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Supplementary verifier · 1 October 2026

TGA 2.1 scaling and membership

TGA 2.1 adds geometric scaling, Aitchison powering, two path-equality certificates, and a six-mark membership test. An admitted relationship still requires a supported witness. This is not a general theorem prover, not a source paper, and not a replacement for TGA 2.0 or the production geometry and field-decoder modules.

Publication status: supplementary release, version 2.1, classified as supplementary work. That sentence is not a mathematical justification and not an execution result.

The graph has 32 admitted relationships in 7 relation categories. Both numbers are taken from the loaded edge list, not from a separate caption. 19 of those relationships are the frozen TGA 2.0 edges. Exact certificates, declarations, trusted analytic schemas, and collision witnesses stay separate labels.

Mathematical justification: rendered proofs and PROOFS.md. Those two files are the frozen 1 October 2026 release members. Their tolerance section uses the single name “band” for two different functions. The dated correction is the tolerance-name erratum (Markdown source). It names gap_band and boundary_band. It does not replace the release bytes and it is not a numerical change. Release files: tga_verifier.py, test_tga.py, graph-report.json, README.md, SHA256SUMS.txt, RELEASE.json, and tga-2.1-download.zip. The download descriptor is tga-2.1-download.json. TGA 2.0 hashes are preserved and are not inside that new container.

Frozen TGA 2.0 remains at its checksum list, tga_verifier.py SHA-256 4339aee9…, and tga-2.0-download.zip. Its author execution record is still execution.json (Python 3.12.14, 2026-09-27T13:12:51Z). The typed-network dossier remains /network. The interactive map is /atlas/index.html. Supplementary workflows outside this release: the witness lab, the exact transfer on sphere average, and the canonical readout on exact field interface. Frame calibration, boundary-field targets, and generated Lean or SMT files are not in that supplement. Formal execution of the supplement was not run. The TGA 2.0 and 2.1 bytes are unchanged.

Execution record execution.json: Python 3.10.21 at 2026-10-01T12:54:41.234993+00:00. python3 -m unittest -v test_tga exited 0. python3 -O -m unittest -v test_tga exited 0. python3 tga_verifier.py --report graph-report.json exited 0. Those outcomes are not the proof status. Numerical comparisons in the report are binary64 checks, not exact certificates.

Separate from the six-mark membership test: a written real-arithmetic corollary for 120 centered coordinates is on the 600-cell signal page under positive-simplex coordinates. It is not a TGA execution result, not a six-mark membership test, and not a 45-byte codec.

Commuting paths

Two supported identities. The general centered-coordinate equivalence is the softmax and Aitchison-power chart. It is not membership in the six-mark family. A numerical residual is not the proof.

The certificate is computed in the browser from the fields above. A previous result is not retained.

Six-mark probability validator

A valid positive probability vector and a probability vector representing the six-mark family are different types. Centered log-ratio coordinates and Aitchison powering explain the general equivalence. The family is the subset whose five gaps match the geometric scale and angle. Numerical agreement is not an exact membership certificate.

Numerical policy: gap comparisons use

∣x−y∣≤10−9+10−8max⁡(∣x∣,∣y∣).|x-y|\le 10^{-9}+10^{-8}\max(|x|,|y|).

Parameter-boundary classification uses

boundary_band(s)=10−9+10−8max⁡(1,∣s∣).\mathrm{boundary\_band}(s)=10^{-9}+10^{-8}\max(1,|s|).

The positive-scale threshold is 1e-9; the probability-sum tolerance is 1e-12. The measured s is never clamped. Results indicate numerical consistency or a boundary status, not exact membership certification. Binary64 evaluation of φ is not an exact arithmetic certificate.

Both policies are named before the rules in the 1 October 2026 tolerance erratum. The frozen proof section still uses one name for both functions.

Membership is computed in the browser. The previous result is not retained.

A full state can evolve by Psi and then be observed by X, or be observed first and then evolve by T_aff. Both routes give the same scalar reading.
X composed with Psi equals T_aff composed with X. Psi and T_aff are invertible on their stated exact domains; X is many-to-one. Floor quantization is a separate operation. Open full-size diagram: Two compatible routes

Admitted relationships

Choose a relationship to read its witness, contract, and inverse conditions.

