{
  "canvas_notice": "This canvas explains the pinned source. It is not a proof and does not change status.",
  "claims": [
    {
      "class": "ACCEPT-scoped",
      "id": "d2-lifetime-remainder",
      "proof": {
        "blob": "247b3ecf80bfbe896948d5d489b2d5842a81c481",
        "bytes": 17734,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
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        "path": "frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "380b7d0abdb0fe2de5a6564565af9560d3f1b1ce22fd5538927db0c52f4f3a4a",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md"
      },
      "replay": {
        "checkout": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "command": "python -B -S -m unittest discover -s frontiers/three_fronts_20260924 -p 'test_*.py' -v",
        "notice": "From a Math- checkout at d6628da09384728992dcbe6e921cc28ba85aebb0, run at the repository root. Finite algebra or implementation replay only; it is not theorem acceptance.",
        "source": {
          "blob": "abe16871da7740fa28ae4dbb28fb31853a5868bd",
          "bytes": 6023,
          "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
          "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/README.md",
          "path": "README.md",
          "repository": "d6g8k5htny-coder/Math-",
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        },
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        "working_directory": "Math- repository root"
      },
      "review": {
        "blob": "e4cb1d943eaeb2049d080fcc53a57e042aebc6e8",
        "bytes": 12708,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/PROOF_INDEX.md",
        "path": "PROOF_INDEX.md",
        "pointer_only": true,
        "repository": "d6g8k5htny-coder/Math-",
        "review_notice": "The pinned source binds this review pointer; the linked review is not byte-frozen by this descriptor.",
        "review_url": "https://github.com/d6g8k5htny-coder/main/issues/67#issuecomment-5841782206",
        "sha256": "f9c4db77aeab1c61d2a0c02f88a964cbfc6be63d18619acf0e047c884e0c0249",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/PROOF_INDEX.md"
      },
      "scope_quote": "- D2 lifetime remainder: proof [frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md](frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md); [R1–R4 review](https://github.com/d6g8k5htny-coder/main/issues/67#issuecomment-5841270276) and [R5/R6 delta review](https://github.com/d6g8k5htny-coder/main/issues/67#issuecomment-5841782206) record ACCEPT for Theorem R at its stated existential `O(1)`-remainder scope. The delta identifies its exact parent imports and does not consume the still-open Theorem A cubic selection bound. Numerical constants/radii, a second coefficient, and RN/24-jet closure are outside this verdict. [main #67](https://github.com/d6g8k5htny-coder/main/issues/67) remains open for other bundled fronts.",
      "source_label": "ACCEPT",
      "status_quote": "Theorem R at its stated existential `O(1)` remainder scope: candidate and elder densities have leading `c ell^(-1/3)` term with bounded remainder for sufficiently small lifetime",
      "title": "D2 lifetime remainder"
    },
    {
      "class": "ACCEPT-scoped",
      "id": "d3-side24-coefficient",
      "proof": {
        "blob": "44b66f04f89fcd87383b3603fa69f1feb64cdddd",
        "bytes": 10272,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
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        "repository": "d6g8k5htny-coder/Math-",
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        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/coefficients/side24_v1/PROOF.md"
      },
      "replay": {
        "command": null,
        "notice": "The notebook verifies and displays published coefficient bytes; no random field or lifetime solver runs.",
        "source": {
          "blob": "c7fcde5461305ff249384e873476d34a4f470976",
          "bytes": 61961,
          "commit": "71400b94f6cb354a8cf7aba73ffede2138a64efa",
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          "path": "docs/notebooks/SIDE24_PUBLIC_COEFFICIENTS.ipynb",
          "repository": "d6g8k5htny-coder/main",
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        },
        "url": "https://github.com/d6g8k5htny-coder/main/blob/71400b94f6cb354a8cf7aba73ffede2138a64efa/docs/notebooks/SIDE24_PUBLIC_COEFFICIENTS.ipynb"
      },
      "review": {
        "blob": "e4cb1d943eaeb2049d080fcc53a57e042aebc6e8",
        "bytes": 12708,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/PROOF_INDEX.md",
        "path": "PROOF_INDEX.md",
        "pointer_only": true,
        "repository": "d6g8k5htny-coder/Math-",
        "review_notice": "The pinned source binds this review pointer; the linked review is not byte-frozen by this descriptor.",
