两旧N425检查点的边来自neutral-component bicliques，W2差540=472+68，c3=583/509；这是既有选定样本的几何解释，不是新人口H4证据。下一项是cut-invariant/covariant component-incidence/overlap的paired population响应，或较长时距vertex reliability。不重复firstcut、图重建或已解释的W2碰撞。交叉matching图、退化商图、最小/有界状态与场识别不由此自动覆盖。
Previous body preserved — historical context
Read against the updated frontier
This starts after main c64bfabde8c8bb13290d4f5d8e5f44c1779d4d30, #403's killed survival certificate, and #401 / PR #415 (d09f925) including the three-step overlap counterexample. It does not propose another c2,c3,... counter. The full survival vector has already been identified as sufficient for the remaining unmarked rank trace at a fixed checkpoint. The question here is different: does that vector, recomputed on the actual configuration after each insertion, itself define an autonomous Markov state?
New exact answer: no, already on the square N16 control. All unbranched future rank distributions can agree while the next predictive-state distribution differs. This is the trace-equivalence / probabilistic-bisimulation distinction made concrete in the existing percolation observable.
1. Equal complete survival law, different branching law
Use the honest square torus P=diag(4,4), with physical site coordinates (x,y) modulo four. Define
A={(0,0),(1,0),(2,0),(3,0),(1,1),(3,1),(0,3),(1,3)},
B={(0,0),(1,0),(2,0),(0,1),(1,1),(2,1),(3,1),(0,3)}.
These are row-major masks 12463 and 4343 under label=x+4*y; the coordinate sets, not the masks, specify the witness.
Both have k=8, ambient rank one and primitive line (1,0). For
b_m(S)=#{U subset vacant(S): |U|=m, r(S union U)=1},
s_m(S)=b_m(S)/C(8,m),
their entire count vector is identical:
b=(1,7,18,20,8,0,0,0,0),
s=(1,7/8,9/14,5/14,4/35,0,0,0,0).
Consequently every future event depending only on the unmarked rank trajectory has the same probability at A and B. This is not a finite-horizon truncation failure.
Now perform a different, operationally explicit experiment:
- make one ordinary, uniformly random vacant insertion, obtaining S';
- clone S' into two independent futures;
- make one uniformly random vacant insertion in each clone;
- ask whether both clones remain rank one. If the common insertion already reaches rank two, score zero.
Write x(S') for the existing one-site exit count. Among the eight possible common insertions the exact distributions are:
| initial state |
already absorbed |
safe, x'=1 |
safe, x'=2 |
safe, x'=3 |
| A |
1 |
3 |
2 |
2 |
| B |
1 |
1 |
6 |
0 |
There are seven vacancies at S', so
B_11(A)=[3*6^2+2*5^2+2*4^2]/(8*7^2)=95/196,
B_11(B)=[1*6^2+6*5^2]/(8*7^2)=93/196,
B_11(A)-B_11(B)=1/98.
Direct enumeration gives 190 versus 186 successful branch choices out of 392. Their unbranched two-step survival is still exactly 9/14 in both cases.
Cloning immediately at the original checkpoint would not detect this: its success probability is a product of the already equal s_m. The shared update before branching is essential.
2. This also fails under the actual uniform-permutation law
The preceding pair disproves strong lumpability. A separate selected-prefix enumeration establishes a history-dependent transition under the actual sampling law, not just an adversarial mixture of starting states.
At k=8, condition on rank one, line (1,0) and the complete count vector above. This stratum contains 192 current subsets and 7,741,440 ordered length-eight prefixes. Let E be the event that the next state remains rank one and has x(S_9)=3. Then
| K1 |
ordered prefixes |
P(E given the full current signature and K1) |
| 4 |
110,592 |
1/6 |
| 5 |
442,368 |
1/6 |
| 6 |
1,198,080 |
2/13 |
| 7 |
2,442,240 |
8/53 |
| 8 |
3,548,160 |
2/11 |
In particular 2/11-1/6=1/66. The first birth K1 is measurable from the past rank records. E is measurable from the next full survival signature. Thus the process of recomputed microscopic survival signatures is not Markov under the uniform-permutation law.
Independent checks already executed locally: the inherited integer-lift BFS and a separately written row-major potential-union-find agree on rank and primitive line for all 65,536 N16 configurations. Integer prefix DP agrees with exhaustive permutation enumeration of all 7,741,440 prefixes in the selected stratum, not all 16P8 prefixes. Subset enumeration independently checks the witness's complete survival counts and the branch probabilities. The reproducible certificate will accompany this issue as an additive commit.
