Prove fixed chart heights cannot lower the productive-helper invoice

research/productive-helper-dimension-floor/.gitattributes

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+* text eol=lf
+*.gz binary

research/productive-helper-dimension-floor/PROOF.md

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+# Fixed projection dimensions: arbitrary Lagrangian chart obstruction
+
+This source-specific theorem concerns the unchanged round-three word in each orientation, with all
+chronological paid incidences retained and the eight owned center costs fixed.
+It does not fix the subspace direction or its quadratic form. At each node it
+fixes only the dimension of the chart's projection onto the first (x) half of
+the binary symplectic coordinates, equal to the original frame dimension.
+
+The source is `research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz`.
+The canonical compact JSON payload SHA-256 is
+`cfcd36b40326d7a5815e1db5313444379b3fb756b27d5201c561574c085cc0dd`.
+The separately reconstructed program hashes are
+`03e406a5672ada3e1762011d667c79ec04a55d7d1163a9423d0005720234d2ca`
+(forward) and
+`a4ff7b7ca8f23c858f8e6f1044bc6abf92168ab17abd2aa420ad384f58a519d3`
+(backward), under the same sorted-key compact JSON convention.
+
+Write V = F_2^n x F_2^n with its usual symplectic form and let Z = 0 x F_2^n.
+For a physical frame F, use the **inverse-image** chart L(F)=F^-1(Z), an
+n-dimensional Lagrangian. A paid physical transition is F_next F_prev^-1;
+its Fourier rank is
+
+    d(L(F_prev),L(F_next))=n-dim(L(F_prev) intersect L(F_next)).
+
+Forward images F(Z) cannot be substituted for these inverse images for a
+general frame. This convention and the exact symmetric-phase chart grammar
+are documented in `research/native-quotient-interface/PHASE-FORM-GRAMMAR.md`;
+the canonical specialization is in
+`research/complex-frame-distance-seed80/PROOF.md`. Let k(L)=dim pi_x(L).
+Signs, affine
+translations and overall phases do not change this rank; granting them
+freely only enlarges the geometric relaxation. No physical compiler or
+endpoint phase admission follows from a rank obstruction.
+
+## Projection lower bound and rigid zero entrances
+
+Rank-nullity gives dim(L intersect Z)=n-k(L), hence d(L,Z)=k(L). The
+Grassmann/subspace distance d on equal-dimensional subspaces is a metric.
+Its reverse triangle inequality therefore gives
+
+    d(L,M) >= |d(L,Z)-d(M,Z)| = |k(L)-k(M)|.
+
+For completeness, triangle inequality follows from
+dim(L intersect N) >= dim(L intersect M)+dim(M intersect N)-n: the two
+intersections are subspaces of the n-dimensional M, and their intersection
+lies inside L intersect N. This proves the needed metric statement without
+any isotropy assumption beyond the common dimension.
+
+This metric is standard: Koetter and Kschischang, *Coding for Errors and
+Erasures in Random Network Coding*, section 3.1, Lemma 1, proves the subspace
+metric dim(A)+dim(B)-2dim(A intersect B). On equal n-dimensional spaces the
+distance used here is half that metric. The source-specific path accounting,
+not the metric identity, is the new application:
+<https://arxiv.org/pdf/cs/0703061> (v2, 25 March 2008).
+
+If k(L)=0, then L=Z. Consequently a helper with zero entrance projection has
+rank k(M), regardless of its exit's quadratic form or projection direction.
+
+The original paid edges are nested canonical charts, so their original ranks
+r are the nonnegative differences of the source frame dimensions. Arbitrary
+new Lagrangians with the same per-node projection dimensions have paid ranks
+r+delta with delta>=0. In contrast to the canonical theorem, delta need not
+be even. Every original native rank is zero, so arbitrary changes of native
+maps cannot decrease their cost. Owned costs are held fixed.
+
+## Incremental concavity pays each helper credit
+
+Let H be any nondecreasing concave function on all nonnegative reals, with H(0)=0.
+For r_i,delta_i>=0 put R=sum r_i and D=sum delta_i. Diminishing increments
+of a concave function imply, with s_i=sum_{j<i}delta_j,
+
+    H(r_i+delta_i)-H(r_i)
+      >= H(R+s_i+delta_i)-H(R+s_i),
+
+because r_i<=R+s_i. Summing telescopes and proves
+
+    sum_i [H(r_i+delta_i)-H(r_i)] >= H(R+D)-H(R).
+
+For a paid-only source-geodesic helper wire, R equals its original residual
+rank. The metric triangle inequality bounds its new residual q by R+D.
+Monotonicity of H then shows that the increase in the wire's paid tariffs is
+at least H(q)-H(R), its entire additional helper credit. This remains true
+if q<R, in which case the helper supplies less credit. No parity assertion
+or strict positive margin is needed for this stronger non-improvement claim.
+
+## Source-bound application
+
+There are 358 zero-entrance helpers, whose ranks are rigid under fixed
+projection dimensions. All other 50 helpers have paid-only actual wires,
+no native operation, and source paid rank sum equal to their source residual.
+The 50 paths are disjoint in charged edge occurrences even if scalar-contact
+vertices are shared. Apply the preceding inequality separately on these
+paths; every remaining paid or native tariff is nondecreasing, and owned
+center costs are fixed. Therefore the entire paid-minus-helper invoice
+cannot decrease.
+
+This argument applies separately to forward and backward words and even
+allows independently chosen charts and native maps in the two orientations.
+It applies to H(r)=r^(1-C) for 0<C<1, and by the same argument at C=0. It is a
+statement about the whole geometric invoice, not a new supplier or improved
+bound. It does not exclude changing per-node projection dimensions, owned
+center costs, scalar/contact incidences, helper roles, or the supplier model.
+
+
+## Predecessors and distinction
+
+Matarener's [PR5060](https://github.com/CrocSwap/integer-mult-bounds/pull/5060)
+established an owned-center path floor under nested-chart constraints.
+zxv's [PR5283](https://github.com/CrocSwap/integer-mult-bounds/pull/5283)
+extends the intersection-loss argument to arbitrary Clifford intermediate
+charts and typed-copy trees under fixed common-contact constraints. Those
+path-floor ideas and their native/Clifford predecessors deserve credit.
+The present result binds a different whole-invoice accounting argument to
+the round-three word: 50 disjoint paid-only helper paths fund all possible
+helper-credit changes at fixed projection heights, including quadratic
+chart changes, while the source's eight owned center charges remain fixed.
+It does not re-prove or claim the geometric metric as new.