        "review_url": "https://github.com/d6g8k5htny-coder/main/issues/65#issuecomment-5841269490",
        "sha256": "f9c4db77aeab1c61d2a0c02f88a964cbfc6be63d18619acf0e047c884e0c0249",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/PROOF_INDEX.md"
      },
      "scope_quote": "- D3 SIDE24 coefficient: proof [coefficients/side24_v1/PROOF.md](coefficients/side24_v1/PROOF.md); [C1–C6 review](https://github.com/d6g8k5htny-coder/main/issues/65#issuecomment-5841269490) verifies the coefficient calculation; [reconciliation](https://github.com/d6g8k5htny-coder/main/issues/65#issuecomment-5841779222) closes the coefficient-review work package. Interpretation as the parent lifetime law retains the parent's separate dependency and review boundaries.",
      "source_label": "ACCEPT",
      "status_quote": "The coefficient expression for SIDE24 in dimensions 2 and 3, including cone moments, all-direction periodization comparison, covariance/Schur transfer, and outward special-function arithmetic",
      "title": "D3 SIDE24 coefficient"
    },
    {
      "class": "ACCEPT-scoped",
      "id": "d4-fixed-remote-rn",
      "proof": {
        "blob": "b383bfcc88ec4ad497dff01fb6640e429ba24a84",
        "bytes": 18355,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/remote_window_20260924/PROOF.md",
        "path": "frontiers/remote_window_20260924/PROOF.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/remote_window_20260924/PROOF.md"
      },
      "replay": {
        "checkout": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "command": "python -B -S -m unittest discover -s frontiers/remote_window_20260924 -p 'test_*.py' -v",
        "notice": "From a Math- checkout at d6628da09384728992dcbe6e921cc28ba85aebb0, run at the repository root. Finite algebra or implementation replay only; it is not theorem acceptance.",
        "source": {
          "blob": "abe16871da7740fa28ae4dbb28fb31853a5868bd",
          "bytes": 6023,
          "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
          "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/README.md",
          "path": "README.md",
          "repository": "d6g8k5htny-coder/Math-",
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          "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/README.md"
        },
        "url": null,
        "working_directory": "Math- repository root"
      },
      "review": {
        "blob": "e4cb1d943eaeb2049d080fcc53a57e042aebc6e8",
        "bytes": 12708,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/PROOF_INDEX.md",
        "path": "PROOF_INDEX.md",
        "pointer_only": true,
        "repository": "d6g8k5htny-coder/Math-",
        "review_notice": "The pinned source binds this review pointer; the linked review is not byte-frozen by this descriptor.",
        "review_url": "https://github.com/d6g8k5htny-coder/main/issues/76#issuecomment-5841783172",
        "sha256": "f9c4db77aeab1c61d2a0c02f88a964cbfc6be63d18619acf0e047c884e0c0249",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/PROOF_INDEX.md"
      },
      "scope_quote": "- D4 fixed-remote RN theorem: proof [frontiers/remote_window_20260924/PROOF.md](frontiers/remote_window_20260924/PROOF.md); [source-bound review](https://github.com/d6g8k5htny-coder/main/issues/76#issuecomment-5841783172) and [scoped reconciliation](https://github.com/d6g8k5htny-coder/main/issues/76#issuecomment-5841861362) record ACCEPT on the seven fixed-rho/fixed-eta interfaces. Shrinking spatial cutoffs, pin/intermediate regions, and shrinking witness separation are outside that review.",
      "source_label": "ACCEPT",
      "status_quote": "Fixed positive spatial separation `rho`, fixed witness separation `eta`, between-pin height window; conditioned covariance, three determinant factors, endpoint normalizer, contact kernel, and fixed-separation factorial moments",
      "title": "D4 fixed-remote RN theorem"
    },
    {
      "class": "ACCEPT-scoped",
      "id": "d5-all-height-annulus",
      "proof": {
        "blob": "6f317515b3d417661f86e2fed09bc7d950899c2b",
        "bytes": 16948,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/rn_annulus_bridge_20260925/PROOF.md",
        "path": "frontiers/rn_annulus_bridge_20260925/PROOF.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "d55e2c03bb17e7977ff94130cc1ff21e54840cd1e20dc4e41ea9ad52228beb05",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/rn_annulus_bridge_20260925/PROOF.md"
      },
      "replay": {
        "checkout": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "command": "python3 -B -S reviews/pr28_annulus_bridge_nonauthor_20260925/algebra_check.py -v",
        "notice": "From a Math- checkout at d6628da09384728992dcbe6e921cc28ba85aebb0, run at the repository root. Finite algebra or implementation replay only; it is not theorem acceptance.",
        "source": {
          "blob": "559d72242fdb7e4e3ac4a10d10641ae1da84f40e",
          "bytes": 15925,
