2026-08-31 团队收敛：结题与后续归属
Bochner/cyclotomic baseline、≥100-mode witness及指定档案分析已经交付，见开放PR #416及档案分析。数值解释采用统一单位修正版。后续可辨识的endpoint预测/干预归 #250；PR416继续开放审阅，不自动扩rank或合并。
Why this is a new question, not another rank vote
Read against the current result chain df57b69 -> 33c557b -> 93eaab1, not the earlier radius-five R2 headline. The rank-eight R2 kernel-plane bridge has already failed. The earlier finite-window rank tests remain valid within their statistical/observer contracts.
The acquisition code reveals a more basic distinction: P250 records a same-field, conjugate-charge spatial autocorrelation, not a measured physical row-transfer or dilation operator. Its natural global completion is therefore a positive spectral measure on a known finite translation group. A larger observed Hankel rank need not mean another CFT field, a Jordan multiplicity, or path memory.
This issue adds an exact result specific to the actual N505 construction and a production-facing convex analysis task. It does not reopen the discarded Alexander-map vote, alter a frozen scorer, authorize production, or impose a new research permission gate.
Code-level source contract
Pinned input semantics: 33c557b9aebed1bc9c07019b9cd5cee6c04be947.
scripts/z5_projective_leg_multiseparation_mc.py: the real root field is +1 for black-NN rank-one component membership, -1 for white-matching rank-one membership, and 0 otherwise.scripts/z5_projective_leg_cross_scale_mc.py: the parent is 10+i, order 101; its two children are 19+12i and 21-8i, each order 505. A parent field is the five-fiber DFT of that real root field.scripts/z5_projective_leg_bivariate_mc.py: the C4 gauge multiplies each charged value by a fifth root of unity; pair_value is F_r(origin)*F_{-r}(target), with F_{-r}=conj(F_r).scripts/score_z5_projective_leg_radius6_flat.py: the fitted matrix uses sum moments G(alpha+beta). It does not impose the finite-group positive spectral completion.
Use the intended uniform parent-anchor average and Bernoulli ensemble. Finite batch estimates, a deterministic pseudorandom sequence, and signed differences between hands are not themselves asserted to be positive-definite kernels.
Exact spatial representation
The parent translation group is
H = Z^2 / <(10,1),(-1,10)> = Z/101,
with coordinate j(a,b)=a-10b mod 101.
For a fixed hand and charge define the complete target
C(d)=E_omega[(1/101) sum_x F_omega(x) conj(F_omega(x+d))].
Writing Fhat(k)=sum_x F(x) exp(-2*pi*i*k*x/101) gives exactly
C(d)=sum_{k=0}^{100} w_k exp(-2*pi*i*k*j(d)/101),
w_k=E[|Fhat(k)|^2]/101^2 >= 0.
Consequences:
- The difference-indexed matrix
T[u,v]=C(v-u) is positive semidefinite; periodicity and C(-d)=conj(C(d)) are exact.
- The canonical spatial translations are commuting unitary operators with
U^101=I, V=U^-10. Their minimal polynomials divide a square-free polynomial, so they have no nontrivial Jordan blocks.
- This says nothing against Jordan structure of a different physical transfer, scale generator, generic-Q confluence or indefinite connectivity pairing. Spatial shifts and dilation must not be conflated.
Stronger result: at least 100 exact spatial modes are present
There is a short cyclotomic proof once one nonconstant field configuration is exhibited.
Every value of the charged field belongs to K=Q(zeta_5), including the /5 DFT normalization and any declared fifth-root site gauge. Since
[Q(zeta_505):Q(zeta_5)] = phi(505)/phi(5) = 100,
Phi_101(t)=1+t+...+t^100 is irreducible over K. For any nonconstant vector f in K^101, its coefficient polynomial P_f(t) cannot vanish at any nontrivial 101st root: otherwise P_f would be a scalar multiple of Phi_101, forcing all 101 entries equal.
Two explicit configurations in the real N505 geometry supply the witness:
- plus
19+12i: occupy the staircase with 19 east steps followed by 12 north steps, excluding the repeated endpoint; 31 occupied sites;
- minus
21-8i: occupy 21 east steps followed by 8 south steps; 29 occupied sites.
An independent integer-lift BFS finds one black rank-one component and one white-matching rank-one component in each configuration. With the valid parent section (j,0) and fibers (j+10f,f), each charge r=1,2 has exactly 29 nonzero and 72 zero parent-field values. Multiplication by arbitrary fifth-root site phases cannot make this vector constant.
Each witness has positive Bernoulli probability for every 0<p<1. Thus every w_k, k!=0, is strictly positive, for both hands and both measured charges. The full 101-by-101 spatial covariance, and equivalently the complete finite-group sum-Hankel matrix, has rank at least 100.
Scope: this excludes an exact global rank-eight translation realization of the complete same-observable endpoint series at this finite geometry. It does NOT exclude an accurate eight-state approximation on the measured window, prove that 100 modes are statistically resolvable, count physical fields, or supply a useful lower bound on the smallest spectral weight. The witness probabilities can be extremely small.