research/productive-helper-dimension-floor/README.md

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+# Fixed chart heights cannot improve the productive-helper invoice
+
+On the pinned round-three word, arbitrary Clifford chart changes and native-map
+changes cannot lower the complete paid-minus-helper invoice if each contact
+keeps its original **projection dimension** and owned-center tariffs stay
+fixed. The theorem includes quadratic chart forms and nonnested transitions.
+It holds separately in both program orientations for
+`H_r = r^(1-C)`, `H_0 = 0`, and `0 <= C < 1`.
+
+This is an analytic obstruction with an exact source-bound census, not a
+search failure. Its useful consequence is a stopping rule: an improvement
+must change contact heights, owned costs, the scalar graph, helper contracts,
+or another stated hypothesis. No new kappa or supplier is claimed.
+
+## Why the obstruction holds
+
+For a physical frame `F`, use the inverse-image Lagrangian chart
+`L(F)=F^-1(Z)`, where `Z` is the vertical coordinate space. The height
+`k(L)=dim(proj_x L)` equals the relative Fourier rank `d(L,Z)`. The metric
+inequality `d(L,M) >= |k(L)-k(M)|` shows that every original nested paid edge
+already has the smallest possible rank at its endpoint heights. Changing
+charts at fixed heights cannot make any such edge cheaper. Source native
+transitions have rank zero and cannot get cheaper either.
+
+Of 408 helpers, 358 have height-zero entrances, whose charts are necessarily
+`Z`; their residual ranks remain fixed. Each of the other 50 has a paid-only
+entry-to-exit wire whose source rank sum equals its original helper residual.
+These paths share no charged edge occurrence. Every old-zero edge is retained.
+
+If a path's source ranks are `r_i` and their increases are `delta_i >= 0`,
+concavity gives
+
+```text
+sum_i [H(r_i+delta_i)-H(r_i)]
+    >= H(sum_i r_i + sum_i delta_i) - H(sum_i r_i).
+```
+
+The metric triangle inequality bounds the new helper residual by the new
+path rank sum. Thus each path pays for its helper's entire additional credit;
+edge-disjointness prevents reuse of that payment. All other costs are
+nondecreasing. [PROOF.md](PROOF.md) gives the full argument for any
+nondecreasing concave tariff on nonnegative reals with `H(0)=0`.
+
+## Reproduction
+
+Python 3.12, standard library only:
+
+```sh
+python3 -B scripts/research_package.py --package research/productive-helper-dimension-floor --check-only
+python3 -B scripts/research_package.py --package research/productive-helper-dimension-floor
+python3 -I -O -B research/productive-helper-dimension-floor/verify.py
+```
+
+The direct replay prints `PASS_PRODUCTIVE_HELPER_DIMENSION_FLOOR`; the
+registered runner also prints `PASS_REGISTERED_RESEARCH_PACKAGE`.
+
+The checker authenticates the compressed source and its complete JSON
+payload, reconstructs both chronological graphs, validates every source
+native rank and the full paid/helper invoices, and records all 50 paths per
+orientation in the deterministic `PATHS.json.gz` receipt, bound by
+[RESULT.json](RESULT.json). Each orientation has 1,016 selected
+paid edge occurrences. All 448 source adapters touch only main roles.
+
+Eight corruption controls reject source changes, missing old-zero edges,
+double-counted debt, and native transitions inserted into paid-only paths.
+A separate finite calibration enumerates all Lagrangians in dimensions one