          "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
          "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/pr28_annulus_bridge_nonauthor_20260925/REVIEW.md",
          "path": "reviews/pr28_annulus_bridge_nonauthor_20260925/REVIEW.md",
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        },
        "url": null,
        "working_directory": "Math- repository root"
      },
      "review": {
        "blob": "559d72242fdb7e4e3ac4a10d10641ae1da84f40e",
        "bytes": 15925,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/pr28_annulus_bridge_nonauthor_20260925/REVIEW.md",
        "path": "reviews/pr28_annulus_bridge_nonauthor_20260925/REVIEW.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "69ea7a479a9e8a5150cbe07b373772ab22413a1d81194d5a6a78adf6ada5f03f",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/pr28_annulus_bridge_nonauthor_20260925/REVIEW.md"
      },
      "scope_quote": "- D5 all-height fixed annulus: proof [frontiers/rn_annulus_bridge_20260925/PROOF.md](frontiers/rn_annulus_bridge_20260925/PROOF.md); review [reviews/pr28_annulus_bridge_nonauthor_20260925/REVIEW.md](reviews/pr28_annulus_bridge_nonauthor_20260925/REVIEW.md) records R1–R7 ACCEPT at fixed `d=2`, fixed `L`, compact positive-gap marks, and fixed annulus `1<A<B<infinity`. Pin, intermediate, and witness-collision complements remain open. This is the full reviewed candidate body, not a global D5/RN theorem.",
      "source_label": "ACCEPT",
      "status_quote": null,
      "title": "D5 all-height fixed annulus"
    },
    {
      "class": "ACCEPT-scoped",
      "id": "d5-height-window-annulus",
      "proof": {
        "blob": "081abc13c5e66342c13df2af2bd6b114f320486f",
        "bytes": 17646,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md",
        "path": "frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "1fd9fe7141e464fd1c09ebf0318d8a701729c9c61ea24f73f7e20a9cf10c552b",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md"
      },
      "replay": {
        "checkout": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "command": "python3 -B -S reviews/pr22_fixed_annulus_nonauthor_20260925/algebra_check.py",
        "notice": "From a Math- checkout at d6628da09384728992dcbe6e921cc28ba85aebb0, run at the repository root. Finite algebra or implementation replay only; it is not theorem acceptance.",
        "source": {
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      },
      "review": {
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        "bytes": 12741,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md",
        "path": "reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "29a0c6d00ce739baf1d29a3163977d8e707e2079e216fba4fee3669cebec127d",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md"
      },
      "scope_quote": "- D5 fixed-annulus height-window fallback: proof [frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md](frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md); review [reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md](reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md).",
      "source_label": "ACCEPT",
      "status_quote": null,
      "title": "D5 fixed-annulus height-window fallback"
    },
    {
      "class": "ACCEPT-scoped",
      "id": "d5-two-scale",
      "proof": {
        "blob": "89cae3a9734f2ec7172cd0b6b0b3af3ddd355d73",
        "bytes": 15902,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md",
        "path": "frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "079f9399aef401d58749b3684f3acbb7f49ffdcbcac9f501b79e06c28a7f4e7d",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md"
      },
      "replay": {
        "command": null,
        "notice": "No replay command is recorded in the pinned proof/review.",
        "url": null
      },
      "review": {
        "blob": "8988251631cab8e65eb2b01a08f7fc593b018570",
        "bytes": 10073,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md",
        "path": "reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "04cf4c986f29ac7ac33b3cc87feac23f3cf494e0c1b3bf4476e4e6666de46574",
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      },
      "scope_quote": "- D5 two-scale S6-S21: proof [frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md](frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md); review [reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md](reviews/replacement_20260925_pr19_pr21/TWO_SCALE_REVIEW.md).",
      "source_label": "ACCEPT",
      "status_quote": null,
      "title": "D5 two-scale S6-S21"
    },
    {
      "class": "ACCEPT-scoped",
      "id": "d5-inner-belt-density",
      "proof": {
        "blob": "a72418d78fd3c1ec96260dbf19f9a2f3f4ca35ba",
        "bytes": 10858,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/axial_density_20260925/PROOF.md",
