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Shock Fit — λ₃ (soccer)

The goal · regular page's step 4 as a standalone calculator. That page starts from a quote board and spends steps 1–3 stripping margins and recovering rates; this lab starts where step 4 starts — the fair probabilities — and runs the fit alone: choose the common shock λ₃, re-solving (λ₁, λ₂) from scratch at every trial so the Handicap and Total anchors never move.

The inputs are exactly step 4's contract, nothing else: the two lines, the two hard anchors (P(home covers h), P(over L)) and the soft 1X2 targets (home and draw; away is derived from Σ = 1 and never scored). Note what is not an input: λₕ and λₐ. The rates are what the fit produces — the baseline pair at λ₃ = 0, then a fresh pair at every trial λ₃ — and the trial table shows each re-solve, row by row: λ₁ + λ₂ giving up exactly what 2λ₃ takes, the draw climbing toward its target, infeasible guesses discarded where the shock alone overshoots the Total.

It runs the same engine function the page runs (fitBP in markets-bp.ts — priceMatchBP is that function behind a margin-stripper), so the two can never disagree. The seed is the page's default quotes' step-1 output at full precision, landing on the same fitted rates: λ = 1.226178 · 0.653620 · 0.439953. Step 4 on that page links here with its current inputs prefilled — whatever quotes you have entered there replay here digit for digit.

One honesty note on precision: near the floor the miss surface is very flat, so hand-shortened inputs (say, anchors retyped at 8 decimals) can land λ₃ within about 10⁻⁴ of the page's value with an equally good miss — the trailing digits of λ₃ carry no pricing meaning. The prefilled hand-off passes full-precision values, which reproduces the page exactly.

The search itself — the 17-point scan, then the 0.618-probe bracket walk — is narrated trial by trial below; the bracket mechanics in isolation live in the Golden-Section Calculator, and the fitted rates paste straight into the Score Box lab.

↑/↓ steps a field (lines by 0.25, probabilities by 0.001), Shift steps bigger.

