[P2 analysis] Wasserstein tangent RG for the full threshold law: separate center/width drift from genuine shape flow

[LightChainr]

Trigger

The repository has repeatedly learned that compressing a near-critical threshold law to one root, one slope, one width or a few derivatives can hide the actual finite-size motion.

Existing work already contains most of the ingredients:

• [P2] Universal threshold-density profile and 4/3 tail test #28 / later threshold-profile work reconstructs the full standardized topological-threshold distribution and already uses Wasserstein distance as a distance diagnostic;
• [P0 analysis] Intrinsic quantile-center spectroscopy of the nonlinear thermal field #101/Score the frozen N145→N290 intrinsic quantile center #236 show that an intrinsic quantile center can transfer cleanly even while width drifts;
• [P2 analysis] Distinguish correction families with multi-u thermal response, not only center derivatives #119 uses several fixed intrinsic u levels rather than one center derivative;
• [Speculative synthesis] Learn a finite-size RG transfer operator from the full threshold curves #180 fits transfer operators after choosing a finite Krawtchouk/jet basis;
• the N100→N400→N900 analyses indicate that the clock/profile itself can widen or redistribute even when a scalar amplitude story weakens.

The missing object is a basis-free finite-size velocity of the entire probability law.

The proposal is to treat the reconstructed threshold distribution as a point in one-dimensional Wasserstein space, where the geometry is exceptionally simple: the quantile function is an isometric coordinate for W2.

This is not another profile-distance score and not an assertion that the physical RG is literally optimal transport.

Canonical distributional coordinate

For each finite geometry/size, use the existing monotone matching/topological profile

F_N(p) = [1 + M_N(p)]/2

(or the exactly corresponding typed threshold CDF when a different channel is declared).

Define its quantile function

Q_N(u) = F_N^{-1}(u),   0<u<1.

For one-dimensional probability measures,

W2(P_N,P_M)^2
 = integral_0^1 [Q_N(u)-Q_M(u)]^2 du.

The finite-size displacement is therefore the actual function

v_(N->M)(u) = Q_M(u)-Q_N(u),

not a fitted collection of moments.

The threshold-rank histograms already reconstruct F_N(p) as a smooth finite-volume polynomial/curve; use that object directly rather than creating an empirical step-CDF when avoidable.

Remove the location/scale orbit geometrically

A key advantage is that “center shift” and “width change” form a precise two-dimensional tangent space.

At a baseline quantile function Q_N, an infinitesimal affine transformation

p -> a p + b

moves Q_N in

span{ 1, Q_N } subset L2(0,1).

Therefore decompose every scale displacement orthogonally:

v = v_affine + v_shape,

v_affine in span{1,Q_N},
v_shape  perpendicular to span{1,Q_N}.

The norm

||v_shape||_L2

is a direct, denominator-free measure of genuine distributional shape flow after the best possible center/width adjustment.

This gives a mechanical version of a question that has appeared in several separate forms in the repository:

Is the finite-size change only a thermal-coordinate/metric drift,
or does the entire threshold law change shape?

No skewness/kurtosis/selected derivative needs to be privileged in advance.

First falsifiable hierarchy

For a fixed scale multiplier m and one lineage define

v_N(u) = [Q_(mN)(u)-Q_N(u)] / log(m),

and its affine-orthogonal piece v_N^shape.

Test the following nested possibilities.

W0 — pure affine flow

v_N^shape = 0

within the full covariance/uncertainty envelope.

Then center/width are sufficient for the full profile at that transition. Any apparent higher-moment drift should be traced to estimator noise or to a different observable contract.

W1 — one stable shape generator

The nonzero functions

v_N^shape(u)

from different sizes/lineages are collinear after covariance whitening, with only their scalar amplitude changing.

This is a distribution-level analogue of a one-generator correction, but the generator is discovered as a full quantile deformation, not chosen from a polynomial/Krawtchouk basis.

W2 — stable two-dimensional shape plane

One direction is insufficient but the shape velocities lie in one stable two-dimensional subspace across lineages/sizes.

This is compatible with a compact multicomponent finite-size flow. Only after this basis-free gate should #180/#218 ask whether the corresponding transfer is ordinary, Jordan-like, or something else.

Wbroad — rotating/growing shape flow

The principal angle between shape-velocity subspaces remains large or the required dimension grows as more held-out transitions are added.

Then a small thermal-jet closure is not an intrinsic property of the threshold law; it is a projection-dependent approximation.

A nontrivial scale-stationarity test

Do not use the tautology

Q_(4N)-Q_N = [Q_(4N)-Q_(2N)] + [Q_(2N)-Q_N]

as an RG test.

Instead ask whether the shape generator itself is stable.

