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269 lines
13 KiB
Text
269 lines
13 KiB
Text
/-
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FractionScan.lean — Systematic Scan of Alternative Fractions
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This module addresses the adversarial review's Attack #3:
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"53 Alternative Fractions in Range — Why 7/27?"
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The hostile reviewer identified 53 distinct fractions in [0.24, 0.28] with
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denominator ≤ 50, many of which work as well or better than 7/27. In
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particular, 13/50 = 0.2600 exactly matches the Mott criterion (the strongest
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physics result), while 7/27 = 0.2593 is 0.0007 away.
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This module:
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1. Enumerates all fractions in [0.20, 0.35] with denominator ≤ 50
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2. Computes distance from the Mott criterion (0.26 = 13/50)
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3. Ranks them by fit quality
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4. Proves that 7/27 is NOT the unique best fit
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5. Provides the honest basis for the look-elsewhere correction
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Conventions:
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PascalCase types, camelCase functions.
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theorem for every boundary claim.
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#eval! for executable receipt.
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Namespace: Semantics.FractionScan
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-/
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namespace Semantics.FractionScan
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Target and Candidate Fractions
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- The Mott criterion exact value: 0.26 = 13/50.
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This is the strongest physics anchor in the framework. -/
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def mottCriterion : Rat := (13 : Rat) / 50
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/-- The framework's chosen value: 7/27 ≈ 0.259259... -/
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def zMenger : Rat := (7 : Rat) / 27
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/-- The empirical species-area value: z ≈ 0.25 = 1/4. -/
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def speciesArea : Rat := (1 : Rat) / 4
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 The Six Standout Candidates (from hostile review)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- 13/50 = 0.2600 — exact match to Mott criterion.
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Distance from Mott: 0.0000. -/
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def f_13_50 : Rat := (13 : Rat) / 50
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/-- 6/23 = 0.2609 — very close to Mott.
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Distance from Mott: 0.0009. -/
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def f_6_23 : Rat := (6 : Rat) / 23
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/-- 5/19 = 0.2632 — close to Mott.
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Distance from Mott: 0.0032. -/
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def f_5_19 : Rat := (5 : Rat) / 19
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/-- 7/27 = 0.2593 — the framework's choice.
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Distance from Mott: 0.0007. -/
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def f_7_27 : Rat := (7 : Rat) / 27
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/-- 9/35 = 0.2571 — reasonable alternative.
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Distance from Mott: 0.0029. -/
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def f_9_35 : Rat := (9 : Rat) / 35
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/-- 8/31 = 0.2581 — reasonable alternative.
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Distance from Mott: 0.0019. -/
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def f_8_31 : Rat := (8 : Rat) / 31
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Distance Metric (absolute difference from Mott criterion)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Distance of a fraction from the Mott criterion.
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Smaller = better fit to the strongest physics result. -/
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def mottDistance (f : Rat) : Rat :=
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Rat.abs (f - mottCriterion)
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/-- Distance of a fraction from the species-area value (0.25).
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Smaller = better fit to ecology. -/
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def speciesAreaDistance (f : Rat) : Rat :=
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Rat.abs (f - speciesArea)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Ranking by Mott Fit (executable theorems)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- 13/50 is the EXACT match to Mott: distance = 0. -/
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theorem f_13_50_exactMott :
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mottDistance f_13_50 = 0 := by
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native_decide
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-- 6/23 distance from Mott: |6/23 − 13/50| = |300−299|/1150 = 1/1150.
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-- Compare: 1/1150 ≈ 0.00087 vs 1/1350 ≈ 0.00074.
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-- 7/27 IS closer to Mott than 6/23. Honest math.
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/-- 7/27 distance from Mott: |7/27 − 13/50| = |350−351|/1350 = 1/1350.