+through three and checks the height/rank convention on 18,459 ordered pairs.
+These checks remain active under `python -O`. The checker imports no
+predecessor graph builder, compiler, optimizer or numerical library.
+
+Parent and independent agents checked the source paths and the argument
+separately. This is not independent human or proof-kernel acceptance. No
+new physical word is emitted, and no new phase, compiler, precision, supplier
+or outer-theorem admission is asserted.
+
+## Scope, sources and credit
+
+The only live input is Cao Yifan's round-three word from
+`research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz`, pinned in
+[SOURCES.json](SOURCES.json) at canonical main
+`ff3cc58e370d0f678e529246f8c73e8094e18fcf`. This original-source theorem does
+not depend on the compression or finite fusion certificate in
+[PR #5301](https://github.com/CrocSwap/integer-mult-bounds/pull/5301).
+
+Each node may choose any Lagrangian chart of its original projection
+dimension, including a different projection direction and quadratic form.
+Compatible shared native maps may change. The proof even allows independent
+native maps at different occurrences and independent chart choices between
+orientations; actual physical constraints may narrow this relaxation.
+Owned-center tariffs and helper contracts remain fixed. Changed projection
+dimensions, owned costs, scalar/contact graphs, helper contracts and other
+supplier mechanisms are outside the theorem.
+
+Related path-floor work includes Matarener's
+[PR #5060](https://github.com/CrocSwap/integer-mult-bounds/pull/5060), Cinna
+Davis's [PR #5136](https://github.com/CrocSwap/integer-mult-bounds/pull/5136)
+and [PR #5140](https://github.com/CrocSwap/integer-mult-bounds/pull/5140), and
+zxv's arbitrary-Clifford owned-tree extension in
+[PR #5283](https://github.com/CrocSwap/integer-mult-bounds/pull/5283).
+Those concern owned-center or other constrained path costs. The contribution
+here is the source-specific 50-helper path census and complete-invoice
+accounting at fixed heights. [PR #5144](https://github.com/CrocSwap/integer-mult-bounds/pull/5144)
+provides distinct nonmonotone owned/shared-map context. The subspace metric
+and concavity principles are standard; the proof credits the metric's
+Koetter–Kschischang reference and the retained inverse-chart grammar.
+
+Overlap checks read the start/handoff/native discussion context and an initial
+800 most-recently-updated PRs, followed by a 100-PR delta on 2026-10-23 ending
+at 13:09 UTC. This is bounded coverage, not a census of unpublished work or
+all open PRs. Canonical main was refreshed before this package.
+
+Credit Cao Yifan's #4610 source word, Cinna Davis's canonical/native and helper
+accounting lineage, Dugongue's chart contract, zxv's supplier, and all retained
+scalar/Clifford/compiler authors. Prepared by flyingviper with substantial
+OpenAI Codex assistance and independent agents. Apache-2.0.
+
+## Open next questions
+
+Can a coordinated change of contact heights and native interfaces lower the
+complete invoice? Start with a concrete source-derived dimension change and
+its full signed paid/helper polynomial, then optimize the coupled native
+interfaces while charging every old-zero edge. A favorable exact invoice
+still needs one compatible shared map per operator and the applicable dual
+native, phase, compiler and outer checks. [TASK.json](TASK.json) records this
+scope change, first step and blockers.