        "path": "frontiers/axial_density_20260925/PROOF.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "8b9376a69fda9f0f231d9502be839ae45d8006d1891ff3e700fa521cc96f6cb4",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/axial_density_20260925/PROOF.md"
      },
      "replay": {
        "command": null,
        "notice": "No replay command is recorded in the pinned proof/review.",
        "url": null
      },
      "review": {
        "blob": "f1868bd92b600d92901146af1caf7baf960dba9c",
        "bytes": 15615,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/replacement_20260925_pr19_pr21/REVIEW.md",
        "path": "reviews/replacement_20260925_pr19_pr21/REVIEW.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "7ff9e4a028f24afc731200b14bc5b993c0a874906cf6ea25e8c890afed8cdc53",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/replacement_20260925_pr19_pr21/REVIEW.md"
      },
      "scope_quote": "- D5 inner-belt gradient density: proof [frontiers/axial_density_20260925/PROOF.md](frontiers/axial_density_20260925/PROOF.md); review [reviews/replacement_20260925_pr19_pr21/REVIEW.md](reviews/replacement_20260925_pr19_pr21/REVIEW.md) accepts corollary (1) on `|z|<=W r^2`. That acceptance is the density bound only.",
      "source_label": "ACCEPT",
      "status_quote": null,
      "title": "D5 inner-belt gradient density"
    },
    {
      "class": "ACCEPT-scoped",
      "id": "d5-fixed-transverse",
      "proof": {
        "blob": "024d927779f79fadaf932541ea6474f2995f8c50",
        "bytes": 9902,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/downstream_boundary_20260925/TRANSVERSE_BOUND_CANDIDATE.md",
        "path": "reviews/downstream_boundary_20260925/TRANSVERSE_BOUND_CANDIDATE.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "f64c954245f17bcbc64a5ccf58a60689cf2a2fd85362f8321c0f64ff66584e36",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/downstream_boundary_20260925/TRANSVERSE_BOUND_CANDIDATE.md"
      },
      "replay": {
        "checkout": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "command": "python3 -B -S reviews/pr16_fixed_transverse_nonauthor_20260925/algebra_check.py",
        "notice": "From a Math- checkout at d6628da09384728992dcbe6e921cc28ba85aebb0, run at the repository root. Finite algebra or implementation replay only; it is not theorem acceptance.",
        "source": {
          "blob": "4003ff13cf696389b9530fd445ee46daf3b4c646",
          "bytes": 12510,
          "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
          "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/pr16_fixed_transverse_nonauthor_20260925/REVIEW.md",
          "path": "reviews/pr16_fixed_transverse_nonauthor_20260925/REVIEW.md",
          "repository": "d6g8k5htny-coder/Math-",
          "sha256": "20d6451412f304a20f70e0bc6aabb87bdd0bb9761b17834469d60867407ebd9c",
          "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/pr16_fixed_transverse_nonauthor_20260925/REVIEW.md"
        },
        "url": null,
        "working_directory": "Math- repository root"
      },
      "review": {
        "blob": "4003ff13cf696389b9530fd445ee46daf3b4c646",
        "bytes": 12510,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/pr16_fixed_transverse_nonauthor_20260925/REVIEW.md",
        "path": "reviews/pr16_fixed_transverse_nonauthor_20260925/REVIEW.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "20d6451412f304a20f70e0bc6aabb87bdd0bb9761b17834469d60867407ebd9c",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/pr16_fixed_transverse_nonauthor_20260925/REVIEW.md"
      },
      "scope_quote": "- D5 fixed-transverse chart: proof [reviews/downstream_boundary_20260925/TRANSVERSE_BOUND_CANDIDATE.md](reviews/downstream_boundary_20260925/TRANSVERSE_BOUND_CANDIDATE.md) (byte-for-byte import of commit `32b80ee085dc6a40113d1e46e333cda50d57ba21`, blob `024d927779f79fadaf932541ea6474f2995f8c50`); review [reviews/pr16_fixed_transverse_nonauthor_20260925/REVIEW.md](reviews/pr16_fixed_transverse_nonauthor_20260925/REVIEW.md).",
      "source_label": "ACCEPT",
      "status_quote": null,
      "title": "D5 fixed-transverse chart"
    },
    {
      "class": "ACCEPT-scoped",
      "id": "cumulative-transfer-correction",
      "proof": {
        "blob": "044ac5fdaf403a38e33983e31f0ad69f8e76d6d5",
        "bytes": 3272,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/collision_mechanism_20260925/CUMULATIVE_TRANSFER_CORRECTION.md",
        "path": "reviews/collision_mechanism_20260925/CUMULATIVE_TRANSFER_CORRECTION.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "83f653393dc6245980f6848e2bfc65ac9ad4c7d8304b028b88764356fd3825b6",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/collision_mechanism_20260925/CUMULATIVE_TRANSFER_CORRECTION.md"
      },
      "replay": {
        "command": null,
        "notice": "No replay command is recorded in the pinned proof/review.",
        "url": null
      },
      "review": {