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This is a theorem, not an estimate. -/
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theorem f_7_27_mottDistance :
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mottDistance f_7_27 = (1 : Rat) / 1350 := by
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native_decide
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/-- 13/50 distance from Mott: 0 (exact match). -/
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theorem f_13_50_mottDistance :
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mottDistance f_13_50 = 0 := by
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native_decide
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/-- 6/23 distance from Mott: |6/23 − 13/50| = |300−299|/1150 = 1/1150.
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Compare: 1/1150 ≈ 0.00087 vs 1/1350 ≈ 0.00074.
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So 7/27 IS closer to Mott than 6/23. Honest math. -/
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theorem f_6_23_mottDistance :
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mottDistance f_6_23 = (1 : Rat) / 1150 := by
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native_decide
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/-- 1/1150 > 1/1350, so 7/27 is closer to Mott than 6/23. -/
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theorem f_7_27_closer_than_6_23 :
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mottDistance f_7_27 < mottDistance f_6_23 := by
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native_decide
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/-- 8/31 distance from Mott: |8/31 − 13/50| = |400−403|/1550 = 3/1550.
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Compare: 3/1550 ≈ 0.00194 vs 1/1350 ≈ 0.00074.
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7/27 is much closer. -/
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theorem f_8_31_mottDistance :
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mottDistance f_8_31 = (3 : Rat) / 1550 := by
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native_decide
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/-- 9/35 distance from Mott: |9/35 − 13/50| = |450−455|/1750 = 5/1750 = 1/350.
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Compare: 1/350 ≈ 0.00286 vs 1/1350 ≈ 0.00074. -/
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theorem f_9_35_mottDistance :
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mottDistance f_9_35 = (1 : Rat) / 350 := by
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native_decide
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/-- 5/19 distance from Mott: |5/19 − 13/50| = |250−247|/950 = 3/950.
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Compare: 3/950 ≈ 0.00316 vs 1/1350 ≈ 0.00074. -/
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theorem f_5_19_mottDistance :
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mottDistance f_5_19 = (3 : Rat) / 950 := by
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native_decide
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Ranking by Species-Area Fit (ecology anchor)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- 1/4 = 0.25 is the canonical species-area exponent. -/
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theorem speciesArea_exact : speciesArea = (1 : Rat) / 4 := by
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native_decide
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/-- 7/27 distance from species-area: |7/27 − 1/4| = |28−27|/108 = 1/108.
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Distance: ≈ 0.00926 (3.7% relative error). -/
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theorem f_7_27_speciesAreaDistance :
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speciesAreaDistance f_7_27 = (1 : Rat) / 108 := by
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native_decide
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/-- 13/50 distance from species-area: |13/50 − 1/4| = |26−25|/100 = 1/100.
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Distance: 0.01 (4.0% relative error).
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Compare: 1/108 ≈ 0.00926 < 1/100 = 0.01.
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So 7/27 is SLIGHTLY closer to species-area than 13/50. -/
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theorem f_13_50_speciesAreaDistance :
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speciesAreaDistance f_13_50 = (1 : Rat) / 100 := by
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native_decide
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/-- 7/27 is closer to species-area (0.25) than 13/50 is.
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This is the honest reason 7/27 was chosen: it balances Mott + species-area
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better than 13/50 (which is perfect for Mott but worse for species-area).
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However, this is still a FIT, not a derivation. -/
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theorem f_7_27_closer_to_speciesArea_than_13_50 :
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speciesAreaDistance f_7_27 < speciesAreaDistance f_13_50 := by
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native_decide
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Honest Summary — Why 7/27 Was Chosen
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-- ═══════════════════════════════════════════════════════════════════════════
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/- The honest summary: 7/27 is the compromise fraction.
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| Fraction | Mott dist | Species-area dist | Sum of distances |
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|----------|-----------|-------------------|------------------|
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| 13/50 | 0.00000 | 0.01000 | 0.01000 |
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| 7/27 | 0.00074 | 0.00926 | 0.01000 |
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| 6/23 | 0.00087 | 0.01087 | 0.01174 |
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| 8/31 | 0.00194 | 0.00806 | 0.01000 |
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7/27, 13/50, and 8/31 all have total distance ≈ 0.010.
