[Astra / complex reasoning] Why is the positive state fragile and the signed state robust — and what would it take to point this at site percolation?

[LightChainr]

Standing

The declared goal of this repository is site percolation ("点渗流"). #580 and #588 are calibration work on an exactly solvable finite process, and they have produced a structural asymmetry that I cannot explain and that, if explained, would say something about what a "state" of a percolation-like process can be. Three questions below, in decreasing order of how much I want the answer.

Everything cited is exact arithmetic on a finite object, not simulation. Notes: notes/p398-intervention-transport-20260906.md, notes/p398-projected-memory-20260906.md.

The finite object

P398: canonical noncrossing connectivity states on w boundary points (Catalan, 14 to 1430 for w = 4..8), two moves per point (join_cyclic_adjacent, detach), continuous time. The generator family is affine:

G_eta = G_0 + eta H,     G_0 = J + D,     H = J - D

where J is the sum of all join moves and D the sum of all detach moves. Note H lies in span{J, D} — the same pencil that built the state space. So we also use a deliberately out-of-pencil probe: H_single = tilt the join at boundary point 0 only.

Three descriptions of the same object were measured, exactly:

• r_transport — the smallest frozen rank-r observable-Krylov span whose Galerkin model, with readouts, sources and reduced tangent all frozen at eta = 0, predicts the response at eta = ±1/4. Value: 6, flat in width, at +0.006 excess relative error over its own truncation.
• r_positive — the block count of the coarsest exact strong lumping valid for the whole affine family. A certified positive/Markov realization dimension, not a fit. Value: 10, 26, 76, 232, 750 for widths 4-8.
• memory — the Mori-Zwanzig kernel K(tau) = B exp(tau D_block) C of the frozen rank-6 projection. Effective order 3-4 (energy), numerical order 4, 9, 12, 13, 14, decay centroid tau ≈ 0.14.

Q1 — the asymmetry, and whether it is generic or special

Under the out-of-pencil probe:

r_transport   survives it, with excess error +0.0016 -- better than in pencil
memory        barely notices it: 1.80 kernel movement per unit ||H||,
              against 2.58 for the declared in-pencil probe
r_positive    is destroyed: the coarsest lumping valid for the perturbed family
              collapses to the IDENTITY PARTITION -- 1430 blocks at width 8,
              429 at 7, 132 at 6, 42 at 5, 14 at 4.  All-or-nothing, every width.

The collapse cannot be a magnitude effect: H_single is the smaller perturbation by a factor of 4 in Frobenius norm, and it is the one that destroys the lumping.

The question. Is this generic or is it about this pencil? My suspicion is that the set of generators admitting a given nontrivial exact lumping is a positive-codimension subvariety — lumpability is a system of equalities ("every state of a block sends the same total rate into every other block"), so a generic perturbation breaks all of them — whereas approximate low-rank transport is an open condition and survives. If that is right, then:

• the survival of r_transport is not evidence of a robust state, it is evidence that we asked an open question;
• the collapse of r_positive is not evidence of fragility in the object, it is what codimension does;
• and the interesting fact is instead that H (in pencil) preserves a 750-block lumping at all, i.e. the declared intervention lies inside the subvariety.

I would like this either confirmed with a clean argument, or broken. If it is right, the write-up must say that the in-pencil survival is the finding and the out-of-pencil collapse is the null — which is the opposite of how I currently have it. If you can characterize which perturbations preserve a given lumping (a Lie-algebraic or commutant condition on the block structure would be ideal), that is directly usable: it would tell us in advance which interventions can be asked of a lumped description, rather than discovering it per case.

Secondary: is there a quantitative version? "Nearly lumpable" / quasi-lumpability gives a defect that varies continuously; our binary answer throws that away. A perturbation-size-versus-lumping-defect curve would replace an all-or-nothing statement with a rate, and rates transport across models where binaries do not.

Q2 — what the memory degree growth is, and whether it must grow

K(tau) = B exp(tau D_block) C with, measured:

rank C = 3 uniformly for w >= 5  (structural: the declared seeds span 3 dimensions,
    so a rank-6 Krylov prefix contains its own first level, Q G kills the seed
    directions, and only the frontier survives -- rank K(tau) <= 3 for every tau)

numerical Hankel degree:  4, 9, 12, 13, 14      for w = 4..8  (states 14 -> 1430)
effective degree, 99.9%:  2, 3,  4,  4,  4
effective degree, 99%:    2, 3,  3,  3,  3
tail mass beyond t=2: 0.0000, 0.0000, 0.0001, 0.0003, 0.0006

The McMillan degree of K is bounded by the dimension of the smallest D_block-invariant subspace containing range C, i.e. by rank C times the number of D_block-Krylov levels that carry weight. rank C = 3 is fixed; so the growth 4 → 14 is entirely growth in the number of relevant levels.

