Prove the sharp P104 price floor for all fixed-contact frame paths

research/p104-fixed-contact-ceiling/INPUTS.json

@@ -0,0 +1,34 @@
+[
+  {
+    "file": "inputs/WORD.json.gz",
+    "sha256": "b2a98c4bd0dbe86c09e9d15ee060dcf1aab09dbef90aa1e2127676f4039f406b",
+    "head": "6efa248cf456d6101d757e9d09ba29711d18a13a",
+    "source": "research/p104-event-contact-step/WORD.json.gz",
+    "credit": "Ali Romman, GamingPuzzled, DreamingOfClouds and the retained P104 authors",
+    "git_blob": "9434855e7dad0d0236d23210b3e71a2b83a9ae64"
+  },
+  {
+    "file": "inputs/local.py",
+    "sha256": "878734dfd64c4b9d362d0ae476cc5c2a19fe93e48b7378763b23f3e0dd0a5a71",
+    "head": "18968423711fd902aaee1099979497d55603e1dc",
+    "source": "research/p104-productive-controller/local.py",
+    "credit": "Credited PR5545 source closure; source notices retained",
+    "git_blob": "dbb3d56a4f5b733bb0c32a91249d1f6a7e1ab5d2"
+  },
+  {
+    "file": "inputs/clifford_exact.py",
+    "sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
+    "head": "18968423711fd902aaee1099979497d55603e1dc",
+    "source": "research/p104-native-channel/vendor/clifford_exact.py",
+    "credit": "Credited PR5545 source closure; source notices retained",
+    "git_blob": "166841a037f069249f6f7f6dedb89248812f59e2"
+  },
+  {
+    "file": "inputs/native_verify.py",
+    "sha256": "3049471905b47b385c88378a1a3be7a18b8858aadd035b8a6da2f4a03b470f38",
+    "head": "18968423711fd902aaee1099979497d55603e1dc",
+    "source": "research/p104-native-channel/vendor/native_verify.py",
+    "credit": "Credited PR5545 source closure; source notices retained",
+    "git_blob": "2c209dfa9cfade3bc20ac5fc9d876b433a147f8c"
+  }
+]