A concrete zero-new-sample next analysis
Replace another arbitrary-root rank escalation with a known-support positive Fourier completion of the archived raw same-hand/charge moments:
observed mean = A w, w>=0,
where A is the exact 101-frequency finite-group Fourier design, realified in the recorded displacement convention. Use all old4/radius5/radius6 covariance blocks with their actual independence/coupling and repeated-mean constraints.
Deliver:
- finite-group endpoint alias table; the radius-six diamond has 85 displacement labels but only 77 parent vertices, so aliases are repeated estimators of the same mean, not new spatial points;
- convex covariance-aware feasibility/distance to the positive Fourier cone;
- conservative bounds on selected spectral mass functionals or held-out endpoints, not an invented unique spectrum when completion is underdetermined;
- effective approximation error versus rank/bandwidth at a stated tolerance, rather than an exact finite-state/field count;
- charge/C4 transport only with the recorded gauge, and separate hand spectra unless a valid independently supplied intertwiner exists.
The periodicity/support/positivity constraints are properties of the raw autocorrelation. Do not impose them on arbitrary signed response matrices, orientation contrasts, or cross-field kernels.
A point-estimate distance or numerical optimizer is not an exact certificate. Strong statistical exclusions still require the existing typed uncertainty conventions; singular covariance directions and exact identities must be retained.
Ordered experiments: separate new physics from projection leakage
For commuting microscopic U,V and a projection P, Q=I-P,
[PUP,PVP] = PVQUP - PUQVP.
A compressed commutator can therefore arise solely from discarded coordinates. A genuine ordered two-morphism experiment remains valuable, but merely relabeling the order of two endpoint translations cannot observe new history. Declare the intervening gate, operation or resampling and its state update explicitly, and compare against this leakage alternative before naming curvature/noncommuting physical transport.
Deliverables and resources
- Complete written Fourier/cyclotomic proof plus two exact N505 witnesses.
- Standalone standard-library witness generator and a separate verifier using independent graph/rank reconstruction; corrupted certificates must fail.
- Small targeted tests, exact period/alias audit and source hashes.
- A later archived-data positive-cone scorer, preserving all frozen prior results.
The exact work is small local CPU only. No server access or new Monte Carlo is required. No production-data fit is claimed in the proof contribution.
Literature boundary
- Yang, Xie, Stoica, arXiv:1505.02510v3: multilevel Toeplitz/Vandermonde spectral methods. Their low-rank uniqueness hypotheses are not silently assumed here; the known finite group gives a direct Fourier proof.
- Widder, Zimmer, Schilling, arXiv:2503.20457v6 (24 April 2026): rigorous Mori-projection/GLE analysis, relevant to projection-induced memory, not a percolation identification.
- Liu, Jacobsen, Saleur, arXiv:2403.19830: physical Potts/loop transfer Jordan structure may emerge in a limit. This is a different operator from finite-group spatial shifts.
- Camia, Feng, arXiv:2508.16047v2 (1 June 2026): triangular-percolation energy/logarithmic fields are a physical control, not contradicted by a positive spatial autocorrelation representation.
Related: #250, #249, #370, #218, #255, PR #385.
Main conjecture after the exact result
The growth from a line-restricted small rank to the present two-dimensional rank ladder is primarily an increase in resolved spatial spectral content, potentially amplified by the chosen fiber-section gauge. This is a testable approximation hypothesis, not established by the exact ≥100-mode theorem. The high-information comparison is positive-spectrum compression versus genuinely intervention-sensitive memory, rather than repeatedly interpreting the next compatible truncation rank as the number of hidden physical fields.
2026-08-31 团队收敛：结题与后续归属
Bochner/cyclotomic baseline、≥100-mode witness及指定档案分析已经交付，见开放PR #416及档案分析。数值解释采用统一单位修正版。后续可辨识的endpoint预测/干预归 #250；PR416继续开放审阅，不自动扩rank或合并。
Why this is a new question, not another rank vote
Read against the current result chain
df57b69 -> 33c557b -> 93eaab1, not the earlier radius-five R2 headline. The rank-eight R2 kernel-plane bridge has already failed. The earlier finite-window rank tests remain valid within their statistical/observer contracts.The acquisition code reveals a more basic distinction: P250 records a same-field, conjugate-charge spatial autocorrelation, not a measured physical row-transfer or dilation operator. Its natural global completion is therefore a positive spectral measure on a known finite translation group. A larger observed Hankel rank need not mean another CFT field, a Jordan multiplicity, or path memory.
This issue adds an exact result specific to the actual N505 construction and a production-facing convex analysis task. It does not reopen the discarded Alexander-map vote, alter a frozen scorer, authorize production, or impose a new research permission gate.