For dyadic or Gaussian-multiplier lineages compute

v_N^shape,
v_(mN)^shape

using only the adjacent transitions, with each transition's affine tangent projected out before comparison.

A simple basis-free curvature statistic is

K_N = || normalized(v_(mN)^shape)
        - sign/alignment * normalized(v_N^shape) ||_L2,

or the corresponding covariance-whitened principal angle for a multidimensional subspace.

For amplitude-bearing models, fit the amplitude law on source transitions and predict the entire held-out quantile function at the next scale. The held-out function, not a selected moment, is the falsification target.

Relation to #180

#180 asks for a finite transfer matrix in a chosen observable/jet basis.

This issue should run before or parallel to that choice:

full threshold law
 -> Wasserstein tangent / shape subspace
 -> only then choose a compact physical basis that spans it.

If the Wasserstein shape subspace is one- or two-dimensional and aligns with known Krawtchouk/thermal coordinates, that is real evidence that the basis is capturing intrinsic profile motion.

If the Wasserstein shape flow is broad while a selected jet looks rank two, then the small rank is an observer-bandwidth statement, exactly the kind of distinction emphasized by #419.

No-new-production first pass

Use only already committed full threshold/profile assets.

1. Exact/synthetic controls

Construct distributions related by:

pure translation,
pure scale,
translation+scale,
one known nonlinear monotone deformation,
two independent shape deformations.

Verify that the affine projection gives exactly zero shape flow for the first three and recovers the declared rank for the latter controls.

2. Exact tiny percolation control

Use N10 or another exactly reconstructible threshold law and verify the numerical quadrature/inverse-CDF implementation against exact integration to controlled precision.

3. Existing lineages

Prefer lineages with full profile reconstruction and clean chronology. Candidate assets include the norm-2/norm-5 full-curve families and the N100→N400→N900 clock/profile work, but do not mix different observable/channel semantics into one flow.

For every transition retain the full delete-one/aligned-batch covariance of the reconstructed quantile grid. Quantile points are correlated coordinates of one distribution, not hundreds of independent tests.

4. Held-out prediction

Fit a one- or two-dimensional shape subspace on source transitions only. Freeze it. Predict one unreused transition's full Q(u) vector and score in the identifiable covariance subspace.

No new samples until a current asset set demonstrates that this test has discriminatory power.

Tail and conditioning boundary

Quantile coordinates avoid arbitrary amplitude denominators, but they are not magically well-conditioned everywhere.

Because

d F^{-1}(u) / dF ~ 1 / f(Q(u)),

extreme tails with tiny density can have large uncertainty.

Therefore freeze a central quantile window from source/count considerations, for example

u_min <= u <= 1-u_min,

and treat tails separately using #28's count/tail protocol. Do not select u_min by whichever value makes the shape rank smallest.

This is the distributional analogue of #579's conditioning discipline.

Possible extension: the joint two-birth law

The one-dimensional F_N forgets the dependence between

K_minus = first ambient-H1 birth,
K_plus  = second ambient-H1 birth.

If the 1D shape flow closes but predictive-state work still requires extra information, the next object should be the joint law / copula of (K_minus,K_plus), not another marginal moment.

A two-dimensional transport analysis is substantially less canonical and more expensive, so it should be attempted only after the 1D program shows that marginal shape has genuinely saturated.

This gives a principled escalation:

marginal threshold law
 -> shape tangent rank
 -> only if needed, joint birth-time dependence.

External frontier and claim boundary

Optimal-transport language is increasingly being used to formulate RG geometrically, but the repository should use it as a mathematical coordinate system, not import a theorem that does not apply.

• Cotler & Rezchikov, arXiv:2202.11737, identify certain exact momentum-space RG equations with Wasserstein/optimal-transport gradient flows. This motivates the language but does not imply that the finite square-site profile follows an OT gradient flow.
• A 2026 APS Global Physics Summit contribution, The Wasserstein Geometry of Real-Space Renormalization, explores real-space coarse graining as motion of Gibbs measures in Wasserstein space. It is current frontier motivation, not a peer-reviewed theorem to cite as evidence for Matching One.
• Vesseron, Cazelles, Le Brigant & Klein, arXiv:2506.04480, develop Wasserstein geodesic PCA for collections of probability measures. The relevant idea here is that modes of distributional variation can be defined geometrically rather than by Euclidean PCA of arbitrary summary statistics.
• Recent real-space RG work also emphasizes that a fixed low-dimensional block transformation can miss information carried across scale boundaries; the present proposal avoids asserting a finite rank until the full-law deformation is measured.

The only exact facts claimed here are finite-distribution/one-dimensional Wasserstein geometry and the repository's own threshold reconstruction.

Decision table

Shape norm consistent with zero

Collapse the relevant full-profile finite-size change to center/width and stop adding higher shape descriptors for that contract.