Fit facts
One knob against two targets (home + draw; away follows from Σ = 1) — a least-squares compromise, never an exact fit of both
The shock cancels in the margin (X − Y = W₁ − W₂), so λ₃ reaches the draw only by trading λ₁ + λ₂ down against the Total anchor
λ₃ ≥ 0 only adds covariance — a target draw at/below the λ₃ = 0 baseline pins the knob at 0
(λ₁, λ₂) are re-solved from scratch at every trial, so both anchors hold to 10⁻⁶ on every accepted point
Inputs — step 1’s outputs, nothing else
Tryexamples
Handicap lineh
Home-team line; quarter lines (±0.25, ±0.75, …) split into their two neighbours.
|h| ≤ 10
P(home covers h)hard
Hard Handicap anchor — step 1’s stripped fair cover probability. Held on every trial.
0 < p < 1
Total lineL
Match-total line, same quarter handling.
0 ≤ L ≤ 20
P(over L)hard
Hard Total anchor — the fair over probability. This is the anchor λ₃ trades against.
0 < p < 1
1X2 target p*_Hsoft
Soft home target — the Shin-stripped 1X2 home leg. Scored in the miss, but the anchor usually pins it.
0 < p < 1
1X2 target p*_Dsoft
Soft draw target — the leg the shock actually chases.
0 < p < 1 and p*_H + p*_D < 1
1X2 target p*_AΣ = 1
Derived, not typed: away = 1 − p*_H − p*_D. It is never scored — adding it would double-count the miss.
Grid sizeN
Scores 0…N with the ≥N tail folded in (Σ = 1). The goal-regular page runs N = 5.
Steps 2–3 — the λ₃ = 0 baseline
λrecover the rates against the two hard anchors — the steps 2–3 nested bisection, λ₃ frozen at 0anchors hit to |Δ| = 3.8e-9 (Handicap) · 7.1e-9 (Total)λₕ 1.608939 · λₐ 1.065121
1X2the win/draw/loss split that grid already implies0.500000 · 0.248599 · 0.251401
Δthe draw gap the shock must close: target − baselinepositive — λ₃ trades λ₁ + λ₂ down against the Total anchor until the draw meets it (least squares)+5.37e-2
Step 4 — every trial, (λ₁, λ₂) re-solved each time
iphaseλ₃λ₁λ₂λ₁+λ₂2λ₃bracketP_DΔ drawmiss
0scan0.0000001.6089391.0651212.6740600.000000—0.248599-5.37e-23.0e-3
1scan0.1875001.4472800.8941622.3414430.375000—0.267404-3.49e-21.3e-3
2scan0.3750001.2831830.7166031.9997870.750000—0.291985-1.03e-21.9e-4
3scan0.5625001.1192630.5330671.6523311.125000—0.324965+2.27e-26.0e-4
4scan0.7500000.9606860.3462461.3069321.500000—0.369787+6.75e-24.6e-3
5scan0.9375000.8151690.1618250.9769941.875000—0.429796+1.27e-11.6e-2
·scan1.125000infeasible — an anchor broke (2λ₃ alone overshoots the Total), trial discarded∞
·scan1.312500infeasible — an anchor broke (2λ₃ alone overshoots the Total), trial discarded∞
·scan1.500000infeasible — an anchor broke (2λ₃ alone overshoots the Total), trial discarded∞
·scan1.687500infeasible — an anchor broke (2λ₃ alone overshoots the Total), trial discarded∞
·scan1.875000infeasible — an anchor broke (2λ₃ alone overshoots the Total), trial discarded∞
·scan2.062500infeasible — an anchor broke (2λ₃ alone overshoots the Total), trial discarded∞
·scan2.250000infeasible — an anchor broke (2λ₃ alone overshoots the Total), trial discarded∞
·scan2.437500infeasible — an anchor broke (2λ₃ alone overshoots the Total), trial discarded∞
·scan2.625000infeasible — an anchor broke (2λ₃ alone overshoots the Total), trial discarded∞
·scan2.812500infeasible — an anchor broke (2λ₃ alone overshoots the Total), trial discarded∞
·scan3.000000infeasible — an anchor broke (2λ₃ alone overshoots the Total), trial discarded∞
6golden0.4739751.1963750.6203591.8167350.947949[0.330737, 0.562500]0.308143+5.85e-31.2e-4
7golden0.3854491.2740090.7065191.9805280.770898[0.330737, 0.473975]0.293573-8.73e-31.6e-4
8golden0.4401611.2259960.6534171.8794140.880322[0.385449, 0.473975]0.302332+3.43e-58.5e-5
9golden0.4530761.2146750.6408111.8554860.906153[0.419263, 0.473975]0.304514+2.22e-39.0e-5
10golden0.4321781.2329960.6611951.8941920.864357[0.419263, 0.453076]0.301006-1.29e-38.7e-5
11golden0.4450941.2216710.6486051.8702760.890188[0.432178, 0.453076]0.303160+8.62e-48.6e-5
12golden0.4371121.2286700.6563901.8850590.874224[0.432178, 0.445094]0.301824-4.74e-48.6e-5
13golden0.4420451.2243440.6515801.8759240.884090[0.437112, 0.445094]0.302648+3.50e-48.6e-5
14golden0.4389961.2270170.6545531.8815700.877992[0.437112, 0.442045]0.302138-1.60e-48.5e-5
15golden0.4408811.2253650.6527161.8780810.881761[0.438996, 0.442045]0.302453+1.55e-48.5e-5
16golden0.4397161.2263860.6538511.8802370.879432[0.438996, 0.440881]0.302258-4.01e-58.5e-5
17golden0.4404361.2257550.6531491.8789040.880871[0.439716, 0.440881]0.302378+8.03e-58.5e-5
18golden0.4399911.2261450.6535831.8797280.879982[0.439716, 0.440436]0.302304+5.90e-68.5e-5
19golden0.4398861.2262370.6536851.8799230.879772[0.439716, 0.440161]0.302286-1.17e-58.5e-5
20golden0.4400561.2260880.6535201.8796080.880112[0.439886, 0.440161]0.302315+1.67e-58.5e-5
21golden0.4399511.2261800.6536221.8798030.879902[0.439886, 0.440056]0.302297-8.09e-78.5e-5
22golden0.4399261.2262020.6536461.8798480.879852[0.439886, 0.439991]0.302293-4.95e-68.5e-5
23golden0.4399661.2261670.6536071.8797740.879932[0.439926, 0.439991]0.302300+1.75e-68.5e-5
24golden0.4399411.2261890.6536311.8798200.879883[0.439926, 0.439966]0.302295-2.39e-68.5e-5
25golden0.4399571.2261750.6536161.8797920.879913[0.439941, 0.439966]0.302298+1.69e-78.5e-5
26golden0.4399471.2261840.6536261.8798090.879894[0.439941, 0.439957]0.302296-1.41e-68.5e-5
27golden0.4399531.2261780.6536201.8797980.879906[0.439947, 0.439957]0.302297-4.35e-78.5e-5
28golden0.4399541.2261770.6536191.8797960.879909[0.439951, 0.439957]0.302298-2.04e-78.5e-5
29golden0.4399521.2261790.6536211.8798000.879904[0.439951, 0.439954]0.302297-5.78e-78.5e-5
30golden0.4399541.2261780.6536191.8797970.879907[0.439952, 0.439954]0.302298-3.47e-78.5e-5
31golden0.4399531.2261790.6536201.8797990.879905[0.439952, 0.439954]0.302297-4.90e-78.5e-5
32golden0.4399531.2261780.6536201.8797980.879906[0.439953, 0.439954]0.302297-4.02e-78.5e-5

Watch λ₁+λ₂ fall as 2λ₃ rises down the scan — the Total anchor trading private goals for shared ones; P_H is omitted because it never moves (the Handicap anchor pins it on every row). The golden miss column is not monotone: the bracket shrinks, and the best trial seen anywhere ships (highlighted). The bracket walk is plain golden-section — see it isolated in the Golden-Section Calculator.

The winner
λ₃the winner of 33 recorded trials (6 feasible scan points, then golden-section)final golden bracket [0.439953, 0.439954] — the best trial seen anywhere ships0.439953
λ₁λ₂the adjustment — λ₁ 1.608939 → 1.226178, λ₂ 1.065121 → 0.653620the Total counts the shock twice, so λ₁ + λ₂ gives up what 2λ₃ takes: 1.879798 + 0.879906 keeps E[total] = 2.759704Σ 1.879798 + 2λ₃ 0.879906
μmarginal goal means of the fitted engineeach side’s mean is its own stream plus the shared one: μ = λᵢ + λ₃μₕ 1.666131 · μₐ 1.093573
1X2achieved split vs the targetsΔ home -9.24e-3 (anchor-pinned book gap — the fit cannot move it) · Δ draw -4.35e-7 · Δ away +9.24e-30.500000 · 0.302297 · 0.197703
✓both hard anchors on the fitted gridheld on every accepted trial — the shape knob never trades them|Δ| 6.0e-11 · 6.6e-11
⧉ — paste straight into the Score Box lab to price the payout ladders off these rates1.226178 · 0.653620 · 0.439953