目标：在进入 N1105 四角 tomography 前，先用更小/更强信号的同圆 pair 确认 post-H4 residual 真的是 angular-scalar
当前 deterministic safe transfer 已经给同一 axis/(3,4) H4-null projector：
ell=5: p_H0-pc ~ -9.81e-6
ell=10: p_H0-pc ~ -7.57e-8
历史 #47 在这些数据之前就预言：若 standard Potts scalar V_<1,4> (x=33/4) 主导 post-H4 block，则同一个 scalar projector应满足
两尺度给 reference-free
pc_hat^(H4,V14)=0.5927460516782668,
C7_hat=-0.7661178948.
但 axis/(3,4) 两角 projector仍携带较大的 H8/H12 geometry weights。需要一个新的 same-circle pair 先把 H8也几乎关掉，而不是立刻去 N1105 追 O(1e-11) 的四角 scalar coefficient。
N377 double-notch geometry
377 = 4^2+19^2 = 11^2+16^2.
两方向都 primitive，故 square-period tori有：
same determinant N=377,
same physical circumference ell=sqrt(377)=19.416...,
same square modulus i,
same Smith class (1,377).
方向：
H4 values：
h1=+0.6748868985...
h2=-0.7435428378...
普通 two-angle H4 projector
p_perp4=(h1 p2-h2 p1)/(h1-h2)
exact annihilates all H4 content。
Because H8=2H4^2-1, its residual H8 gain after H4 projection is
C8=-(1+2 h1 h2)=+0.0036146395...
only about 0.36%. Later harmonics are not simultaneously nulled (C12≈-0.1378, C16≈-0.9811), so this is a specific H4/H8 double-notch rather than a trivial all-angle blind spot.
Expected scalar signal scale
Using the reference-free C7_hat only as planning input,
This is ~43x larger than the expected N1105 scalar signal (~1.7e-11).
The raw H4 root shifts at this ell remain O(1e-6), so root accuracy/covariance must be reported explicitly; the scientific object is a cancellation of large deterministic roots.
Phase 0 — specialized exact state/resource pilot only
Use the committed #771 lifted-gain safe semantics. Do not use generic Python BFS as the production engine; it already exceeded 200k-state cap.
A specialized width=1 / large-memory C++ implementation is appropriate. Exact transformed-edge memories are approximately:
(4,19): G4 memory 19, G8 memory 23
(11,16): G4 memory 16, G8 memory 27
First deliverable only:
safe-state counts for all 4 automata,
aggregated transition counts,
peak RAM,
wall seconds,
Perron solve residual at one p,
root precision estimate,
checkpoint/restart if needed.
stop rule
If certified/semi-certified root differences cannot realistically reach <<1e-10 with current resources, stop after the cost model. Do not substitute MC, larger N, or lower-precision extrapolation.
Phase 1 — only if Phase 0 is numerically comfortable
Compute charge roots p1,p2 from safe Perron equality with no external pc.
Primary outputs:
p_perp4_N377,
H4 amplitude,
physical thermal slopes D1,D2,
Perron/root numerical errors.
Then compare after construction with the reference-free two-stage predictor
pc_hat^(H4,V14)=0.5927460516782668,
C7_hat=-0.7661178948.
The V14-bottom prediction is
p_perp4_N377
≈ pc_hat + C7_hat * 377^(-7/2)
up to H12/higher and further scalar powers. Do not use the high-precision external pc to locate roots or tune amplitudes.
Decision
residual follows scalar ell^-7 within numerical/higher-harmonic allowance
Promote
POST_H4_ODD_SCALAR_x33/4: strong deterministic evidence.
Then N1105 (#807) is justified as stage-2 H0/H8/H12 tomography; generic-Q #586 can address V_<1,4> bottom vs W(2,2)-related logarithmic collision.
residual much smaller / incompatible in sign or size
The axis/(3,4) two-scale ell^-7 agreement was accidental/mixed. Reopen H12/higher or nonlinear angular channels before spending N1105.
H8 leakage unexpectedly dominates despite C8≈0.0036
Then current H8 amplitude estimate/mechanism is qualitatively wrong; report the counterexample rather than adding fit terms.
Secondary, only if cheap
At each root evaluate physical thermal slope D=|u| Theta_p ell^(1/4) as another denominator-scalar regression. Do not compute full near-critical curves unless root result itself is informative.
Boundaries
Related: #47, #61, #586, #768, #802, #807.
目标：在进入 N1105 四角 tomography 前，先用更小/更强信号的同圆 pair 确认 post-H4 residual 真的是 angular-scalar
当前 deterministic safe transfer 已经给同一 axis/(3,4) H4-null projector：
历史 #47 在这些数据之前就预言：若 standard Potts scalar
V_<1,4>(x=33/4) 主导 post-H4 block，则同一个 scalar projector应满足两尺度给 reference-free
但 axis/(3,4) 两角 projector仍携带较大的 H8/H12 geometry weights。需要一个新的 same-circle pair 先把 H8也几乎关掉，而不是立刻去 N1105 追 O(1e-11) 的四角 scalar coefficient。
N377 double-notch geometry
两方向都 primitive，故 square-period tori有：
方向：
H4 values：
普通 two-angle H4 projector
exact annihilates all H4 content。
Because
H8=2H4^2-1, its residual H8 gain after H4 projection isonly about 0.36%. Later harmonics are not simultaneously nulled (
C12≈-0.1378,C16≈-0.9811), so this is a specific H4/H8 double-notch rather than a trivial all-angle blind spot.Expected scalar signal scale
Using the reference-free
C7_hatonly as planning input,This is ~43x larger than the expected N1105 scalar signal (
~1.7e-11).The raw H4 root shifts at this ell remain O(1e-6), so root accuracy/covariance must be reported explicitly; the scientific object is a cancellation of large deterministic roots.
Phase 0 — specialized exact state/resource pilot only
Use the committed #771 lifted-gain safe semantics. Do not use generic Python BFS as the production engine; it already exceeded 200k-state cap.
A specialized width=1 / large-memory C++ implementation is appropriate. Exact transformed-edge memories are approximately:
First deliverable only:
stop rule
If certified/semi-certified root differences cannot realistically reach
<<1e-10with current resources, stop after the cost model. Do not substitute MC, larger N, or lower-precision extrapolation.Phase 1 — only if Phase 0 is numerically comfortable
Compute charge roots
p1,p2from safe Perron equality with no external pc.Primary outputs:
Then compare after construction with the reference-free two-stage predictor
The V14-bottom prediction is
up to H12/higher and further scalar powers. Do not use the high-precision external pc to locate roots or tune amplitudes.
Decision
residual follows scalar ell^-7 within numerical/higher-harmonic allowance
Promote
Then N1105 (#807) is justified as stage-2 H0/H8/H12 tomography; generic-Q #586 can address
V_<1,4>bottom vs W(2,2)-related logarithmic collision.residual much smaller / incompatible in sign or size
The axis/(3,4) two-scale ell^-7 agreement was accidental/mixed. Reopen H12/higher or nonlinear angular channels before spending N1105.
H8 leakage unexpectedly dominates despite C8≈0.0036
Then current H8 amplitude estimate/mechanism is qualitatively wrong; report the counterexample rather than adding fit terms.
Secondary, only if cheap
At each root evaluate physical thermal slope
D=|u| Theta_p ell^(1/4)as another denominator-scalar regression. Do not compute full near-critical curves unless root result itself is informative.Boundaries
V_<1,4>from its Q=1 logarithmic collision block.Related: #47, #61, #586, #768, #802, #807.