research/p104-fixed-contact-ceiling/PROOF.md

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+# A sharp price floor for fixed scalar contacts
+
+## Claim and domain
+
+Fix the ordered ADD, COPY, SCALE, INIT and CLEAR records and their frames.
+Fix the main entry and exit frames. Each helper has an independent public
+input. Its scalar endpoint is its own input, with no other input term.
+Keep the source frame interface: a register carries one field in one
+canonical chart. A transition between two charts is a paid Gaussian job
+between monomial maps. No joint query on several registers is introduced.
+
+Allow every binary subspace as an intermediate chart, any number of
+transitions between contacts, and arbitrary public helper entry and exit
+charts. No nesting condition is imposed. Grant every added ordinary map
+for free. The result is therefore a favorable bound for this larger model.
+
+For every power tariff r^t with 0 < t < 1, the PR5580 word attains the
+maximum of helper credit minus paid transition cost in this model.
+Thus changes confined to these paths or endpoints cannot improve its
+complete recursive moment. The scalar contacts themselves can change
+this bound. This is not an optimum over all P104 words.
+
+## Exact native rank for two charts
+
+For binary subspaces U,V of F2^n define
+
+    d(U,V) = dim(U) + dim(V) - 2 dim(U intersection V).
+
+This is the subspace distance. It is a metric. In particular, an arbitrary
+path from U to V has sum of edge distances at least d(U,V).
+
+Use the canonical chart G_U in the pinned Clifford source. Ignore Pauli
+signs only in this rank calculation. Its image of the Z Lagrangian is
+
+    L_U = {(x,z): x in U, x+z in U-perp}.
+
+To check the formula directly, put the basis of U first in an invertible
+matrix G and put H=G^-1. The source chart conjugates the first dim(U)
+coordinate C gates by D_Q P_G, where Q=I+H^T H. A Z vector z has image
+x=G P G^T z and z'=z+Qx, where P is the first-coordinate projector.
+Then x is in U and x+z' is orthogonal to U. Both spaces have dimension n,
+so the inclusion is equality. The zero and full charts give the same rule.
+The symplectic action of each chart is an involution. The square of the
+chart is a Pauli, not necessarily the identity including its phase.
+
+The intersection L_U intersection L_V is parameterized by x in U
+intersection V and x+z in (U+V)-perp. Its dimension is
+
+    n - dim(U) - dim(V) + 2 dim(U intersection V).
+
+The X block of G_V G_U^-1 therefore has rank d(U,V). Monomial maps
+preserve the Z Lagrangian. They cannot reduce this rank. The retained
+Gaussian normal form gives monomial left and right factors and one
+rank-d Gaussian. All scalar phases belong to the factors. The price is
+d(U,V), not abs(dim(U)-dim(V)) for a nonnested pair. For example, two
+distinct lines cost rank two even though their dimensions agree.
+
+These identities concern one address column. For width f, lift each
+native operator columnwise, with its scalar phase raised to the f-th
+power. Each rank-d block then has d*f address axes and remains a rank-d
+job in the source width convention. Owner scalar coefficients are not
+tensored. No identity on a sum is inferred by tensoring that sum.
+
+## Internal paths
+
+For nonnegative a,b and 0<t<1, (a+b)^t <= a^t+b^t. Combine this with
+the metric triangle inequality. Every transition path between two
+consecutive contacts A and B has cost at least d(A,B)^t. A direct native
+transition attains the bound. This applies separately to each register
+and to every gap, including the fixed main endpoints and owned contacts.
+It requires no search over intermediate subspaces and no finite cutoff.
+
+## Helper endpoints
+
+Let A and B be the first and last contacts of a public helper. Let E,F
+be any proposed entry and exit charts. Put a=d(E,A), b=d(B,F), and
+h=d(E,F). The metric inequality and subadditivity give
+
+    h^t - a^t - b^t <= d(A,B)^t.
+
+Paths with more transitions can only reduce the left side. Thus helper
+credit, after the endpoint path cost, is at most the credit obtained by
+starting at A and ending at B. This conclusion also permits a rank-zero
+helper in the favorable bound. No extra productive field is credited
+for such a helper. A helper with no contact has credit at most its path
+cost, so its net contribution is at most zero.
+
+The endpoint normalization mechanism is inherited from PR5550 and
+PR5567. If the helper's physical map is M_left H_r M_right, put
+M_right^-1 before the word and M_left^-1 after it. The resulting job is
+H_r on an independent input. These are two ordinary monomial maps per
+helper, not two new Gaussian calls. The proof above grants all such
+ordinary costs for free; it does not assert that they vanish physically.
+
+## Binding and equality for PR5580
+
+The verifier replays all 735 public inputs and eight owned lifetimes with
+the unchanged source scalar checker. It then reads every contact in the
+literal word and computes every consecutive distance. All required source
+chart pairs also pass the native X-block rank test.
+
+The paid floor is
+
+    {1:2211, 2:459, 3:494, 4:94, 5:163, 6:18, 7:14}.
+
+The corresponding helper core is
+
+    {4:83, 5:4, 8:15, 9:424}.
+
+The source differs from these two profiles by exactly one rank-five paid
+job and one rank-five helper target. Both belong to register 711, which
+has no scalar contact. They cancel for every tariff. Every other required
+interval is already at the bound. This proves equality of the complete
+signed local profiles, not just equality at one numerical exponent.
+
+The source caller uses 18 copies of this local word. All other caller
+terms are fixed in this comparison. Multiplication by 18 and addition
+of those fixed terms preserve the bound. No root approximation is needed.
+
+## Verification boundary and next step
+
+The finite checks bind the written general proof to the exact source.
+They are not a formal-kernel proof. The retained source checker, native
+compiler and framework assumptions retain their prior review status.
+The full physical caller, ordinary budgets, outer ledger and rounded
+multiplier are not re-run here. No new kappa or supplier is admitted.
+
+A useful next change must alter a contact, scalar realization, query
+interface, or productive target model. The active changed-contact and
+basis searches remain outside this result. The distance formula can be
+used as a lower bound before a new candidate is physically compiled.
+
+Credit Matarener for PR5567, Chris R/Clear1nsight for PR5550,
+DreamingOfClouds for PR5561/5568, GamingPuzzled for PR5569, and the
+PR5580, PR5544/5545 and retained P104/native/compiler authors. This is an
+application of standard subspace distance and concavity; no general
+mathematical priority is claimed. Ali Romman with substantial OpenAI
+Codex assistance. Apache-2.0.

