Gaussian cache reuse for cross-frame ADDs and paid history (no new κ)

research/gaussian-cache-reuse/PLAN.json

@@ -0,0 +1,13 @@
+{
+  "at_UTC": "2026-10-23T13:12:01.622494+00:00",
+  "base_commit": "8544ddf488e4ecd2f8c85f79a5b2277d1380aee0",
+  "canonical_main": "ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c",
+  "scope": "Exact cache primitive and source-bound generalized center floor, not a complete supplier search or new kappa.",
+  "candidate": "One stable arbitrary source at zero frame; two arbitrary targets in planes <1,2> and <1,4>. Retain a line-<1> owned cache and reuse a second owned field for both output updates.",
+  "predicted_recursive_change": "2 H2 -> 3 H1, with zero new productive credit and two owned cache fields.",
+  "predicted_gain_at_selected_exponent": "0.991635124775620904301391563105821801159734270612346654",
+  "center_question": "Does the owned-center intersection floor extend to non-nested or arbitrary Clifford transport paths and typed-copy branching trees while persistent common-chart contact caps remain fixed?",
+  "claimed_new_application": "Constructive exponent-aware cache sharing plus its source-bound applicability limit. Metric, memoization and subadditivity are classical, not inventions claimed here.",
+  "stop": "Verify literal arbitrary-input fanout and raw dual, phases, invalidation controls and complete module tariff. Reconstruct all16 center blocks before asserting an 8-rank bound. Do not substitute the toy saving into the selected supplier without an actual changed caller.",
+  "geometry_sha256": "20c217d44893d2cc3b517af826936fc8a4c58f498f5bf76b833e5599a25d9084"
+}