Stable one-dimensional shape velocity

Promote one full-function shape generator and ask which existing thermal/Krawtchouk coordinate represents it. Make a held-out full-profile prediction before field naming.

Stable two-dimensional shape plane

Use it as a basis-free input to #180/#370. Test ordinary vs confluent/Jordan action only after the plane survives held-out scale transport.

Broad/rotating flow

Stop interpreting a small set of moments as a closed RG state. Retain full-profile or multiscale-process descriptions.

Tail-dominated apparent rank

Narrow the claim to the central profile and hand the tails back to #28's explicit tail/count analysis; do not call tail conditioning a new physical mode.

Deliverable

A first delivery should contain:

1. exact/synthetic affine-null controls;
2. a typed threshold-CDF -> quantile reconstruction utility;
3. covariance-aware affine tangent projection;
4. shape-flow functions and norms for existing lineages;
5. principal-angle/rank diagnostics across transitions;
6. one frozen held-out full-profile prediction;
7. one decision: affine, rank-1 shape, rank-2 shape, broad, or underpowered.

Related: #28, #101, #119, #122, #180, #182, #236, #367, #373, #379, #383, #419, #579.

[LightChainr]

The affine tangent here is #579's ray statistic with a second column — first pass delivered

scripts/wasserstein_shape_tangent.py, tests/test_wasserstein_shape_tangent.py (9 tests), results/wasserstein-shape-tangent-controls/latest.json, notes/wasserstein-shape-tangent-controls-20260906.md, on claude/matching-one-workspace-pwr5pv. Synthetic only — no threshold data touched.

The statistic already exists

An affine map p → a p + b sends Q to aQ + b, so the affine tangent at Q is span{1, Q} — a two-dimensional subspace. The covariance-weighted distance from the displacement to it,

D = min_{α,β} (v − α·1 − β·Q)ᵀ S⁺ (v − α·1 − β·Q),

is projective_inference.subspace_residual with a two-column basis, the same routine that scores a model ray with one column. So deliverable item 3 (covariance-aware affine tangent projection) needed no new implementation, and it inherits the χ² calibration, the pseudo-inverse rank reporting and the df bookkeeping already tested under #579. Nine levels minus two tangent columns leaves 7 df.

One amendment to the proposal, and it is about the tail section

The issue proposes ‖v_shape‖_L2, then correctly notes that a quantile's uncertainty carries 1/f(Q(u)) and therefore freezes a central window u_min — with the warning not to choose it by whichever value makes the rank smallest.

The weighted form does not need the window to carry that. Control:

displacement = affine, plus a ONE-SIGMA excursion in the noisiest quantile
               (that coordinate's sigma is 400x the others')

unweighted ‖v_shape‖_L2  =  310.2          → "large shape flow"
weighted  D              =  1.00 on 7 df   → 0.006 sigma

The unweighted norm reports shape where there is none; the weighted statistic knows that excursion was worth 1σ. The window is still worth declaring (rank pathologies are real), it just stops being where the conditioning lives. That is the same point as #579's, one level up.

Controls (deliverable item 1)

control	result
pure translation	D = 0 exactly
pure scale	6.5e−35
translation + scale	1.6e−33
negative scale	9.3e−34
one declared generator	0.310 on the affine basis → 1.0e−32 once it joins
two declared generators	0.402 alone, 0.326 with only the first, 1.2e−32 with both

That last row is the one your decision table turns on: a rank-1 method applied to a rank-2 flow would close it with one generator and report every flow as rank 1. It does not.

What is left, and where the work actually is

The remaining first-pass items are all data-side: the typed threshold-CDF → quantile utility, the exact tiny control, and the aligned delete-one covariance of a quantile grid for one lineage. The statistic is ready; the covariance is the work.

One warning worth restating from #579, because it bites harder here. S has to be the covariance of one random object across all quantile coordinates — the lineage's transitions sharing a stream, and the delete-one deleting a batch from every level at once. A quantile grid whose levels were estimated independently has a diagonal S by construction, and a diagonal S on a hundred correlated coordinates of one distribution will make almost any displacement look significant.

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Generated by Claude Code

[LightChainr]

#584's primary coordinate stays this contract: F_N(p) = [1 + M_N(p)] / 2, or whichever typed CDF was actually used to produce g_N and r_N. Gate 2's ranking of channels for (A4, δ) does not retarget it.

Note: notes/literature-officer-20260906-gate2-gate3-channel.md. Protocol on #584. Does not close this ticket. Does not start the screen.

[LightChainr]

Same as #276: draft #716 separates median/q_w from the two-birth law and from IQR→1. Quartiles at width 4 from matrix powers, not MC. Independent Bernoulli snapshots at one p are the wrong tool for the exponentially rare central sector. Not a Newman–Ziff complexity bound.