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baseline", "full complex coefficients" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "FourierMarkedBaseline" }, "target": { "convention": [ "real=c=cos(pi/1296000)", "sum(imag)=0", "dimensionless" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "ComplexMarkedBaseline" } }, { "alignment": { "contract": { "domain": "full complex spectrum", "exact": true, "loses": [ "Fourier_phase" ], "preserves": [ "Fourier_magnitudes" ], "requires": [ "n=6", "fixed Fourier convention" ], "reversible": false }, "left": "FullComplexDFT", "relation": "LOSSY_PROJECTION", "right": "FourierMagnitudes", "witness": { "checker": "phase_collision", "claim_digest": "23075c09eb6f49327b256584e4aef4e54ac4e1af3d213b5b60014949d0d07f35", "payload": "{\"inputs\":\"v and -v for v=(1,0,0,0,0,0)\",\"same_magnitude\":true}", "support": "EXACT_COLLISION" } }, "arithmetic": "binary64 preview", "formula": "m_k=abs(F_k)", "id": "fourier_magnitude", "implementation": "fourier_magnitude", "introduced_in": "2.0", "inverse": null, "source": { "convention": [ "exp(-2*pi*i*j*k/6)", "unnormalized forward", "fixed input labels" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "FullComplexDFT" }, "target": { "convention": [ "DFT convention fixed", "phase discarded" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "FourierMagnitudes" } }, { "alignment": { "contract": { "domain": "centered real six-vectors, r>0", "exact": true, "loses": [], "preserves": [ "label_order", "centered_coordinates", "coordinate_units" ], "requires": [ "r>0", "finite centered input", "declared beta>0 retained", "dimensionless units", "fixed label order" ], "reversible": true }, "left": "M_r(z)", "relation": "EQUIVALENCE", "right": "r z", "witness": { "checker": "scale_centered", "claim_digest": "8dfb4a0cbf34c11dd8300bb825035d215e13652dd5c9360486d99f3a9844a405", "payload": "{\"derivation\":\"PROOFS.md#softmax-scale-power\",\"heading\":\"softmax-scale-power\",\"schema\":\"scale_centered\"}", "support": "TRUSTED_ANALYTIC_SCHEMA" } }, "arithmetic": "binary64 preview", "formula": "M_r(z) = r z for r > 0 on centered six-vectors. beta, dimensionless units, and label order stay in the configuration.", "id": "scale_centered", "implementation": "tga.2.1/M_r", "introduced_in": "2.1", "inverse": { "domain": "CenteredMarks satisfying its image constraints", "left_inverse": "M_{1/r} on centered marks", "map_id": "scale_centered", "right_inverse_on_image": "M_r on the centered image", "source": { "convention": [ "sum=0", "dimensionless" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "CenteredMarks" }, "target": { "convention": [ "sum=0", "dimensionless" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "CenteredMarks" } }, "source": { "convention": [ "sum=0", "dimensionless" ], "dimension": 6, "labels": [ "d2", 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default beta=lambda.", "id": "softmax_beta", "implementation": "tga.2.1/S_beta", "introduced_in": "2.1", "inverse": { "domain": "PositiveProbabilityVector satisfying its image constraints", "left_inverse": "L_beta on the positive simplex", "map_id": "centered_log", "right_inverse_on_image": "S_beta on centered marks", "source": { "convention": [ "sum=1", "strictly positive", "dimensionless", "fixed label order", "beta is not part of this type" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "PositiveProbabilityVector" }, "target": { "convention": [ "sum=0", "dimensionless" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "CenteredMarks" } }, "source": { "convention": [ "sum=0", "dimensionless" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "CenteredMarks" }, "target": { "convention": [ "sum=1", "strictly positive", "dimensionless", "fixed label order", "beta is not part of this type" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "PositiveProbabilityVector" } }, { "alignment": { "contract": { "domain": "positive probability vectors", "exact": true, "loses": [], "preserves": [ "label_order", "centered_contrast_information", "coordinate_units" ], "requires": [ "beta>0", "strictly positive normalized probabilities", "fixed label order" ], "reversible": true }, "left": "L_beta(p)", "relation": "EQUIVALENCE", "right": "centered coordinates", "witness": { "checker": "centered_log", "claim_digest": "ef5474fcf6a53ff9d54d54977bcac3fd865d25d484dbdafc433daa59e7d1eb9b", "payload": "{\"derivation\":\"PROOFS.md#centered-log-ratio\",\"heading\":\"centered-log-ratio\",\"schema\":\"centered_log\"}", "support": "TRUSTED_ANALYTIC_SCHEMA" } }, "arithmetic": "binary64 preview", "formula": "L_beta(p)_i = (log p_i - mean log p) / beta. General centered equivalence, not six-mark membership.", "id": "centered_log", "implementation": "tga.2.1/L_beta", "introduced_in": "2.1", "inverse": { "domain": "CenteredMarks satisfying