32 of 32 admitted relationships shown.

phi identity(1+sqrt5)/2 → 1/2+sqrt5/2IDENTITYEXACT_CERTIFICATEDomain: the two named golden expressions
Relation
IDENTITY
Release
2.0
Support
EXACT_CERTIFICATE
Mathematical status
exact algebra certificate checked
Domain
the two named golden expressions
Requirements
  • rational coefficients
  • positive sqrt5
Preserves
  • algebraic_value
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
exact rational quadratic algebra
Reversible flag
true
Inverse contract
  • Map: phi_identity
  • Domain: NamedScalar satisfying its image constraints
  • Left inverse: inverse(forward(x))=x on source
  • Right inverse on image: forward(inverse(y))=y on declared image
  • Types: NamedScalar → NamedScalar
Implementation
identity
Formula
normalize(left)=normalize(right) in Q(sqrt(5))
lambda betalambda → optimizer_betaPARAMETER_IDENTIFICATIONEXPLICIT_DECLARATIONDomain: named parameter assignment only
Relation
PARAMETER_IDENTIFICATION
Release
2.0
Support
EXPLICIT_DECLARATION
Mathematical status
declaration only; no optimizer equivalence
Domain
named parameter assignment only
Requirements
  • explicit parameter choice
Preserves
  • chosen_scalar_value
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
false
Inverse contract
No inverse contract. Reversibility is not certified.
Implementation
declaration
Formula
declare beta := lambda
lambda approximationlambda → Q/SAPPROXIMATIONEXACT_CERTIFICATEDomain: the named constants
Relation
APPROXIMATION
Release
2.0
Support
EXACT_CERTIFICATE
Mathematical status
exact arithmetic certificate checked under documented analytic identities
Domain
the named constants
Requirements
  • fixed Q,S and lambda definitions
Preserves
  • stated_error_bound
Loses
  • exact_real_value
Exact formula
exact formula flag is false
Arithmetic
exact interval certificate
Reversible flag
false
Inverse contract
No inverse contract. Reversibility is not certified.
Implementation
approximation
Formula
0 < lambda-Q/S < 10^-15
floor agreementfloor(lambda*d) → floor(Q*d/S)DOMAIN_RESTRICTED_EQUIVALENCEEXACT_CERTIFICATEDomain: 0 <= d < 42189065
Relation
DOMAIN_RESTRICTED_EQUIVALENCE
Release
2.0
Support
EXACT_CERTIFICATE
Mathematical status
exact arithmetic certificate checked under documented analytic identities
Domain
0 <= d < 42189065
Requirements
  • integer 0<=d<42189065
Preserves
  • floor_output_on_certified_domain
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
exact integers after certified rewrite
Reversible flag
false
Inverse contract
No inverse contract. Reversibility is not certified.
Implementation
certified_floor
Formula
left(d)=right(d) for integer 0<=d<42189065
floord → floor(Q*d/S)LOSSY_PROJECTIONEXACT_COLLISIONDomain: nonnegative integers
Relation
LOSSY_PROJECTION
Release
2.0
Support
EXACT_COLLISION
Mathematical status
exact counterexample to injectivity
Domain
nonnegative integers
Requirements
  • d>=0 integer
Preserves
  • nonnegative_integer_output
Loses
  • preimage_identity
Exact formula
exact formula flag is true
Arithmetic
exact Python integers
Reversible flag
false
Inverse contract
No inverse contract. Reversibility is not certified.
Implementation
floor
Formula
F(d)=Q*d//S
paird → (d_next,rho)EMBEDDINGTRUSTED_ANALYTIC_SCHEMADomain: NonnegativeInteger
Relation
EMBEDDING
Release
2.0
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied, not machine-checked universal proof
Domain
NonnegativeInteger
Requirements
  • fixed Q,S
  • remainder bounds
  • exact inverse divisibility
  • valid pair image
Preserves
  • integer_seed_information
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
exact Python integers
Reversible flag
true
Inverse contract
  • Map: unpair
  • Domain: QuotientRemainderImage satisfying its image constraints
  • Left inverse: inverse(forward(x))=x on source
  • Right inverse on image: forward(inverse(y))=y on declared image
  • Types: QuotientRemainderImage → NonnegativeInteger
Implementation
pair
Formula
Q*d=S*d_next+rho; inverse=(S*d_next+rho)/Q; iterate in ledger order
unpair(d_next,rho) → dEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: QuotientRemainderImage
Relation
EQUIVALENCE
Release
2.0
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied, not machine-checked universal proof