3. Exact bounded state-refinement census
For each tested quotient, group rank-one states by (k,line,complete b-vector). Separately compute the coarsest strong Markov partition preserving k and line by backward refinement of the distributions over successor classes, with rank-two configurations collapsed to one absorbing cemetery state.
| quotient |
rank-one configurations |
full-survival classes |
strong Markov classes |
split survival classes |
| 2+i, N5 |
10 |
2 |
2 |
0 |
| 3, N9 |
162 |
10 |
10 |
0 |
| 3+i, N10 |
310 |
16 |
16 |
0 |
| 3+2i, N13 |
2,340 |
62 |
62 |
0 |
| 4, N16 |
19,932 |
210 |
214 |
4 |
| 4+i, N17 |
38,896 |
346 |
390 |
42 |
Counts exclude the common cemetery state. This is only this six-quotient census, not an all-HNF minimality theorem. No asymptotic rank-growth conclusion follows from it.
4. The correct autonomous object is an observable algebra, not only a linear row space
Include a cemetery state so the transition operator P is stochastic. Begin with the rank/k/line observation functions. On a finite state space, a partition is strongly lumpable precisely when its block-constant function algebra is invariant under P.
This gives a constructive target:
A_pred = smallest unital pointwise algebra containing the declared observations
and invariant under P.
Why an algebra? A linear span of P^m 1_alive predicts single traces, but it need not contain nonlinear functions of successor predictions. Independent future forks introduce pointwise products. An ordinary transition introduces P. Alternating these operations generates a branching-sensitive test language.
For a family of future survival functions f_i at the successor layer, the matrix
Gamma_ij(S)=P(f_i f_j)(S)-P f_i(S)*P f_j(S)
is exactly the covariance matrix of successor predictions and hence positive semidefinite. In the witness, the relevant means coincide while P(f_1^2) differs by 1/98.
This is an application of finite probabilistic state minimization, not a new abstract bisimulation theorem. Nor should this pointwise function algebra be identified with #398's Q-adic or Hankel multiplication algebra: a finite real function algebra is reduced; it does not by itself manufacture a physical nilpotent or an LCFT module.
5. Preserve the useful weaker model
The full survival vector remains exactly sufficient for its original rank-only future language. A belief state conditioned only on continued survival can update by shifting and normalizing the survival curve. That filtered belief is not the same object as recomputing the signature of the actual, unobserved microscopic successor.
Similarly, the process-Hankel suggestion in #401 is valid for its declared future language. The new result does not refute minimal linear realization there. It shows that branching, intermediate state readout or interventions require a larger language and potentially a different state quotient.
If the scientific objective is only to estimate remaining lifetime, no branching-aware enlargement is necessary. It becomes necessary when claiming an autonomous coarse process or composing state-conditioned operations.
6. Next experiment: branch after a common, scaled continuation
Reuse the current-state snapshot/pilot machinery, without changing its frozen primary score. Freeze a small set of shared-prefix lengths a and subsequent horizons m,n. For layered killed kernels, measure
B_(a;m,n)(S)=P_k ... P_(k+a-1)[s_m s_n](S),
with survival functions evaluated at the correct successor layer. At a=0 this reduces to the existing independent-future product; a>0 probes missing update structure.
At larger N, choose both shared and branch horizons on a declared near-critical scale. N^(5/8) is a conditional square-site candidate, not a theorem here; an independently calibrated intrinsic/pivotal clock is preferable when available. Keep the early checkpoint, shared continuation and cloned futures as nested clusters in covariance estimation. Do not treat clones as independent initial configurations.
An exact tiny verifier should return a smallest distinguishing branching test when a proposed state quotient fails. For approximate closure, compare successor laws with an explicitly chosen total-variation or behavioural-distance tolerance and propagate error over the frozen horizon. Small mean residuals or a low Hankel rank alone are not such a guarantee.
Risky hypothesis: a compact geometry-aware state may predict both traces and branching laws after rescaling even though the raw full-survival signature does not. The alternative is sharper than another age coefficient: the branching-sensitive state complexity grows, so an autonomous low-dimensional description is the wrong target even when single-trace prediction stays cheap.
7. arXiv connections checked
- Turkenburg, Beohar, van Breugel, Kupke, Rot, 2504.08639v2, revised 13 October 2025: constructive witnesses for lower bounds on behavioural distances. This motivates returning a distinguishing experiment rather than only reporting a failed rank or residual.
- Spork, Baier, Katoen, Klueppelholz, Piribauer, 2505.15587v2: approximate CTMC bisimulation separates transition-probability error from clock-rate error and bounds timed reachability. Our process is layered discrete growth; their CTMC bounds cannot be imported without the clock/model map.
- Chen, Clerc, Panangaden, 2511.21621: transition-based versus path-based behavioural pseudometrics. These are useful distinct validation targets, not a percolation result.
- Garban, Pete, Schramm, 1305.5526: near-critical percolation is constructed on a full geometric state space. It does not imply a finite rank/line/reliability compression is Markov.
