Boundary research: spectra, profiles, anomalies and certified response

Separate tests help investigate a proposed GHU boundary mechanism: roots of a flat fermion boundary problem, the numerical rank of overlapping profiles, exact parity-inserted spinor traces and a restricted six-dimensional SU(2) tensor condition. The fifth compares two wall profiles after common pole and residue matching, using replayed continuum certificates. The sixth counts finite neutral-complex classes on exact strata. Section 6b adds a continuous mass enclosure and isolation test for the light neutral vector at three exact Wilson angles.

Getting started All guides Glossary Download manual (PDF)

Neutral and Z supplement · 28 October

Open tool Video chapter

Control names are kept as they appear in the application. Start with a reference, change one input at a time, and retain the assumptions with your result.

What this tool is for

Use these tools to find which requirement fails and why before attempting a full model. A mass gap addresses unwanted paired modes; Gram rank tests linear independence; inserted traces expose localized anomaly terms; the tensor condition tests a restricted quantization requirement. Passing these tests does not prove a viable Standard Model embedding or a complete quantum theory. Matched response asks whether fixing two local conditions removes all finite-width profile dependence in the declared quadratic model.

First steps

Open Anomalies & proton and find Boundary research. Select English in the card. Press Research baseline to restore all inputs; Demo runs the worked examples and ends with this guide.
Set Boundary strength to 0, then 1. Compare the first normal root and the paired curve at the same imposed Wilson angle.
Set Profile example M to 0, then 1. Compare the Gram eigenvalues and numerical rank.
Switch the parity insertion between C1, C0 and C0 W. Read the exact trace table, keeping its degree and support dimension in mind.
With lepton chirality Opposite to Q, compare one and three identical 6D families. Read the signed doublet excess and its residue.
Select Certified response query: t = 1/4, then t = 1/2. Read section 5 and the positive lower bound of uniform minus triangular. Download the proof and certificate ZIP and follow its README to replay the independent Sage verification.
Select Exact neutral stratum: s ≠ 0 and s = 0. In section 6 the quotient dimension changes from 0 to 1. Download the ZIP and run python verify.py certificate.json with SymPy.
Select Certified neutral angle: 1/10, 1/5, then 1/4. Read section 6b: compare the rational mass interval and positive separation from the next active mode. Download the continuous proof and replay it with SymPy.
Expand assumptions and provenance. Pin a comparison, export the card JSON and save the application link to repeat the calculation.

Controls and inputs

Control	Meaning	Try this
Boundary strength κ²L	Nonnegative dimensionless coupling a of the uniform flat boundary action, from 0 to 100. L = πR.	Use 0 and 1; the first kR changes from 0 to approximately 0.273853.
Imposed Wilson θ [rad]	Supplied angle from 0 to π for the periodic transverse branch; this is not a computed vacuum.	At a = 1 compare θ = 0 and θ = 0.2; the unpaired mass changes while the normal root stays fixed.
Profile example M	Mass parameter from 0 to 2 on a separate unit rectangle. It is not a bulk mass inserted in the Spin(20) action.	Compare M = 0 and M = 1: rank 2 becomes rank 3.
Spin(20) parity insertion	C1, C0 or C0 W sign insertion on the positive half-spin representation of dimension 512. The Cartan probe is fixed and displayed.	Compare C1 and C0 W. Six-dimensional anomalies use degrees 4 and 2; four-dimensional ones use degrees 3 and 1.
Identical 6D families	Integer from 0 to 12 in a separate family-count example, assuming identical 6D chirality assignments.	One family fails; three pass this restricted condition. Six also pass: it does not select exactly three.
6D lepton chirality	Same as Q or opposite to Q, with quark chirality fixed positive and three colours.	Switch the sign and compare the signed doublet excess; retain the identical-family assumption.
Mass landscape view / 3D perspective	The 2D map and 3D surface use the same sampled paired masses and fixed colour range. Perspective is a display angle, not a physical parameter.	Drag on the 2D map and release to apply the values. In 3D click a mesh node; adjust perspective between 15 and 75 degrees. Arrow keys move the selected point.
Certified response query	Select one of the two certified probes, t = 1/4 or t = 1/2. The unit-square model, source, epsilon = 1/8, alpha = 1/20, c = 38, kL = 2, kT = 8, pole s = 1/100 and residue R = 1/2 stay fixed.	Select both probes. Their response-difference intervals exclude zero; no uncertified parameter is editable.
Exact neutral stratum	Select s ≠ 0 or s = 0 independently of the Wilson slider.	Compare rank M, stabilizer and quotient; no tolerance decides the stratum.
Certified neutral angle	Exact neutral-vector query, with L5 = L6 = muOmega = muPhi = 1 and eta = 0. Independent of the fermion landscape angle and the finite scalar stratum.	At 1/5, the interval for mZ squared times L5 squared is [244/8425, 4/125]; the separation is at least 337/4000.