Domain
QuotientRemainderImage
Requirements
  • fixed Q,S
  • remainder bounds
  • exact inverse divisibility
  • valid pair image
Preserves
  • integer_seed_information
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
exact Python integers
Reversible flag
true
Inverse contract
  • Map: pair
  • Domain: NonnegativeInteger satisfying its image constraints
  • Left inverse: inverse(forward(x))=x on source
  • Right inverse on image: forward(inverse(y))=y on declared image
  • Types: NonnegativeInteger → QuotientRemainderImage
Implementation
unpair
Formula
Q*d=S*d_next+rho; inverse=(S*d_next+rho)/Q; iterate in ledger order
ledgerd → extinction_ledgerEMBEDDINGTRUSTED_ANALYTIC_SCHEMADomain: NonnegativeInteger
Relation
EMBEDDING
Release
2.0
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied, not machine-checked universal proof
Domain
NonnegativeInteger
Requirements
  • fixed Q,S
  • remainder bounds
  • exact inverse divisibility
  • canonical unpadded ledger
Preserves
  • integer_seed_information
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
exact Python integers
Reversible flag
true
Inverse contract
  • Map: unledger
  • Domain: CanonicalExtinctionLedger satisfying its image constraints
  • Left inverse: inverse(forward(x))=x on source
  • Right inverse on image: forward(inverse(y))=y on declared image
  • Types: CanonicalExtinctionLedger → NonnegativeInteger
Implementation
ledger
Formula
Q*d=S*d_next+rho; inverse=(S*d_next+rho)/Q; iterate in ledger order
unledgerextinction_ledger → dEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: CanonicalExtinctionLedger
Relation
EQUIVALENCE
Release
2.0
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied, not machine-checked universal proof
Domain
CanonicalExtinctionLedger
Requirements
  • fixed Q,S
  • remainder bounds
  • exact inverse divisibility
  • canonical unpadded ledger
Preserves
  • integer_seed_information
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
exact Python integers
Reversible flag
true
Inverse contract
  • Map: ledger
  • Domain: NonnegativeInteger satisfying its image constraints
  • Left inverse: inverse(forward(x))=x on source
  • Right inverse on image: forward(inverse(y))=y on declared image
  • Types: NonnegativeInteger → CanonicalExtinctionLedger
Implementation
unledger
Formula
Q*d=S*d_next+rho; inverse=(S*d_next+rho)/Q; iterate in ledger order
geometry marksSixMarkGeometry → CenteredMarksLOSSY_PROJECTIONEXACT_COLLISIONDomain: specified six-mark family with recorded origin
Relation
LOSSY_PROJECTION
Release
2.0
Support
EXACT_COLLISION
Mathematical status
definitional coordinate-projection collision
Domain
specified six-mark family with recorded origin
Requirements
  • scale>0
  • 0<epsilon<pi
  • trusted labels
Preserves
  • label_order
  • centered_contrast_information
Loses
  • baseline_origin
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
false
Inverse contract
No inverse contract. Reversibility is not certified.
Implementation
geometry_marks
Formula
a*(-A,-B,-s,s,B,A); drop the recorded baseline origin
marks probabilityCenteredMarks → PositiveProbabilityVectorEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: CenteredMarks
Relation
EQUIVALENCE
Release
2.0
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied, not machine-checked universal proof
Domain
CenteredMarks
Requirements
  • beta=lambda>0
  • fixed labels
  • zero-sum centering
  • strictly positive probabilities
Preserves
  • label_order
  • centered_contrast_information
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: probability_marks
  • Domain: PositiveProbabilityVector satisfying its image constraints
  • Left inverse: inverse(forward(x))=x on source
  • Right inverse on image: forward(inverse(y))=y on declared image
  • Types: PositiveProbabilityVector → CenteredMarks
Implementation
marks_probability
Formula
p_i=exp(beta*z_i)/sum_j exp(beta*z_j); z_i=(log(p_i)-mean(log p))/beta
probability marksPositiveProbabilityVector → CenteredMarksEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: PositiveProbabilityVector
Relation
EQUIVALENCE
Release
2.0
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied, not machine-checked universal proof
Domain