- Widder, Zimmer, Schilling, 2503.20457: projection-induced memory and the full dynamics must be distinguished. The finite counterexample here is direct, with no GLE approximation.
- Alves, Baldasso, Moreira, Teixeira, 2606.11503, submitted 9 June 2026: hierarchical percolation offers a separately defined recursive control for state compression and scaling. Its graph-replacement theorems are not square-site universality results.
Related: #401, #403, #249, #370, #398, #400, #419. Exact finite results and a proposed acquisition only; no new Monte Carlo production, continuum-memory law, physical field identification or repository-wide CI claim.
Research delivery — 2026-08-31 e / completed million-sample response and physical next readout
新的uniform-blockade结果已完整读入，固定commit d53db2f，open PR #484。
所有均匀独立封锁的均值剂量曲线都由完整birth clock经二项变换给出，B_a B_b=B_ab；均匀固定封锁数的均值同样闭合。因此更密的mean-dose曲线无法区分isoclock几何。新增信息在空间标记：singleton条件响应方差由最终触发site的collision probability决定。构造的double-star与C4+inert例子具有完全相同时钟/所有均值，k=4空间方差却为14/625与4/625。
下一项可区分机制的读数是spatial conditional variance/final-site collision，再接paired、stratum-weighted总体加载；不再做首次uniform semigroup或重复mean-dose。这里的方差不是suffix抽样噪声，构造例子也不是147真实前缀中已找到的等时钟pair。全部147 clocks、noise budget和canonical crossings的旧成果继续保留。当前注意力。所有并行路线仍开放。
Previous body preserved — historical context
Research delivery — 2026-08-31 d / completed physical source and updated handoffs
又一批完成成果已回填，避免重复派首次工作：实际报告与精确来源。
下一物理范围是paired、stratum-weighted总体加载和source/readout耦合；用已经求出的完整law与noise预算，不再从首次147扩展、首个variance或两例longer-horizon开始。12包含在147中，均依赖原N425源；上述branch_only/open_pr状态详见来源。优先级只分配注意力。
Previous body preserved — historical context
Context reconciliation — 2026-08-31 c / completed work before next assignments
本线已远超“解释两例W2后做longer-horizon”。完整来源/档案交接记录：
下一问题是这些完整结构在更广的已存前缀总体/配对方向上如何分布、耦合H4、带来多少计算信息增益。22选择图、12同向同age/line前缀、20k/100k总体档案分别记录，不把选择例子当总体比例。已有counter可重构，但本轮没有运行重放、DP或新采样。引用讨论不等于结果已合入Draft/main；优先级不锁任务。
Previous body preserved — dated historical context
Scientific output update — 2026-08-31 second continuation
Open PR #491 / ab90201 已提交occupied-cut机制。在所述embedded rank-one torus范围，切occupied essential cycle后，rank2 continuation等价于两端点vertex connectivity；保留初始cut并activate/contract得到update-closed网络。
两旧N425检查点的边来自neutral-component bicliques，W2差540=472+68，c3=583/509；这是既有选定样本的几何解释，不是新人口H4证据。下一项是cut-invariant/covariant component-incidence/overlap的paired population响应，或较长时距vertex reliability。不重复firstcut、图重建或已解释的W2碰撞。交叉matching图、退化商图、最小/有界状态与场识别不由此自动覆盖。
Previous body preserved — historical context
Read against the updated frontier
This starts after main
c64bfabde8c8bb13290d4f5d8e5f44c1779d4d30, #403's killed survival certificate, and #401 / PR #415 (d09f925) including the three-step overlap counterexample. It does not propose anotherc2,c3,...counter. The full survival vector has already been identified as sufficient for the remaining unmarked rank trace at a fixed checkpoint. The question here is different: does that vector, recomputed on the actual configuration after each insertion, itself define an autonomous Markov state?New exact answer: no, already on the square N16 control. All unbranched future rank distributions can agree while the next predictive-state distribution differs. This is the trace-equivalence / probabilistic-bisimulation distinction made concrete in the existing percolation observable.
1. Equal complete survival law, different branching law
Use the honest square torus
P=diag(4,4), with physical site coordinates(x,y)modulo four. DefineThese are row-major masks 12463 and 4343 under
label=x+4*y; the coordinate sets, not the masks, specify the witness.Both have
k=8, ambient rank one and primitive line(1,0). Fortheir entire count vector is identical:
Consequently every future event depending only on the unmarked rank trajectory has the same probability at A and B. This is not a finite-horizon truncation failure.
Now perform a different, operationally explicit experiment:
Write
x(S')for the existing one-site exit count. Among the eight possible common insertions the exact distributions are:There are seven vacancies at S', so
Direct enumeration gives 190 versus 186 successful branch choices out of 392. Their unbranched two-step survival is still exactly 9/14 in both cases.
