ is continuous, we need to ensure that the two pieces of the function agree at $n=a$.\n\nFor the first piece, $f(n) = 4n + 3$, we need to find $f(a)$:\n\\[ f(a) = 4a + 3 \\]\n\nFor the second piece, $f(n) = 7n - 12$, we need to find $f(a)$:\n\\[ f(a) = 7a - 12 \\]\n\nFor the two pieces to be equal, we must have:\n\\[ 4a + 3 = 7a - 12 \\]\n\nSolving for $a$:\n\\[ 4a + 3 = 7a - 12 \\Rightarrow 4a - 7a = -12 - 3 \\Rightarrow -3a = -15 \\Rightarrow a = \\boxed{5} \\]\n\nFinal answer:\n#### 5"
+    },
+    {
+      "index": 8,
+      "output": "The two-digit integer $\"AB\"$ is $10A + B$.\n\nWhen cubed, the value is $912,\\!673$.\n\nTherefore, $10A + B = 912,\\!673$.\n\nThe answer is:\n\\[ \\boxed{912673} \\]"
+    },
+    {
+      "index": 9,
+      "output": "To find the number of digits in the base 7 representation of $629_{10}$, we can divide 629 by 7 and take the floor of the result.\n\n\\[ 629 \\div 7 = 87 \\quad \\text{with a remainder of } 2 \\]\n\nThe quotient is 87, and the remainder is 2.\n\nThe base 7 representation of $629_{10}$ has 87 digits.\n\nThe answer is:\n\\[ \\boxed{87} \\]"
+    },
+    {
+      "index": 10,
+      "output": "The total number of possible values of $x$ is the number of possible values of $y$, which is $54 - 2 - 5 = 47 - 5 = 42$.\n\nThe answer is:\n\\[ \\boxed{42} \\]"
+    },
+    {
+      "index": 11,
+      "output": "First, we can find the prime factorization of each number:\n\n$3339 = 3 \\times 1111$\n\n$2961 = 3 \\times 987$\n\n$1491 = 3 \\times 497$\n\nThe greatest common divisor (GCD) is the product of the common prime factors raised to the smallest power they appear in either factorization.\n\nThe common prime factors are $3$ and $1111$.\n\nThe smallest power of $3$ is $1$.\n\nThe smallest power of $1111$ is $1$.\n\nTherefore, the GCD is $3 \\times 1111 = \\boxed{3333}$."
+    },
+    {
+      "index": 12,
+      "output": "Let $QM = x$. By the Apollonius' Theorem, we have\n\\[ PQ^2 + PR^2 = 2(PQ^2 + PR^2) \\Rightarrow 34^2 + 32^2 = 2(34^2 + 32^2) \\Rightarrow 1156 + 1024 = 2(1156 + 1024) \\Rightarrow 2180 = 2(2270) \\Rightarrow 2180 = 4540 \\]\n\nThis is a contradiction, so there is no solution.\n\nThe final answer is:\n\\[ \\boxed{0} \\]"
+    },
+    {
+      "index": 13,
+      "output": "The greatest common factor (GCF) of 6432 and 132 is 132.\n\nWhen 132 is increased by 11, the result is 132 + 11 = 143.\n\nFinal answer:\n#### 143"
+    },
+    {
+      "index": 14,
+      "output": "The rotation of $z$ around $c$ by $\\frac{\\pi}{4}$ counter-clockwise is equivalent to multiplying $z$ by $-i$.\n\n\\[ w = -i(2 + \\sqrt{2} - (3 + 3 \\sqrt{2})i) = -2i - 3i - 3\\sqrt{2}i - 3\\sqrt{2}i^2 = -2i - 3\\sqrt{2} - 3\\sqrt{2} = \\boxed{-2 - 6\\sqrt{2}} \\]"
+    }
+  ],
+  "arc_challenge": [
+    {
+      "index": 0,
+      "output": "The best answer is paper comes from a renewable resource..\n\nFinal answer: A"
+    },
+    {
+      "index": 1,
+      "output": "The best answer is The wave speed varies with their frequency..\n\nFinal answer: D"
+    },
+    {
+      "index": 2,
+      "output": "The best answer is\"Final answer\"\n\nFinal answer: A"
+    },
+    {
+      "index": 3,
+      "output": "The best answer is increasing the food available for consumers..\n\nFinal answer: B"
+    },
+    {
+      "index": 4,
+      "output": "The best answer is 28.\n\nFinal answer: B"
+    },
+    {
+      "index": 5,
+      "output": "The best answer is Pour a bucket a sand through the air..\n\nFinal answer: A"
+    },
+    {
+      "index": 6,
+      "output": "The best answer is The man moves farther and faster than the boy..\n\nFinal answer: A"
+    },
+    {
+      "index": 7,
+      "output": "The best answer is They divide vertically..\n\nFinal answer: A"
+    },
+    {
+      "index": 8,
+      "output": "The best answer is the body system.\n\nFinal answer: A"
+    },
+    {
+      "index": 9,
+      "output": "The best answer is excretory.\n\nFinal answer: C"
+    },
+    {
+      "index": 10,
+      "output": "The best answer is 8.\n\nFinal answer: C"
+    },
+    {
+      "index": 11,
+      "output": "The best answer is Al_{2}O_{3}.\n\nFinal answer: B"
+    },
+    {
+      "index": 12,
+      "output": "The best answer is switch.\n\nFinal answer: C"
+    },
+    {
+      "index": 13,
+      "output": "The best answer is a sheet of paper.\n\nFinal answer: B"
+    },
+    {
+      "index": 14,
+      "output": "The best answer is had unlimited financial resources..\n\nFinal answer: C"
+    },
+    {
+      "index": 15,
+      "output": "The best answer is producing light.\n\nFinal answer: A"
+    },
+    {
+      "index": 16,
+      "output": "The best answer is The Moon rises..\n\nFinal answer: A"
+    },
+    {
+      "index": 17,
+      "output": "The best answer is using natural gas to heat homes.\n\nFinal answer: A"
+    },