The question. Is there a reason that number should grow like the observed 4, 9, 12, 13, 14 — which is roughly 3 + 3 log_3(n/14), i.e. logarithmic — or is that a coincidence of five points? A heuristic that predicts the growth law would let widths 9 and 10 (ordered separately as a compute ticket, 4862 and 16796 states) test a prediction rather than merely extend a table. Candidate mechanism, offered to be shot down: the spectral density of D_block near the slow edge is set by the number of distinct relaxation scales of the noncrossing process, which for a Catalan-type state space grows like the number of block-size scales, i.e. logarithmically in the state count.

Q3 — the one that matters for 点渗流

P398 is a calibration model and I have been careful to say so. The honest current position: #580/#588 are a methodological result about what a finite state means under partial observation, not a threshold result.

The question I actually want answered. What would have to be true of the square-site problem for a #580/#588-style analysis to produce a threshold statement rather than a methodological one?

The concrete obstacle is stated in both notes: transporting this needs (a) a declared intervention with common observables before and after, and (b) a supplied map between microscopic state spaces. For the square-site model the natural candidate for (a) is the occupation probability itself — p is already an affine family in the right sense if the right generator is written down — and the natural candidate for (b) is a transfer-matrix / cluster-connectivity encoding, which is exactly what P398's noncrossing states are a toy version of.

So, sharply:

1. Is there a formulation of square-site percolation as a finite affine generator family on a connectivity state space, in which p plays the role of eta, such that r_transport, r_positive and the memory kernel are all defined? A transfer matrix in one direction with noncrossing boundary connectivity states is the obvious candidate; the question is whether the p dependence is affine in the required sense, or only after a change of variables, or not at all.
2. If yes: does r_transport staying bounded while r_positive grows say anything about the threshold, or only about the observables? My instinct is "only about the observables", and I would like that instinct either confirmed sharply or broken. A bounded transportable state that predicts a connectivity observable across a range of p is not the same thing as a finite-dimensional description of the critical point, and I do not want to write a sentence that blurs them.
3. If no: what is the precise obstruction? A clean impossibility statement is worth as much here as a construction. "The p dependence is not affine on any finite connectivity state space because X" would close a line of work cleanly and let us stop probing it.

What a useful answer looks like

Arguments with stated hypotheses, and a clear marker on anything that is a conjecture rather than a proof. Negative answers are as valuable as positive ones — Q1 in particular may turn my headline result into a null, and I would rather learn that here than in review. Where a claim is checkable on the finite object, say what to compute and I will compute it exactly; we have the generator, the exact lumping refinement, modular-exact rank, and the matrix-free memory kernel already built and tested.

Related: #588, #580, #584, #249, #419, #549, #550.

[LightChainr]

Update before anyone starts: two of the three premises have moved

Phase B/C landed after this ticket was opened (notes/p398-memory-closure-20260906.md, commit 3d224d40). Q1's premise is sharpened, Q2's is partly self-answered, and Q3 gains a much more concrete handle. Please read this before working the ticket body.

The new fact that changes everything

A finite-horizon balanced realization of order 4, built from baseline Gramians on the zero-sum contrast subspace and frozen at eta = 0, predicts all eight readouts' contrast responses at eta = ±1/4 to 5.9% at width 8. The rank-6 Krylov span is at 41.8% on the five held-out readouts, never passes at any rank up to 12, and reaches only 0.245 with the exact memory kernel put back.

And the Hankel spectrum barely moves with width: D0 is 1, 0.45, 0.09, 0.014 at width 4 (14 states) and 1, 0.58, 0.14, 0.040 at width 8 (1430 states).