Cloning immediately at the original checkpoint would not detect this: its success probability is a product of the already equal
s_m. The shared update before branching is essential.2. This also fails under the actual uniform-permutation law
The preceding pair disproves strong lumpability. A separate selected-prefix enumeration establishes a history-dependent transition under the actual sampling law, not just an adversarial mixture of starting states.
At k=8, condition on rank one, line
(1,0)and the complete count vector above. This stratum contains 192 current subsets and 7,741,440 ordered length-eight prefixes. Let E be the event that the next state remains rank one and hasx(S_9)=3. ThenIn particular
2/11-1/6=1/66. The first birth K1 is measurable from the past rank records. E is measurable from the next full survival signature. Thus the process of recomputed microscopic survival signatures is not Markov under the uniform-permutation law.Independent checks already executed locally: the inherited integer-lift BFS and a separately written row-major potential-union-find agree on rank and primitive line for all 65,536 N16 configurations. Integer prefix DP agrees with exhaustive permutation enumeration of all 7,741,440 prefixes in the selected stratum, not all 16P8 prefixes. Subset enumeration independently checks the witness's complete survival counts and the branch probabilities. The reproducible certificate will accompany this issue as an additive commit.
3. Exact bounded state-refinement census
For each tested quotient, group rank-one states by
(k,line,complete b-vector). Separately compute the coarsest strong Markov partition preserving k and line by backward refinement of the distributions over successor classes, with rank-two configurations collapsed to one absorbing cemetery state.Counts exclude the common cemetery state. This is only this six-quotient census, not an all-HNF minimality theorem. No asymptotic rank-growth conclusion follows from it.
4. The correct autonomous object is an observable algebra, not only a linear row space
Include a cemetery state so the transition operator P is stochastic. Begin with the rank/k/line observation functions. On a finite state space, a partition is strongly lumpable precisely when its block-constant function algebra is invariant under P.
This gives a constructive target:
Why an algebra? A linear span of
P^m 1_alivepredicts single traces, but it need not contain nonlinear functions of successor predictions. Independent future forks introduce pointwise products. An ordinary transition introduces P. Alternating these operations generates a branching-sensitive test language.For a family of future survival functions f_i at the successor layer, the matrix
is exactly the covariance matrix of successor predictions and hence positive semidefinite. In the witness, the relevant means coincide while
P(f_1^2)differs by 1/98.This is an application of finite probabilistic state minimization, not a new abstract bisimulation theorem. Nor should this pointwise function algebra be identified with #398's Q-adic or Hankel multiplication algebra: a finite real function algebra is reduced; it does not by itself manufacture a physical nilpotent or an LCFT module.
5. Preserve the useful weaker model
The full survival vector remains exactly sufficient for its original rank-only future language. A belief state conditioned only on continued survival can update by shifting and normalizing the survival curve. That filtered belief is not the same object as recomputing the signature of the actual, unobserved microscopic successor.
Similarly, the process-Hankel suggestion in #401 is valid for its declared future language. The new result does not refute minimal linear realization there. It shows that branching, intermediate state readout or interventions require a larger language and potentially a different state quotient.
If the scientific objective is only to estimate remaining lifetime, no branching-aware enlargement is necessary. It becomes necessary when claiming an autonomous coarse process or composing state-conditioned operations.
6. Next experiment: branch after a common, scaled continuation
Reuse the current-state snapshot/pilot machinery, without changing its frozen primary score. Freeze a small set of shared-prefix lengths a and subsequent horizons m,n. For layered killed kernels, measure
with survival functions evaluated at the correct successor layer. At a=0 this reduces to the existing independent-future product; a>0 probes missing update structure.
At larger N, choose both shared and branch horizons on a declared near-critical scale.
N^(5/8)is a conditional square-site candidate, not a theorem here; an independently calibrated intrinsic/pivotal clock is preferable when available. Keep the early checkpoint, shared continuation and cloned futures as nested clusters in covariance estimation. Do not treat clones as independent initial configurations.An exact tiny verifier should return a smallest distinguishing branching test when a proposed state quotient fails. For approximate closure, compare successor laws with an explicitly chosen total-variation or behavioural-distance tolerance and propagate error over the frozen horizon. Small mean residuals or a low Hankel rank alone are not such a guarantee.
Risky hypothesis: a compact geometry-aware state may predict both traces and branching laws after rescaling even though the raw full-survival signature does not. The alternative is sharper than another age coefficient: the branching-sensitive state complexity grows, so an autonomous low-dimensional description is the wrong target even when single-trace prediction stays cheap.
7. arXiv connections checked
Related: #401, #403, #249, #370, #398, #400, #419. Exact finite results and a proposed acquisition only; no new Monte Carlo production, continuum-memory law, physical field identification or repository-wide CI claim.