research/gaussian-cache-reuse/PROOF.md

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+# Cache tree identities and the selected center intersection bound
+
+## Frames and actual paid ranks
+
+Let a field have \(2^n\) binary-address positions. A Clifford frame F has
+exact unitary operator \(G_F\). Set
+\(L_0=\{(0,z):z\in\mathbb F_2^n\}\) and
+\(\Lambda_F=F^{-1}L_0\), using the induced binary symplectic action.
+For the actual transport \(G_B G_A^{-1}\), its Fourier rank is
+
+\[
+r(A,B)=n-\dim(\Lambda_A\cap\Lambda_B).
+\tag{1}
+\]
+
+Indeed its B block measures the dimension outside \(L_0\) of
+\(BA^{-1}L_0\). Its kernel is the intersection of \(L_0\) with that
+image. Pulling the intersection back through B gives (1).
+Phases and monomial factors do not change this rank, but must still be
+implemented. This rank is an integer metric on Lagrangian charts; different
+Cliffords with the same chart can have distance zero and nontrivial paid
+ordinary monomial work.
+
+For the repository's canonical subspace frame \(C_U\), direct substitution
+gives
+
+\[
+\Lambda(U)=\{(x,x+z):x\in U,\ z\in U^\perp\}.
+\]
+
+Consequently
+\(r(U,V)=\dim U+\dim V-2\dim(U\cap V)\), including non-nested pairs.
+`primitive.py` independently checks this against exact lowered Clifford
+operators for every pair of the 16 subspaces of \(\mathbb F_2^3\), including
+all input columns and absolute phase. The binary chart formula is the
+algebraic statement; these finite tests check its convention in the retained
+implementation.
+
+## A cache is a snapshot with a representation
+
+Fix a logical column x. At each cache node v maintain physical value
+\(z_v=G_vx\). An owned child w is created by copying \(z_v\), then applying
+the fully paid transport \(G_wG_v^{-1}\). Thus
+
+\[
+G_wG_v^{-1}z_v=G_wx.
+\tag{2}
+\]
+
+Induction proves the invariant on any typed-copy tree. At a terminal,
+\(y_i\leftarrow y_i+\alpha_i z_i\) supplies precisely the requested
+contribution. Internal caches carry no productive-owner credit. Leaves can
+be processed in their original order using a reusable owned field if the
+retained parent cache remains unchanged.
+
+The logical snapshot is an operator row on the independent caller inputs.
+For a cached row R, a later required row S may use the cache when
+\(S=qR\) for an explicitly known scalar q, by changing the ADD coefficient.
+Writes to the original owner are harmless only when this row relation still
+holds. The invariant concerns expressions, not register identifiers. In a
+longer caller the row algebra must include all address operators, coefficients
+and phases; the small scalar-row control here is not such a caller analysis.
+
+Let the desired main endpoint be
+\((x,y_1,y_2)\mapsto(x,y_1+A_1x,y_2+A_2x)\).
+Its raw inverse transpose is
+
+\[
+(x,y_1,y_2)\mapsto
+(x-A_1^Ty_1-A_2^Ty_2,y_1,y_2).
+\tag{3}
+\]
+
+Since G is unitary, \(G^T=\overline G^{-1}\). Reversing the fanout into a
+gather therefore uses conjugate transports in the reversed directions.
+`dual_program` implements (3), not a forward move with a reversed name.
+The inverse transpose is of the main endpoint; owned CLEARs make the full
+scratch-inclusive map noninvertible. Arbitrary owned entries are erased under
+the retained owned-erasure contract and both owned fields leave zero.
+
+Equations (2) and (3) hold for arbitrary field columns. Lifting each block by
+\(I_b\otimes G\) proves the same identities with b idle positions, and block
+composition proves arbitrary repetitions. Tensoring address operators does
+not merge owner roles or turn their sum into independent output credit.
+This is an all-width algebraic identity under the retained Gaussian and
+owned-erasure primitives. A bit-cost implementation still needs those
+primitives' precision, storage, normalizer and tape contracts.
+
+## A sharp three terminal example
+
+Take \(U=0\), \(V_1=\langle1,2\rangle\),
+\(V_2=\langle1,4\rangle\). The three terminals have pairwise distance 2.
+Their shared line \(M=\langle1\rangle\) has distance 1 to each.
+Copy x to owned Z and move U to M; copy Z to owned W, move M to V1,
+ADD to Y1 and clear W; repeat for V2; clear Z. The program has three
+rank-1 moves and the ordinary passes recorded in RESULT.json.
+
+For \(0<s\leq1\), the minimum recursive moment over typed-copy cache trees
+connecting these terminals is exactly
+
+\[
+\min\{2\,2^s,3\}.
+\tag{4}
+\]
+
+To prove the lower bound, remove nonterminal leaves and suppress degree-two
+vertices. Suppression does not increase moment: the metric triangle
+inequality and \((a+b)^s\leq a^s+b^s\) allow replacement of a path by
+one direct edge. Contract zero-rank edges for this rank-only argument.
+A reduced tree connecting three terminals is either a path through one
+terminal or a three-edge tree with one nonterminal junction. The former
+costs \(2\,2^s\). In the latter all three positive integer edge ranks are
+at least 1, hence the cost is at least 3. These bounds apply to arbitrary
+Clifford intermediate charts. The separate transfers and the shared line
+attain them. Ordinary monomial costs are outside this rank-only optimum.
+
+At \(s=1-0.00302015\), the frozen gain is
+\(2\,2^s-3=0.9916351247756209043\ldots\).
+The exact rational interval engine certifies its positive sign and the
+negative sign after charging one more H1. In fact the repaired saving is
+\(4(2^{-\kappa}-1)<0\) for every \(\kappa>0\).
+At \(s=1/2\) the separate
+transfers win. This sign reversal is a useful control on treating rank mass
+as a sufficient cost statistic.
+
+If Z is already a caller-supplied observation \(C_Mx\), omit its creation
+and final erasure. One owned W then suffices for two marginal H1 moves.
+Z is returned unchanged. Its relation to x and all costs outside the module
+are preconditions; the module does not manufacture that observation.
+
+## Intersection loss bounds any typed tree
+
+For a rooted tree of n-dimensional charts, in parent-before-child order let
+\(I_k=\bigcap_{v\leq k}\Lambda_v\). When child w of p is added,
+\(I_{k-1}\subseteq\Lambda_p\), so the quotient map injects
+
+\[
+I_{k-1}/(I_{k-1}\cap\Lambda_w)
+\hookrightarrow
+\Lambda_p/(\Lambda_p\cap\Lambda_w).
+\]
+
+The lost dimension is at most \(r(p,w)\). Summing proves
+
+\[
+\sum_e r_e\geq n-\dim\!\bigcap_v\Lambda_v.
+\tag{5}
+\]
+
+Matarener's nested path argument is the predecessor of this proof. The
+extension uses the actual inverse-image Clifford chart and permits branching
+and arbitrary intermediate Lagrangians. For \(0<s\leq1\), subadditivity gives
+\(\sum_e r_e^s\geq(\sum_e r_e)^s\).
+
+For canonical terminal frames \(U_i\), set \(I=\bigcap_iU_i\) and
+\(J=\sum_iU_i\). The explicit chart formula gives
+
+\[
+\dim\bigcap_i\Lambda(U_i)=\dim I+n-\dim J.
+\tag{6}
+\]
+
+Adding arbitrary intermediate vertices can only shrink that intersection.
+
+## Application to all selected center blocks
+
+Keep #4470's chronological scalar demand, coefficients, fixed warps and helper
+boundaries. Persistent L roles retain the original monotone subspace-frame
+rules. Every source/target scalar contact is in a common canonical frame.
+The predecessor's interval propagation forces a source frame R to contain
+\(A=\operatorname{Low}(\mathrm{source})\), and every target frame Fi to
+lie in \(B_i=\operatorname{Up}(\mathrm{target}_i)\).
+
+Allow the owned payload to travel along arbitrary Clifford intermediates and
+to branch by typed copies. It carries the same specified snapshot; combining
+independent payloads is outside this contract. From (5) and (6), its rank mass
+is at least
+
+\[
+\begin{aligned}
+\dim(R+\sum_iF_i)-\dim(R\cap\bigcap_iF_i)
+&\geq\dim R-\dim(R\cap\bigcap_iB_i)\\
+&\geq\dim A-\dim(A\cap\bigcap_iB_i).
+\end{aligned}
+\tag{7}
+\]
+
+The last inequality is the injection
+\(A/(A\cap X)\hookrightarrow R/(R\cap X)\).
+For a dual gather, apply the same argument to the transposed typed-copy tree
+with conjugate charts; intersection dimensions and edge ranks are preserved.
+
+`center.py` rebuilds the selected local word, recovers paid profile
+`{1:2237,2:388,3:494,4:147,5:36,8:8}` in both orientations, and propagates
+all original contact intervals. For all eight blocks in each orientation,
+it finds \(\dim A=8\) and \(A\cap\bigcap_{i=1}^{22}B_i=0\).
+Thus every such tree pays rank mass at least 8 and moment at least \(8^s\).
+The existing single H8 attains the bound. The actual terminal-chart
+intersection independently gives the same loss eight.
+
+This closes the predecessor's stated non-nested-C question under unchanged
+persistent/contact constraints. It does not cover changed cross-frame contact
+interfaces, changed scalar graphs or helper boundaries, new observations, or
+general circuits mixing independent payloads. Such changes alter the premises
+of (7). In particular the bound must not be transferred unchanged to the
+split-contact or paid-history continuations.