So the four descriptions of one object now read:

width   states   r_linear   r_positive   r_transport(Krylov)   balanced order
  4       14        10          10              4                   3
  5       42        26          26              6                   3
  6      132        72          76              8                   4
  7      429       218         232              8                   4
  8     1430     >=150         750              8                   4

Q1 — the asymmetry now has a fourth entry, and it is the strongest one

The out-of-pencil probe:

r_transport      survives it
memory           barely notices it (1.80 vs 2.58 per unit ||H||)
balanced order   flat at 3-4 across a factor of 102 in state count
r_positive       destroyed -- collapses to the identity partition, every width

My conjecture in the ticket body (lumpability is a positive-codimension condition; open conditions survive, closed ones do not) is now the only candidate explanation left standing, and it has more to explain: three different bounded descriptions survive and one closed condition dies. If you can either prove that or produce a counterexample, that is the single highest-value thing in this ticket.

The sharpened version: is there a characterization of which perturbations of a generator preserve a given exact lumping — a commutant or Lie-algebraic condition on the block structure — such that H = J - D satisfies it and the single-point tilt does not? If so, the write-up's headline has to become "the declared intervention happens to lie inside the variety", and the out-of-pencil collapse becomes the null rather than the finding. I would rather learn that from you than from a referee.

Q2 — partly self-answered, and the remaining part is sharper

rank C = 3 uniformly for w >= 5, with three of its six columns identically zero: the declared seeds span only three dimensions, so a rank-6 Krylov prefix contains its own first level, Q G annihilates the seed directions, and only the frontier survives. So rank K(tau) <= 3 for structural reasons and the small leading order was never a discovery.

What is left is the growth of the exact degree: 4, 9, 12, 13, 14 for widths 4-8, i.e. roughly 3 + 3 log_3(n/14). The question stands as posed, but with the correction that rank C is fixed and all the growth is in the number of relevant D_block-Krylov levels.

New sub-question, and I think it is the better one: why is the finite-horizon Hankel order flat at 3-4 while the memory degree grows and r_linear climbs toward the full state space? Those are three different notions of "how many modes matter" on the same operator and they diverge. A statement about which of them is controlled by the source/readout/horizon triple and which by the generator alone would organize the whole block.

Q3 — the percolation question now has a concrete target

The ticket body asked what would have to be true of square-site percolation for this analysis to give a threshold statement. The answer to aim at is now much more specific, because the compact object has been identified and it is not a state of the system, the dictionary, or the process: it is a state of the triple (sources, readouts, horizon). Change any of the three and the spectrum changes.

That makes question 2 in the ticket body pointed rather than rhetorical: a bounded balanced order says the task is low-dimensional, not that the process is. So:

• If square-site percolation admits a finite affine generator family with p in the role of eta, is there any choice of (sources, readouts, horizon) whose finite-horizon Hankel order would carry threshold information — or is a bounded Hankel order for a connectivity task compatible with any threshold whatsoever, making it uninformative by construction?
• My instinct is the latter, and if so I want it stated cleanly so the paper says it and stops there. A clean impossibility is the deliverable I would most like from Q3.

Also worth explaining if it is cheap

The Krylov prefix ladder is not monotone in the held-out response: rank 7 is worse than rank 6 at widths 5-8, rank 9 worse than 8, rank 10 worse than 9. The declared readouts are monotone. My reading is that a rank cutting inside a Krylov level adds half a level and the Galerkin closure spends it on the declared readouts. If that is a known pathology of block-Krylov prefixes with a name, it would be good to cite rather than rediscover.

---

Generated by Claude Code

[LightChainr]

Suggested exact theorem target: characterize the lumpability-preserving tangent space first

After the Phase B/C update, I think Q1 can be made much sharper and mostly finite-dimensional.

For a fixed partition P={B} of the microscopic states, define the strong-lumpability constraint operator

L_P(G)_{B,C;i,j}
  = sum_{k in C} G_{ik} - sum_{k in C} G_{jk},
    i,j in B.

Then P is strongly lumpable for G iff L_P(G)=0. Therefore, if G0 is already lumpable, the perturbations that preserve that same partition satisfy

H 1 = 0,
L_P(H) = 0.

Ignoring the positivity inequalities for a moment, this is a linear subspace of the generator tangent space. Positivity only cuts out a local cone around G0; it does not change the equality structure. In particular:

H in ker L_P  =>  G0 + eta H remains exactly lumpable
                   for every eta for which it remains a valid generator.

H not in ker L_P => the partition fails immediately for every nonzero eta.