PositiveProbabilityVector
Requirements
  • beta=lambda>0
  • fixed labels
  • zero-sum centering
  • strictly positive probabilities
Preserves
  • label_order
  • centered_contrast_information
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: marks_probability
  • Domain: CenteredMarks satisfying its image constraints
  • Left inverse: inverse(forward(x))=x on source
  • Right inverse on image: forward(inverse(y))=y on declared image
  • Types: CenteredMarks → PositiveProbabilityVector
Implementation
probability_marks
Formula
p_i=exp(beta*z_i)/sum_j exp(beta*z_j); z_i=(log(p_i)-mean(log p))/beta
marks complexCenteredMarks → ComplexMarkedBaselineEMBEDDINGTRUSTED_ANALYTIC_SCHEMADomain: CenteredMarks
Relation
EMBEDDING
Release
2.0
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied, not machine-checked universal proof
Domain
CenteredMarks
Requirements
  • fixed c=cos(pi/1296000)
  • labels
  • centered imaginary coordinates
Preserves
  • label_order
  • centered_contrast_information
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: complex_marks
  • Domain: ComplexMarkedBaseline satisfying its image constraints
  • Left inverse: inverse(forward(x))=x on source
  • Right inverse on image: forward(inverse(y))=y on declared image
  • Types: ComplexMarkedBaseline → CenteredMarks
Implementation
marks_complex
Formula
w_i=c+i*z_i; inverse z_i=Im(w_i) on Re(w_i)=c
complex marksComplexMarkedBaseline → CenteredMarksEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: ComplexMarkedBaseline
Relation
EQUIVALENCE
Release
2.0
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied, not machine-checked universal proof
Domain
ComplexMarkedBaseline
Requirements
  • fixed c=cos(pi/1296000)
  • labels
  • centered imaginary coordinates
Preserves
  • label_order
  • centered_contrast_information
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: marks_complex
  • Domain: CenteredMarks satisfying its image constraints
  • Left inverse: inverse(forward(x))=x on source
  • Right inverse on image: forward(inverse(y))=y on declared image
  • Types: CenteredMarks → ComplexMarkedBaseline
Implementation
complex_marks
Formula
w_i=c+i*z_i; inverse z_i=Im(w_i) on Re(w_i)=c
dftComplexVector → FullComplexDFTEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: ComplexVector
Relation
EQUIVALENCE
Release
2.0
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied, not machine-checked universal proof
Domain
ComplexVector
Requirements
  • n=6
  • retain every complex coefficient
  • fixed Fourier convention
Preserves
  • full_labelled_complex_state
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: idft
  • Domain: FullComplexDFT satisfying its image constraints
  • Left inverse: inverse(forward(x))=x on source
  • Right inverse on image: forward(inverse(y))=y on declared image
  • Types: FullComplexDFT → ComplexVector
Implementation
dft
Formula
F_k=sum_j v_j exp(-2*pi*i*j*k/6); v_j=sum_k F_k exp(+2*pi*i*j*k/6)/6
idftFullComplexDFT → ComplexVectorEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: FullComplexDFT
Relation
EQUIVALENCE
Release
2.0
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied, not machine-checked universal proof
Domain
FullComplexDFT
Requirements
  • n=6
  • retain every complex coefficient
  • fixed Fourier convention
Preserves
  • full_labelled_complex_state
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: dft
  • Domain: ComplexVector satisfying its image constraints
  • Left inverse: inverse(forward(x))=x on source
  • Right inverse on image: forward(inverse(y))=y on declared image
  • Types: ComplexVector → FullComplexDFT
Implementation
idft
Formula
F_k=sum_j v_j exp(-2*pi*i*j*k/6); v_j=sum_k F_k exp(+2*pi*i*j*k/6)/6
marks dftComplexMarkedBaseline → FourierMarkedBaselineEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: ComplexMarkedBaseline
Relation
EQUIVALENCE
Release
2.0
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied, not machine-checked universal proof
Domain
ComplexMarkedBaseline
Requirements
  • n=6
  • full complex spectrum
  • fixed labels and c
  • DFT image constraint
Preserves
  • label_order
  • centered_contrast_information
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: dft_marks
