Conditional κ = 0.0030217758: joint helper interfaces and Lean compiler tightening

research/kappa-interface-round2-20261022/.gitattributes

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+* -text whitespace=blank-at-eol,blank-at-eof,space-before-tab,cr-at-eol

research/kappa-interface-round2-20261022/ARITHMETIC.json

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+{
+  "compiler": {
+    "H": 31385200000000,
+    "N": "2^31385200000000",
+    "eta": {
+      "lower": "0.000000204791629522697050728486",
+      "upper": "0.000000204791629522697050728487"
+    },
+    "gap": "1/700000000",
+    "integer_test_exponent": 44836,
+    "logarithm_power": 1000,
+    "theta": "63/64",
+    "uniform_saving": "211507/70000000",
+    "word_materialized": false
+  },
+  "gain_over_PR4535": "271/200000000",
+  "kappa": "37769/12500000",
+  "local_helpers": {
+    "4": 28,
+    "7": 7,
+    "8": 47,
+    "9": 326
+  },
+  "local_paid": {
+    "1": 2201,
+    "2": 412,
+    "3": 506,
+    "4": 135,
+    "5": 36,
+    "8": 8
+  },
+  "margin_at_old_root_upper": [
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+    "1077939473810483326168879543482116908465746557861384173649/3138550867693340381917894711603833208051177722232017256448"
+  ],
+  "native_receipt_sha256": "c6f16840b3f7bed8686ad494b479df599b18b420c239dc6167c214130b78eb64",
+  "negative_controls": {
+    "closed_epsilon": [
+      "epsilon_below_one",
+      "global_gamma_power_slack",
+      "actual_logarithmic_gamma_guard",
+      "source_axis_superpolynomial",
+      "gaussian_above_main",
+      "new_semantic_guard_with_C1_one"
+    ],
+    "closed_uniform": [
+      "new_complex_uniform_saving_above_b"
+    ],
+    "old_BIT": [
+      "actual_bit_above_effective"
+    ],
+    "unfunded_endpoint": [
+      "main_transform_above_kappa",
+      "global_coordinate_router_above_kappa",
+      "fused_gaussian_above_kappa",
+      "group_bit_router_above_kappa",
+      "boundary_algebraic_matrix_above_kappa"
+    ]
+  },
+  "nonrecursive": {
+    "local_routes": 4802000,
+    "local_scalar": 158168,
+    "raw_scalar_upper": 181098496
+  },
+  "ordinary_BIT": {
+    "admitted": "17/5000",
+    "after_cover": "69/20000",
+    "atom": "1/250",
+    "before_cover": "3459/1000000",
+    "ordinary_saving": "8590884599/2500000000000",
+    "prime_rule": "odd q > 2^26*N^5 avoiding all fixed denominators and rank witnesses"
+  },
+  "outer": {
+    "functional_checks": {
+      "actual_new_BIT_stock_sublinear": "174299999253/175000000000",
+      "fixed_halo_explicit_metadata_branch": "13475000001/175000000",
+      "new_BIT_row_stock_sublinear_in_active_group": "697899997009/700000000000",
+      "new_semantic_group_guard": "3/700000000"
+    },
+    "margins": {
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+      "fused_gaussian": "1480547593654799/490000000000000000",
+      "global_coordinate_router": "528767/175000000",
+      "group_bit_router": "370137073413699/122500000000000000",
+      "group_scans": "349999999/350000000",
+      "main_transform": "1480546897884933/490000000000000000",
+      "microbox_face_corrections": "13475000001/175000000",
+      "polynomial_products": "699999999/700000000",
+      "sparse_union_movement": "200543047597884933/490000000000000000"
+    },
+    "parameters": {
+      "a": "528767/175000000",
+      "b": "2115069/700000000",
+      "beta": "699999997/700000000",
+      "bit_saving": "17/5000",