its image constraints", "left_inverse": "S_beta", "map_id": "softmax_beta", "right_inverse_on_image": "L_beta on the centered image", "source": { "convention": [ "sum=0", "dimensionless" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "CenteredMarks" }, "target": { "convention": [ "sum=1", "strictly positive", "dimensionless", "fixed label order", "beta is not part of this type" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "PositiveProbabilityVector" } }, "source": { "convention": [ "sum=1", "strictly positive", "dimensionless", "fixed label order", "beta is not part of this type" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "PositiveProbabilityVector" }, "target": { "convention": [ "sum=0", "dimensionless" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "CenteredMarks" } }, { "alignment": { "contract": { "domain": "positive probability vectors, r>0", "exact": true, "loses": [], "preserves": [ "label_order", "positive_simplex", "coordinate_units" ], "requires": [ "r>0", "strictly positive normalized probabilities", "no clipping of underflow" ], "reversible": true }, "left": "P_r(p)", "relation": "EQUIVALENCE", "right": "powered probability vector", "witness": { "checker": "power_probability", "claim_digest": "0e0fbf8e3e11e2fab0121490452082c6bd57f0a68ee03f3ba97c9c1dce6b1c5a", "payload": "{\"derivation\":\"PROOFS.md#softmax-scale-power\",\"heading\":\"softmax-scale-power\",\"schema\":\"power_probability\"}", "support": "TRUSTED_ANALYTIC_SCHEMA" } }, "arithmetic": "binary64 preview", "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}.", "id": "power_probability", "implementation": "tga.2.1/P_r", "introduced_in": "2.1", "inverse": { "domain": "PositiveProbabilityVector satisfying its image constraints", "left_inverse": "P_{1/r} on the positive simplex", "map_id": "power_probability", "right_inverse_on_image": "P_r on the positive simplex", "source": { "convention": [ "sum=1", "strictly positive", "dimensionless", "fixed label order", "beta is not part of this type" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "PositiveProbabilityVector" }, "target": { "convention": [ "sum=1", "strictly positive", "dimensionless", "fixed label order", "beta is not part of this type" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "PositiveProbabilityVector" } }, "source": { "convention": [ "sum=1", "strictly positive", "dimensionless", "fixed label order", "beta is not part of this type" ], "dimension": 6, "labels": [ "d2", "d1", 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"marked Fourier image, r>0", "exact": true, "loses": [], "preserves": [ "label_order", "complex_phase", "dc_bin", "coordinate_units" ], "requires": [ "r>0", "marked image", "phase retained" ], "reversible": true }, "left": "D_r(W)", "relation": "EQUIVALENCE", "right": "scaled marked spectrum", "witness": { "checker": "scale_marked_fourier", "claim_digest": "9f16a24e9eda1bd19c08f4ef957aa100b00297abe94706e5665218a90815f1e2", "payload": "{\"derivation\":\"PROOFS.md#fourier-mark-scale\",\"heading\":\"fourier-mark-scale\",\"schema\":\"scale_marked_fourier\"}", "support": "TRUSTED_ANALYTIC_SCHEMA" } }, "arithmetic": "binary64 preview", "formula": "D_r leaves W0 unchanged and multiplies W1..W5 by r>0. 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DC stays 6 c0. Off-DC bins scale by r. Phase is kept.", "id": "path_fourier_scale", "implementation": "alignment", "introduced_in": "2.1", "inverse": null, "source": { "convention": [ "sum=0", "dimensionless" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "CenteredMarks" }, "target": { "convention": [ "W0=6*c0", "W_{6-k}=-conjugate(W_k) for k=1..5", "full complex phase retained", "unnormalized DFT" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "MarkedFourierImage" } }, { "alignment": { "contract": { "domain": "positive probability vectors", "exact": true, "loses": [], "preserves": [ "positive_simplex", "label_order" ], "requires": [ "r>0", "t>0", "strictly positive probabilities" ], "reversible": false }, "left": "P_r \u2218 P_t", "relation": "IDENTITY", "right": "P_{rt}", "witness": { "checker": "power_composition", "claim_digest": "aa190702266fccc081e36656c0385866e6fc78b2d4c1b5615da675d4b917c5ce", "payload": "{\"derivation\":\"PROOFS.md#power-composition\",\"heading\":\"power-composition\",\"schema\":\"power_composition\"}", "support": "TRUSTED_ANALYTIC_SCHEMA" } }, "arithmetic": "not a path-equality certificate schema", "formula": "P_r \u2218 P_t = P_{r t} for r>0 and t>0, by cancellation of the power denominator.", "id": "power_composition", "implementation": "alignment", "introduced_in": "2.1", "inverse": null, "source": { "convention": [ "sum=1", "strictly positive", "dimensionless", "fixed label order", "beta is not part of this type" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "PositiveProbabilityVector" }, "target": { "convention": [ "sum=1", "strictly positive", "dimensionless", "fixed label order", "beta is not part of this type" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "PositiveProbabilityVector" } }, { "alignment": { "contract": { "domain": "positive probability vectors", "exact": true, "loses": [], "preserves": [ "positive_simplex", "label_order" ], "requires": [ "normalized strictly positive probabilities" ], "reversible": false }, "left": "P_1(p)", "relation": "IDENTITY", "right": "p", "witness": { "checker": "power_unit", "claim_digest": "b9d446f934e7af2d4e4956ff9faf6c3440ba52d0d8d512d30fad3cc97f986658", "payload": "{\"derivation\":\"PROOFS.md#power-composition\",\"heading\":\"power-composition\",\"schema\":\"power_unit\"}", "support": "TRUSTED_ANALYTIC_SCHEMA" } }, "arithmetic": "not a path-equality certificate schema", "formula": "P_1 is the identity on normalized positive probabilities.", "id": "power_unit", "implementation": "alignment", "introduced_in": "2.1", "inverse": null, "source": { "convention": [ "sum=1", "strictly positive", "dimensionless", "fixed label order", "beta is not part of this type" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "PositiveProbabilityVector" }, "target": { "convention": [ "sum=1", "strictly positive", "dimensionless", "fixed label order", "beta is not part of this type" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "PositiveProbabilityVector" } }, { "alignment": { "contract": { "domain": "positive probability vectors", "exact": true, "loses": [], "preserves": [ "positive_simplex", "label_order" ], "requires": [ "r>0", "strictly positive probabilities", "no clipped underflow" ], "reversible": false }, "left": "P_r inverse", "relation": "IDENTITY", "right": "P_{1/r}", "witness": { "checker": "power_inverse_law", "claim_digest": "1be56189abb349c41cb73516e07540ae688a9b11d6eef965d101e45ad2d3f3d0", "payload": "{\"derivation\":\"PROOFS.md#power-composition\",\"heading\":\"power-composition\",\"schema\":\"power_inverse_law\"}", "support": "TRUSTED_ANALYTIC_SCHEMA" } }, "arithmetic": "not a path-equality certificate schema", "formula": "The inverse of P_r on the positive simplex is P_{1/r}.", "id": "power_inverse_law", "implementation": "alignment", "introduced_in": "2.1", "inverse": null, "source": { "convention": [ "sum=1", "strictly positive", "dimensionless", "fixed label order", "beta is not part of this type" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "PositiveProbabilityVector" }, "target": { "convention": [ "sum=1", "strictly positive", "dimensionless", "fixed label order", "beta is not part of this type" ], "dimension": 6, "labels": [ "d2", "d1", "d", "b", "b1", "b2" ], "name": "PositiveProbabilityVector" } }, { "alignment": { "contract": { "domain": "exact positive probability vectors; not a binary64 certificate", "exact": true, "loses": [], "preserves": [ "necessity_and_sufficiency_on_exact_positive_reals" ], "requires": [ "a>0", "00", "strictly positive probabilities", "fixed label order" ], "reversible": false }, "left": "five geometric gap conditions", "relation": "EQUIVALENCE", "right": "six-mark family probability", "witness": { "checker": "six_mark_gap_criterion", "claim_digest": "26086a4bf6571d6c13d733db48fbcbf8b2b7b297efb22df782de75774da6e4a3", "payload": "{\"derivation\":\"PROOFS.md#six-mark-membership\",\"heading\":\"six-mark-membership\",\"schema\":\"six_mark_gap_criterion\"}", "support": "TRUSTED_ANALYTIC_SCHEMA" } }, "arithmetic": "exact statement; binary64 membership is a separate numerical report", "formula": "g2=g4>0, g1=g5=phi g2, and 0 0; declared beta > 0", "support": "TRUSTED_ANALYTIC_SCHEMA" }, "condition": null, "detail": "certificate admitted by the implemented schema", "reversible_badge": true } ], "note": "Only these two schemas admit a certificate. Power laws are proved and checked, but they are not additional certificate schemas.", "schema": "TGA_PATH_EQUALITY_CERTIFICATE/2.1", "supported_schemas": [ "tga.2.1/proof/softmax-scale-power", "tga.2.1/proof/fourier-mark-scale" ] }, "proposals_only": [ { "admitted": false, "left": "phi_A", "reason": "same declared kind and shared symbols; not proof", "right": "phi_B" }, { "admitted": false, "left": "phi_A", "reason": "same declared kind and shared symbols; not proof", "right": "wrong_phi" }, { "admitted": false, "left": "phi_B", "reason": "same declared kind and shared symbols; not proof", "right": "wrong_phi" } ], "publication": { "classified_as": "supplementary work", "date": "2026-10-01", "not": [ "source paper", "general theorem prover", "replacement of TGA 2.0", "replacement of production geometry or field decoders", "exact binary64 membership certificate" ], "preserves_tga_2_0": { "note": "Published TGA 2.0 files are inputs. 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