  • Domain: FourierMarkedBaseline satisfying its image constraints
  • Left inverse: inverse(forward(x))=x on source
  • Right inverse on image: forward(inverse(y))=y on declared image
  • Types: FourierMarkedBaseline → ComplexMarkedBaseline
Implementation
marks_dft
Formula
DFT restricted to the complex-mark image; retain all six complex coefficients; inverse IDFT
dft marksFourierMarkedBaseline → ComplexMarkedBaselineEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: FourierMarkedBaseline
Relation
EQUIVALENCE
Release
2.0
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied, not machine-checked universal proof
Domain
FourierMarkedBaseline
Requirements
  • n=6
  • full complex spectrum
  • fixed labels and c
  • DFT image constraint
Preserves
  • label_order
  • centered_contrast_information
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: marks_dft
  • Domain: ComplexMarkedBaseline satisfying its image constraints
  • Left inverse: inverse(forward(x))=x on source
  • Right inverse on image: forward(inverse(y))=y on declared image
  • Types: ComplexMarkedBaseline → FourierMarkedBaseline
Implementation
dft_marks
Formula
DFT restricted to the complex-mark image; retain all six complex coefficients; inverse IDFT
fourier magnitudeFullComplexDFT → FourierMagnitudesLOSSY_PROJECTIONEXACT_COLLISIONDomain: full complex spectrum
Relation
LOSSY_PROJECTION
Release
2.0
Support
EXACT_COLLISION
Mathematical status
exact counterexample to injectivity
Domain
full complex spectrum
Requirements
  • n=6
  • fixed Fourier convention
Preserves
  • Fourier_magnitudes
Loses
  • Fourier_phase
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
false
Inverse contract
No inverse contract. Reversibility is not certified.
Implementation
fourier_magnitude
Formula
m_k=abs(F_k)
scale centeredM_r(z) → r zEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: centered real six-vectors, r>0
Relation
EQUIVALENCE
Release
2.1
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied in PROOFS.md; not a binary64 certificate
Domain
centered real six-vectors, r>0
Requirements
  • r>0
  • finite centered input
  • declared beta>0 retained
  • dimensionless units
  • fixed label order
Preserves
  • label_order
  • centered_coordinates
  • coordinate_units
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: scale_centered
  • Domain: CenteredMarks satisfying its image constraints
  • Left inverse: M_{1/r} on centered marks
  • Right inverse on image: M_r on the centered image
  • Types: CenteredMarks → CenteredMarks
Implementation
tga.2.1/M_r
Formula
M_r(z) = r z for r > 0 on centered six-vectors. beta, dimensionless units, and label order stay in the configuration.
softmax betaS_beta(z) → positive probability vectorEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: centered real six-vectors
Relation
EQUIVALENCE
Release
2.1
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied in PROOFS.md; not a binary64 certificate
Domain
centered real six-vectors
Requirements
  • beta>0
  • centered finite input
  • fixed label order
  • strictly positive image
Preserves
  • label_order
  • centered_contrast_information
  • coordinate_units
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: centered_log
  • Domain: PositiveProbabilityVector satisfying its image constraints
  • Left inverse: L_beta on the positive simplex
  • Right inverse on image: S_beta on centered marks
  • Types: PositiveProbabilityVector → CenteredMarks
Implementation
tga.2.1/S_beta
Formula
S_beta(z)_i = exp(beta z_i) / sum_j exp(beta z_j), beta>0, default beta=lambda.
centered logL_beta(p) → centered coordinatesEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: positive probability vectors
Relation
EQUIVALENCE
Release
2.1
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied in PROOFS.md; not a binary64 certificate
Domain
positive probability vectors
Requirements
  • beta>0
  • strictly positive normalized probabilities
  • fixed label order
Preserves
  • label_order
  • centered_contrast_information
  • coordinate_units
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: softmax_beta
  • Domain: CenteredMarks satisfying its image constraints
  • Left inverse: S_beta
  • Right inverse on image: L_beta on the centered image
