Probe MID — Location without shape: explicit toy families
Assign to: ordinary (adequate) reasoning, ordinary CPU. Analysis + numpy/mpmath. No percolation engine, no arXiv, no N=725, no #612 rescoring.
This is a construction probe. The deliverable is families you invent, not a match to a repository JSON.
Standing
Frontier: claude/matching-one-workspace-pwr5pv @ 8b5f9d1a.
Theorem L as a cited fact: notes/p613-quantile-convergence-20260907.md — location, not rate. You do not re-prove it.
North-star
Exhibit explicit sequences of CDFs F_N (or quantile functions Q_N) on [0,1] such that
Q_N(u) → p_* uniformly on every compact [ε, 1−ε] ⊂ (0,1)
for some p_* ∈ (0,1), while the Aff(1)-invariant shape
Z_N(u) = (Q_N(u) − Q_N(a)) / (Q_N(b) − Q_N(a))
(with declared anchors, default a=0.2, b=0.8) does one of the following, by construction:
| target |
what you must exhibit |
| T1 |
Z_N fails to converge (oscillation, two accumulation points, or N-dependent anchors that still oscillate after a fixed-anchor definition) |
| T2 |
Z_N converges, and the limit Z_∞ can be prescribed: for any continuous strictly increasing ζ with ζ(a)=0, ζ(b)=1, there is a family with that Z_∞ |
| T3 |
Z_N converges, but the limit depends on the anchors (a,b) in a way that is not a reparametrisation (then “the” shape is not well-posed even for toys) |
If T2 is achieved, W1 of #622 becomes a theorem about location statements in general: Theorem L cannot pin percolation’s shape. That is the point of this probe.
If you believe T2 is false for monotone CDFs with a single crossing of every u∈(ε,1−ε), prove that restriction and still do T1.
Constraints on the toys (so they are fair analogues)
Each family must be:
- a CDF in
p for each N: F_N(0)=0, F_N(1)=1, nondecreasing (cadlag allowed; say so);
- strictly increasing on a neighbourhood of
p_* for large N, so Q_N(u) is a singleton for u∈[ε,1−ε];
- not a mixture of two distant jumps that “cheats” location by parking mass at
0 and 1 only — at least one of the families must have a window of width w_N → 0 about p_* that carries all interior quantiles (this is what Theorem L actually looks like);
- parametrised in closed form, or by a 20-line sampler of a named distribution. No black-box neural CDF.
Write the two Aff(1) actions and say which one your Z quotients:
on p: p ↦ αp+β
on Q: Q(u) ↦ α Q(u)+β
Z as written quotients the Q-action. One family should also be inspected under a warp of the p-axis, to show the two quotients disagree (a numerical example is enough; the theorem is #622 W5).
Directions (all three targets attempted; T2 is the prize)
D1 — Logistic / probit window (baseline that does pin a shape)
F_N(p) = σ( (p − p_*) / w_N ), w_N → 0, σ a fixed logistic or Gaussian CDF.
Compute Z_N. It will converge to a fixed logistic/probit shape independent of w_N. This is the family people silently imagine when they say “sharp threshold ⇒ universal shape”. Record it as the non-example: location plus a fixed window profile pins Z. Theorem L does not give you a fixed profile.
D2 — Prescribed shape (T2)
Replace σ by a sequence σ_N, or by a mixture of two window profiles whose weights oscillate, or by an N-dependent skew (e.g. GEV / skew-logistic with a parameter γ_N).
Goal: pick any target ζ (give three: symmetric logistic; a strongly skew one; a piecewise-linear “kink”), build F_N with window w_N = N^{-1} or N^{-3/4} (the exponent is free in a toy), prove Q_N(u)→p_* uniformly on [ε,1−ε], and prove Z_N → ζ.
If a monotone one-parameter exponential family cannot do T2, say so and use a two-parameter window (location + skew), still with width → 0.
D3 — Oscillation (T1)
Two profiles σ and τ with different Z_∞, switched on even/odd N, or a skew parameter γ_N = sin(log N). Prove location still holds. Then Z_N has no limit. This kills “sharp ⇒ shape converges”.
D4 — Anchor pathology (T3)
Using one D2 family, recompute Z with anchors (0.1,0.9) vs (0.3,0.7) vs (0.2,0.8). If the three Z_∞ differ by more than reparametrisation of u (make that precise: they do not lie on one orbit under increasing maps that fix {a,b} — or they do). Either “anchors are gauge” or “the invariant is not well-posed”. One page.
D5 — What extra input would pin ζ in percolation
After T1/T2, one page, no literature search. Named extra inputs already in the #613/#606 notes (RSW, four-arm / F1, self-duality). For each: would it pin σ (the window profile), or only the width? Write Y/N/unknown. Unknown is allowed. Do not quote new papers.
Compute
Grid u = 0.05(0.05)0.95, N in {2^k} over at least two decades of w_N. JSON of Q_N, Z_N for each family. Plots optional as committed png under results/, not required.
No Monte Carlo of percolation. No L=3 enumerator.
Stop rules
Deliverables
notes/probe-location-without-shape-YYYYMMDD.md
formulas, proofs of location, T1/T2/T3 verdict, D5 table
scripts/probe/toy_cdf_families.py
results/probe-location-without-shape/latest.json
Interface
#622 uses this as the W1 laboratory. #618 may cite T2 as “location does not imply a common ω for the 9-vector” — only if you actually proved T2. #620 is not involved.