research/p104-fixed-contact-ceiling/README.md

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+# Fixed P104 contacts already attain the all-frame price floor
+
+PR5580 has no remaining recursive moment gain from frame detours or
+public helper endpoint changes with its scalar contacts fixed. This
+includes arbitrary nonnested charts and any number of intermediate
+transitions. It holds for every power tariff r^t, 0<t<1. No new kappa.
+
+[PROOF.md](PROOF.md) gives the exact native rank law and the complete
+bound. The source word attains it. Register 711 has no scalar contact;
+its one rank-five target and paid job cancel exactly. All other contact
+intervals are at the floor. Thus the result does not suggest a better
+numerical root for the unchanged invoice.
+
+This directs further kappa work toward changed scalar contacts, words or
+query interfaces. It supplies a computable lower bound for those searches.
+The active contact/order and basis searches are outside this fixed-contact
+model. INIT and CLEAR frames remain fixed as part of the contact list.
+
+## Evidence
+
+The package pins PR5580 head 6efa248cf456d6101d757e9d09ba29711d18a13a and
+unchanged files from PR5545's executable source closure. INPUTS.json gives
+the byte hashes and credit. The complete scalar replay uses the retained
+source checker. New checks derive the contact floor, verify the native
+rank on all source pairs, and exhaust 25, 256 and 4,489 ordered subspace
+pairs in dimensions two, three and four. Three nonnested examples also
+pass the retained independent phase interpreter. Changed scalar, changed
+contact and false dimension-difference controls are included.
+
+    python3.12 -B scripts/research_package.py --package research/p104-fixed-contact-ceiling
+
+Expected marker: PASS_P104_FIXED_CONTACT_CEILING in normal and optimized
+Python. The standard library is sufficient. RESULT.json is checked in
+full. These are author-run checks with credited source tools. No new
+independent human or formal-kernel acceptance is claimed.
+
+## Scope and overlap
+
+The refreshed review found that PR5567 already answers PR5568's nested
+helper endpoint question. That audit is credited, not repeated as a new
+result. This package proves a sharp bound over the larger nonnested path
+model and binds it to PR5580. Coordination is in
+[Discussion 5589](https://github.com/CrocSwap/integer-mult-bounds/discussions/5589).
+PR2013 concerns the older selected k8 word and also varies gate frames;
+this result concerns the current P104 word with all contacts fixed.
+
+Canonical base: 5e0052ca0aa3036a3afa8b0438927377c8d97d53, priority 7.
+The entry overlap scan covered all 7,611 local main notes, 329 open PRs
+and 4,853 Discussion records with complete reported pagination. It did
+not search arbitrary diffs or unpublished work. The selected record is
+unchanged. The full caller, precision/tape/compiler and outer checks are
+not repeated. No universal P104 obstruction follows.
+
+## Open next questions
+
+- Can a changed contact or scalar realization beat this floor? Start with
+  one literal source-bound change and its complete signed profile. Observe
+  the active reservations in Discussions 5575 and 5125.
+- Can the frame distance serve as a pruning bound for a new interface?
+  Establish that the candidate still has the one-register chart contract
+  before applying it. A joint query is outside the proof.
+
+Credit Matarener, Chris R/Clear1nsight, DreamingOfClouds, GamingPuzzled,
+thu-nmrc, Gareth Davidson, David Arias, Cinna Davis and the retained native,
+Clifford and compiler lineage. Imported files and notices are unchanged.
+Prepared by Ali Romman with substantial OpenAI Codex assistance.
+Apache-2.0. Standard subspace geometry and concavity are not claimed as
+new general theorems.

research/p104-fixed-contact-ceiling/RESULT.json

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+{
+  "controls": [
+    "false_dimension_difference",
+    "changed_scalar",
+    "changed_contact"
+  ],
+  "helper_core": {
+    "4": 83,
+    "5": 4,
+    "8": 15,
+    "9": 424
+  },
+  "interval_sha256": "544092a4c732d94b7895689745b92c4ea0cafda665cdd6652b9bee352ffbb378",
+  "neutral_target_paid_pair": {
+    "5": 1
+  },
+  "nonzero_intervals": 3453,
+  "owned_lifetimes": 8,
+  "paid_floor": {
+    "1": 2211,
+    "2": 459,
+    "3": 494,
+    "4": 94,
+    "5": 163,
+    "6": 18,
+    "7": 14
+  },
+  "public_inputs": 735,
+  "scope": "All nonnested paths and helper endpoints, fixed scalar contacts; every 0<t<1.",
+  "small_pair_counts": {
+    "2": 25,
+    "3": 256,
+    "4": 4489
+  },
+  "source_pairs": 2921,
+  "source_pr": 5580,
+  "status": "PASS_P104_FIXED_CONTACT_CEILING",
+  "unused_helpers": [
+    711
+  ]
+}