+      "cap_xi": "445543046897884933/490000000000000000",
+      "complex_uniform_saving": "211507/70000000",
+      "d": "1/700000000",
+      "epsilon": "699999999/700000000",
+      "kappa": "37769/12500000",
+      "lambda_": "39879139/40000000",
+      "lambda_prime": "697884933/700000000",
+      "layout": "1/700000000",
+      "q": "2115067/700000000",
+      "r0": "1/7000000000",
+      "stop": "1/700000000",
+      "tested_saving": "302153/100000000"
+    },
+    "strict_checks": {
+      "actual_bit_above_effective": "66233/175000000",
+      "actual_logarithmic_gamma_guard": "1/700000000",
+      "actual_logarithmic_group_guard": "44456954152115067/245000000000000000",
+      "alpha_logarithm_absorption": "1/7000000000",
+      "boundary_algebraic_matrix_above_kappa": "2797884933/490000000000000000",
+      "cap_above_band": "200543046897884933/490000000000000000",
+      "cap_below_cell": "44456952402115067/490000000000000000",
+      "cap_below_phase_cell_power": "534456953102115067/490000000000000000",
+      "complex_above_effective": "1/700000000",
+      "complex_below_one_over_32": "19759931/700000000",
+      "complex_endpoint": "1/350000000",
+      "epsilon_below_one": "1/700000000",
+      "epsilon_positive": "699999999/700000000",
+      "fine_prime_halo_umbrella": "7/2",
+      "fused_gaussian_above_kappa": "2793654799/490000000000000000",
+      "gaussian_above_main": "347884933/245000000000000000",
+      "global_coordinate_router_above_kappa": "1/175000000",
+      "global_gamma_power_slack": "9/7000000000",
+      "group_below_dimension": "1/350000000",
+      "group_bit_router_above_kappa": "873413699/122500000000000000",
+      "group_gauge_power_slack": "44456954117115067/245000000000000000",
+      "group_key_fits_record": "1/350000000",
+      "group_positive": "699999997/700000000",
+      "group_scans_above_kappa": "348942467/350000000",
+      "halo_covers_generic_inverse_radius": "5",
+      "improvement_over_user_baseline": "645274790400418148631/500000000000000000000000",
+      "lambda_above_internal": "1/1400000000",
+      "lambda_above_sigma": "3/1400000000",
+      "lambda_above_tau": "1/1400000000",
+      "lambda_prime_above_lambda": "1/1400000000",
+      "lambda_prime_above_leaf": "1397884931/490000000000000000",
+      "leaf_saving_above_effective": "697884931/490000000000000000",
+      "main_transform_above_kappa": "2097884933/490000000000000000",
+      "membership_arithmetic_fits_record": "1/700000000",
+      "microbox_face_corrections_above_kappa": "2694894247/35000000",
+      "microbox_halo_volume": "55300000003/700000000",
+      "new_complex_uniform_saving_above_b": "1/700000000",
+      "new_semantic_guard_with_C1_one": "1/700000000",
+      "polynomial_products_above_kappa": "139576987/140000000",
+      "positive_layout_slack": "1/700000000",
+      "q_below_effective": "1/700000000",
+      "q_below_internal": "1/700000000",
+      "q_below_leaf": "1397884931/490000000000000000",
+      "q_positive": "2115067/700000000",
+      "short_record_fallback": "697884931/700000000",
+      "source_axis_superpolynomial": "1/700000000",
+      "sparse_union_movement_above_kappa": "199062502797884933/490000000000000000",
+      "stop_below_one": "699999999/700000000",
+      "stop_positive": "1/700000000",
+      "strassen_integer_power_gap": "1951441519231",
+      "support_router_span": "3/700000000",
+      "theta_positive_polynomial_power": "2",