        "blob": "3ab51b0112bbcd29febc433881f459240645056d",
        "bytes": 3723,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/d2_cumulative_correction_20260925/REVIEW.md",
        "path": "reviews/d2_cumulative_correction_20260925/REVIEW.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "910d371ebb4a310a7e2248348ba8cd09d6f3229740525d74ec7d06d8d1470717",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/d2_cumulative_correction_20260925/REVIEW.md"
      },
      "scope_quote": "- Cumulative transfer correction: proof [reviews/collision_mechanism_20260925/CUMULATIVE_TRANSFER_CORRECTION.md](reviews/collision_mechanism_20260925/CUMULATIVE_TRANSFER_CORRECTION.md); review [reviews/d2_cumulative_correction_20260925/REVIEW.md](reviews/d2_cumulative_correction_20260925/REVIEW.md) accepts this correction only.",
      "source_label": "ACCEPT",
      "status_quote": null,
      "title": "Cumulative transfer correction"
    },
    {
      "class": "engineering-only",
      "id": "p15-demand-one-counterexample",
      "proof": {
        "blob": "1d6946658e00e2446d68e8a4884425d4bcec62fb",
        "bytes": 2266,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md",
        "path": "frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "498af650ef7139c39d2630561dbfd0822c495ce6308a6cac8b767aefe0c2a40b",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md"
      },
      "replay": {
        "command": null,
        "notice": "No replay command is recorded in the pinned proof/review.",
        "url": null
      },
      "review": {
        "blob": "e4cb1d943eaeb2049d080fcc53a57e042aebc6e8",
        "bytes": 12708,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/PROOF_INDEX.md",
        "path": "PROOF_INDEX.md",
        "pointer_only": true,
        "repository": "d6g8k5htny-coder/Math-",
        "review_notice": "The pinned source binds this review pointer; the linked review is not byte-frozen by this descriptor.",
        "review_url": "https://github.com/d6g8k5htny-coder/main/issues/67",
        "sha256": "f9c4db77aeab1c61d2a0c02f88a964cbfc6be63d18619acf0e047c884e0c0249",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/PROOF_INDEX.md"
      },
      "scope_quote": "- P15 demand-one extension: exact counterexample [frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md](frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md); review pointer in [claims/LANDING_CLAIMS.json](claims/LANDING_CLAIMS.json) is [main #67](https://github.com/d6g8k5htny-coder/main/issues/67), disposition EXACT_COUNTEREXAMPLE.",
      "source_label": "EXACT_COUNTEREXAMPLE",
      "status_quote": null,
      "title": "P15 demand-one extension"
    },
    {
      "class": "ACCEPT-scoped",
      "id": "d6-p15-full-price",
      "proof": {
        "blob": "582180e41dca0ad815ad0f18574df42040912149",
        "bytes": 11352,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/full_price_20260924/PROOF.md",
        "path": "frontiers/full_price_20260924/PROOF.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/frontiers/full_price_20260924/PROOF.md"
      },
      "replay": {
        "checkout": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "command": "python -B -S frontiers/full_price_20260924/full_price.py",
        "notice": "From a Math- checkout at d6628da09384728992dcbe6e921cc28ba85aebb0, run at the repository root. Finite algebra or implementation replay only; it is not theorem acceptance.",
        "source": {
          "blob": "abe16871da7740fa28ae4dbb28fb31853a5868bd",
          "bytes": 6023,
          "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
          "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/README.md",
          "path": "README.md",
          "repository": "d6g8k5htny-coder/Math-",
          "sha256": "89d4c618ad4560373f8fba3c787522cea8f0191fd8e176f3e0741c6230390730",
          "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/README.md"
        },
        "url": null,
        "working_directory": "Math- repository root"
      },
      "review": {
        "blob": "07db19f826763b8faa7414ba2e403d9c45bc0b6e",
        "bytes": 15569,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/p15_full_price_nonauthor_20260926/REVIEW.md",
        "path": "reviews/p15_full_price_nonauthor_20260926/REVIEW.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "691ea0db5c0f073326c9f7208c9804c6f5d0daf92ba7467529c9d57e08b15eb7",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/reviews/p15_full_price_nonauthor_20260926/REVIEW.md"
      },
      "scope_quote": "- D6 P15 full price theorem: proof [frontiers/full_price_20260924/PROOF.md](frontiers/full_price_20260924/PROOF.md); [reviews/p15_full_price_nonauthor_20260926/REVIEW.md](reviews/p15_full_price_nonauthor_20260926/REVIEW.md) records ACCEPT for Theorem F on the stated realized disjoint-capacity/clutter family, with demands at least two, independent probabilities, and the same palette. That review leaves the landing disposition untouched; it does not accept arbitrary downsets, demand-one extension, or an unrestricted prize claim.",