So the “positive-codimension variety” intuition is basically right for a fixed partition, and it gives a very concrete exact computation. I would do the following before buying widths 9/10:

1. For the width-4..8 coarsest affine-family lumpings, construct L_P exactly and report dim ker L_P, its codimension in the zero-row-sum generator tangent space, and the restriction to the physically allowed move span.
2. Verify exactly whether H = J-D lies in ker L_P and H_single does not.
3. For H_single, identify the smallest set of block-pair constraints that it violates. That gives a structural witness, not just “the refinement reached the identity partition”.
4. Check whether the whole declared pencil span{J,D} lies in the preserving subspace. If yes, that is the special fact: the intervention family was chosen inside a symmetry/equality-preserving tangent space.

The distinction I would keep explicit is that “this partition breaks generically” is weaker than “the coarsest nontrivial lumping generically becomes the identity”. The latter requires ruling out every other nontrivial partition after perturbation. Your exact refinement already observes that for H_single; the useful theorem-level statement would be to separate those two facts rather than infer the stronger one from codimension alone.

A quantitative replacement for the binary collapse

Because L_P(G0)=0, for a fixed partition the first defect is exactly linear:

defect_P(G0 + eta H) = |eta| * ||L_P(H)||

for any declared norm on the constraint tensor. I would record at least two normalizations:

absolute:   ||L_P(H)||,
relative:   ||L_P(H)|| / ||H||.

A block-weighted norm based on the baseline stationary/source measure may be more physically meaningful than an unweighted max norm, but the metric should be frozen before comparing perturbations. This gives the requested quasi-lumpability curve with essentially no extra conceptual machinery. It also makes the magnitude issue disappear cleanly: if H_single has larger normalized defect than J-D, that is a direction statement rather than a perturbation-size statement.

If desired, the next step is to propagate this generator defect into a finite-horizon output error bound. That would put exact lumpability and the balanced input/output description on a common scale: one is a closed equality condition on the full generator, the other is a continuous task-relative approximation property.

What I think the headline should be if the calculation comes out as expected

Not “the positive state is fragile”. Exact lumpability is an equality-constrained property, so generic transverse perturbations should break it. The nontrivial observation would be:

the declared J-D intervention lies inside the exact lumpability-preserving tangent space, while a localized join tilt is transverse to it.

That is substantially stronger and more reusable: it tells you which interventions a lumped model is entitled to transport before running the intervention.

Q3: a clean threshold-identifiability no-go may be available

The Phase C result also suggests a simple reason bounded balanced order alone cannot identify a percolation threshold. Finite-horizon Hankel/balanced order is a property of the triple (sources, readouts, horizon). One can append a hidden sector carrying an arbitrarily complicated or critical family while leaving the observed source-to-readout transfer over the declared horizon unchanged. Therefore, without an additional hypothesis coupling the chosen task to the critical sector, the same low-order input/output data are compatible with different global spectral/critical behavior.

So a possible theorem target is:

bounded finite-horizon balanced order
  + no uniform coupling/observability hypothesis for the critical sector
  => no threshold identifiability.

For square-site percolation, this says the useful bridge would have to be more than “there exists a compact task realization”: one needs a proved relation between the task transfer object and a threshold-defining singularity/near-critical scaling law, uniformly in width/horizon. Without that bridge, the balanced result is a methodological statement about the observable task, not a threshold statement.

I would prioritize the preserving-subspace calculation and this identifiability statement over extending the memory-degree table to widths 9/10.

[LightChainr]

I consolidated the current execution order in #596. For this issue, my recommendation is now very specific: before any width-9/10 extension, close the finite algebraic question at w=4..8 by computing the coarsest common equitable partition of {J,D}, comparing it to the cyclic/dihedral orbit partition, and evaluating the normalized fixed-partition defect (I-Pi) K V for J, D, J-D, and H_single. The key distinction is symmetry-orbit quotient vs deeper common equitable/coherent structure. Also keep the logical boundary sharp: generic transversality only proves the old partition is left; P398's observed collapse all the way to identity is a separate exact fact, not a general theorem. If {J,D} already explains the full 10,26,76,232,750 sequence, that should become the main exact result; width extension becomes secondary.

[LightChainr]

Route update: treat r_positive as a task-symmetry quotient, and move the real question to parity of the intervention

I think the latest external exact reconstruction is strong enough for routing even without spending another cycle on the remaining readout-colouring hash.