Code-level source contract
Pinned input semantics:
33c557b9aebed1bc9c07019b9cd5cee6c04be947.scripts/z5_projective_leg_multiseparation_mc.py: the real root field is +1 for black-NN rank-one component membership, -1 for white-matching rank-one membership, and 0 otherwise.scripts/z5_projective_leg_cross_scale_mc.py: the parent is10+i, order 101; its two children are19+12iand21-8i, each order 505. A parent field is the five-fiber DFT of that real root field.scripts/z5_projective_leg_bivariate_mc.py: the C4 gauge multiplies each charged value by a fifth root of unity;pair_valueisF_r(origin)*F_{-r}(target), withF_{-r}=conj(F_r).scripts/score_z5_projective_leg_radius6_flat.py: the fitted matrix uses sum momentsG(alpha+beta). It does not impose the finite-group positive spectral completion.Use the intended uniform parent-anchor average and Bernoulli ensemble. Finite batch estimates, a deterministic pseudorandom sequence, and signed differences between hands are not themselves asserted to be positive-definite kernels.
Exact spatial representation
The parent translation group is
H = Z^2 / <(10,1),(-1,10)> = Z/101,with coordinate
j(a,b)=a-10b mod 101.For a fixed hand and charge define the complete target
C(d)=E_omega[(1/101) sum_x F_omega(x) conj(F_omega(x+d))].Writing
Fhat(k)=sum_x F(x) exp(-2*pi*i*k*x/101)gives exactlyC(d)=sum_{k=0}^{100} w_k exp(-2*pi*i*k*j(d)/101),w_k=E[|Fhat(k)|^2]/101^2 >= 0.Consequences:
T[u,v]=C(v-u)is positive semidefinite; periodicity andC(-d)=conj(C(d))are exact.U^101=I,V=U^-10. Their minimal polynomials divide a square-free polynomial, so they have no nontrivial Jordan blocks.Stronger result: at least 100 exact spatial modes are present
There is a short cyclotomic proof once one nonconstant field configuration is exhibited.
Every value of the charged field belongs to
K=Q(zeta_5), including the /5 DFT normalization and any declared fifth-root site gauge. Since[Q(zeta_505):Q(zeta_5)] = phi(505)/phi(5) = 100,Phi_101(t)=1+t+...+t^100is irreducible over K. For any nonconstant vectorf in K^101, its coefficient polynomialP_f(t)cannot vanish at any nontrivial 101st root: otherwiseP_fwould be a scalar multiple ofPhi_101, forcing all 101 entries equal.Two explicit configurations in the real N505 geometry supply the witness:
19+12i: occupy the staircase with 19 east steps followed by 12 north steps, excluding the repeated endpoint; 31 occupied sites;21-8i: occupy 21 east steps followed by 8 south steps; 29 occupied sites.An independent integer-lift BFS finds one black rank-one component and one white-matching rank-one component in each configuration. With the valid parent section
(j,0)and fibers(j+10f,f), each charge r=1,2 has exactly 29 nonzero and 72 zero parent-field values. Multiplication by arbitrary fifth-root site phases cannot make this vector constant.Each witness has positive Bernoulli probability for every
0<p<1. Thus everyw_k,k!=0, is strictly positive, for both hands and both measured charges. The full 101-by-101 spatial covariance, and equivalently the complete finite-group sum-Hankel matrix, has rank at least 100.Scope: this excludes an exact global rank-eight translation realization of the complete same-observable endpoint series at this finite geometry. It does NOT exclude an accurate eight-state approximation on the measured window, prove that 100 modes are statistically resolvable, count physical fields, or supply a useful lower bound on the smallest spectral weight. The witness probabilities can be extremely small.
A concrete zero-new-sample next analysis
Replace another arbitrary-root rank escalation with a known-support positive Fourier completion of the archived raw same-hand/charge moments:
observed mean = A w,w>=0,where A is the exact 101-frequency finite-group Fourier design, realified in the recorded displacement convention. Use all old4/radius5/radius6 covariance blocks with their actual independence/coupling and repeated-mean constraints.
Deliver:
The periodicity/support/positivity constraints are properties of the raw autocorrelation. Do not impose them on arbitrary signed response matrices, orientation contrasts, or cross-field kernels.
A point-estimate distance or numerical optimizer is not an exact certificate. Strong statistical exclusions still require the existing typed uncertainty conventions; singular covariance directions and exact identities must be retained.
Ordered experiments: separate new physics from projection leakage
For commuting microscopic U,V and a projection P, Q=I-P,
[PUP,PVP] = PVQUP - PUQVP.A compressed commutator can therefore arise solely from discarded coordinates. A genuine ordered two-morphism experiment remains valuable, but merely relabeling the order of two endpoint translations cannot observe new history. Declare the intervening gate, operation or resampling and its state update explicitly, and compare against this leakage alternative before naming curvature/noncommuting physical transport.
Deliverables and resources
The exact work is small local CPU only. No server access or new Monte Carlo is required. No production-data fit is claimed in the proof contribution.
Literature boundary
Related: #250, #249, #370, #218, #255, PR #385.
Main conjecture after the exact result
The growth from a line-restricted small rank to the present two-dimensional rank ladder is primarily an increase in resolved spatial spectral content, potentially amplified by the chosen fiber-section gauge. This is a testable approximation hypothesis, not established by the exact ≥100-mode theorem. The high-information comparison is positive-spectrum compression versus genuinely intervention-sensitive memory, rather than repeatedly interpreting the next compatible truncation rank as the number of hidden physical fields.