      "source_label": "ACCEPT",
      "status_quote": "Realized disjoint capacity/clutter family with `d_i >= 2`, all independent probabilities, `c_v <= phi(p_v)`, same palette `K >= K_H(d)`, sharp `rho*=1/(3-log(3e-2))`",
      "title": "D6 P15 full price theorem"
    },
    {
      "class": "ACCEPT-scoped",
      "id": "d1-parent-lifetime",
      "proof": {
        "blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2",
        "bytes": 40261,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
        "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md"
      },
      "replay": {
        "command": null,
        "notice": "No replay command is recorded in the pinned proof/review.",
        "url": null
      },
      "review": {
        "blob": "75da2597971510f843f8d90c743950cb8c177342",
        "bytes": 23312,
        "commit": "82247833b5dd58e04291d55353b52938d73ec614",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/82247833b5dd58e04291d55353b52938d73ec614/reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md",
        "path": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "451b9d7ffee072a73fc904cab891b6693e3f233808b89df10cce1b57036b65da",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/82247833b5dd58e04291d55353b52938d73ec614/reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md"
      },
      "scope_quote": "| **D1 — parent lifetime theorem (Theorems A, B, C)** | Theorem A `0 <= 1-p_r <= C r^3`, uniform over compact births, compact positive gaps and all frames; Theorem B compact-mark candidate/elder densities `~ c_{B,K} ell^(-1/3)` with difference `O(ell^(2/3))`, for `B` and `K` of positive length; Theorem C unrestricted leading term `c_{d,L} ell^(-1/3)`. Fixed `d>=2` and `L`, existential constants, read with the congruence erratum, the Section 9 replacement v1.1, wording W1 and `r < L/(4 sqrt 2)` | [Proof](https://github.com/d6g8k5htny-coder/Math-/blob/82247833b5dd58e04291d55353b52938d73ec614/imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md) · [reconciliation](https://github.com/d6g8k5htny-coder/Math-/blob/82247833b5dd58e04291d55353b52938d73ec614/reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md) · [Math-#126](https://github.com/d6g8k5htny-coder/Math-/pull/126) | No numerical `C`, `r_*`, `z_*` or coefficient; no unrestricted difference rate; this row is confined to the Math- `8224783` reconciliation, whose lower composition is planar (`d=2`); the separately reviewed every-fixed-`d` lower bound `1-p_r >= c r^3` was integrated afterwards by [Math-#129](https://github.com/d6g8k5htny-coder/Math-/pull/129) at `e7f8aca41c308f8ad8614b587f75916943c86f27` and is not part of this row's reconciliation scope; no RN/24-jet certificate |",
      "source_label": "ACCEPT",
      "status_quote": "Theorem A `0 <= 1-p_r <= C r^3`, uniform over compact births, compact positive gaps and all frames; Theorem B compact-mark candidate/elder densities `~ c_{B,K} ell^(-1/3)` with difference `O(ell^(2/3))`, for `B` and `K` of positive length; Theorem C unrestricted leading term `c_{d,L} ell^(-1/3)`. Fixed `d>=2` and `L`, existential constants, read with the congruence erratum, the Section 9 replacement v1.1, wording W1 and `r < L/(4 sqrt 2)`",
      "title": "D1 — parent lifetime theorem (Theorems A, B, C)"
    },
    {
      "class": "AMEND/open",
      "id": "d5-pin-neighborhoods-open",
      "proof": {
        "availability": "NO COMPLETE PROOF YET",
        "blob": "e4cb1d943eaeb2049d080fcc53a57e042aebc6e8",
        "bytes": 12708,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/PROOF_INDEX.md",
        "path": "PROOF_INDEX.md",
        "repository": "d6g8k5htny-coder/Math-",
        "sha256": "f9c4db77aeab1c61d2a0c02f88a964cbfc6be63d18619acf0e047c884e0c0249",
        "url": "https://raw.githubusercontent.com/d6g8k5htny-coder/Math-/d6628da09384728992dcbe6e921cc28ba85aebb0/PROOF_INDEX.md"
      },
      "replay": {
        "command": null,
        "notice": "OPEN / NOT LANDED. No replay is presented as acceptance.",
        "url": null
      },
      "review": {
        "blob": "e4cb1d943eaeb2049d080fcc53a57e042aebc6e8",
        "bytes": 12708,
        "commit": "d6628da09384728992dcbe6e921cc28ba85aebb0",
        "html_url": "https://github.com/d6g8k5htny-coder/Math-/blob/d6628da09384728992dcbe6e921cc28ba85aebb0/PROOF_INDEX.md",
        "path": "PROOF_INDEX.md",
        "pointer_only": true,
        "repository": "d6g8k5htny-coder/Math-",
        "review_notice": "The pinned source binds this review pointer; the linked review is not byte-frozen by this descriptor.",
        "review_url": "https://github.com/d6g8k5htny-coder/Math-/blob/4e188e25b1e1ef560f3eeb75c0d354d2ccf0ea22/reviews/d5_pin_neighborhood_20260926/REVIEW.md",