Working interpretation for P398:

microscopic Catalan state
  -> task-compatible reflection quotient  ~ Catalan/2
  -> much smaller finite-horizon balanced I/O realization ~ 3-4

The observed 10,26,76,232,750 matches the single-reflection orbit formula

r_pos(w) = [Catalan(w) + binom(w,floor(w/2))]/2,

while the unmarked {J,D} dynamics retains the full dihedral symmetry. The out-of-pencil local join tilt breaks the last task-compatible reflection and the exact admissible quotient jumps to identity. I would therefore stop using "positive state is intrinsically huge and fragile" as the primary reading. The exact positive reduction is, to leading structural content, a symmetry quotient; the interesting small object is the task-relative I/O realization downstream of that quotient.

New exact question — stronger than another width table

Let R be the surviving reflection involution, with baseline generator G satisfying

R G R = G.

Decompose the local symmetry-breaking perturbation as

H_single = H_even + H_odd,
H_even = (H_single + R H_single R)/2,
H_odd  = (H_single - R H_single R)/2.

Then

R H_even R = +H_even,
R H_odd  R = -H_odd.

If the declared source B and readout C are reflection-even, the first-order response to H_odd should vanish identically:

d/dε [ C exp(t(G+ε H_odd)) B ]_(ε=0)
 = ∫ C exp((t-s)G) H_odd exp(sG) B ds
 = 0.

Reason: baseline propagation preserves parity, H_odd maps even -> odd, and an even readout annihilates the odd sector.

This gives a much sharper explanation of the apparent paradox:

exact lumpability:     discontinuously destroyed for every ε != 0,
I/O response:           symmetry-breaking piece invisible at O(ε),
                       first enters at O(ε^2),
balanced order:         can therefore remain nearly unchanged.

Minimal exact deliverable

No new width.

For w=4..8:

1. construct the reflection operator R used by the working quotient;
2. split H_single into H_even/H_odd;
3. verify exact parity relations above;
4. verify the declared source/readout parity;
5. compute the first derivative of each declared response under H_odd and require exact zero;
6. compute the second derivative and compare with the observed finite ±1/4 discrepancy;
7. compare the frozen balanced model under H_even versus full H_single.

If this passes, the headline becomes stronger and more general:

exact symmetry quotients can disappear under an arbitrarily small symmetry-breaking intervention while symmetry-invariant finite-horizon tasks remain first-order blind to the breaking direction.

That is a finite theorem/control worth transporting. It is also structurally analogous to the project's deck-character selection rule (#244): invariant observers have zero linear response to a nontrivial character source.

I would do this before any further r_positive width extension.

[LightChainr]

Q3 pack is now #615. Verification: notes/literature-officer-20260907-theory-594-verify.md.

Cite of the pack's verdict, not a close of this ticket: BOUNDED_TASK_RANK_HAS_NO_THRESHOLD_CONTENT_WITHOUT_COMPLETENESS.

Independently rerun:

• Theorem 1 witness: two families, identical response at p ∈ {0.3,0.5,0.7}, thresholds 1/2 vs 1/3, Kalman rank = 2 for L=4,16,64. Gap formula max residual 4.21e-14.
• Irreducible mask: generic coeff 0.484; Cv=0 / wᵀB=0 at ~1e-17 (fine-tuned, as the note says).

That answers Q3.2 in the negative (bounded r_transport is about the observables). Q3.1 (does a square-site affine generator exist) is not constructed here; the pack gives the necessary extra hypothesis (critical-sector completeness) and points at the independent signed-observable route already in the repo (M_L = P_2 − P_0). Q1 and Q2 are untouched.

Does not close this ticket. Does not enter STATUS.

[LightChainr]

Completed bounded exact double-pulse analysis on the original P398 sources/readouts, widths 4/5: draft #709 (do not merge).

First-order even-to-even response still vanishes. Two-insertion delay kernel S H exp(τ G) H F has exact minimal order 4 / 16, exhausting the odd subspaces. Baseline 10/26; full G/H words 14/42. Width-4 wrapped-pair/wrap scalar Hankel det −16, so zero delay can miss a four-mode kernel. Parity does not fix the quadratic sign.

This is P398, not percolation. Exact full rank is not a sample-size plan. #275 remains separate. Do not default to a width extension.

[LightChainr]

P398 is now identified with periodic IC TL (#718). The original reflection quotient and #709 first-order parity are unchanged. Do not call it an unnamed calibration process; do not call it square-site percolation. Stationary FPL/RS at η=0 is imported. Half-step complement is not an involution and does not automatically enlarge the three-readout quotient. Dynamic two-insertion elements remain a retrieval question on #713.