Probe MID — Location without shape: explicit toy families
Assign to: ordinary (adequate) reasoning, ordinary CPU. Analysis + numpy/mpmath. No percolation engine, no arXiv, no N=725, no #612 rescoring.
This is a construction probe. The deliverable is families you invent, not a match to a repository JSON.
Standing
docs/STATUS.md. Does not close [P0 proof] One probability package: arbitrary-period balance roots and the sharp axial full-law boundary #613, [P2 long-horizon probe] After Q_N(u)→p_c: which rates are theorems from named inputs? #618, [P1 CPU / existing data] After #702: within-model quantile-shape symmetry and full-vector finite-size motion #622, [P2 exact/probability] Matching root as a homological balance point: prove p_N→p_c before fitting its shift #276.S([P1 CPU / existing data] After #702: within-model quantile-shape symmetry and full-vector finite-size motion #622); you supply the analytic counterweight that [P1 CPU / existing data] After #702: within-model quantile-shape symmetry and full-vector finite-size motion #622’s W1 asked for.claude/matching-one-workspace-pwr5pv. Comment the URL on the issue. Leave the issue open. Not against docs: four post-#612 probe briefs (gauge / rate / exact / literature) #616.Frontier:
claude/matching-one-workspace-pwr5pv@8b5f9d1a.Theorem L as a cited fact:
notes/p613-quantile-convergence-20260907.md— location, not rate. You do not re-prove it.North-star
Exhibit explicit sequences of CDFs
F_N(or quantile functionsQ_N) on[0,1]such thatfor some
p_* ∈ (0,1), while the Aff(1)-invariant shape(with declared anchors, default
a=0.2,b=0.8) does one of the following, by construction:Z_Nfails to converge (oscillation, two accumulation points, orN-dependent anchors that still oscillate after a fixed-anchor definition)Z_Nconverges, and the limitZ_∞can be prescribed: for any continuous strictly increasingζwithζ(a)=0,ζ(b)=1, there is a family with thatZ_∞Z_Nconverges, but the limit depends on the anchors(a,b)in a way that is not a reparametrisation (then “the” shape is not well-posed even for toys)If T2 is achieved, W1 of #622 becomes a theorem about location statements in general: Theorem L cannot pin percolation’s shape. That is the point of this probe.
If you believe T2 is false for monotone CDFs with a single crossing of every
u∈(ε,1−ε), prove that restriction and still do T1.Constraints on the toys (so they are fair analogues)
Each family must be:
pfor eachN:F_N(0)=0,F_N(1)=1, nondecreasing (cadlag allowed; say so);p_*for largeN, soQ_N(u)is a singleton foru∈[ε,1−ε];0and1only — at least one of the families must have a window of widthw_N → 0aboutp_*that carries all interior quantiles (this is what Theorem L actually looks like);Write the two Aff(1) actions and say which one your
Zquotients:Zas written quotients the Q-action. One family should also be inspected under a warp of thep-axis, to show the two quotients disagree (a numerical example is enough; the theorem is #622 W5).Directions (all three targets attempted; T2 is the prize)
D1 — Logistic / probit window (baseline that does pin a shape)
Compute
Z_N. It will converge to a fixed logistic/probit shape independent ofw_N. This is the family people silently imagine when they say “sharp threshold ⇒ universal shape”. Record it as the non-example: location plus a fixed window profile pinsZ. Theorem L does not give you a fixed profile.D2 — Prescribed shape (T2)
Replace
σby a sequenceσ_N, or by a mixture of two window profiles whose weights oscillate, or by anN-dependent skew (e.g. GEV / skew-logistic with a parameterγ_N).Goal: pick any target
ζ(give three: symmetric logistic; a strongly skew one; a piecewise-linear “kink”), buildF_Nwith windoww_N = N^{-1}orN^{-3/4}(the exponent is free in a toy), proveQ_N(u)→p_*uniformly on[ε,1−ε], and proveZ_N → ζ.If a monotone one-parameter exponential family cannot do T2, say so and use a two-parameter window (location + skew), still with width → 0.
D3 — Oscillation (T1)
Two profiles
σandτwith differentZ_∞, switched on even/oddN, or a skew parameterγ_N = sin(log N). Prove location still holds. ThenZ_Nhas no limit. This kills “sharp ⇒ shape converges”.D4 — Anchor pathology (T3)
Using one D2 family, recompute
Zwith anchors(0.1,0.9)vs(0.3,0.7)vs(0.2,0.8). If the threeZ_∞differ by more than reparametrisation ofu(make that precise: they do not lie on one orbit under increasing maps that fix{a,b}— or they do). Either “anchors are gauge” or “the invariant is not well-posed”. One page.D5 — What extra input would pin
ζin percolationAfter T1/T2, one page, no literature search. Named extra inputs already in the #613/#606 notes (RSW, four-arm / F1, self-duality). For each: would it pin
σ(the window profile), or only the width? Write Y/N/unknown. Unknown is allowed. Do not quote new papers.Compute
Grid
u = 0.05(0.05)0.95,Nin{2^k}over at least two decades ofw_N. JSON ofQ_N,Z_Nfor each family. Plots optional as committed png underresults/, not required.No Monte Carlo of percolation. No L=3 enumerator.
Stop rules
Deliverables
Interface
#622 uses this as the W1 laboratory. #618 may cite T2 as “location does not imply a common ω for the 9-vector” — only if you actually proved T2. #620 is not involved.