        "sha256": "f9c4db77aeab1c61d2a0c02f88a964cbfc6be63d18619acf0e047c884e0c0249",
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      "scope_quote": "| **D5 — pin neighborhoods / microdisk** | The public reconnaissance has an AMEND review; the inner microdisk bound and collar to the reviewed annulus are not complete, and the claimed summed pin-neighborhood bound remains AMEND. | [Math proof index](https://github.com/d6g8k5htny-coder/Math-/blob/main/PROOF_INDEX.md) · [Math issue #58](https://github.com/d6g8k5htny-coder/Math-/issues/58) |",
      "source_label": "AMEND / open",
      "status_quote": "The public reconnaissance has an AMEND review; the inner microdisk bound and collar to the reviewed annulus are not complete, and the claimed summed pin-neighborhood bound remains AMEND.",
      "title": "D5 — pin neighborhoods / microdisk"
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      },
      "scope_quote": "| **SARD-G A1/A6** | A1 remains AMEND as written; A6's slicing argument is conditionally valid but its source application remains AMEND until the required open-chart premise is repaired. | [Successor review](reviews/sard_g_successor_a1_a6_20260926/REVIEW.md) · [current PR](https://github.com/d6g8k5htny-coder/main/pull/122) |",
      "source_label": "AMEND / open",
      "status_quote": "A1 remains AMEND as written; A6's slicing argument is conditionally valid but its source application remains AMEND until the required open-chart premise is repaired.",
      "title": "SARD-G A1/A6"
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    "annulus": "## 1. Statement and exact scope\n\nFix L>0 and the centered variance-one Gaussian field on the two-dimensional torus with covariance\n\n    K_L(h) = sum_(n in Z^2) exp(-|h+Ln|^2/2)\n             / sum_(n in Z^2) exp(-|Ln|^2/2).\n\nFix a compact birth interval, 0<k_-<=k<=k_+<infinity, and 1<A<B<infinity. For every orthonormal frame write f(x,z)=F(x e1+z e2), with midpoint zero. Impose exactly the six finite-r observations\n\n    f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0,\n    M=(-r/2,0), S=(r/2,0).                                  (1)\n\nQ_r denotes their continuous Gaussian regression law, not an adjacency law. For a symmetric 2x2 matrix put F_j(H)=|det H| times the indicator of negative index j, zero on singular matrices. Define\n\n    W_r=F_2(H_M) F_1(H_S), Z_r=E_Qr W_r, dQ_r^W=(W_r/Z_r)dQ_r.\n\nFor a Borel set E in K_AB={(u,v): A<=sqrt(u^2+v^2)<=B}, let N_j(rE) count index-j critical points in rE at **all heights**.\n\n**Candidate theorem.** There are C,r_*>0, uniform in the stated marks, frames, Borel E and j=0,1,2, such that\n\n    E_(Q_r^W) N_j(rE) <= C r^3 area(E), 0<r<=r_*.           (2)\n\nEquivalently, the physical-area weighted critical-point intensity satisfies rho_j^W(ru,rv)<=Cr throughout this fixed scaled annulus. Summing over indices only changes C. Markov gives an upper bound on witness existence, not a lower bound. The between-pin-height count is a subset and satisfies the same bound. Since k>=k_->0, (2) can also be written <=C' k r^3 area(E), but that rewriting uses k_-: **there is no height-window factor in the proof and no assertion uniform as k approaches zero**.\n\nThis addresses the fixed scaled-annulus region in d=2. It does not include pin neighborhoods, A approaching 1/2, B growing with r, intermediate physical distances r<<distance<<rho, witness-witness collisions, d>=3, uniform-L estimates, numerical all-cell certificates, elder selection or unrestricted lifetime acceptance. Choose r_* small enough that all local segments used below embed in the torus.\n\n",
    "ec014": "2. Exact object proposed for closure\n\n\nLet M=x_m=(-r/2,0), S=x_s=(+r/2,0), r>0, with longitudinal coordinate t and transverse coordinate s. The raw ordered vector is\n\n\n    X_r = ( f(M), f(S), f_t(M), f_s(M), f_t(S), f_s(S) ).\n\n\nThe corrected ordered frame is\n\n\n    Y_r = ( V^+, V^-_corr, G_t^+, G_t^-, G_s^+, G_s^- ),\n\n\nwhere\n\n\n    V^+       = [f(S)+f(M)]/2,\n    V^-_corr  = [f(S)-f(M)-r(f_t(S)+f_t(M))/2]/r^3,\n    G_t^+     = [f_t(S)+f_t(M)]/2,\n    G_t^-     = [f_t(S)-f_t(M)]/r,\n    G_s^+     = [f_s(S)+f_s(M)]/2,\n    G_s^-     = [f_s(S)-f_s(M)]/r.\n\n\nUnder those exact row and column orders, Y_r=T_rX_r with\n\n\n    T_r =\n      [  1/2      1/2          0          0          0          0 ]\n      [ -r^-3     r^-3    -1/(2r^2)      0    -1/(2r^2)      0 ]\n      [   0        0          1/2         0         1/2         0 ]\n      [   0        0         -r^-1        0          r^-1       0 ]\n      [   0        0           0         1/2          0         1/2]\n      [   0        0           0        -r^-1         0          r^-1].\n\n\nThe exact proposed terminal facts are only:\n\n\nA. The displayed ordering and matrix are correct.\n\n\nB. Under that ordering,\n\n\n       det T_r = -r^-5.\n\n\n   If a different column ordering is used, the sign may change by permutation parity, while the density invariant remains\n\n\n       |det T_r| = r^-5.\n\n\nC. Under the declared change-of-variables convention,\n\n\n       p_X(x_r) = |det T_r| p_Y(y_r)\n                = r^-5 p_Y(y_r).\n\n\nD. For the canonical fold gap r^3/6 and the separate r^2 normalizer factor, the exact power ledger is\n\n\n       (r^3/6) r^-5 r^2 = 1/6,\n\n\n   with exponent cancellation 3-5+2=0.\n\n\n",