  • Types: CenteredMarks → PositiveProbabilityVector
Implementation
tga.2.1/L_beta
Formula
L_beta(p)_i = (log p_i - mean log p) / beta. General centered equivalence, not six-mark membership.
power probabilityP_r(p) → powered probability vectorEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: positive probability vectors, r>0
Relation
EQUIVALENCE
Release
2.1
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied in PROOFS.md; not a binary64 certificate
Domain
positive probability vectors, r>0
Requirements
  • r>0
  • strictly positive normalized probabilities
  • no clipping of underflow
Preserves
  • label_order
  • positive_simplex
  • coordinate_units
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: power_probability
  • Domain: PositiveProbabilityVector satisfying its image constraints
  • Left inverse: P_{1/r} on the positive simplex
  • Right inverse on image: P_r on the positive simplex
  • Types: PositiveProbabilityVector → PositiveProbabilityVector
Implementation
tga.2.1/P_r
Formula
P_r(p)_i = p_i^r / sum_j p_j^r for r>0. Aitchison power. P_1 is the identity and P_r inverse is P_{1/r}.
marked fourierF(z) → marked Fourier imageEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: centered real six-vectors
Relation
EQUIVALENCE
Release
2.1
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied in PROOFS.md; not a binary64 certificate
Domain
centered real six-vectors
Requirements
  • centered real input
  • W0=6 c0
  • conjugate symmetry
  • full complex coefficients
Preserves
  • label_order
  • complex_phase
  • coordinate_units
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: marked_fourier_inverse
  • Domain: MarkedFourierImage satisfying its image constraints
  • Left inverse: inverse DFT then imaginary part on the marked image
  • Right inverse on image: F on centered marks
  • Types: MarkedFourierImage → CenteredMarks
Implementation
tga.2.1/F
Formula
F(z) = DFT(c0 + i z), c0=cos(pi/1296000), unnormalized, phase retained.
marked fourier inversemarked Fourier image → centered coordinatesEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: marked Fourier image
Relation
EQUIVALENCE
Release
2.1
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied in PROOFS.md; not a binary64 certificate
Domain
marked Fourier image
Requirements
  • marked image constraints
  • centered imaginary part
Preserves
  • label_order
  • complex_phase
  • coordinate_units
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: marked_fourier
  • Domain: CenteredMarks satisfying its image constraints
  • Left inverse: F
  • Right inverse on image: inverse DFT on the marked image
  • Types: CenteredMarks → MarkedFourierImage
Implementation
tga.2.1/F_inverse
Formula
Inverse DFT, require real part c0, return the centered imaginary part.
scale marked fourierD_r(W) → scaled marked spectrumEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: marked Fourier image, r>0
Relation
EQUIVALENCE
Release
2.1
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied in PROOFS.md; not a binary64 certificate
Domain
marked Fourier image, r>0
Requirements
  • r>0
  • marked image
  • phase retained
Preserves
  • label_order
  • complex_phase
  • dc_bin
  • coordinate_units
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
binary64 preview
Reversible flag
true
Inverse contract
  • Map: scale_marked_fourier
  • Domain: MarkedFourierImage satisfying its image constraints
  • Left inverse: D_{1/r}
  • Right inverse on image: D_r on the marked image
  • Types: MarkedFourierImage → MarkedFourierImage
Implementation
tga.2.1/D_r
Formula
D_r leaves W0 unchanged and multiplies W1..W5 by r>0. Phase is not replaced by magnitude.
path softmax scaleS_beta ∘ M_r → P_r ∘ S_betaIDENTITYTRUSTED_ANALYTIC_SCHEMADomain: centered real six-vectors to positive probabilities
Relation
IDENTITY
Release
2.1
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied in PROOFS.md; not a binary64 certificate
Domain
centered real six-vectors to positive probabilities
Requirements
  • implemented proof schema
  • r>0
  • beta>0
  • positive intermediate image
Preserves
  • identity_on_the_declared_domain
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
not a state execution; numerical execution is a separate record