+      "union_header_fits_record": "3/700000000",
+      "union_tag_work_per_visit_fits_record": "3/700000000"
+    }
+  },
+  "paid": {
+    "1": 41298,
+    "16": 448,
+    "2": 7416,
+    "24": 448,
+    "3": 9108,
+    "32": 448,
+    "4": 2430,
+    "40": 448,
+    "48": 448,
+    "5": 648,
+    "56": 448,
+    "64": 448,
+    "72": 224,
+    "8": 144
+  },
+  "paid_calls": 64404,
+  "productive_fields": 10928,
+  "rank_mass_surplus": 752,
+  "raw_root_interval": [
+    "30215551/10000000000",
+    "236059/78125000"
+  ],
+  "scope": "Exact changed-profile arithmetic only. Requires separately admitted changed native programs, paid transports and absolute phases. Retained compiler/native/precision/tape/analytic/recovery assumptions.",
+  "source_sha256": {
+    "research/fresh-helper-boundary/native.py": "9b55a2d3c3972148fae9d112e745d9244968166bd89bd900c938f429fc76ab29",
+    "research/higher-moment-strictification/verify.py": "0ce7e7137422d5ffeda22085c81fa88471bbe963100fdad62e659f1e3bf89326",
+    "research/joint-helper-lattice/CONTROLLER.json": "1f5d6aa267e4e66048818714b5d3b5eae17a58abe91d15deeaa34574f7391a5f",
+    "research/joint-helper-lattice/native_joint.py": "50869b1bcad7493e6b4cb2a4cd5d2fd9a193a5921100483fa2c3f2c999ddfd46",
+    "research/joint-helper-lattice/verify.py": "8eb13f77a94315f811612cb557a17edeecb6b9d9a75dfa238b112dd6520c336f",
+    "research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/bit-compression/carrier-reuse/reference/joint_pair_dag.py": "8a0de761642d21919e969475c3f13e5d01a4bae7343c1e1f37c586d94b676a85",
+    "research/kappa-interface-round2-20261022/price_profile.py": "adde619968d3ac37147d239e25145b28150650e9ce0f76035f680f22589adcbb",
+    "research/kappa-rebuild/architecture/mixed_target_probe.py": "011c3c9d0cf81103dc84d709a0e36793eb7c5ae3f23c85eac27650ef13afa04a",
+    "research/kappa-rebuild/bit/complex_controller_factor.py": "1de9aaf918e6a4f2f5448754c1f3c3da3b85e6257849ebd097af09859c4e5497",
+    "research/kappa-rebuild/bit/mixed_productive_bit.py": "b32291b0db7fc0d332ec444c70746ab0fb03351d2e50546f8bcb64060e3ff95e",
+    "research/pr4359-p7-padding/dependencies/compiler-refill/endpoint.py": "d097e55f06c3f244f2074ffd7c1a0a5f3c4b4b6a6accf9619d1dde5c0ae85e15",
+    "research/query-thinned-controllers/RESULT.json": "6ebfc8c0f09166f4834e9d7427410cb31aefff2cd4d4ce588c8e52c9c70e9a0f",
+    "research/query-thinned-controllers/controller.py": "8b462d660487b1be4940357c4c7522967b1bfaede3c02b1c66109b85acae0fd2"
+  },
+  "status": "PASS_CHANGED_PROFILE_ARITHMETIC_NATIVE_ADMISSION_SEPARATE",
+  "strictification": {
+    "alpha": {
+      "lower": "0.999999973838823307324189327075",
+      "upper": "0.999999973838823307324189327076"
+    },
+    "cutoff": 20,
+    "delta": {
+      "lower": "0.958285176191435812121414033913",
+      "upper": "0.958285176191435812121414033914"
+    },
+    "floor": 21,
+    "gamma": {
+      "lower": "0.956204373997172367485448051318",
+      "upper": "0.956204373997172367485448051319"
+    },
+    "normalized_paid_upper": "2097152001953127/2097154147483648",
+    "power": 1024,
+    "tail_offset": "1/5",
+    "tested": "302153/100000000"
+  },
+  "targets": {
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+    "17": 448,
+    "25": 448,
+    "33": 448,
+    "4": 504,
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+    "49": 448,
+    "57": 448,
+    "65": 448,
+    "7": 126,
+    "73": 224,
+    "8": 846,
+    "9": 5868
+  }
+}