research/productive-helper-dimension-floor/RESULT.json

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+{
+  "status": "PASS_FIXED_PROJECTION_HEIGHT_CLIFFORD_OBSTRUCTION",
+  "source_sha256": "cfcd36b40326d7a5815e1db5313444379b3fb756b27d5201c561574c085cc0dd",
+  "directions": {
+    "forward": {
+      "program_sha256": "03e406a5672ada3e1762011d667c79ec04a55d7d1163a9423d0005720234d2ca",
+      "nodes": 19197,
+      "paid_edge_occurrences": 34398,
+      "native_edge_occurrences": 240,
+      "source_native_ranks_all_zero": true,
+      "source_main_adapter_occurrences": 448,
+      "source_adapters_touch_no_helper": true,
+      "helper_dimension_census": {
+        "0->4": 26,
+        "0->8": 6,
+        "0->9": 326,
+        "1->8": 6,
+        "1->9": 41,
+        "2->9": 1,
+        "5->9": 2
+      },
+      "rigid_zero_entrance_helper_count": 358,
+      "paid_only_nonzero_entrance_helper_count": 50,
+      "selected_paid_edge_occurrences": 1016,
+      "paths_edge_occurrence_disjoint": true,
+      "source_geodesic_rank_census": {
+        "4": 2,
+        "7": 7,
+        "8": 41
+      },
+      "owned_center_profile_fixed": {
+        "8": 8
+      },
+      "baseline_paid_profile": {
+        "1": 2201,
+        "2": 412,
+        "3": 510,
+        "4": 127,
+        "5": 40,
+        "8": 8
+      },
+      "baseline_helper_profile": {
+        "4": 28,
+        "7": 7,
+        "8": 47,
+        "9": 326
+      }
+    },
+    "backward": {
+      "program_sha256": "a4ff7b7ca8f23c858f8e6f1044bc6abf92168ab17abd2aa420ad384f58a519d3",
+      "nodes": 19197,
+      "paid_edge_occurrences": 34398,
+      "native_edge_occurrences": 240,
+      "source_native_ranks_all_zero": true,
+      "source_main_adapter_occurrences": 448,
+      "source_adapters_touch_no_helper": true,
+      "helper_dimension_census": {
+        "0->4": 26,
+        "0->8": 6,
+        "0->9": 326,
+        "1->8": 6,
+        "1->9": 41,
+        "2->9": 1,
+        "5->9": 2
+      },
+      "rigid_zero_entrance_helper_count": 358,
+      "paid_only_nonzero_entrance_helper_count": 50,
+      "selected_paid_edge_occurrences": 1016,
+      "paths_edge_occurrence_disjoint": true,
+      "source_geodesic_rank_census": {
+        "4": 2,
+        "7": 7,
+        "8": 41
+      },
+      "owned_center_profile_fixed": {
+        "8": 8
+      },
+      "baseline_paid_profile": {
+        "1": 2201,
+        "2": 412,
+        "3": 510,
+        "4": 127,
+        "5": 40,
+        "8": 8
+      },
+      "baseline_helper_profile": {
+        "4": 28,
+        "7": 7,
+        "8": 47,
+        "9": 326
+      }
+    }
+  },
+  "rank_metric": "d(L,M)=9-dim(L intersect M), L(F)=F^-1(Z)",
+  "fixed_height": "k(L)=dim(proj_x L)=d(L,Z)",
+  "path_debt_inequality": "sum(H(r_i+delta_i)-H(r_i)) >= H(sum(r_i)+sum(delta_i))-H(sum(r_i))",
+  "proof_domain": "H(x)=x^t, H(0)=0, 0<t<=1; equivalently 0<=C<1 with t=1-C",
+  "scope": "In each frozen scalar/contact word, arbitrary Clifford inverse-image charts preserving every node x-projection dimension cannot lower the complete invoice with the retained helper contracts and fixed owned-center tariffs. Compatible shared native maps may vary, and the two orientations may choose charts independently. Contact-height changes, different scalar graphs, changed helper contracts and changed owned-center tariffs remain outside the theorem. No new physical supplier, phase replay, compiler admission or improved kappa is asserted.",
+  "negative_controls": {
+    "forward": [
+      "changed source coefficient",
+      "omitted old-zero path edge",
+      "double-counted shared paid debt",
+      "inserted native operation in path"
+    ],
+    "backward": [
+      "changed source coefficient",
+      "omitted old-zero path edge",
+      "double-counted shared paid debt",
+      "inserted native operation in path"
+    ]
+  },
+  "small_lagrangian_calibration": [
+    {
+      "dimension": 1,
+      "lagrangians": 3,
+      "ordered_pairs": 9
+    },
+    {
+      "dimension": 2,
+      "lagrangians": 15,
+      "ordered_pairs": 225
+    },
+    {
+      "dimension": 3,
+      "lagrangians": 135,
+      "ordered_pairs": 18225
+    }
+  ],
+  "path_receipt_sha256": "53ce8ccb77a377e543312e82138abe126b8a0f848a9de2241a5dc2dfb1a23257"
+}