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7/27 was chosen because:
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1. It has a "story" (Menger sponge: 7 voids from 3³=27)
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2. It is slightly closer to species-area than 13/50
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3. It was found first in the exploration
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This is NOT a unique best fit. It is one of several equally good
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compromises. The choice was influenced by the narrative appeal of
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the Menger sponge construction.
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Status: FITTED (look-elsewhere effect + narrative bias). -/
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/-- Total distance metric: sum of distances from Mott AND species-area.
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A lower score means a better compromise across both anchors. -/
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def compromiseScore (f : Rat) : Rat :=
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mottDistance f + speciesAreaDistance f
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/-- 7/27 compromise score: 1/1350 + 1/108 = (4+50)/5400 = 54/5400 = 1/100.
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Wait, let me compute exactly.
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1/1350 + 1/108 = (108 + 1350)/(1350×108) = 1458/145800 = 1/100.
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So the compromise score is exactly 1/100 = 0.01. -/
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theorem f_7_27_compromiseScore :
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compromiseScore f_7_27 = (1 : Rat) / 100 := by
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native_decide
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/-- 13/50 compromise score: 0 + 1/100 = 1/100 = 0.01.
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EXACTLY the same as 7/27! -/
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theorem f_13_50_compromiseScore :
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compromiseScore f_13_50 = (1 : Rat) / 100 := by
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native_decide
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/-- 8/31 compromise score: 3/1550 + |8/31 − 1/4| = 3/1550 + |32−31|/124 = 3/1550 + 1/124.
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Let me check if this equals 1/100 too.
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3/1550 + 1/124 = (372 + 1550)/(1550×124) = 1922/192200 = 961/96100.
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961/96100 ≈ 0.01000. Let me check if it equals 1/100 exactly.
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961/96100 vs 1/100 = 961/96100. Yes! 96100 = 100 × 961.
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So 8/31 ALSO has compromise score = 1/100 = 0.01. -/
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theorem f_8_31_compromiseScore :
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compromiseScore f_8_31 = (1 : Rat) / 100 := by
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native_decide
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/-- THREE fractions (7/27, 13/50, 8/31) all have the SAME compromise score.
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7/27 is NOT uniquely optimal. It is one of (at least) three equally good
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compromise fractions. This is the formal proof of the look-elsewhere
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effect demanded by the adversarial review. -/
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theorem threeFractionsTied :
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compromiseScore f_7_27 = compromiseScore f_13_50 ∧
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compromiseScore f_13_50 = compromiseScore f_8_31 := by
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constructor
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· native_decide
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· native_decide
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 The Look-Elsewhere Effect (formalized)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Number of distinct fractions in [0.20, 0.35] with denominator ≤ 50.
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This is the number of alternative hypotheses that were implicitly tested.
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The hostile reviewer estimated 53 in [0.24, 0.28].
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In [0.20, 0.35] the count is higher.
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Formal note: we do not enumerate all 53+ fractions in Lean because the
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list is long and unilluminating. The key insight is captured by the
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`threeFractionsTied` theorem: even among the TOP candidates, 7/27 is not
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unique. The look-elsewhere penalty is at least a factor of 3. -/
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def lookElsewhereFactor : Nat := 3
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/-- The effective significance of the 7/27 match after look-elsewhere
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correction: divide the apparent significance by the number of equally
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good alternatives (at least 3). -/
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def lookElsewhereCorrectedSignificance (apparentSigma : Rat) : Rat :=
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apparentSigma / (lookElsewhereFactor : Rat)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Executable Receipts
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval! mottCriterion
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#eval! mottDistance f_7_27
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#eval! mottDistance f_13_50
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#eval! mottDistance f_6_23
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#eval! speciesAreaDistance f_7_27
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#eval! speciesAreaDistance f_13_50
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#eval! compromiseScore f_7_27
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#eval! compromiseScore f_13_50
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#eval! compromiseScore f_8_31
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end Semantics.FractionScan
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