Reversible flag
false
Inverse contract
No inverse contract. Reversibility is not certified.
Implementation
alignment
Formula
S_beta ∘ M_r = P_r ∘ S_beta. Proof: the common softmax denominator cancels after exponentiation.
path fourier scaleF ∘ M_r → D_r ∘ FIDENTITYTRUSTED_ANALYTIC_SCHEMADomain: centered real six-vectors to the marked Fourier image
Relation
IDENTITY
Release
2.1
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied in PROOFS.md; not a binary64 certificate
Domain
centered real six-vectors to the marked Fourier image
Requirements
  • implemented proof schema
  • centered source
  • marked intermediate image
  • r>0
Preserves
  • identity_on_the_marked_image
  • complex_phase
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
not a state execution; numerical execution is a separate record
Reversible flag
false
Inverse contract
No inverse contract. Reversibility is not certified.
Implementation
alignment
Formula
F ∘ M_r = D_r ∘ F. DC stays 6 c0. Off-DC bins scale by r. Phase is kept.
power compositionP_r ∘ P_t → P_{rt}IDENTITYTRUSTED_ANALYTIC_SCHEMADomain: positive probability vectors
Relation
IDENTITY
Release
2.1
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied in PROOFS.md; not a binary64 certificate
Domain
positive probability vectors
Requirements
  • r>0
  • t>0
  • strictly positive probabilities
Preserves
  • positive_simplex
  • label_order
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
not a path-equality certificate schema
Reversible flag
false
Inverse contract
No inverse contract. Reversibility is not certified.
Implementation
alignment
Formula
P_r ∘ P_t = P_{r t} for r>0 and t>0, by cancellation of the power denominator.
power unitP_1(p) → pIDENTITYTRUSTED_ANALYTIC_SCHEMADomain: positive probability vectors
Relation
IDENTITY
Release
2.1
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied in PROOFS.md; not a binary64 certificate
Domain
positive probability vectors
Requirements
  • normalized strictly positive probabilities
Preserves
  • positive_simplex
  • label_order
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
not a path-equality certificate schema
Reversible flag
false
Inverse contract
No inverse contract. Reversibility is not certified.
Implementation
alignment
Formula
P_1 is the identity on normalized positive probabilities.
power inverse lawP_r inverse → P_{1/r}IDENTITYTRUSTED_ANALYTIC_SCHEMADomain: positive probability vectors
Relation
IDENTITY
Release
2.1
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied in PROOFS.md; not a binary64 certificate
Domain
positive probability vectors
Requirements
  • r>0
  • strictly positive probabilities
  • no clipped underflow
Preserves
  • positive_simplex
  • label_order
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
not a path-equality certificate schema
Reversible flag
false
Inverse contract
No inverse contract. Reversibility is not certified.
Implementation
alignment
Formula
The inverse of P_r on the positive simplex is P_{1/r}.
six mark gap criterionfive geometric gap conditions → six-mark family probabilityEQUIVALENCETRUSTED_ANALYTIC_SCHEMADomain: exact positive probability vectors; not a binary64 certificate
Relation
EQUIVALENCE
Release
2.1
Support
TRUSTED_ANALYTIC_SCHEMA
Mathematical status
trusted analytic schema; derivation supplied in PROOFS.md; not a binary64 certificate
Domain
exact positive probability vectors; not a binary64 certificate
Requirements
  • a>0
  • 0<epsilon<=pi
  • beta>0
  • strictly positive probabilities
  • fixed label order
Preserves
  • necessity_and_sufficiency_on_exact_positive_reals
Loses
none recorded
Exact formula
exact formula flag is true
Arithmetic
exact statement; binary64 membership is a separate numerical report
Reversible flag
false
Inverse contract
No inverse contract. Reversibility is not certified.
Implementation
alignment
Formula
g2=g4>0, g1=g5=phi g2, and 0<g3<=2 g2 are necessary and sufficient for the positive-scale six-mark family. Zero mean fixes the six coordinates. Canonical angle adds g3=2 sin(pi/1296000) g2.

Proposals only

These candidates share a declared kind and symbols. They are not admitted relationships and they are not in the graph above.

What this release does not do