research/kappa-interface-round2-20261022/PROFILE.json

@@ -0,0 +1,9 @@
+{
+  "status": "CANDIDATE_PROFILE_NATIVE_VALIDATION_PENDING",
+  "helpers": {"4": 28, "7": 7, "8": 47, "9": 326},
+  "paid": {"1": 2201, "2": 412, "3": 506, "4": 135, "5": 36, "8": 8},
+  "native": {
+    "forward": {"real_scalar_allowance": 158120},
+    "backward": {"real_scalar_allowance": 158168}
+  }
+}

research/kappa-interface-round2-20261022/PROOF.md

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+# Jointly changing native maps gives a smaller complete invoice
+
+Prepared with substantial OpenAI Codex assistance. Apache-2.0. This is an
+AI-assisted conditional construction and executable certificate, without
+independent human review or a formal proof of the multiplication theorem.
+
+## The changed degree of freedom
+
+The PR4470 supplier proves its logical endpoint identity in the free group
+algebra of 72 independently numbered invertible warp operators. Its two
+literal evaluators retain all 632 arbitrary persistent inputs and the owned
+center input. They require no equality between distinct warp generators.
+Consequently a generator can be replaced consistently, at every occurrence,
+by another invertible operator, with negative occurrences bound to its actual
+inverse. Substitution preserves the entire endpoint identity, including the
+autonomous dirty-helper outputs. In the backward orientation the substituted
+physical operators and frame transports use the retained conjugate convention.
+
+This permits changing the binary map G itself. It goes beyond moving a warp
+with G held fixed. For 60 operators the only nontrivial frame occurrence is
+one forward B warp; other occurrences have zero or full frames. If U and V
+have equal dimension, extending bases constructs an invertible G with GU=V.
+The remaining 12 operators are kept fixed by this search. All occurrences of
+each changed operator use the same new G and exact phase convention.
+
+For a frame U in F2^9 put
+
+    L_U = {(x,x+p): x in U, p in U-perp}.
+
+The native symplectic action is
+
+    S_G(x,z) = (Gx, Gx + G^(-T)(x+z)).
+
+Thus S_G(L_U)=L_(GU). Exact Clifford lifts show the physical warp
+T_(GU) K_G T_U^-1 is monomial. This is the transported-frame lemma already
+proved in `../transported-warp-frames/PROOF.md`; it alone does not determine
+the absolute phase. The new certificate explicitly binds every changed G to
+its native K_G, checks all 512 columns by independent integer formulas, and
+lowers every used physical warp including its absolute phase.
+
+The joint scalar-contact/map search chooses 422 changed scalar frame labels.
+It frees 60 maps simultaneously; 34 actually differ from the original maps.
+All 17,453 scalar opcodes, their order and coefficients remain unchanged.
+Owned-center jobs and every external main/helper interface remain unchanged.
+The forward/backward formal prefix L1 bounds remain 457/169, with maximum
+denominators 32/64. The substituted native warps have unit-modulus monomial
+entries, so these formal bounds continue to control their scalar prefixes.
+The emitted word, rather than obsolete source A/B metadata, is the authority.
+The frozen actual payload SHA-256 is
+`4409d2d9328fffbd276bb03fb800f59a33ec3b2ab15f79a330a7af66e75110cc`.
+
+The numerical MILP is only a discovery method. Neither its numerical
+optimality nor completeness of its frame catalogue is needed to verify this
+specific construction. The verifier needs no optimization package.
+
+## Exact productive and paid lists
+
+The 408 strictly productive arbitrary helper fields have the unchanged list
+
+    H = 28 H4 + 7 H7 + 47 H8 + 326 H9.
+
+The complete new paid list in either orientation is
+
+    Pnew = 2201 H1 + 412 H2 + 506 H3 + 135 H4 + 36 H5 + 8 H8.
+
+The predecessor's list is
+
+    Pold = 2237 H1 + 388 H2 + 494 H3 + 147 H4 + 36 H5 + 8 H8.
+
+Therefore
+
+    Pold - Pnew = 12(3 H1 - 2 H2 - H3 + H4).
+
+There are 12 fewer paid jobs per full call, with unchanged paid rank mass and
+unchanged helper credit. The full p16 controller uses 18 full calls and 15
+query-only shears, and pays all retained idle windows. Its target list is
+unchanged; its paid-call count decreases from 64,620 to 64,404. Its rank-mass
+surplus remains 752. Every field and every child is counted once in the
+literal physical event inventory.
+
+This improvement is strict for every moment exponent 0<t<1. Define
+
+    g(t) = 3 + 4^t - 2*2^t - 3^t.
+
+We have g(1)=0 and
+
+    g'(t)/3^t = 2[(4/3)^t - (2/3)^t] log(2) - log(3).
+
+The bracket is increasing in t, so on [0,1] the right side is at most
+(4/3)log(2)-log(3), which is strictly negative because 16<27. Hence g(t)>0
+for 0<t<1. The local paid-moment reduction is 12g(t), and the complete
+controller's reduction is 216g(t). This is a changed invoice, not additional
+digits of the old profile's root.
+
+## Recursive and outer admission
+
+`price_profile.py` substitutes the freshly admitted native list into the
+literal retained p16 word, reconstructing all full calls, thin calls, idles
+and main diagonal jobs. Two rational interval implementations check the
+moments independently. The changed profile has strict opposite signs at
+the rational endpoints `.0030215551` and `.0030215552`; this is an enclosure
+of a raw moment crossing, not a claim of global optimality.
+
+We use the smaller rational saving
+
+    kappa = 37769/12500000 = .00302152,
+    tested saving = .00302153,
+    compiler gap = 1/700000000.
+
+Both charged tail ratios, the productive-completion tail, splitting and
+rounding fit the inherited strictification with power 1024, cutoff 20,
+floor 21 and offset 1/5. The logarithmic majorant uses theta=63/64 and
+power 1000. Its explicit uniform threshold is N=2^31385200000000; the
+astronomical algorithm-defined template is not materialized. The actual
+scalar word and new rank-zero routes fit the inherited larger fixed caps.
+All 54 strict and four functional outer inequalities pass with the retained
+ordinary BIT saving and stock conversion. Four incompatible outer patches
+are rejected.
+
+These are the same analytic, exact-recovery, native-movement, precision,
+stock and fixed-tape assumptions as the repository framework. The new work
+supplies a conditional improvement within that framework. It does not turn
+the framework into an unconditional integer-multiplication theorem or a
+practical multiplication implementation.
+
+## Credits and provenance
+
+The supplier, helper retiming and native certificate lineage are Cinna
+Davis's PR4470 and its attributed predecessors. Productive mixed-job
+recursion is due to Jacob Alan Sussman; the retained completion/logarithmic
+compiler is from zxv's lineage. The PR4535 public proposal supplies the
+comparison value `.003020165`; this construction changes the invoice rather
+than reusing that proposal's extra root digits. The repository's detailed
+community credits and inherited notices remain in place. Source hashes and
+fresh replay receipts accompany this result. No selected record is edited.