research/p104-fixed-contact-ceiling/SOURCES.json

@@ -0,0 +1,18 @@
+{
+  "schema_version": 1,
+  "base_commit": "5e0052ca0aa3036a3afa8b0438927377c8d97d53",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "fc3d4ba105e1f097d0d7d051a308281e86ab1446",
+      "sha256": "66974efdced4083c00025f97d95d83909a226cdf801edb9301c195c6641f6903"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    }
+  ]
+}

research/p104-fixed-contact-ceiling/TASK.json

@@ -0,0 +1,95 @@
+{
+  "schema_version": 1,
+  "task_id": "formal-interface-lemma",
+  "priority_revision": 7,
+  "base_commit": "5e0052ca0aa3036a3afa8b0438927377c8d97d53",
+  "original_card": {
+    "id": "formal-interface-lemma",
+    "question": "Can one uncovered interface implication be formalized with exact caller and certificate bindings?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/gate-operand-parity-lean/README.md",
+      "research/staged544-certified-endpoint/README.md",
+      "research/unconditional-batched-crt/README.md"
+    ],
+    "methods": [
+      "Formalize a missing boundary lemma identified by the selected review.",
+      "Audit a formal theorem-to-certificate binding independently."
+    ],
+    "acceptance_gate": "State the exact formalized proposition and remaining assumptions; avoid duplicate arithmetic formalizations already present.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Inventory the gate-operand parity lemma (#3807) and certified endpoint arithmetic (#3857). Neither is a full multiplication proof. For #4100, audit the compact-switch statement and its hypotheses before formalizing it.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Write one missing statement with quantified hypotheses, actual caller and existing coverage. A compact-switch lemma is a candidate only after its mathematical contract is reviewed. Establish a bounded proof plan before a separately budgeted proof-assistant build; do not rerun large Lean jobs as orientation.",
+    "bottleneck_link": "A missing interface proof can reduce the conditional gap even if it does not change kappa.",
+    "allowed_variations": [
+      "Narrow to one formal lemma or one theorem-to-certificate binding."
+    ],
+    "scope_change_triggers": [
+      "Formalizing already-covered arithmetic and claiming the full multiplication theorem."
+    ],
+    "overlap_terms": [
+      "hyperplane end",
+      "formal interface",
+      "Lean binding",
+      "3807",
+      "3857",
+      "4100",
+      "gate-operand-parity-lean",
+      "staged544-certified-endpoint",
+      "unconditional-batched-crt"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #3807: scoped gate-operand parity formalization.",
+        "evidence": "research/gate-operand-parity-lean/README.md",
+        "remaining": "Exact theorem-to-caller binding and genuinely uncovered interface lemmas.",
+        "status": "author-checked",
+        "source_commit": "4c3530f42a6abcc48e7a6bcffd6c901911dc8269",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #3857: certified real-power moment/endpoint arithmetic.",
+        "evidence": "research/staged544-certified-endpoint/README.md",
+        "remaining": "Uniform algorithmic implications, not another endpoint digit.",
+        "status": "author-checked",
+        "source_commit": "ef4315b8caa605e47a2515004845eb2d2ed71a86",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4100: compact-switch formalization proposed as future validation.",
+        "evidence": "research/unconditional-batched-crt/README.md",
+        "remaining": "Audit the statement and assumptions before a formal proof campaign.",
+        "status": "author-checked",
+        "source_commit": "337d04939571a637aeab54a0f58f54fde811663a",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Prove a sharp power-moment ceiling for every frame path and helper endpoint with the PR5580 scalar contacts fixed.",
+  "scope_change_reason": "PR5567 already resolves nested endpoint admissibility. This result admits arbitrary nonnested paths and proves a sharp global bound in the fixed-contact model.",
+  "bottleneck_link": "Remove all fixed-contact detours and endpoint-only variants from the kappa search without excluding changed contacts or words.",
+  "budget_minutes": 90,
+  "stop_rule": "Prove the all-frame metric bound and show that the current word attains it. Do not optimize the unchanged numerical root.",
+  "outcome": "completed",
+  "evidence": "PROOF.md; exact contact certificate and source replay in RESULT.json.",
+  "author": "Ali Romman",
+  "ai_assistance": "Substantial OpenAI Codex assistance.",
+  "open_questions": [
+    {
+      "question": "Can changed scalar contacts or a changed scalar realization beat the fixed-contact floor?",
+      "why_it_matters": "Such changes escape the proved model and can improve kappa.",
+      "first_step": "Use the floor as a lower bound in a source-bound changed-contact search; preserve the active claims.",
+      "blocked_by": "No new search or changed word is supplied here."
+    }
+  ]
+}