Reading the results

Result	Meaning
Root table and mass curves	Six roots of x tan(x) = a, with x = kL, and scaled residuals. Curves show the lightest paired and unpaired branches versus the imposed angle; they are not the full Kaluza–Klein spectrum.
Gram eigenvalues and rank	Eigenvalues of the complex Hermitian Gram matrix for profile angles 0, π/2 and π. Rank counts eigenvalues above the displayed numerical threshold; overlap alone does not establish dependence.
Exact inserted moments	Tr(λ Fᵈ) for degrees 0 through 4 at one integer Cartan probe, evaluated with exact rational arithmetic. A nonzero value disproves an identically vanishing polynomial; a zero at one probe does not prove it.
Doublet excess and residues	The signed complex-doublet excess must be a multiple of 6 in the stated standalone 6D test. Twice that coefficient counts half-doublets and must be a multiple of 12.
JSON, comparison and saved link	JSON records tool version, schema, all parameters, roots, Gram results, exact moments, assumptions, sources and unresolved requirements. A pinned comparison overlays the earlier paired curve; a saved link restores controls.
2D map, 3D surface and Gram bars	Mass colour is linear from 0 to 0.7071 in mR. The red slice fixes the current coupling; the white point is the selected calculation. Surface cells visualize sampled data, while the selected value is recomputed. Gram bars have their own labelled adaptive linear scale and numerical eigenvalues.
Certified matched response and ZIP	Outward intervals enclose each continuum response and their difference. The ZIP contains the functional proof, polynomial witnesses, displayed reports, standalone verifier and its 228-check Sage 10.8 receipt. This is a computational certificate with a written proof, not a Lean formalization.
Finite neutral quotient and certificate	Exact dimensions 0/1 and ZIP with witnesses, written proof and symbolic verifier. Continuous reduction and physical matching are external.
Continuous neutral eigenvalue interval and certified separation	Section 6b bounds the continuous eigenvalue, including the natural boundary conditions. Positive lower mass and gap prove a simple isolated mode within the active block. Three decoupled zero directions remain separate.

Read each result with its own assumptions. The normal gap is dimensionless kR; a Gram rank is numerical and threshold-dependent; the traces are exact scalar evaluations without local weights; a zero residue passes only the stated standalone six-dimensional condition. Full anomaly cancellation stays unknown. In section 5, positive difference bounds establish separation at the two admitted points for this model. They do not establish a complete GHU prediction or a thin-layer rate. In section 6, dim ker A = 5 in both strata; rank M = 5 or 4, and dim(ker A/im M) = 0 or 1. The stabilizer ker M is a different space. The exact selector is independent of the Wilson slider. Section 6b bounds the continuous eigenvalue, including the natural boundary conditions. Positive lower mass and gap prove a simple isolated mode within the active block. Three decoupled zero directions remain separate.