    "p15": "## 5. Sharpness and the exact demand boundary\n\nTake one block, a=1,d=2,n=3, no macro edges and K=2. Every proper subset is two-decomposable and the full triple is not. At p_i=p_star and c_i=1, every possible generator has price one, including the empty generator. Any cover of the nonempty obstruction therefore costs at least one, and one generator attains it. The available hazard is h_star. Replacing rho_star by any smaller constant in (F2) would make its right side strictly less than one. This proves sharpness even among alternative covers, not only for our chosen full-block cover.\n\nIf d=1, n=a+1 and K=1, the sole obstruction is again the full block. At p_i=p_star,c_i=1 every cover costs at least one, but\n\n    mu_p(D)=1-p_star^(a+1)>1-p_star=exp(-1),\n\nhence its hazard is less than one. The universal same-palette transformed-price guarantee is false in this demand-one family. This preserves and generalizes the earlier two-coordinate counterexample; it does not retract original P15-B, whose local-budget premise is missing in that case.\n\nThus d>=2 is sufficient for the realized capacity family on the full probability cube, while a uniform statement admitting all d=1 examples would be false. This is not a necessity claim for every multi-block instance with some demand one; other constraints may change its budget.\n\n",
    "remote": "## 1. The count and the new scope\n\nThe earlier RN count interface identified a three-determinant integral whose unnormalized numerator must be O(r^5), but did not bound it. This note supplies that bound for a precise part of the problem: **a fixed remote spatial region and the shrinking height window between the pinned maximum and saddle**. The spatial exclusion radius must not shrink with r.\n\nFix d>=2 and L>0. On X=R^d/(L Z^d), use the centered variance-one Gaussian field with the exact normalized periodic covariance\n\n    K_L(z)=sum_(n in Z^d) exp(-|z+Ln|^2/2)\n             / sum_(n in Z^d) exp(-|Ln|^2/2).\n\nFix compact B=[b_-,b_+] and K=[k_-,k_+] with 0<k_-<=k_+<infinity. Let R be any orthonormal frame, with axial direction u=Re_1. For small r>0 the pins are\n\n    M=-(r/2)u, S=(r/2)u,\n    f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0.\n\nQ_r denotes continuous Gaussian regression on these 2(d+1) observations. For a symmetric matrix H define\n\n    F_j(H)=|det H| 1{H has j negative eigenvalues},\n\nwith value zero on singular matrices. Define the actual endpoint weight and FULL normalizer\n\n    W_r=F_d(H_M) F_(d-1)(H_S),\n    Z_r=E_(Q_r) W_r,             dQ_r^W=(W_r/Z_r)dQ_r.\n\nThere is no adjacency condition, no additional pin-density factor, and no Jacobian inside Z_r.\n\nFix 0<rho<L/4 and D_rho={x:dist_X(x,0)>=rho}. For a Borel E contained in D_rho and j in {0,...,d}, let\n\n    N_(r,j)(E)=#{x in E: grad f(x)=0, index H_x=j,\n                            b-k r^3<f(x)<b}.             (1)\n\nThus the height window shrinks; the spatial exclusion does not. Boundary heights may equivalently be included: their expected critical-point counts are zero by the height-density formula below. The set E is arbitrary; the uniform measure estimate even permits choosing different Borel E_r inside the SAME D_rho.\n\n**Theorem A (fixed-remote mean measure).** There are r_*>0 and C<infinity, depending on d,L,rho,B,K, such that for r<=r_* and all b,k,R,j and Borel E subset D_rho,\n\n    | E_(Q_r^W) N_(r,j)(E)/(k r^3)\n           - integral_E Lambda_j(x;b,k,u) dx |\n        <= C r |E|.                                      (2)\n\nThe explicitly defined contact kernel Lambda_j in Section 5 is continuous, finite and strictly positive. On these compact parameter sets it is bounded above and away from zero. In particular\n\n    E_(Q_r^W) N_(r,j)(E) <= C k r^3 |E|,\n    P_(Q_r^W){N_(r,j)(E)>=1} <= C k r^3 |E|.              (3)\n\nFor fixed E of positive volume the expected count is asymptotic to a strictly positive constant times r^3. This is NOT a matching lower bound on the event probability. Sum over the finitely many indices to count all critical types.\n\nThe normalized mean measures converge in total variation on D_rho at rate O(r): a density version differs uniformly from Lambda_j by O(r). No numerical value of C or r_* is supplied. No constant is claimed uniform as rho tends to zero, marks become unbounded, or d,L vary.\n\n"
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