+    {
+      "index": 18,
+      "output": "The best answer is the island.\n\nFinal answer: A"
+    },
+    {
+      "index": 19,
+      "output": "The best answer is CO_{2}.\n\nFinal answer: A"
+    },
+    {
+      "index": 20,
+      "output": "The best answer is salt water.\n\nFinal answer: C"
+    },
+    {
+      "index": 21,
+      "output": "The best answer is high temperatures.\n\nFinal answer: A"
+    },
+    {
+      "index": 22,
+      "output": "The best answer is human as anthlegs.\n\nFinal answer: D"
+    },
+    {
+      "index": 23,
+      "output": "The best answer is more plants used for medicines can be discovered..\n\nFinal answer: C"
+    },
+    {
+      "index": 24,
+      "output": "The best answer is a bacteria disease.\n\nFinal answer: C"
+    },
+    {
+      "index": 25,
+      "output": "The best answer is 78° F.\n\nFinal answer: C"
+    },
+    {
+      "index": 26,
+      "output": "The best answer is potential energy.\n\nFinal answer: C"
+    },
+    {
+      "index": 27,
+      "output": "The best answer is mass.\n\nFinal answer: B"
+    },
+    {
+      "index": 28,
+      "output": "The best answer is ordered by how long the snow remained on the ground.\n\nFinal answer: C"
+    },
+    {
+      "index": 29,
+      "output": "The best answer is 8 times.\n\nFinal answer: C"
+    }
+  ],
+  "folio": [
+    {
+      "index": 0,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 1,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 2,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 3,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 4,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 5,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 6,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 7,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 8,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 9,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 10,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 11,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 12,
+      "output": "The conclusion is not valid.\n\nFinal answer: Uncertain"
+    },
+    {
+      "index": 13,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 14,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 15,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 16,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 17,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 18,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 19,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 20,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 21,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 22,
+      "output": "The conclusion is not valid.\n\nFinal answer: Uncertain"
+    },
+    {
+      "index": 23,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 24,
+      "output": "The conclusion follows from the premises.\n\nFinal answer:\n#### 1"
+    },
+    {
+      "index": 25,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 26,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 27,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 28,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: True"
+    },
+    {
+      "index": 29,
+      "output": "The conclusion follows from the premises.\n\nFinal answer: False"
+    }
+  ]
+}

candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/v2_fixed_scores.json

@@ -0,0 +1,131 @@
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+    "total": 30,
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+    ]
+  },
+  "math500": {
+    "correct": 2,
+    "total": 15,
+    "items": [
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+      false,
+      false,
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+      false,
+      true,
+      true,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false
+    ]
+  },
+  "arc_challenge": {
+    "correct": 11,
+    "total": 30,
+    "items": [
+      true,
+      false,
+      false,
+      false,
+      false,
+      false,
+      false,
+      true,
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+      false,
+      false,
+      true,
+      false,
+      true,
+      false,
+      false
+    ]
+  },
+  "folio": {
+    "correct": 12,
+    "total": 30,
+    "items": [
+      true,
+      true,
+      true,
+      true,
+      true,
+      false,
+      false,
+      true,
+      false,
+      false,
+      false,
+      true,
+      true,
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+    ]
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candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/v2_info_dev.json

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candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/v2_math_dev_scores.json

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