Worked example

Find three different reasons a construction can fail

Load Research baseline, then Uncoupled boundary: the first normal root is zero.
Restore boundary strength 1: the first kR is approximately 0.273853.
Load Massless profile example: the three chosen profiles span only two independent directions.
Restore M = 1: the eigenvalues are approximately 0.172999, 2.783424 and 10.686355, and the numerical rank is three.
Select One 6D family with Opposite to Q: excess 2 gives residue 2. Select Three 6D families: excess 6 gives residue 0.
Switch C1 to C0 W and inspect the trace table, then export JSON.
Select each certified response query, download the certificate, then save the shared JSON and link. The selected probe, fixed model and certificate identifier travel with the result.
Compare Exact neutral stratum: s ≠ 0 and s = 0. Observe rank M = 5 and 4, and quotient dimension 0 and 1; download the finite proof and verifier.
Select Certified neutral angle: 1/10, 1/5, then 1/4. Read section 6b: compare the rational mass interval and positive separation from the next active mode. Download the continuous proof and replay it with SymPy.

What to expect: The boundary gap, profile independence and restricted family condition respond to different changes. None of them settles localized or global anomaly cancellation of the full construction. The quotient jump is a certified finite result; it determines neither continuous modes nor a Higgs mass. The continuous neutral mass stays positive and isolated at all three admitted angles; its Wilson response and experimental interpretation remain unproved.

Formulas and conventions

Q(t) = F(t) - F(s) - (t-s) F′(s); C(t) = -1 / [(t-s)/R + Q(t)]

Exact matching identity for F(t)=t/D(t), with the same source and fixed s,R. The certified denominators exclude zero at the selected probes.

rho = 4 theta²/5; beta = 5/32 - theta²; rho - (4 theta⁴/25)/(beta-rho) ≤ lambdaZ ≤ rho

Temple enclosure of the first active vector eigenvalue; beta bounds the second one. Valid for the declared walls and 0 < theta <= 1/4. The UI offers three exact queries.

Scope and limitations

The flat spectrum assumes uniform coupling, no additional boundary kinetic terms and the periodic transverse branch. It does not determine a Wilson vacuum, realistic flavour or neutrino masses.
The profile calculation is a separate mathematical example, not a derivation of three physical families. Numerical rank depends on tolerance; exact zero modes need an analytic boundary-domain analysis.
Inserted traces omit projector and covering-space normalization, consistent descent, mirror contributions and inflow. The scalar probe is not the full anomaly polynomial.
The SU(2) criterion applies to a specified standalone six-dimensional tensor setting. It cannot be transferred automatically to an eight-dimensional orbifold corner coupled to bulk inflow.
Local coefficients, integral tensor charge lattice, global gauge group, new boson zero modes and full quantum consistency remain unresolved.
Certification is restricted to the declared unit-square domain and two fixed probes at finite layer width. Changing the source, forms, matching conditions or width requires a new certificate. Software tests and hashes alone are not mathematical certification.
The finite quotient in section 6 certifies the specified matrices A,M. Section 6b has its own continuous proof for the declared vector block and fixed walls. It does not certify the Wilson derivative, RZW, vacuum, currents or physical Higgs direction. Never differentiate the interval endpoints to infer a derivative bound.

If something is unclear

The first normal root is zero.

Check boundary strength. At zero coupling this is the expected zero mode; it is not a numerical solver failure.

Overlapping profiles still have rank three.

Overlap measures nonorthogonality. Dependence requires a null Gram direction; inspect the eigenvalues and threshold.

An inserted moment is zero.

Check its degree and insertion. Evaluate the polynomial or more probes before claiming cancellation; this interface shows only one fixed probe.

Three families pass but full cancellation is unknown.

That is intentional: the modular condition is necessary in its stated setting, not a complete local/global anomaly calculation.

An invalid input leaves the old result visible.

The controls report the error and keep the last valid calculation. Restore the allowed range or press Research baseline.

Why can I not change layer width or import a response?

Only independently certified instances are admitted. New computations belong in the external mathematical engine until their witnesses and functional premises are verified.

I cannot find section 6 or see it in the video.

Open Anomalies & proton, use the Boundary research shortcut and find Exact neutral stratum. The current Demo includes section 6. The 17 October videos do not; consult this guide and the 26 October demo.

Changing the landscape angle does not change the neutral enclosure.

Use Certified neutral angle. The landscape and neutral vector describe different stated problems; their controls are intentionally independent.

Sources and attribution

Current instrument guide. Archived editions retain their own scope.